JULIUS WMGEMMEIM "Q7 \ Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/collectionofelemOOchririch MATHEMATICAL BOOKS. Works by Barnard Smith, M.A. Arithmetic and Algebra in their Principles and Applications; with numerous systematically arranged Examples taken from the Cambridge Examina- tion Papers. Ninth Edition. Crown 8vo., cloth, 10s. Gd. Arithmetic for the use of Schools. New Edition. Crown 8vo., cloth, is. 6d A Key to the Arithmetic for Schools. Fourth Edition. Crown 8vo., cloth, 8s. 6d, Exercises in Arithmetic. Crown 8vo. 2s. ; or, with Answers, 2s. 6d. Also, sold separately, in Two Parts, Is. each. Answers 6d. The School Class-Book of Arithmetic. Parts I. and IT., 18mo., limp cloth, price lOd. each. Part III. Is. ; or Tliree Parts in One Volume, price 3s. ISmo., cloth (forming one of Macmillan's Elementary School Class-Books). Key to Class-Book of Arithmetic. Complete, 18mo., cloth, price 6s. 6d. ; or separately. Parts I. 11. and III. 2s. M. each. A Shilling Book of Arithmetic, for National and Elementary Schools. Part I. 2d. Part II. 3d Part III. Id. In One Vol Is. or, with Answers, cloth. Is. 6d. Works by Isaac Todhunter, M.A. F.E.S. Euclid for Colleges and Schools. Second Edition. 18mo., bound in cloth, 3s. 6d. Algebra for Beginners. With numerous Examples, 18mo., bound in cloth, 2s. Gd. A Treatise on the Differential Calculus. With numerous Exam- pies. Fourth Edition. Crown Svo., cloth, 10s. Gd. A Treatise on the Integral Calculus. Second Edition. With numerous Examples. Crown 8vo., cloth 10s. Gd. A Treatise on the Analytical Statics. With numerous Examples. Second Edition. Crown 8vo., cloth, 10s. Gd. A Treatise on the Conic Sections. With numerous Examples. Third Edition. Crown 8vo., cloth, 7s. Gd. Algebra for the use of Colleges and Schools. Third Edition. Crown 8vo., cloth, 7s. Gd. Plane Trigonometry for Colleges and Schools. Third Edition. Crown 8vo., clotli, 5s. A Treatise on Spherical Trigonometry for the use of Colleges and Schools. Second Edition. Crown 8vo., cloth, 4s. 6d. Critical History of the Progress of the Calculus of Variations during the Nineteenth Century. Svo. cloth, 12s. Examples of Analytical Geometry of Three Dimensions. Second Edition. Crown Svo., cloth, 4s. A Treatise on the Theory of Equations. Crown 8vo., cloth, price 7&6d. MACMILLAN AND CO. LONDON. MATHEMATICAL BOOKS. An Elementary Treatise on DiTerential Equations. By the late Gkorgk Boole, D.C.L., F.R.S., Professor of Mathematics in the Queen's University, Ire- land. A new Edition, revised by I. Todhunteb, F.R.S. Crown 8vo., cloth, 14s. Supplementary Volume. Crown 8vo., cloth, 8s. 6rf. A Treatise on the Dynamics of a Particle. With numerous Ex- ples. By Petfr Guthrie Tait, M.A., late Fellow of St Peter's College, Professor of Natural Philosophy in the University of Edinburgh ; and the late William John Stejle, B.A., Fellow of St Peter's College. Second Edition. Price 10s. 6d. The Cambridge Course of Elementary Natural Philosophy; being the Propositions in Mechanics and Hydrostatics in which those Persons who are not Candidates for Honours are examined for the Degree of B.A. Originally compiled by J. C. Snowball, M.A., late Fellow of St John's College, Cambridge. New Edition, revised and enlarged, and adiipted for the Middle-Class Examinations. Bv Thomas Lund, B.D., late Fellow and Lecturer of St John's College, Editor of Wood's "Alge- bra." Fifth Edition. Crown 8vo., cloth, 5s. The Elements of Plane and Spherical Trigonometry. By J. C. Snowball, M.A., Fellow of St John's College, Cambridge. Tenth Edition- Crown Svo., cloth, 7s, 6d. An Elementary Treatise on Plane Trigonometry. With a nume- rous Collection of Examples. By R. D. Beaslkt, M.A., late Fellow'of St John's College, Cambridge, Head Master of Grantham Grammar-schooL Second Edition. Crown 8vo., cloth, 3s. 6d. A Treatise on Elementary Mechanics. For the use of the Junior Classes at the University and tlie Higher Classes in Schools. With a Collection of Examples. By S. Parkinson, D.D., President of St John's College, Cambridge. Third Edition, revised. Crown 8vo., cloth, 9s. 6ci. Elementary Hydrostatics. By J. B. Phear, M.A., Fellow of Clare College, Cambridga Third Edition, accompanied by numerotxs Examples, with the Solutions. Crown 8vo., cloth, 5s. 6d. A Geometrical Treatise on Conic Sections, with Copious Exam- pies from the Cambridge Senate-House Papers. By W. H. Drew, M.A., of St John's College, Cambridge. Third Edition. Crown 8vo., cloth, 4s. &d. Solution to Problems contained in Mr Drew's "Treatise on Conic Sections." Crown 8vo., cloth, 4s. 6d. Geometrical Conies, including Anharmonic Ratio and Projection. With numerous Examples. By C. Taylob, B.A., Scholar of St John's CoUege, Cam- bridge. Crown 8vo., cloth, 7s. 6a. An Elementary Treatise on Conic Sections and Algebraic Geo- metry. With a numerous collection of Easy Examples progressivelv arranged, espe- cially designed for the use of Schools and Beginners. By G. Hale Pccklk, *I.A„ Prin- cipal of Windermere College, fcecond Edition, enlarged and improved. Crown 8vo. cloth, 7s. 6cf. A Treatise on the Calculus of Finite Differences. By the late Gkorge Boole, D.C.L.. Professor of Mathematics in the Queen's University, Ireland- Crown 8vo., cloth, 10s. 6d. MACMILLAN AND CO. LONDON. ELEMENTARY TEST QUESTIONS US PURE AND MIXED MATHEMATICS, WITH ANSWERS. CTambnlrgc : 'RINTED BY C. J. CLAY, M.A. AT TRK UNIVERSITY PEB8S. A COLLECTION OF ELEMENTARY TEST QUESTIONS PURE AND MIXED MATHEMATICS WITH ANSWERS. AND APPENDICES ON SYNTHETIC DIVISION AND ON THE SOLUTION OF NUMERICAL EQUATIONS BY HORNER'S METHOD. BY JAMES R. CHRISTIE, F.R.S, F.R.AS. LATE FIRST MATHEMATICAL MASTER AT THE EOYAL MILITARY ACADEMY, WOOLWICH. ILonUoit anlr ©ambtiljgc: MACMILLAN AND CO. 1866 o^-^ yy^ (f ^- fff^i^'ruu.r^A^^ PREFACE. The Series of Mathematical Exercises here offered to the pubUc is collected from those which the author has, from time to time, proposed for solution by his pupils during a long career at the Royal Military Academy: they are, in the main, original; and having well fulfilled the purpose for which they were first framed, it is hoped that they may be made still more widely useful. The aim in proposing them was not so much to set before the pupil intricate and puzzling questions, as to determine, from the form of solution, whether his mind had fairly grasped the fundamental principles of the particular subject, and was capable of applying those principles : so that a stu- dent who finds that he is able to solve the larger portion of these exercises, may consider that he is thoroughly well grounded in the elementary principles of Pure and Mixed Mathematics. It has not been considered desirable to place the ques- tions strictly in order of presumed difficulty; first, because, on such a point, no two opinions would always agree; and secondly, because a student should be exercised to pass from one style of solution to another with as little effort as his mental capacity will allow. It has been thought advisable to place the answers at «()9823 VI PREFACE. tlie end of the volume, in a form whicli the author hopes will preclude the loss of time which such an arrangement usually entails. For this purpose the numbering of the questions is continuous throughout; and at the head of each page of answers are placed the index numbers of the solu- tions which commence and terminate the page. Increasing attention is now being paid to the Method of Synthetic Division and to the Solution of Numerical Equa- tions by Horner's Method, neither of which processes is given, in a form comprehensible by any but advanced students, in any of the treatises of the day on Elementary Algebra. It has therefore been found necessary to give a short and very prac- tical outline of the principles upon which these methods are based; and the author is not without hope that his Second Appendix may lead to the introduction of the general nu- merical solution of equations in its natural position in every course of Elementary Algebra, immediately after the subject of Quadratic Equations, where it would tend to develope in the mind of the student a knowledge of the nature of alge- braic functions, which would be of the utmost service throughout his subsequent course. For the single purpose of the determination of the numerical roots of an equation, a vast amount of unnecessary and somewhat intricate investigation of the properties of equations is always entered into ; and it is a real boon to the learner to clear away all redundancies and reduce the theory of solution to its simplest elements. This has been done in Appendix II. by basing the determination of a root, first upon the law of signs, for hypothetical position, and, ulti- mately, upon the test-fact of its satisfying the condition implied by the equation ; this fact being shown in actual sub- stitution by a process of sjrnthetic division originated by PEEFACE. VI I Horner for this very purpose. The simple logic of solution then becomes this : " If the roots be all real, we have found by the law of signs that one of them lies between a and h ; treating this assumed root by Horner's process of develope- ment, we find that the developed root, a + &c., satisfies the equation more and more nearly according to the extent of its developement. There can be no doubt therefore that this value is a root, wdiether the remaining roots contain among them imaginary forms or not." By this means the mass of difficulties involved in the various theorems of Newton, Sturm, Fourier and others, regarding the limits of the roots, may be safely ignored until the learner is better pre- pared to undertake their study. The method of dealing with approximately equal roots by a system of reciprocal equations, was first published by the author in 1842, and he still believes it to be the most efficient algorithm yet proposed for application to those deli- cate cases in which tAvo or more roots are identical to more than one or two places of decimals, supposing this fact to be unknown to the worker. The exercises in Practical Mechanics will, it is hoped, assist in calling attention to this important, though much neglected subject. To prevent the necessity of reference to other books on mere matters of memory, a few pages of the more useful tables and elementary formulae in all the subjects comprised in the present work have been given in a form which, while perfectly intelligible to students who have really studied the particular subjects, will probably be of little or no use to those who have merely acquired a smattering. The collection of Examination Papers, also from original sources, will be useful to Instructors in ascertaining pro- VIU PEEFACE. gress from time to time; and, with this view, the answei-s to them have been omitted. In selecting and arranging questions from a mass of papers, spread over many years, it is very difficult to avoid the insertion of dupHcates : although every effort has been made to do this, the author regrets that he has not been, in all cases, successful. JAMES R CHRISTIE. 9, Arundel Gardens, NoTTiNG Hill, 9 Feb. 1866. CONTENTS. PAGK Arithmetic 1 Algebra 10 Equations 19 Problems producing Equations 24 Progressions 31 Horner's Solution of Equations 37 Continued Fractions .... 48 Indetemiinate CoeflBcients 49 Binomial Theorem . 51 Permutations and Combinations ....... 53 Piles of Shot . 55 Logarithms .60 Geometrical Deductions 67 Application of Algebra to Geometry 70 Descriptive Geometry 72 Plane Trigonometry 73 Mensuration 87 Spherical Trigonometry 98 Practical Astronomy 103 Coordinate Geometry . . .111 Statics 121 Practical Mechanics 153 Dynamics 159 Hydrostatics . . 180 X CONTEXTS. PAGE DiflFercntial Calculus 187 Integral Calculus 193 Moments of Inertia 198 Centre of Oscillation 199 Motion in a Resisting Medium 200 Examination Papers 203 Answers 303 Tables and Formulae 354 Appendix 1 375 Appendix II 381 MATHEMATICAL EXAMINATION QUESTIONS. I. AEITHMETIC. 1. Supposing tliat a person can count two hundred in a minute, and that after counting incessantly for 30 years he dies, and his son goes on counting for 30 years and then dies, and so on; how many generations must elapse before one billion be counted ? 2. A party of five persons agree to travel abroad, and they find that, after they have been absent from England a fortnight, they have expended 1000 francs; two of them are then obliged to return home ; how long can the other three continue their tour with their remaining fund of 1 500 francs ? 3. An estate is to be divided between A, B and (7, in the ratios of 1 : 2 : 3, subject to the proviso that each of them pay to D one-eleventh of the whole; what would D receive, supposing ^'s share to be £550 ? 4. If a watch be two minutes and a half too fast by the clock of the K M. Academy when the bell rings at 4 p.m. on Wednes- day, what will be the time by that watch when the bell rings at noon on Monday % the daily rate of the watch being - 20 sec, and that of the clock + 18 sec. 5. If 6580 shot of 24 lbs. cost £725, how many 42 lb. shot can be purchased for £2100 when iron is five per cent, dearer % = • c. 1 2 MATHEMATICAL EXAMINATION QUESTIONS. 6. If 30| miles of iron rail^s. cost £15580 when iron was at £10. 5s. 6d. per ton, what would be' ihe .expense of 157| miles of the same rails >heD ivCfii. m aii£l'^-'.4iS..23?. per ton ? 7. If a battery of 6 guns, firing three rounds in ten minutes, will breach a certain work in 60 houra, how many guns must be employed for the same purpose, firing two rounds in five minutes, in order that the breach may be made practicable in 15 hours? 8. A can run at the rate of eight miles an hour and B at the rate of seven and a half miles an hour; what is the greatest num- ber of yards start that A may give ^ so as to beat him in a race of 440 yards? 9. If two 32-pounders can render a breach practicable in fifty hours, in how many hours can three 24-pounders, two 18-pounders, and one 12-pounder render practicable a breach requiring double the quantity of battering, supposing the effects of the difierent kinds of shot to be as 6, 5, 2 and 1 ? 10. A garrison of 1000 men was victualled for 30 days, but after 10 days it was reinforced, and the provisions were, in con- sequence, exhausted in 5 days : what was the number of men in the reinforcement 1 11. A wall 1236 yards in length was to have been built by 60 men in 21 days, but at the end of 15 days it was found that only 824 yards had been completed; how many extra men must be employed in order that the work may be completed in the given time? 12. The proportions of sulphur, charcoal and nitre, in the composition of gunpowder being as 2, 3 and 15 respectively; what is the value of 5460 lbs. of powder, independent of the expense of manufacture, when sulphur is £l. 4s. 6d. per cwt., charcoal £1. 15s. 2d. per cwt., and nitre £l. l6s. 8d. per cwt. ? 13. What is the value of five pounds Troy of an alloy of silver and gold, in which the weight of the gold is f the weight of the silver, gold being £4. 5s. and silver 6s. per oz. ? AEITHMETIC. 3 14. Gunpowder being composed of 75 per cent, of nitre, 12-5 of charcoal, and 12-5 of sulphur, how much of each of these sub- stances is there in 5 tons of powder 1 15. If two ounces of gun-cotton produce the same effect as five ounces of gunpowder, and the spaces occupied by the same weight of each be as 10 to 3 (the cotton being the more bulky); what must be the length of a musketry cartridge, filled with cotton, which is to be used instead of a gunpowder cartridge two inches long 1 16. If 25 labourers can dig a trench 220 yards long, 3 feet 4 inches wide, and 2 feet 6 inches deep, in 32 days of 9 hours each; how many would it require to dig a trench half-a-mile long, 2 feet 4 inches deep, and 3 feet 6 inches wide, in 36 days of 8 hours each? 17. Supposing the rates of marching of two columns of in- fantry to be as 4 to 3, and the one to be three miles in advance of the other and marching at the rate of 2^ miles per hour ; in what time will the column in the rear overtake the other ? 18. If five bricklayers can lay 45 bricks in five minutes, and the bricks made use of be 10 inches long, 4 inches broad, and 3 J inches thick ; how long (with such bricks) will it take 8 brick- layers to build a wall which would contain 95000 bricks 8 inches long, 5 inches broad, and 3 inches thick; the day being 12 hours long? 19. Six thousand of a certain kind of shot weigh 64 tons 5 cwt. 2 qrs. 24 lbs., how many 9 lb. shot will weigh as much as 3810 such shot? 20. What is the time when the hour and minute hands of a watch are exactly together between 9 and 10 o'clock? 21. Find the time, between three and four o'clock, when the hour and minute hands of a watch are (l) coincident, (2) in ex- actly opposite directions, and (3) at right angles to each other. 1—2 4 MATHEMATICAL EXAMINATION QUESTIONS. 22. By what part of its present value must the farthing be increased or diminished if a decimal coinage were introduced in which 1 sovereign = 10 royals, 1 royal = 10 groats, and 1 groat = 10 farthings; the value of the sovereign remaining unaltered ? 23. The French Metre being the ten-millionth part of the distance, at the sea-level, from the pole to the equator of the earth, find the length of an entire meridian in miles, the metre being 39*370089 English inches. 24. The French Gramme is the weight of a cubic CetUimetre of water, and the superior denominations, ascending decimally, are the Decagramme^ Hectogram/me^ KUogram,me and Myriagramme ; find the difierence of weight of a French gun weighing 263 Myria- grammes and an English gun weighing 50 cwt. 45 lbs., supposing a cubic foot of water to weigh 1000 oz., and the centimetre to be •3937 of an English inch. 25. A reservoir of water, which will hold 2641500 gallons, is to be filled by a pipe which discharges 1000 French litres per minute; how long will it take to fill it? 26. If 50 sappers can dig a trench 500 yards long in 25 days; how long will it take 10 sappers to dig a trench 75 hectometres long, of the same depth, but twice the width of the former % 27. It being required to re-cast a quantity of captured shot and guns, weighing 105730 kilogrammes, into the form of 13-inch shells weighing 1 96 lbs. each; what number of shells will be obtained, allowing 6 per cent, for waste; the kilogramme being 2 '2047 of the pound avoirdupois? 28. If <£l sterling be worth 25 francs, 60 centimes; and also worth 6 thalers, 20 silber groschen; how many francs and cent- imes is a thaler worth, when 1 thaler is equal to 30 silber groschen, and 1 franc equal to a 100 centimes'? 29. The ditch of a fortress can be filled by one sluice alone in 12 hours, and by another in 15 hours: in what time will it be filled by both open together? ARITHMETIC. 5 30. A cistern wliicli holds S20 gallons is filled in 20 minutes by three pipes, one of which conveys 10 gallons more, and the other 5 gallons less per minute than the third ; how much water flows through each pipe per minute ? 31. The imperial gallon contains 277-274- cubic inches, and the hectolitre contains 6102-379 cubic inches; determine the value of the litre in terms of the quart. 5 6 "^2 ,_ 32. Reduce the fraction ^^ ' ..^ , ., ^ ^ ^ to its 4 simplest form. 1 /2 _ 1 \ _ 6 /4 1 \ 2 U V 7\5^ 9/ \8 2/ 3\7 10/ 1 5 2 _ 2 ' fi S * 8 33. Reduce //^ qn t^ ^t^ lowest terms. 3 / T" 7_2/l_5\ • 8 5 U 7/ \3 2/ 9\4 28/ 7 2247 774 , 1017 6^5 ^ „ .1 J • 1 r 34. Find the value of -^^ x — ^ of a £ as the decimal of 339 565 a £50 note. 30. Express •'^ cwt. as the decimal of a ton. 36. Reduce ^ of a £ to the fraction of a shilUng. 93+ * 2 ^« 2f 37. Add together ? of a guinea, A of a £, | of 7^. 6d., and subtract - of 2d. from the result. 4 6 MATHEMATICAL EXAMINATION QUESTIONS. 38. Keduce ^ of 7s. Od. ij/ to the fraction of a half-crown. 39. What part of five shillings is five-sixths of sevenpence- halfpenny ? 40. What part of 4^. Qd.\s (^- -^^ ~ + ~) of £l. 4s. ? ^ \15 120 4<0 24/ 41. What part of 58. Ad. is two-sevenths of 17s. 6c?. % 42. Reduce 4s. S^d. to the decimal of a crown ; and to the fraction of a Napoleon of 20 francs; the franc being ^th of a £. 43. What is the value of 100 shells, each weighing 1 96 lbs., when cast iron is 5s. 6^d. per cwt., and the expense of manu- facture Is. Sd. per shell ? 3 42 1 44. A person by selling a certain horse lost 77 of -^ of — of 1 28 18 £60, and by selling another he gained - of — of — of £2, when 15 24 he found that the price received for both together was -^ of — 42 14 of ~ of -^ of £17 j what did he give for the two 1 45. From an ammunition waggon containing 27 cwt. of ara- 3-— \0w 77 munition -^ of —j- of -—— of the whole was delivered to Battery Af and | of the remainder to Battery B, what quantity was delivered to each 1 (1 2 1 3 1 \ 2 "^ 3 ~ 6 '*' 8 ~ 12) ^^ *^ ®^***® ^® ^^^^ £(52000, what will be the value of 1-^ of ? of ?| of — of it ? 12 8 26 75 1 47. What part of ^ of 17 shillings is ? of - of ^ of l6s. SdA o 3 5 o ARITHMETIC. 14 3 2 5 1 15 48. If - of - of - of a ton of coals cost - of r— of a £, what 5 3 4 ^ _L 10 is tlie price per cwt. 1 4 3 2 400 49. If ^ — of — of - of a mass of metal weighed —— lb., 12. 2 what was the weight of j| of - of the same mass I 4 3 2 50. Owning — of a ship, I sold — of - of my share for 400 1^ 2 £> -^TTT ; what was the value of -4 of - of the vessel at the same 33 ' 4;| 5 rate? 51. Multiply 162-5473 by 8726-47231, contracting the operar tion to four places of decimals. 52. Multiply 1854-362 by -000087931, contracting to six places of decimals. 53. Multiply -0073654281 by 37584*26, contracting the work to six places of decimals. 54. Multiply 13-50629 by -0036472, contracting to six places of decimals ; and divide the second by the first, contracting to eight places of decimals. o5. Divide 15-63214725 by -0057123, contracting to three places of decimals; and multiply 730-6581 by -08652, contracting also to three places of decimals. 5Q, Divide 634-7538292 by -0657391, contracting to three places of decimals; and multiply the quotient by the divisor, con- tracting to four places of decimals. 57. Divide 6-38572164 by -00752681, contracting the work so that the quotient may contain four places of decimals. 8 MATHEMATICAL EXAMINATION QUESTIONS. 58. What place in the decimal scale will the first figure of the quotient of -0003279634 by 286-3471 hokU State how the result is obtained. 59. Divide -0073628439 by -000265847, contracting the work to five places of decimals ; and then multiply the quotient by the dividend, contracting the work to six places of decimals. 60. Divide -00034984 by 37627*15, giving the quotient to twelve places of decimals. 61. Find the value of the recurring decimal 2-54312, &c. 62. Find the vulgar fraction equivalent to the recurring decimal -9243. 63. Find the values of -13888 &c. of a shilling ; -23 of a £ ; and -04 of a yard. 64. Divide 2-0505 by 31-2; and show the correctness of the result by reducing it and each of the above recurring decimals to its eqiiivalent fraction. 65. Divide the recurring decimal •0357 by the recurring decimal 25-84, giving the quotient to eight places of decimals. 66. Reduce each of the above decimals to its equivalent frac- tion ; and prove the correctness of the division by means of these fractions. 67. Extract the square root and also the cube root of 534-267351, giving the root in both cases to five places of decimals. 68. Extract the square root of -00000475850596. 69. Extract the square root and also the cube root of 2 to six places of decimals. 70. Extract the cube root of 5-76 to five places of decimals. ARITHMETIC. 9 71. Extract the cube root of 17 to nine places of decimals. 72. Extract the cube root of 28 to ten places of decimals. 73. Find the cube root of 7 to ten places of decimals. 74. The population of Great Britain in 1851 was 21,121,967, and the increase during the previous half century had been 93-47 per cent. What was the population in 1801 1 75. The Income Tax upon a certain income at I5. 4-d. in the pound amounts to £20; find the sum invested in the 3 per cents, from which the income is derived. 76. The simple interest on a certain sum for 9 months, at 5 per cent, per annum, is £150 less than the simple interest on the same sum for 15 months, at 4 per cent, per annum. Find the principal. 77. An estate purchased at X84. 7^. 6d. per acre was sold at £90. 14s. Ofc?. per acre; find the gain per cent. 78. The investment of a certain sum at 3 per cent, produces an income of .£501. 15s. A portion of this, sufficient, when re-in- vested at 5 per cent., to produce the same income as the whole formerly did, is called in. Find the amount of income derived from the whole when this re-investment has been efiected. 79. Multiply together lift. 4 in. and 4 ft. 7 in., by duo- decimals, stating clearly the value of the superficial unit in each denomination. 80. Multiply 3 feet 2 inches 8 lines by 1 foot 8 inches 10 lines, using the duodecimal method; a line being a twelfth part of an inch; and state what each term of the result expresses. 81. Multiply 6 feet 5 inches by 4 feet 9 inches, according to the duodecimal notation, and exhibit the result in the decimal scale, the unit of measure being a square foot. 10 MATHEMATICAL EXAMINATION QUESTIONS. 82. Multiply 17 feet 9 inches 5 lines by 3 feet 2 inches 7 lines by duodecimals, and state the value of the superficial unit in each denomination. Find the result also by decimals, and shew that the two agree. 83. Multiply 5237684 by 4539, in the duodecimal scale, using a to express ten, and /? to express eleven. 84. Multiply 1432312 by 31422 in the quinary scale; and transform the former to the denary scale. 85. Express the undenary number 57a4685 in the octenary scale : where a expresses 10. 8Q. Convert the senary number 5321425 to the quinary scale. 87. Transform the quinary number 3241342 to the octenary scale. II. ALGEBRA. Introductory Operations. 88* Eind the numerical value of 3a' - 26 {«* - 3c (6" - 2a) + c'} - 4c (a - 6)^ when a = 7, 6 = 5, and c = 2. 89. Find the numerical value of ' M'J{h-a)^{cd(d-e)(h-d)-f} V a+/{c + d-d{a + b){c-b)\ to five places of decimals ; when a = 10, 6 = 2, c = 7, c?= 5, c = ~ 3 and /= 6. ALGEBRA. 11 90. Find the numerical value of ab^ -'c\a-{cA- 6)} when a = 6, h — % and c-^, 91. Give the numerical values of the following expressions, when a = 5, 6 = '25, c=l, 6?= 7, e = and /= J : 2.-6(l-2^+3/)-5».|^^ (1), 12 (ct' - c") - 5e \/36 - 10c? /2\ 4 {(a + cf — 4ac} 2 (a + 5) . (5c - 6/) 8 (5c? ~ a^dfy .^. 5jl5af-{Sb'd + 9Y ^ '' 8/f^i2^/ 4 / X) '{a>'-hd).e (4), ^{5b'd(a*-6ac)} ^ ^ abe>J{5ar{c + bd)} 92. Find the numerical value of the expression V a-b ^ a + b V a + b + c ' when a= 16, 6 = 2, and c=9; having previously reduced it to its simplest form. 93. Reduce (4a.f 1^-64.^-48^(2^- 1)-1 ^ ^ ^ /o^_.) 12 ^ to the form |^(8»-5)(a + l); and give this result in terms of b when 6 = a — 1. 12 MATHEMATICAL EXAMINATION QUESTIONS. 94. Find the value of to six places of decimals, having previously reduced it to its simplest form. 95. Find the value of iV24- ^§ + 2.73375.^3^75 to five places of decimals. 96. Eeduce J a, IJb^, ^c^, ^Sa'^ and d~^ to a common index. 97. Subtract 2 /?^' from ^I /1^[, and add the -sy suit " - 81y re- 50x' 98. Prove that ^(- 1) + J{- J{- 1)} = ± J2. 99. Prove that s/[2 {a + J{a^ + h^)}] =J{a^h J{- l)}^J{a-h J{- 1)}. Simplify the following expressions : 100. 102. X y v/18 101 — + xy x + y 2J2-J3' X y x-y x + y 1 1 n.q - J /3 7 7 20 3 + 2^2 104 ■y( ^+y) + N/(^-y) _ sl{^+y)-sl{^-y) 'J(^+y)-J{^-y) J(^+y)+J(x-y)' ic' + 2/ ^(«^V) - 2 ^(ccV) • -^ "^^ a^- b' '^a'+b' a'-b' ' ALGEBRA. 13 107. y(.-,2,)^(.^_2,^)_2(.V + a.2/^^^^,. 108. '/ ^"{^{^ + ^y)-{x+Sy)} + 4.xi/ f ' x'^-l+x'^'-x'''' Jx + (a-h)J{-l) Jx-{a-h)J{-l) ''^- Jx-(a + b)J{-l)^Jx + (a + b)J(-l)' 111 ^ ( x + Jx'-l x-Jx'-l ) Jx'-l ' \x - Jx^- 1 cc + Jx^'-lS ' 112. ^"' T 3 /92^ y{-'^i^K-5-^ n^ V I ^ \ ^v ; 115. £C^ + £C - 1 n+l 116. 117. 3a3^ - 2a;y - y^ ^x^-^x'y-Sxy^ + y'"' 119. 120. 1 - Sa-h^ + 3a-^6^ - a"^^>'^" /(a -ly+^ahia-hf + l 6a'b' (a - bf 14 MATHEMATICAL EXAMINATION QUESTIONS. 1+^5 ^ 1-^5 — ^x-^ — ^a;-2 J a?-x' x' 124. 1 a — x x-y ^ z-x ^ y~z xy xz yz ^+Jo^-y' . 121. I . x' f-x x' ^x J (a' + b') . {a + by - 4>a'b -4^" 123. 125. , .. , . .. x-J^-f y io« J^Ta + J^^ Ja^ + a-Jx^-a 120. — . . + ■ — , ^£c* + a- Jx^ - a Jx' + a + Jx^ - a A(a - by + 4a{a + b)- 4>a'\ . |a'(l - ^) + ^'j 127. 3 . ^ a'Ub'{b' --a') +a*\-b' x -y x + yl{x~yf x-y) 129. n:^xx\x-yy 131. ^+y>/^ I ^-yj^l , x-yj-l x + yj-l' 132. /n/4 (^ct V6)"-1 6 s/ab+^l6 {a\b'-h4>sf^b(a+b)+ 6ab} ALGEBRA. 15 V a? + Sx^y + Sxy^ + y^ x'^2xy + 'if - Sx'y + 3ix^f -f ' 'J'^y ^""'f' 134. Simplify the expression : (\-¥x)h + {l-x)^ (1 + x)k - (1 - xf. ' first by rationalizing the numerator, and then by rationalizing the denominator j and show that the sum of the results is, as it ought to be, the double of the original expression. 135. Multiply 7£c' -Sx^'-Bx + S by Saf - 1 la; - 9, using de- tached coefficients j and divide the result by Sx^-^x — Q by the same method. 136. Write down the coefficient of x^ in the product of ax^ + hx^ — cx^ ^ex^ -\-fx — h by kx^ + Ix^ + mn - n without actual multiplication ; and state how the result is obtained. 137. Multiply together 7^- + ^«2/"+2/% 4^^^\ 7a;--a;V + 2/" and V^» + v/". 138. Multiply together 111 » '{a?-¥oh^V)\ {a-hy, {a-hy and {a' + ab + by. 139. Multiply 4a^b-2a%'-6ab^ + 3h' by ab^-2a-'b', using detached coefficients. 140. Square the expression ^-1+n/-^-1. 141. Cube the expression a^Jx-JbaJy. 142. Extract the square root of - y'z"" - Sy'z^ + 34y V + 1 Syz' + ? z'. 16 MATHEMATICAL EXAMINATION QUESTIONS. 143. Extract the square root of 144. Extract the square root of 41 3 + 3a f- 9 ^"■Te" — 2 — '^*"- 145. In any equation x + Jy = a-¥ Jh, which involves ra- tional quantities and quadratic surds, show that the rational parts on each side are equal, and also the irrational parts : and extract in the form of a binomial surd the square root of 146. Extract the square root of 3 J3 - 2j6, without using a formula : show that 2 + J3 is the reciprocal of 2 - J 3, and state generally what must be the connexion between the two terms w and ^, so that their sum may be the reciprocal of their difference. Find the square roots of the following binomial surds, with- out using a formula : 147. ^27-2^6. 148. J20-J15. 149. 56-2J55. 150. 43-30^2. 151. 7 + 2^10. 152. 21+12^3. 153. Divide x' + {a' - 2¥) x* - (a* - b') x'-a'- 2a*b' - a'b* by x^-a^-b% by Horner's or the ''synthetic'' method. 154. Arrange 6(a;'+y^) + (l8ajy-4)(a:+?/)-8(a;'+2/')-l6a3^-120 and x' + 2/^ + 2a;(l + y) + 2?/ + 6 according to the powers of {x + y), and divide the former by the latter, by the synthetic method. ALGEBRA. 17 155. Divide - 10a Vy - QOaxY + 6«*^* - 9a V/ by Sx^y* - 5ax^y^ + aVy, by the synthetic method, giving six terms of the quotient, an-anged according to the ascending powers of y. 4 156. Divide Sx^-9x + — by 4a;^ + 8, using the synthetic rm- tJiod, and giving the quotient as far as the term involving x~^. 157. Divide x^-2x^-31x'+25x'+3x'-15x'-8x^+19x^+3x+lO by 3x* - 21x^ + 9^ - 6f by the synthetic method; showing the ^^ final remainderj^ and then carrying on the quotient to twelve terms. 158. Expand :; z and — ^ — -p^-^ — — in series by synthetic division. rji^ X -V \ 159. Expand — in a series to five terms, by Homer's mode of division ; and show the arithmetical equality of the frac- tion and resulting series, when x = 3, 160. Divide x" - 4a;« -x^ ->r\^x^- lOo;' - 21x^ + 22a;' - 8£c' - 10a; + 3 by X* — 4a;' + 2a;^ - 2, showing the final remainder, by the synthe- tic method. 161. Find the remainder left after dividing V - 372/* - ^y' - 27^ + 7/ - 1 Oy - 6 by 4y-28/-l6y^-24j^ + 8, synthetically ; the quotient involving only positive powei-s of y. 162. Divide 2a-'b'+10b'-4>ab'-32a'b^+l6a%'-37a*b + 2a'+66a%-''-8a'b-' by 2a-'+4>a~*b-'-6a-^b~'-8a-%~^, by Horner's method, showing the remainder. c. 2 18 MATHEMATICAL EXAMINATION QUESTIONS. 163. Divide 4>xhj-2Sx^y^z- 36xYz^+ 26xYz^- 52xYz*- 30xY^'+'^6xy'^z^+ 35]f^ by ^x^yz'"^ — Sx'ifz'^ — ly^ to such extent that the quotient may involve no negative powers of x, and give the remainder. 164. Divide 2a%~^ + ^a*h~^ -Qa^ -kc, thQ remaining coeffi- cients being - 21 + 69 - 74- + 68 - 15 - 36 + 29, by ^a-'h' + Sa-^h' - Sa'^^ + ^a-'h\ giving the remainder whose first term involves a~^b^. 165. Divide 2xY^ - 14^3' + 22a;y - 22?/' + l6x~^y^ + paj'V - 12fl;~V by 2cc~'2/' - 4a}~y - 2a;~'*2/'*', by Horner's method, and give the remainder. 166. Divide Sx^yz-' + 4£c' + C}x*y-'z - 21xV"V + l6a;VV -llaiV^^^-Sa'V'^'-^icV^'-SSa.-'VV + SSaj^y-V by Sx'yz"^ - 9.x^z~^ + x^y~^z~^ — 5x'y~^, and give the final remainder. 167. Divide 1 8a;-y + 6a;-y + 20a;-»y' + 3 1 ar*y + 2a;-^+ 6x~Y^ + 39iK~\?/~^-64y~^ + 20ic?/"* by 3x~Sj~''-x~^y~^+bx'^y-*-9.x"^y-^, by Horner's method, and give the remainder. 168. Divide %a-\V + Vla-%'' + a'^^ - Qa-'b' + Sa'^b* - W + 6ab'-5a% + 4>a^-4>a'b-' + a'b-' by a' + 3a'b-' + 2a'b-^ - a'.b-'' by Homer's method. 1 69. Divide Sax'y + 7y' - Sa'^xy' + SSa'^xY - 22a-^xY + 18a-'xY + 30a- Vy - 60^- V+ ISa" Vy"' + 5a-'xY' by 3x-^y^-2a~^x~*y + 7a~^x~^-5a~^x~Y^-(^~*^~^y~'f ^7 Homer's method; giving the remainder whose first term involves a"V. x^ +2 170. Expand -^ — ^i in a series, and show the arith- X "T /iX — 3 — X metical equality of the fraction and its expansion, when x=l ; taking four terms of the series. ALaEBRA. 171. Find the greatest common measure of x'^-x^-x+l and 5x^ - ^x^ - 1. 172. Find the greatest common measure of 6a V - 9a Vy - t Oa\y^ + 1 5ay^ and lOa'xY- 15aV2/'+ SaV?/*- ISaV^/'. 173. Find the greatest common measure of a;® + 4a£c* - 3a V - l6aV + 1 la^x"" + ISa'o; - 9a* and Qa^'x^ + 20aV - 12aV - 48aV + 22a*a; + 12a^ 19 Equations. Solve the following equations 174 1-ag 2-3a; 3 - a; _ 3 (1 - a;) (1 + 2a;') 4a; 4a; (3 - 2a;) l7o. 3aj-2.— — - + 5. i-=6a;-7 t- 5 3 4 --. 3a;+7 „ 2a;-14 5 a;+l6 176. — r-: — + 7 14 21 12 4 ,-^ 7a;+5 9a;- 1 x-9 2a; - 3 _ Tg ^^^* ~W'*'"i:o 5"^ 15 "3* 178. ^(i+x) + J{l+x + J{l-x)} = J{l-x). ' x-2 10-x 2/-10 ' 179. 5,3 4 2y— 4 2x + i/ _x+13 ~~3 8"" ~ 4 2—2 20 MATHEMATICAL EXAMINATION QUESTIONS. ^ K^-O-ie-) !-^ 4 a; 183. a:-l = 24--^. 184. ^(24 + a;) + ^(24 - a:) = 5. 185. 2;y(a3''-5a; + 2)-a:»+8a; = 3a;-78. a-J(a^-x) 1 186. 187. a + ^(a"* - re) a' 4a; -22 4a;' - 1 _ 5a;' + Sa; + 1 35 5x~ ~ ~7x 2/-1 (y-l)(2/-2)-^>-^2 189. Jx + J{a-x) Jx-J(a-x) Jx* 190. ^ + ^±1 = :^. 191. »_^^^f = i. a;+ 1 a; o x— 2 a;— 1 192 a;"- /(a;^-QW21 193 n /(^+^) _ v/(l - ^) 194. V«^-4(^a; + 13)^ + 7 = £c-2;ya;-9. 1Q^ <^ + ^ . _ a-a; ■»«■ y'V((-F)('-£)}='- 197. 4a;' + 12a; . ^(1 + «) = 27 (1 + a;). EQUATIONS. 21 198. -^-±^^9)_, , 199. \l-Sx^ + 5x' = 9x^ + 299 - x\ 1 1 200. x-J{2-x') x + ^{2~x')'^'^' 201. x' + ^,-a^-\^o. 202. J,; + -4_ = i? aj a' ^ 24/a; 4 203. a; = 7_^_, 4?/ —- = 5. 7 5 204. /^'-2^' = ^^y(^-2/) + 8,) U^y + a??/ = 4 (a; 4- 1). j 205. 206. ( x-'2, 10-£C _ y- 10 1 ~5 3~"^4~~' l2y+4 2x-\-y x + \S [r~~S 8~~~ 4 ' 207 /(^'+2/')(«^ + 2/) = 2336,) • ((a;^- 2/0 (a. -2/):. 576. / 208. ^x"" - 6xy + Sx-ly^-9,y=\'2. and Sx-2y= 7. 209. (a; + 2/)^ - £c + 2/ = 4iC2/ + 20- ^1^- y^ + y = '^^-x^+x, aj + 2/=12. a;y = 4. 211. L-l~i^Tl~ ^'l U + 2/=5. / [ y{x + 2) = l. J 213. { ^ -"j ; 2U. \ j{y-x)^J(a-x) _!^ I 22 MATHEMATICAL EXAMINATION QUESTIONS. 215. \x-y^h. ) 216 ^ ^' 12^ + ^ = 1 + ^. ^a c 217. (r^.^"^'- ] 218. K + ^ = 89.\ 219. (^-^' = f) 220. K-3^^ = 36+^-.y,| la;-y = 5. j ""''' \ xy = 36{x + y). j 223. «Vy^=8, andi+i=l. 224 / «^ + 2/' = 34, ) * (2iB»-3aJ2/ = 23-2/.j 905 ({x + yT-3a'{x + y)--=2a,l \{x-yy + 5b'{x-y)-^ = 6b.) nox + y ^^ 226. J xy [9{y-x) = 18. 227 r«'' + 2/'-«'-y=78,j 1 xy + x + y = 39.) ^^^' { xKy^ = 9.] U-.y^ = 155oJ ( {x' + f)xy = 7S,\ (x-^y+x' + y'=l8^ ^^^' \af{x' + 6y') + y* = 313.j '^'^^' { xy = 6. j r a;y= 1225,-1 fl+a . 1+y ''■^*- Ua!+^/2/ = 12. / 235. ^^-y '-'' \+X 1 -V 7 1+2/ 1 -ic EQUATIONS. 23 ix + y x — y 236. \^-y x + y 241 237. (^'V + V = 3V2,| ix y + xy^ I x^^y' x'' + y^ = ^0. ooo / x^ + xy = %\ ^^^' Ux^+f = 9j 240. [ ^ + ^ = 720 239. f / ^^ /x + y 3 ] ((aJ + 2/) V(^2/) = 510.J 94.4. K"^ ^2/ +2/'=lS,| ^^"^^ W + a;y + 2/^ = 91.J r x + 2y + 3z= 17,' 246. J2ir + Sy+ ;s = 12,' l3a;+ y + 2^=13] 243. f ax+ by + cz -d, 248. - a^x + h;y + c^« = ^,, «//«' + ^/.2/ + c.« = ^./ - + - = 5, a; 2/ 250. - 1 1 ^ Vz X ) ' 251. 2^% 2/ X 2' ■ ^ _ y ^ 3 2^ a; 2 * J \xy-y' =20.) {a; + 2y+3;2;=l6,1 2a; + 3y + 4« = 23,' 3£C + 4y + 2«=18.J 247. \y{x + z) = 65A 249. la^x + h^y + c^z = e^, \2x + 3y-\-4fZ=l9, 3x+4fy-2z=9, ^5x-2y+Sz = -33. lax +by- cz = df 'bx-cy + az = -ef ^cx + ay-bz=/. 24 MATHEMATICAL EXAMINATION QUESTIONS. 253. 256. 258. 260. X y z _+ ii -f _ 12 3 32. -^+1+7 = 22,^ 2 3 4 z _l — 5 17. r a; + 2y+35;=17/ 2a; + 3?/+ «=12, \sx-\- 2/ + 2«=13.! |aj(2/4-^) = 20,| [«(a;+y)=27.J " t; + 2a; + 3y = 6a, "" aj + 2y + 35; = 65, y + 22; + Sv = 6c, ^ « + 2v + 3x = 6c?. 254. 255. 257. 259. iax+ br/ + cz hx + cy + az cx + ay + hz {x + 2y + 3z 2x + 5y+ 7z 3x-4^y + 2z f x~3y-\-^z y-3z + 2x [3x + 3y + 3z = m.] = 14, = 33, = 5. \ = 5, =-11 = 24. PROBLEMS PRODUCING EQUATIONS. 261. In a battery of 2 guns A and B, it is found that A fires 2 rounds more than JB in half an hour, and if A then cease firing for 10 minutes, the number of rounds fired by A is to the number by B, in the ratio of 4 to 5; what is the number of rounds fired by each 1 262. A light six-pounder gun weighs two-thirds as much as the carriage, and both together weigh half as much again as the limber when filled with ammunition ; but the weight of the gun is three-fourths that of the empty limber; what are the re- spective weights of the gun, the carriage, and the empty limber, supposing the weight of the ammunition to be 4 cwt. ? PEOBLEMS PRODUCING EQUATIONS. 25 263. Two companies of sappers were ordered to throw up a field-work, each company to do one-half of the work ; No. 1 com- menced operations half an hour before No. 2, and both stopped to rest for one hour at 12 o'clock, when it was observed that the work was just half finished; No. 2 completed its portion at 7 p.m., but No. 1 had not finished until a quarter to 10. At what time did each company commence work in the morning ? 264. Having bought as many muskets as cost £8500, I re- served 800, and sold the remainder for £8400, gaining Ws. on each; how many muskets did I purchase, and what did each cost? 265. A railway-train runs a certain distance at a certain rate ; had the rate been increased by 5 miles an hour, the distance would have been performed in i of the time; but had the rate been diminished by 5 miles an hour, the time would have been increased by 2| hours. Find the time and distance. 266. A battalion of 820 men is raised in twenty weeks from three recruiting districts, one of which supplies 10 men more, and the other 5 men less, per week, than the third : how many men are supplied per week by each district ? 267. An officer was sent by government to expend £10,000 in the purchase of cavalry horses. Being despatched to the various depots, 25 of the horses died on the road; and this had the effect of increasing the average price of each surviving horse by £1. 13s. 4c?. What was the number of horses purchased ? 268. Seven Enfield rifle bullets weigh 210 grains less than eight of the old spherical bullets, and each of the former weighs 40 grains more than each of the latter; find the weight of each. 269. Two companies of sappers could have completed a cer- tain work in 16 days, but, after 4 days, one of them is ordered off to other duty, and the remaining company completed the work m 40 days from the commencement : in what time could each com- pany have completed the work separately? 26 MATHEMATICAL EXAMINATION QUESTIONS. 270. A train leaves B at 9 A.M. and runs to C at tlie rate of 1 5 miles an hour, and another train leaves A at noon and, run- ning, through B, to C at 25 miles an hour, arrives half an hour later than the train from B, Find the distance from A to C, the distance from ^ to ^ being 15 miles. 271. A detachment from an army received a reinforcement containing one-third as many men as the front of the detachment consisted of when drawn up "four deep"; had the depth how- ever been increased by 2, and the whole been di-awn up together, the number of men in front would have been 125 fewer than it was before : find the number of men in the original detachment. 272. After a cavalry action, the commanding officer found that the number of his men killed was seven-ninths of the number of horses he had lost, and that the number of men still mounted was five-sixths of his original force ; with how many men did he commence the action, supposing that he now requu'ed 48 horses to restore to the ranks his dismounted troopers? 273. A contractor supplied 3560 muskets and 1650 cutlasses for £10,000, and found that if 50 were added to the number of cutlasses sold for £100, it would be the same as the number of muskets for X500: what was the price of each musket and cutlass 1 274. A trench was dug by A and B, whose rates of working were as 9 to 8 ; after having worked together for 8 days, the re- mainder was completed by A alone in one-sixth of the time that B would have taken to do the whole : how long would each have taken to do the work separately ? 275. At a review, the infantry were drawn up in close column 40 deep, and it was found that there were just one- fifteenth as many men in the fronts of all the columns together as there were cavalry and artillery with the army : and that had the depth of each column been increased by five, and the troopers and artillerymen been drawn up with the infantry, the number of PROBLEMS PRODUCING EQUATIONS. 27 men in the whole front would have been 100 more than before. What was the whole number of men in the army 1 276. A retreating enemy is ten miles in advance of the pur- suing army, which marches at the rate of three miles an hour; how far will the enemy be able to retreat before being overtaken, supposing that their rate per hour is j^jV of the number of hours they are on their march 1 and what is their rate of march 1 277. What number consisting of three digits, is greater by 99 than when its digits are inverted; greater by 126 than the sum of its digits; and greater by 27 than when its second and third digits are transposed ? 278. A troop of horse-artillery on the march from Wool- wich to Chatham met a company of marines marching to Wool- wich at the rate of three miles an hour; four miles further on, and four hours after leaving Woolwich, two stragglers from this company were met hastening on at the rate of four miles an hour, which pace would just enable them to join the company as it marched into Woolwich : at what rate was the troop marching, and at what distance from Woolwich did it meet the company of marines ? Give an interpretation of the negative result. 279. Three gunners. A, B and (7, were discharged from the regiment; ^'s pension was treble that of B^ and, if increased by Id. per diem, would have been double those of B and G together; but if (7's had been increased by the same sum, it would have been only three-fourths of those of A and B together. What was the pension of each per week ? 280. A detachment of laeld-artillery had to receive 216* rounds of ammunition, but before it arrived three of the guns were disabled, and the rest, in consequence, received each six rounds more than they otherwise would have done; how many guns were there at first ? 281. If a cwt. of sulphur and h cwt. of nitre were bought for £c, and it were found that d cwt. more of sulphur were bought 28 MATHEMATICAL EXAMINATION QUESTIONS. for £e than of nitre for £/; what would be the price per cwt. of each ? 282. Find five numbers, having equal differences, and such that their sum shall be 25, and the sum of their cubes 1225. 283. Two steamers A and B started from Woohvich at 11 o'clock to proceed, with the tide, to Greenwich; and it was found that their rates of speed were then as 3 to 2. A arrived at Greenwich at five and twenty minutes to 12; and, after remaining there five minutes, commenced her return, and met £ just half a mile fi-om Greenwich. At what rate was the tide running, supposing the distance between "Woolwich and Greenwich to be 5 miles ? 284. A detachment of 5000 men, ten miles to the rear of the British army, to which it belonged, was ordered to the front, that it might take part in an action just commenced with the French, whose numbers exceeded the British originally engaged by 2700 men. At what rate was the detachment marched if it arrived just in time to cause a similar preponderance on the side of the British, supposing the men to have fallen at the rate of 1130 British and 1250 French per hour? 285. Two railway-trains start, the one from London for Bristol at 6 a.m., travelling at the rate of 32 miles per hour; the other from Bristol for London at 5 A. m. How far from London will the trains meet, supposing the number of miles per hour the Bristol "up train" travels to be less by 2 than ten times the number of hours the London train takes to meet it, and the distance of Bristol from London to be 118 miles? (State under what circumstances the negative result is explicable.) 286. A and B are 300 miles distant from each other and start at the same time on foot to meet; A goes 6 miles the first day, 10 the next, 14 the next, and so on; B, on the contmry, goes 27 miles the first day, 22 the next, I7 the next, and so on; find how far they were from one another at the end of the PROBLEMS PRODUCING EQUATIONS. 29 eleventli day; and shew how it happens that B is so short a dis- tance from his starting-point. 287. Supposing that at the termination of a journey it were known that each ordinary wheel of the locomotive engine had made ten thousand more revolutions than the driving-wheel, and that the circumferences of these wheels were 25 feet and 8 feet respectively; what would be the distance passed over? 288. A cistern has three taps, two of which are of equal dimensions : when all three are open they will empty the cistern in 9 hrs. 36 min. And if one of the equal taps be stopped, the others will empty seven-ninths of the cistern in 10h.rs. 40 min. In how many hours would each tap empty the cistern by itself? 289. A column of infantry is known to contain 264 rneri more than a column of cavalry which has the same number of men in front, but six men less in depth : had the infantry, how- ever, been increased by having ten men more in depth, with the same front as before, the number of men would have been double that of the cavalry: what was the number of men in each column ? 290. Two parties of sappers, A and B, were ordered to work at two distinct saps; A commenced at 5 o'clock and ,5 at 11, and, when ordered to cease, A had completed S60 yards and B 250 yards : now if B had commenced work at 5 and ^ at 1 1 o'clock, one party would have done just the same quantity of work as the other. At what time did they leave off working, and how many yards of sap did each complete in an hour ? 291. A requisition for 35,000 lbs. of bread for the consump- tion of an army was made, in equal portions, upon a certain number of villages, but two of them having afterwards fallen into the hands of the enemy, it was found that each of the others had to supply 2000 lbs. more than the original share in order to make up the deficiency ; what was the original number of villages ? 292. There are 15 guns of a certain kind in the fort A, and 12 guns of another sort in the fort B : all the guns in A with one 30 MATHEMATICAL EXAMINATION QUESTIONS. from B are worth the same sum as all the guns in B with one from A^ viz. £1969. Find the value of a gun in each fort. 293. A triangular piece of garden ground, measuring 180 feet, 240 feet, and 300 feet in the sides, is to have a path of uniform width, and to contain one-tenth of the area of the ground, cut round it. Find the width of the path. 294. Two contractors engage to supply, equally, a certain quantity of stores, in equal weekly deliveries, in three months; but one of them fails at the end of the first month, and the other, in consequence, supplies half as much again as before: in what time will the whole of the stores have been delivered % 295. Of 20 sheep purchased by a farmer, he lost one and sold the remainder at two shillings a head more than they cost, thereby gaining ten shillings on the bargain. Find the cost price. 296. Two squads of sappers can throw up a breast- work in h hours when working together; but, when working separately, No. 1 can do the same work in k hours less than thrice the time No. 2 can do it in : how many hours will each take to do the work 1 297. A troop of horse-artillery and a company of sappers are ordered to march to Chatham, and the sappei-s start at half- past seven and the horse-artillery at eight ; they both rested one hour at noon, and it appeared that the two together had then marched as much as the whole distance; at what time would they severally arrive at their destination, supposing the artillery to arrive one hour before the sajjpers 1 298. A wine-merchant bought a cask of wine, holding 38 gallons, for £25 ; after drawing off 8 gallons for his own use, he sold the remainder at such a price as to gain 8 per cent, upon the whole; at what price per gallon did he sell the wine ] PROGEESSIONS. 31 PEOGEESSIONS. 299. If a be the first term, h the common difference, and n the number of terms of an arithmetical progression, find the value of s the sum of the series ; and apply this formula to the summa- tion of the series 5 - 2 - 9 - 16 - &c. to eight terms, and x/o'^'J^'^^x/'q'^^^' *^ twenty terms. 300. If a be the first term, I the last term, b the common difference, n the number of terms, and s the sum of a series of numbers in arithmetic progression, show that 2a + (n-l)b , l-a + b s= ^^ ^—.n and n— — = , 2 b 301. The first term of an arithmetical progression is 5, and the fiftieth is 95 ; find the series and the sum of the fifty terms. 302. Find an expression for the sum of n terms of the series 5 + 11 + 1 8 + 26 + 35 + 45 + 56 + &c. 303. Find the tenth term of the Arithmetic Progression whose first and sixteenth terms are 3 and 48 : and the sum of those eight terms the last of which is 60. 304. The sum of the pth. terms of the two series 1 + 4 + 7 + 10 + &c. and 3 + 7 + H + 15 + &c. is 172; determine those terms. 305. Insert five arithmetic means between 10 and 8. 306. Insert four aiithmetic means between - 2 and - 16. 13 307. Find the sum of 20 terms of the series 7 + -5- + ^ + *^'c. 32 MATHEMATICAL EXAMINATION QUESTIONS. 308. The sum of the first three terms of an arithmetic pro- gression is 15, and the sum of their squares is 83; find an expres- sion for the sum of n terms. 309. Find the sum of the series 27 + 22^ + 18 + ] 3^ + &c. to sixteen terms; and also in infinitum, 310. "What is the expression for the sum of n terms of an 3 Arithmetical Progression whose first term is - , and the difference of whose third and seventh terms is 3 ] 311. The first and ninth terms of an arithmetic progression are 5 and 22; find the sum of 21 tei-ms. 312. The difference between the first and tenth terms of an increasing arithmetic series is 3, and the sum of ten terms is 45 ; determine the series. 313. Find the sum of thirty terms of the series 7 + -— + 6 + n terms, shew that the ratio of the sum of the first n terms and last n terms is to the sum of the remaining 2n terms as ?♦"" - r" + 1 to r". 347. Prove that if quantities be in geometrical progression their differences are also in geometrical progression, having the same common ratio as before. 348. The first and sixth terms of a geometric series are I and 243 ; find the sum of six terms, commencing at the third. 349. The sum of a geometric series is 117185, the first term 6, and the common ratio 5 ; determine the number of terms, by- logarithms. 350. Show that the sum of ii terms of the geometric series whose common ratio is r and first term a, when divided by the sum of its first two terms, is equal to of the sum of n terms ^ r+ 1 of the series whose first term is 1 and common ratio r; whatever be the value of a. 351. If S^ and aS^^ represent the sum of n terms, and the sum of p terms of two geometric progressions having the same first term, and whose common ratios are respectively r and t, show that the ratio of aS',, to «S"^ may be expressed by the fraction r"-' + r"-' + r"-^ +r' + r-i-l f-' + t^-" + f--' +t'' + t+\' 352. If four quantities of the same kind be proportionals, the greatest and least of them together are greater than the other two together. 353. Prove that if a, 5, and c be three quantities in harmonic progression, -, ^ and - are in arithmetic progression. Horner's solution of numerical equations. 37 354. If an ordinary train leave the terminus at 4 p. m, at the rate of ten miles an hour, and increase its speed at the rate of two miles an hour every ten minutes ; and an express train leave the same terminus at 4 hrs. lOmin. p.m. starting at the same speed as the other, but increasing its speed every ten minutes, in the ratio of 5 to 7 j what would be the relative position of these trains at the end of fifty minutes after the starting of the express ? 355. Find the number by which the numbers 20, 50, and 100 being severally increased, the results may be (1) In geometrical progression : (2) In harmonical progression. 356. Shew that three different quantities cannot be in har- monic progression and also in arithmetic progression; but that, if they could, they must also be in geometric progression. 357. If a, b, c be in harmonic progression, show that 111 1 „ - + - + 7+ J = 0. a c a—b c-b HORNEE'S SOLUTION OF NUMERICAL EQUATIONS. 358. Show that if a polynomial in z be divided by z-v, the final remainder will be the same as the original polynomial, only having v in the place of z. 359. Apply the property stated in the last question to the determination of the value of the expression v^ - 5v^ - 3v^ -5v + 2, when v = ~'5. 38 MATHEMATICAL EXAMINATION QUESTIONS. 360. Find, by synthetic division, the values of the expression w^ - ^V3^ - 12i«' + w> - 53, when t^j = 3, w = 6, w- '58, and w = - 2*73. 361. State the principle upon which the method of deter- mining the value of an algebraical expression by synthetic divi- sion depends, and apply it to the determination of the value of y*-4y' + 9y + 3, wheny = --3. 362. Find, by synthetic division, the values of the following fractions, when x = 3: a;* - 3a;' - 7a;' + 25a; -12 a;'- 10a;* + 60a;* + 10a; + 2 a;' + 3a;' -4.0a; + 66 ' a;* + 2a;'- 15a;*- 2a; + 6 ' 363. Find the value of a;' - 7x^ + 5a; + 8 ; when a; = - 1 -2. 364. What is'the numerical value of when y = - -16? 365. Find the value of a'-6a*+12a*+3a+8 when a = - 1*1 ; by synthetic division. 366. Determine the value of a;* - 2a;* + 5a; + 10 when a; ="12, by synthetic division. 367. Find the value of a;' - 17a;' + 2a;* + 20, when a; = 4 ; by division. 368. Show that if any equation be divisible without re- mainder by a; -a, a is a root. 369. Show that every equation has the same number of roots as there are units in the highest exponent of the unknown quan- tity in it. Horner's solution of numerical equations. 39 370. Form the equation whose roots are - 3, 2 + 3 J{- 1), ^~Sj(—l) and 5, without multiplication, and prove its cor- rectness bj multiplication. 371. Find the middle term of the equation whose roots are 1, - 2, 3 and — 4 ; without determining any other term. 372. Determine by inspection the roots of the equation ax^ — {b + ac- a^d) x^ + (he - ahd - a^cd) x + abed = 0. 373. Determine the last term but one of the equation whose roots are I, - 2, 3, - 4, 5 and - 6, without determining any other term. 374. Of how many products is the coefficient of the middle term of an equation having 8 roots, the sum ] 375. Form the equation whose roots are 2, —5, and **^/(-7). 376. Find the last term but one of the equation whose roots are 1, 2, — 3, - 4 and 5, 377. The second teim of an equation is 9aJ*j its last term is 2520, and three of its roots are 5,-6 and — 7 ; find the remain- ing roots. 378. Two roots of the equation X* - 35x'' +90x-56==0 are 1 and 2 ; find the remaining roots. 379. The roots of the equation are in arithmetical progression : find them. 380. Show that if the roots of the equation ax* + bx^+cx^ ■\-dx + e = be of the form a, ft - and ^ , then a = e and b = d. 40 MATHEMATICAL EXAMINATION QUESTIONS. 381. Two roots of an equation whose second and last terms fire respectively - llic^ and + 30, are 1 and 5 ; what are its other roots ? 382. Determine the last term but three of the equation whose roots are 1,2, 3, - 1, - 2, and - 3 ; without finding any other term. 383. Assuming that if an equation in x be divisible by a?- r, r is a root of it ; state how it appears that if p, q, 5, t, v, &c. he also roots of such an equation, the successive quotients arising from the division of the original equation by any number of the binomials x—p, x — q, x—r, x — s, &c. will, when equated to 0, be equations whose roots are, in each case, the second terms of the binomials by which it has not been divided. 384. Find all the roots of the equation x^- 2x*- I0x^+20x'+9x-18 = 0. 385. Determine whether any of the digits less than 6 are roots of the equation £C° - 6a;' + 9a;* + 20?" - ISx* + 26a; - 15 = 0. 386-. Find all the roots of the equation X* - 8x^ + 24a;' - 32a; - 9 = 0. 387. For what value of n will the roots of the equation 2a;^ + 8a; + ^ = be equal ? 388. Show that if an equation have p roots each equal to ?•, and q roots each equal to s, the limiting equation will have p- I roots each equal to r, and q~l each equal to s : and state the law of the formation of the coefficients of the limiting equation from those of the original. 389. Supposing (y-py '{!/- gf '{2/ -ry . {y-sf . {^-ty . {y -v) to be the form into which the greatest common measure between an equation and its limiting equation can be transformed ; state HORNER S SOLUTION OF NUMERICAL EQUATIONS. 41 wliat you would infer therefrom with regard to the roots of that equation. 390. The equation 4x^ - 20x^ + 25x^ + lOic^ - 20aj -8 = lias equal roots ; find them. 391. Determine the equal i-oots of the equation x^- 5x^ + 5x + 2 = 0; and, by means of them, complete its solution. 392. Determine the equal roots of the equation x' -l6x^+ 90x^ - W8x + 169 = 0. 393. By means of its equal roots solve the equation x^-x''-l6x-20 = 0. 394. Solve the equation x*-6x^+15x'- 18x + 9 = 0, which has equal roots. 395. Detei-mine the equal roots of the equation x' + 5x' - 5x^ - 4'5x' + 108 = 0. 396. By means of its equal roots, solve the equation 2^* - %' + 1 5^ - 20y + 1 2 = 0; 397. The two equation* a'- 6x'+llx- 6=0, and ic'-14a;^ + 6Sic-90 = 0, have one root common, find it j and thenee determine the re- maining roots of both. 398. Prove that if the alternate signs of an equation be changed, the roots of the new equation will be the same as those of the original with contrary signs. 399. Prove that if we wish to form, from any equation, another, whose roots shall be greater, or less, by a given quantity' 42 MATHEMATICAL EXAMINATION QUESTIONS. s, than those of the original, the coefficients of the new equation will be the successive final remainders left after dividing the original continually by x + s, or by x — s; and state when the positive, and when the negative sign is to be used. 400. From the equation a;' — 12a;— 15 = form two others, one whose roots are greater by 3*97, and another whose roots are less by 1*5 than the roots of the original. 401. Determine the equation whose roots are the roots of the equation x^ + 10a;* + 40.'c' + 80a;'' + 80a; + 32 = 0, increased by 2 ; and from the form of the result state what are the roots of the given equation. 402. Eliminate the second term from the equation a;*_3a;»-7a;» + 6a;-2 = 0. 403. Supposing the roots of the equation ax" + bx"'^ + cx''~' +^a;' + ^ + wi = to be j(?, q, r, «, &c., show that the equation whose roots are :;;' ;;» z^ 7' ^^' ^iUbe p q r 8 rnx" + lx*~^ + kx"~^ + ca;' -I- 6a; + a = 0. 404. If the roots of the equation ax* + bx^ + cx' + dx + e = be r, s, t, and v, what is the equation whose roots are -, -, -, 7* 8 t and - ? V 405. Give the principle upon which the reciprocal equation may be made available for the determination of the imaginary roots of an equation. 406. Define a recurring equation; state what is the charac- teristic property of the roots of such an equation ; and also why this property enables us always to assign the value of one root of a recurring equation of an odd degree. HORNER'S SOLUTION OF NUMERICAL EQUATIONS. 43 407. Solve completely the recurring equations (1) x^ + Sx^'+^x" + 5x+l = 0. (2) x' - 8x* + 9x' -9x'+8x-l=0. 408. Solve the following, as recurring equations: (1) x* + 2x^ + Sx' + 2x+l=0, (2) x'-Sx^ + Jx^-Jx^'+Sx -1 = 0. 409. Solve completely the recurring equations (1) a;*-10a:^ + 26x*-10a;+l=0. (2) x'-6x'+7x^-7x'' + 6x-l=0. 410. Solve completely the recurring equation x' + 9x^ + iGx^'-lGx'-gx -1 = 0. 411. Solve completely the recurring equation x^-9x* + 16a;' +-[6x'-9x+\=Oy and show that the roots are of the recurring form. 412. Solve the recurring equation x'-Ux* + 36x^-36x'+Ux- 1=0, and show that to each root there is a corresponding one which is its reciprocal. 413. Solve completely the recurring equation x' -I3x* + 36x^ + 36x' - ISx+l =0; and show tkat the roots are of the recurring form. 414. Solve the recurring equation x' + ^x^-Sx^' + Bx'-^x-l = 0, giving all the roots, and showing that half of them are the reci- procals of the other half. 415. Solve completely the recurring equation x'-2x* + 5x^'-5x' + 2x-l=0. 44 MATHEMATICAL EXAMINATION QUESTIONS. 416. Solve completely the recurring equation x^ - 2x* + 3x^ - &c. = 0. 417. If+- + + - + +be the series of signs in any equa- tion, show that if a positive root be introduced there will be at least one more variation than before in the result, and if a nega- tive root be introduced there will be at least one more permanence than before; and state the rule of signs to which this gives rise. 418. State and show the truth of De Gua'a criterion of the presence of imaginary roots in an equation. 419. What is the smallest and what the greatest number of imaginary roots which can exist in an equation of the form A^x' + A.^x' - A^x' + A^^OI 420. Apply De Gua's criterion to the determination of the number of imaginary roots of the equation x'' + A-x'" -7x^ + Sx" - 2x' - 6x* - 8x + 5 = 0, 421. Determine the number of imaginaiy roots indicated by the absent terms of the equation x' + 3x^ - 2x' + 1=0. 422. Can a cubic equation have all its roots imaginary? Give a reason for your answer, and state generally what class of equations must have at least one real root. 423. What is the least number of imaginary roots which an equation of the form x' -ex^ + h = can have 1 424. Two roots of the equation x^ + 2x' - lA^x"- - 12x'' + 4^7 x^ + 2lx- 70-^0 are 2 and - 5 ; find all the other roots. 425. One root of the equation x^ - 4^x^ + 5x^ + 8x - 14i = is 2 + J(- 3) ; find all the other roots. hoener's solution of numerical equations. 4o 426. Determine the positions of the roots of the equation Sx" - 104ic^ + 465a;' - 806x + 388 = 0, by showing that each root lies between numbers different from those which limit any other root. 427. Show that the two positive roots of the equation X* — 4£c^ + x^ + 6x + 2 = are real, and find the first decimal of each of them. 428. Solve the equation x^ + 2x^ -3x+ 4f = 0, by Horner's method; giving such roots as are real to six places of decimals. 429. Find the character and positions of all the roots of the equation x* - 4cc^ + Hx^ - 5x~6 = 0, and find the negative root to five places of decimals. 430. Solve the equation x^- 3x^ + 5x + 10 = 0j by Horner's method, giving the roots to six places of decimals. 431. Solve the equation x^ -Ux + 12 = 0,hj Horner's method, giving the smallest positive root to five places of decimals, and the integer figures of the other roots. 432. Find the real roots of the equation x*-llx^ + 33x'-40x+19=-0 to seven places of decimals. 433. One root of the equation x* - 4>x^ +5x-+2x + 52 = is 3-2 J(- 1) ; determine the remaining roots. 434. Solve the equation, giving all the roots, x^+17x'-4>6x + 29 = 0. 435. Solve the equation a;' - 8aJ + 12 = 0, by Horner's method, giving the real root to eight places of decimals; and show that the other two roots are imaginary. 436. Determine the position and character of the roots of the equation x* - 6x^ - 14a;^ - l6a; + 8 = 0, and find the greatest root to nine places of decimals. 46 MATHEMATICAL EXAMINATION QUESTIONS. 437. Solve the equation x^ — 4>a? + 3a; + 5 = 0, by Homer'a method, giving the real root to five places of decimals. 438. Detei-mine the positions of the roots of the equation a;*- 14a:' + 55x^ - 12a;- 184 = 0, and find the least positive root to eight places of decimals. 439. Solve the equation y* — 2^ + Sy*— 4y — 5 = 0, giving such of the positive roots as are real, to seven places of decimals; and determining merely the position of the negative root. 440. Solve the equation a;* - 3a;* — 4a; + 7 = 0, by Horner's method, giving all the real roots to six places of decimals. 441. Find such of the positive roots of the equation a;*- 20;" + 9a;- 14 = as are real, commencing the conti-action at the third decimal; and show how it appears that the others are imaginary. 442. Determine all the roots of the equation X* - 12a;' + 44a;' - 48a; + 16 = 0. 443. Determine the characters and positions of the roots of the equation x*~19a?+ 132a;'' - 302a; + 200 = 0. 444. Solve the following equations by Horner's method, giving the roots to eight places of decimals : (1) a;* - 6a;' + 8a;' + 12a; -20 = 0. (2) a;* - 1 2a;' + 48a;' -72a; +36 = 0. 445. Solve the equation 4a;* - 20a;' + 133a;' - 1 20a; + 29 = 0, giving all the roots, 446. Solve completely the equation x* - 2a;' + 7a; - 6 = 0. 447. Find the positions of the roots of the equation a;' + 2a;' - 12a;' + 15a; 4- 5 = 0; and determine the least negative root, by Homer's method, to eight places of decimals. HORNER S SOLUTION OF NUMERICAL EQUATIONS. 47 448. Solve the equation x^ - 3x^ + 9i»^ + 2a; - 1 = 0, giving all the real roots. 449. Find the roots between 4 and 5 of the equation X* ± Ox^ - 75x^ + S18X- 322 = by the method of reciprocals. 450. In the equation z^ + 3cz-.2d=0j show that the value of z is c U{d+j(d'+<^] md+^{i+l). 588. Sum the series a + 1 + 2 (a + 2) + 3 (a + 3) + 4(a + 4) + {9.n + zf 5{2n + zy j in order to obtain the Napierian logarithm of 12 ] 614. Assuming the "Exponential Theorem," show that , 1 +x ^ / x^ x^ x^ , \ And prove from first principles that . log. X log,aj=-^^. ^ log, a ■ession steps you would take to calculate the numerical value of 615. "Write down the expression for \ogi^-\-z) and state the 1 the modulus of the common system of logarithms. 616. Given log, 5 = 1 -60944, calculate log, 7 to five places. 617. Find the value of a; in the equation 5' = 300. 618. Find the values of x and y in the equations 2^=72/; 3"=10y. 619. Find the values of a; and y from the equations |' = 4and7'=3''. 64 MATHEMATICAL EXAMINATION QUESTIONS. 620. Find the values of x and y in tlie equations o 621. Given 2". T = 80000 and 3^ = 500, find x and y. 622. Find the value of x in the equations : (1) 3^-729. (2) a^6^=c. 623. Find the values of x and y in the simultaneous equa- tions a'^ }f^~' = c'^ and d^ = h". 624. If a series of numbers be in geometric progression, prove that their logarithms are in arithmetic progression. 625. What tenn of the series 2 8 32 128 . 2097152- 5^15^45^r35^^^^-^^ 295245 ' 626. Find, by logarithms, the value of ^{bc) + /(^) , when a = 6340-5 18, 6 = 7-36'0591, c = -003758426. 627. Point out the principle and use of the ^^ ^Arithmetic Compliment'' in logarithmic calculations and calculate, by log- arithms, the value of J- y -00754326' X 78-34295 x 8172-371* x -00052 64285-7P X 154-27* x -001 x 586-7983^ ' taking care to arrange the work in a proper form. 628. Find the value of j^832^x 5793-64* X -7842613 •000327* X 768-94' x 3015-28 x -OO7' ' bj the aid of logarithms. 7: LOGARITHMS. 65 629. Find the value of 3-5724^ y -00753286 x 3426-002' x 7*854 X V 30'-; 408-62 V 30-47269' x ^'-03278971 x -5163084' by logaritliins. 630. Find the value of V-00357 X 628-4931' X 73-0875' V-' \J 49-3284' X -031264' x -00529 ~ V ' by logarithms. 0047238' 5423701 631. Find, by the aid of logarithms, the value of ^ ^ 7-189546 X 4764-2' x -00326' 7^ •0004895361 X 457' x 5764-387' * 632. Find, by logarithms, the value of 7 3-14159 X 4771-213 x 2-718282' 30-103* X -4342945^ x 69-897* 633. Find, by logarithms, the value of ^ '-03271' X 5fi'4<29SC^ X -7754231' 7- 32-76894 x -000371'' 634. Find, by logarithms, the value of ' '732-0561' X -0003572* X 8979306 7 4227 -984" x 3-457391 x -0026518' * .o^ T,. , ,, , . V/ 7932 X -00657 x -8046392 \ 635. Find the value of ^^ :^ 3274 x -6428 j' ^^ logarithms ; and find the values of cc and y in the equations a = b and ar^ = c. 636. Find the value of Vf 7812-934' X ^(18-5374) X ^(-7526821) x -6173958 ) V 17(59-60827) X 7(756-0083) x 1726-953 x7(-0733)/' by the use of logarithms. c. 5 66 MATHEMATICAL EXAMINATION QUESTIONS. •. 637. Find the value of 7-120635 X ^(-13274) x -05738921 7: ^(•4346829) X 17-3854 x7(-0096372)' by the use of logarithms. 638. Find, by the aid of logarithms, the value of r3-075526' X 5771-213* x -0036984^ x 7-74)^ rw \ 7225851 X 327*9341^ x '8697003 639. Find the value of V7056421 X 301-572' x 7*830721^ x -54836 7 •416327^x6284731 by logarithms. 640. Find the value of the following expression by log- arithms : y 3-5 76428 X -00 8630472 x 598163-2 x^y(2li) 286-9734 X i/(-0069847) x 2708637^^(^76) 7i 641. Calculate by logarithms the value of the expression when a= 574-3268, /= -03572, = 21-3641, p= 32-6841, k= 3-87964, r = 78561-32, m = 76-49573, s= 5736-058, w = 12-86097, 642. Find, by the aid of logarithms, the value of ' ^-0057146 X 73-24981 x 2793-468 a\c.d' e.A ' a = 574-3268, 6- •0372, c = 73-5486, d = 1807-463, e = 179-6284. 7 774-2605 X -0000829 x 3461 '724 * 643. Find, by logarithms, the value of 7(36-51942)^ X (-005327481)" x 863157*6 7 (-07136529)* X (73-69421)' GEOMETRICAL DEDUCTIONS. G7 644. If a person be entitled to receive an annuity of £6i> for ten years ; what is the sum which he should receive in one present payment in lieu of the annuity, assuming 4 per cent, compound interest 'I Probabilities. 645. The probabilities of three Cadets -4, J^, and C qualifying 11 7 at this examination, are respectively j » -r , and -— ; what is the probability of one at least of them qualifying 1 646. A man makes six *' outers," four "centres" and two "bull's-eyes" in twenty rounds rifle practice; what is the pro- bability of his hitting the target every time in the next four rounds 1 647. "What are the odds in favour of throwing *'aces" at least once in three throws with a pair of dice 1 GEOMETRICAL DEDUCTIONS. 648. Prove that if the points of bisection of the sides of a triangle be joined, the triangle so formed will be equal to one- fourth of the whole triangle. 649. Divide a triangle into two parts which shall have the same ratio to one another as two of the sides of the triangle, by a straight line drawn through the angular point common to those sides. 650. It is required to cut oflf from a given triangle one-fourth part of it, by a straight line drawn parallel to one of its sides. 651. If a circle be described upon a side of a triangle as diameter, and its points of section with the other two sides be joined; the triangle so formed will be similar to the whole tri- angle. 5—2 68 MATHEMATICAL EXAMINATION QUESTIONS. 652. If two straight lines be drawn from tlie same point, cutting a circle, and the alternate points of section be joined; the triangles so formed will be similar to one another. 653. If through the angles of an isosceles triangle which has each of the angles at the base double of the third angle, and is inscribed in a circle, straight lines be drawn touching the circle, an isosceles triangle will be formed which has each of the angles at the base one-third of the angle at the vertex. 654. From one of the angles C, at the base of an isosceles triangle AJ3C, of which the vertex is -4, it is required to draw a straight line CD to meet AB in D, so that CD shall be a meiin proportional between AC and BD. 655. If, at any point D, in the side AG of the triangle ABC, a straight line DE be cU-awn making an angle ADD equal to the an^le ABC, and meeting AB in D, prove that if DB and DC be drawn, the angle ADB is equal to the angle ADC. I 656. If from any point in the common tangent to two circles drawn through the point where they touch each other externally, as a centre, a circle be described cutting those circles; and straight lines be di*awn from that centre through these points of intersec- tion; then the other points of intersection of the straight lines with the circles will also be in the circumference of a circle. 657. If from the angle ^ of a triangle ABC, a straight line AD be drawn to bisect that angle, and to cut the op^iosite side in D; and from D, two straight lines be drawn respectively parallel to the remaining sides; the parallelogram thus formed will have all its sides equal, and each of them will be a mean proportional between the remaining segments of the two sides of the triangle. 658. If from the angles A and C, of the triangle AJ3C, per- pendiculars, AD and CD, be let fall upon the opposite sides, prove that DD being joined, the angle CAD is equal to the angle CDD; and that the rectangle CB, BD is cq-ial to the rectangle AB, BD. GEOMETRICAL DEDUCTIONS. GO 659. Find a mean proportional between two given straight lines. Also construct an arithmetic mean, and a harmonic mean between two given straight lines. 660. If two contiguous chords be dra^vn in a circle, and per- pendiculars be let fall upon them from the centre, the angle con- tained by these perpendiculars will be equal to the angle in the remaining segment formed by joining the extremities of the chords. 661. Divide geometrically a given straight line into two parts, so that the square described on the greater part shall be double of the square described on the other part. 662. Describe a circle which shall touch a given circle, and also touch a given straight line in a given point. 663. Bisect a given triangle by a straight line drawn through a given point in any one of its sides. 664. If a straight line be drawn to bisect the angle formed by two straight lines, one of which is drawn to the focus from any point in the parabola, and the other perpendicular to the directrix, that line will be a tangent to the parabola at that point. 6Q5. Describe a circle which shall touch a given straight line in a given point and also touch a given circle. Q66. Prove that the three straight lines drawn from the angles of a triangle to the points of bisection of the opposite sides pass through the same point. 667. From a given point without a circle draw a straight line such that the part of it intercepted within the circle shall be equal to the side of the inscribed equilateral pentagon. 668. If two circles cut one another, all parallel straight Unes drawn through the points of section and intercepted by the outer circumferences of the circles will be equal. 70 MATHEMxVTICAL EXAMINATION QUESTIONS. 669. If a circle be described about a triangle, and a perpen- dicular from tlie centre upon one of the sides be produced to meet the circle, the line which joins this intersection with the opposite angle of the triangle will bisect that angle. 670. If two circles touch each other and also touch a straight line, the part of the straight line between the points of contact is a mean proportional between the diameters of the circles. 671. If any point be taken in the circumference of a circle, and lines be drawn from it to the three angles of an inscribed equilateral triangle, prove that the middle line so drawn is equal to the sum of the other two. APPLICATION OF ALGEBRA TO GEOMETRY. 672. Given r and r^, the radii of two concentric circles, and I the distance from their common centre of a point -i, (exterior to both of them) ; find the point K in the circumference of the outer circle, such that AK produced shall have its segment within that circle divided by the circumference of the other into three equal parts. 673. Given the segments of the hypothenuse of a right-angled triangle made by the sti*aight line which bisects the right angle ; determine the length of that line, the sides of the triangle, and the perpendicular from the right angle on the hypothenuse. 674. Show that the length of the perpendicular let fall from the opposite angle of a triangle whose sides are a, 6, c upon the side c, is expressed by 2 where s =r - (a + & + c). APPLICATION OF ALGEBRA TO GEOMETRY. 71 675. Given tlie area of a triangle = a^, and that of it^ in- scribed square =s^j find the perpendicular and the base of the triangle. 676. Find the point in the side of a square, through which, if a straight line be drawn to the opposite angle, the triangle so formed will be one-third of the square. 677. Given the ratio of the sides of a triangle, together with both the segments of the base made by a perpendicular from the opposite angle; find the sides. 678. Find the radius of the circle which circumscribes a tri- angle whose sides are a, h and c. 679. From a given point without a given circle draw a straight line which shall be divided in extreme and mean ratio by the circumference. 680. Given the base of an isosceles triangle, determine the sides when the diameter of the circumscribing circle is d. 681. In a triangle ABC, given AG = CB, AB = b, CD (the perpendicular on AB) = a, and BU a segment of the perpendicular = - : it is required to draw through B a straight line FBG, which n shall bisect the triangle. 682. Given the two sides a and 6 of a triangle and the length of the line d, which bisects their contained angle, find the seg- ments into which it divides the base. 683. One angle of a triangle, the perpendicular from that angle upon the opposite side, and one of the segments into which the perpendicular divides that side, being given, find the other segment. 684. The base of a triangle is 25, the perpendicular let fall upon it from the opposite angle is 12, and the rectangle contained by the sides is 300 ; find fclie sides. Explain the meaning of the 4 answers. 72 MATHEMATICAL EXAMINATION QUESTIONS. DESCRIPTIVE GEOMETEY. OrTHOGKAPHIC AND HORIZONTAL PROJECTIONS. 685. Define the terms "line of level," "planes of projection," "trace," and "plan." 686. Sliow that the straight line which joins the projections of the same point is perpendicular to the line of level. 687. If a plane be perpendicular to one of the planes of pro- jection, prove that its ti*ace upon the other plane is perpendicular to the line of level. 688. Prove that the horizontal and vertical projections of a point are in the line drawn through either of them, perpendicular to the line of level. 689. State the principles upon which the projections of the intersection of two planes whose horizontal and vertical traces are given, may be determined. 690. Prove that the horizontal trace of the vertical project- ing plane of a straight line is perpendicular to the line of level. 691. The plan and elevation of a straight line being given, find its traces. 692. The traces of a straight line being given, determine its plan and elevation. 693. Find the intersection of a given line with a given plane. 694. Pind the point of intersection of three given planes. 695. The plan of a straight line in a given plane being given, find the elevation of that line. 696. Through a given point draw a plane parallel to two given straight lines. PLANE TRIGONOMETRY. 73 697. From a given point draw a straight line perpendicular to a given plane, and find its true length. 698. Construct the angle of inclination of two given planes. 699. The true representation of any plane rectilineal figure upon a given plane, being given, find the plan and elevation of the figure. 700. If (<^jg5jjj) be the horizontal projection of a straight Hue; state clearly the principles on which you woiild determine its absolute length, and its inclination to the horizon. 701. The horizontal projections and indices -of two points being given, state how the inclination to the horizontal plane of the straight line joining these points may be determined. 702. If the horizontal projections of the points a^^, h^^ be twenty units of the scale of the plan distant from one another, at what distance from the former will be the horizontal trace of the line joining those points ? 703. The scales of slope of two planes being given, determine the projection of the line of intersection of those planes. 704. The scale of slope of a plane being given; show how its trace upon the horizontal plane may be found. 705. Show that in a system of horizontal contours the in- creasing proximity of the lines indicates an increasing degree of slope; and vice versd. PLANE TEIGONOMETEy. 706. Define *'the chord," "the tangent," and "the secant" of an arc ; and show that chd 2a = 2 sin a. 707. Express the "tangent" and ''secant" of an arc in terms of its " sine." 74 I^rATHEMATICAL EXAMINATION QUESTIONS. 708. Show that if a" be the angle in a segment of a circle whose radius is r, the base of the segment is 2r sin a". 709. The difference of two arcs of a circle is 20 grades, and their sum is 48 degrees ; find the arcs. 710. Find, geometrically, the values of tan 30", tan 45", and tan 60". 711. Find the angle whose circular measure is |. 712. Show that sin 75" = '^^—, and sinl5" = '^^^\"\ 713. Give accurate line-definitions of the sine, tangent, se- cant, and cosine of an arc of a circle; and detennine the numerical values of cos 30", sin 45", sec 45", and tan 1 20" ', the radius of the circle being imity in the first two examples, and 10 in the latter two. 714. Trace by a figure the changes in the cosine of an arc of a circle, as the arc varies from to 360". 715. The radius of a circle being 16 feet, find the number of degrees, &c. in an arc of 6 feet, and also the circular measure of the subtending angle at the centre. 716. Find the values of tan30", tan 60", tan45" and tan 7° 30', and thence determine tan 15" and tan 75". 717. Find the length of the arc whose circular measure is 2. 718. Prove, geometrically, that c,;^ ^ X 1 1 ^ sin a sm a = tan a = , and tan a = ; cosec a cot a cos a and thence show that 1 sin a . cos a tan a + cot a PLANE TRIGONOMETRY. 75 719. In a right-angled triangle ABC^ express the ratios AB AB AG BG AG' BG' CB' AB' as trigonometrical functions of the angle A, when G is the right angle. 720. The length of an arc in inches is expressed numerically by five times its circular measure, and by ten times the reciprocal of its angular measure in degrees; find the length of the radius, and the number of degrees in the arc. Prove the relations: 721. cosec a . cos^ a = cosec a - sin a. 722. cota.coseca = . 723. cos2^= ,— - '— 2:^- sec a — cos a 1 + tan tf ^ _ , cos a + sin a , 724. -. = tan 2a + sec 2a. cos a — siu a Investigate the following relations : 725. sin ^ + sin 5 = 2 . sin ^(A + B) cos J (i - B). 726. cos 30" - cos 70" = 2 sin 50" . sin 20". ^^„ ^ A 1-cos^ 727. tan — = — -. — j- . 2 am A 728. tan^ a + cot^ a = 2 + 4 cot' 2a. Prove the relations : 729. tani= ^'"^ 2 1 + cos A ^_- tan^ + tan^S ^ ^ x z? 730. i ^ = tan ^ . tan B. cot A + cot B 731. cos {A + B). cos {A-B) = cos' A - sin' B. 732. An arc of 40" on a circle whose radius is 6 inches is found to measure a inches; and this length applied to the en- cnmference of another circle is found to cover an arc of 25"; what is the radius of the latter circle? 733. Give the algebraic signs of the sine, cosine, tangent and secant of each of the arcs 70", 110", 195", and 280"; illustrating the result given by a geometrical figure. 734. Prove that sin 30" = -, and sin 45" =^ cos 45" = - ^72 to 2 2 radius unity; and thence find the values of tan 30", sin 75", and cos 25" 30', to radius 10. 735. Prove that sin a = - cos r— - a) and tan a = sec a • cosec a. Prove the relations : 736. sin f- + aj = cosa. 737. cos (- + aj = -sina. 738. tan (tt- a)-- tan a. 739. tan (tt + a) = tan a. 740. cot a + tan a = 2 cosec 2a. 1 — 2 sin^ a 1 — tan a 741. 1 + sm 2tt 1 4- tan a cot ^ + tan 743. sec 26 = cot <{> — tan <^ 744. 1 + cos 2$ . cos 2<^ = 2 (sin' . sin' + cos' . cos' ). 745. tan-^l + tan-^^ = 45". 4 5 746. sin2(9 = tan0.(l+cos2^). 747. cos 2a=]~^'^,^ ^ ' I + tan* a 748. tan' a + cot' a + 2 = sec' a cosec' a. 749. 'J'^^^^^oot'^.cot^-l. cos a — cos p 2 2 50. tan^ - PLANE TEIGOXOMETRY. 77 2 sin (9 - sin 20 2 2 Bin + sin 2d ^^- tan a + sec a ^ /.^ o-N, a / 01. = tan 45 + - tan - . cot a + cosec a \ 2/ 2 /it 1 \ 752. Prove that tan f ~- - - j = sec i - tan i ; and show that if cos (a - /3) . cos (0 + cjj,) := cos (a + /5) cos {0 - ), then cot a . cot y8 = cot ^ . cot 0. Show that 753. cos Sa = 4 cos^ a — 3 cos a. 754. sec 2a = 1 + tan 2a . tan a. 755. tan^ a - sin^ a — tan^ a . siu^ a. 75C. cos-'y~-cos-'4±f^ = 6o". 757. .sec ± tan ^ - cot 45% " . 758. sec^^ = '"'' -^^ . \ 2/ 1 + sec 20 759. 2 760. cos* (9 - sin' ^ = cos 261. tan^.cosec^^ ^q-^ sin a a ^ -^ '"^' 1-cosa 2' 762. 2 cosec 2(9 - sec ^ ^o /, ,o ^\ = cot ( 45 + 2 cosec 26^ + sec t^ \ 2/ 763. sin^ a 2 ^ cos - 764. 2 sin (a — jS). cos a = sin (2a — P) — sin y8. 765. 2 cosec 2^ = sec ^ . cosec 6. 766. ABC is a right-angled triangle, B being the right angle; prove by means of a figure, and Eucl. vi. 4, that , AB ^ , BG cos ^ = -rjz , tan ^ = — — ; . AC Ali 78 SrATHEMATICAL EXAMINATION QUESTIONS. Prove that ^^^ /. a + /3 a+jSW. a-13 a-/?> 7G7. I sm + cos ) . ( sm — -^ + cos \ 2 2 / \ 2 2 , = sin a + cos /S. 768. tan-i.sin-^^ = ta.-.^. 61 . 2 tan 769. ^±^^ = tan»(45Via). 770. sin^ = 5-. 1 - sm a ^ ^ ' . 9 ^ 1 + tan' - 771. tan 50"+cot 50"=2 sec 10^ 772. tan~^ - + tan"' - + tan"' - + tan"' - = — • 3 5 7 8 4 773. Given cosec 2a — sin 2a = tan a ; find sin a. 774. Find the values of a; and y in the simultaneous equation 2 . sin a: = 3 cos y and 3 cot y = 5 sec x. 775. If sin 6 = sin 2^, find cos 6 ; and find cos when tan = cosec 2^. 776. Show that, in any plane triangle, tan = tan [

and 6-, and thence show that 3 sin ^9 = 5 sin 2<^ j interpret the negative signs wherever they appear in your result. 778. Find x from the equation cos (/8 + x) _ m sin p cos (a — a;) n sin a 779. Find the value of 6 in the equation tan (45° + ^) = 3 tan (45°-^ PLANE TRIGONOMETRY. 79 780. If COS (a-p-O) cos (a + /3) + cos (a + p + 6) cos (a - ^) = 0, find 0. 781. Show that if sin a + sin y = 2 sin ^ . cos (jB — a), then a, (3 and y are in arithmetical progression. 782. If tan (a + 6) = n cosec 2a - cot 2ct ; find 0. 783. If tan ^ = /x j find sin 2/3. 784. Given sin ^ . cos <^ =^ sin <^ (2 cos ^ + 7 sin <^), and sin ^ = 3 sin <^ ; find tan . 785. Find the value of the tangent of the arc tan~^ (a + b) + tan~^ (a — h). 786. Given sin {A+B)=^^ sec {A - B), and sin {A-B)= — sec {A + B) ; find A and ^. 787. A and jB are two arcs of a circle, radius unity; find them, if sin A . , tan A . -r— ^ = ^2, and ^ = ^S. smi? ^ tani? ^ 788. Show that, in a plane triangle, -4-5(7 : sin A sin ^ sin G _ /s.{8 — a) ^ /s.is-a) , a + 6 + c and cos — - = A / — ^ ^^ when s 80 MATHEMATICAL EXAMINATION QUESTIONS. 789. The sides of the triangle ABC being denoted by «, 6, c, show that a^ = h^ -k-c^-9.hc cos J, A /s (s — a) cos 2 "v' 6c Also, find the least and greatest angles of the triangle ABC^ to the nearest second, by means of the formula for the tangent of half an angle, when AB = 2264>, BC = 1854>, AC=^0l6. 790. Two sides of a plane triangle are 10 and 20 and the included angle is 60"; find the remaining angles and side without the use of tables. 791. In a plane triangle given a = 5847328, 6 = 367-4001, and B = ST 42' 15" ; find the remaining angles and sides. 792. A^ B, C ; a, b, c being the angles and sides opposite to them in a plane triangle, prove the relations cos A and thence prove that 2 V b' + c'- a' 2bc i '(.. -b)(s -c) {a -a) and a= {b-c) sec 6, /) X -1 (2 sin IC .,, .) and also c={a+b) cos <^, Q 2 cos - - if : Bin<^ = --~V(«i)- Prove also that the perpendicular let fall from C upon c may be expressed by a^ sin B + b^ ^in A a + b ' PLANE TRIGONOMETRY. 81 793. The three sides of a plane triangle are 3050, 7854? and 5398 j find its angles. 794. Wishing to know the height of the crest of the parapet of an inaccessible fortress above the spot on which I stood, I measured its angle of elevation above the horizontal plane through that point, and retreating 350 yards up a slope inclined 10° to the horizon, I measured the corresponding angle of elevation: what was the height of the observed point above the horizontal plane passing through the first station, supposing the measured angles of elevation to have been respectively 15" 27' 40" and 12" 32' 10"? 795. Wanted to know the distance between the flanked angles of two adjacent bastions C and i), of a besieged fortress; I measured a base line AB = 250 yards, and then found that the angles subtended at each extremity of the base by the line CD, were CJi) = 28" 37' 40", Ci?2> = 25" 18'50", and that the angles CAB and ABB were respectively QQ" 10' 20" and 92"17'10"; what was the distance CD'i 796. Wanting to find the breadth of a river having a straight course; I measured a distance of 500 yards along its bank, and then found that the angle between the point from which I started and an object immediately opposite on the other bank was 30 ; what was the breadth of the river? 797. At what distance from the foot of the escarp must the lower end of a scaling ladder, 40 feet long, be placed, in order that the other end may just reach the top of the escarp which is inclined at an angle of 85" to the horizon; the vertical depth of the ditch being 35 feet ? *& 798. Show that if r be the radius of the earth, h the height of the eye above the sea, and D the depression or "dip" of the sea horizon below the plane tangential to the surface of the earth at the place of observation ; then tanZ> = ^^^ ~''^^ '— ^> and, pmc- C. 82 . MATHEMATICAL EXAMINATION QUESTIONS. tically, h being very small compared with r, tan i> = / ^ ; or if h be taken in yards, and r be assumed = 396O miles, then **°^- 2640- 799. The top-gallant-mast truck, 120 feet above the water- line of a man-of-war coming into port at the rate of 10 miles an hour, was first seen on the horizon at S** 45™ a.m. by a person swimming near the water's edge; and at lO'^G™ a.m. she cast anchor : find an approximate value for the radius of the earth. 800. The angle subtended at a certain point *S' by the line joining the flanked angles of two alternate bastions A and C of & fortress was found to be 38" 17' 40", and the angle between A and the flanked angle B of the intermediate bastion was 26° 45' 10": and by a plan of the place it was found tliat the distances of these three points were AB = BC = 360 yards and AC = 6OO yards ; what was the distance of JS from B ? 801. Show that if, in a regular fortification, the length of the curtain be c, that of the flank of the bastion /, the distance be- tween the angles of the shoulder of two adjacent bastions cl, and the angle which the lines of defence make with the curtain a ; then, when the lines of defence are perpendicular to the flanks, /= —-. — and d = cosec a — 2 sin a cos 2 a 802. At a point P, near the top of Woolwich Common, and in a line with the flagstafl* at the mortar battery F and the east chimney in the dockyard ^, the horizontal angular distance of these from the west chimney W was observed to be 16° 55' 25"; find the distance of F from each of the three objects, supposing their mutual distances to be i^^=1108, i^ir= 1120 yards, and i^'jr= 645 yards. 803. At each extremity of a base AB = 758 yards, the angles between the other extremity and two remarkable objects C and PLANE TRIGONOMETRY. 83 D were observed, viz. C^^= 103" 50' 41", i>^5= 53M 7' 24.", DBA = 85' 47' 30", and CBA = 46" 13' 27"; find CD. 804. Having measured a base AB = 5038*127 feet, I took at each station the angles subtended by the other and each of two re- markable objects C and D, viz. (7^^= 74" 16' 30", DAJB=23"17' 52'\ J)BA = 87' 21' 13", and CBA = 4>1' 13' 20"; find the distances CD, AD and AC. 805. At a station A, T found that the angle of elevation of the top of a tower which I knew to be 200 feet above the hori- zontal j)lane passing through A, was 52" 21' 30", and afc another station B, in the vertical plane through the tower and A, I found the angle of elevation to be 28" 15^41", and the angular distance between A and the top of the tower to be 19" 36' 20". What was the direct distance from A to B, and their difference of height ? 806. Wishing to determine the distance between two build- ings A and C (each of which, as well as every part of the line joining them, is inaccessible to me), and also the distance of a particular station B from the line AC, 1 measure a base BD = 650 yards in a perpendicular direction towards AC ; 1 observe also the angles CBD = 42" 27' 1 5", CDB =:: 1 1 4" 1 6' 30" and ADB=1 26" 38' 20" : what are the required distances ? 807. The elevation of a tower standing on a horizontal plane is observed, and, at a station p feet nearer to it in a direct line, the elevation is found to be the complement of the former. On advancing q feet nearer still, the elevation is found to be double the first ; show that the horizontal distance of the steeple from n the last station is ^ and its height 6—2 84 MATHEMATICAL EXAMINATION QUESTIONS. 808. Find the horizontal distance between the flanked angles of two bastions C and D of & besieged foi-tress, from the following horizontal angles measured at the extremities of the horizontal base AB, 800 yards in length : via. CAB = 97" 32' W; DAB = 53' 51' 20"; CBA = 32° 2' 10", and DBA = 68'1'15", 809. From the top of a cliff* 108 feet high, the angles of de- pression of the top and bottom of a cliff" which forms the opposite bank of a river are observed to be 30° and 60° respectively j find the height of the opposite cliff", and the breadth of the river. 810. From the bottom of a martello tower 50 feet high, the angle of depression of a ship at anchor was found to be 25° 17' 20", and, from the top, 31° 12' 30"; find the direct distance of the ship from the top of the tower, and the height of the cliff" on which the tower is built, above the sea- level. 811. From the top y( of a tower AB, standing upon the summit of a hill, the slope of which, BCD, is inclined to the horizon at an angle a, the depressions of two objects C and I) were observed to be yS and y respectively. Find the distance be- tween the objects ; the height of the tower being h. 812. From the top of a hill the angles of depression of two objects in the plain at its base were observed to be 45° and 30°, and the horizontal angle between them was also 30°; find the height of the hill in terms of the distance between the objects. 813. Wishing to know the breadth of a river from A to B, I measured from A a distance of 150 yards, to a point C on the prolongation of the line BA, and then measured 200 yards to a point i> on a line at right angles to BAC ; from this point the angle BBA was found to be 37° 18' 30". Find the breadth of the river. PLANE TRIGONOMETRY. 85 814. Two conspicuous headlands are observed from the deck of a ship, sailing due east, to bear 63^ 9.Y 10" and 24" 17' 30", respectively, to the northward of the ship's course; and, after sailing 8 miles, the corresponding angles were observed to be, 151" 16' 20" and 97" 12' 15". Pind the distance of the headlands from one another, and from the ship at the first observation. 815. A river flows between two towers, one of which is 40 feet high : from its summit the angle of elevation of the top of the other is found to be 2" 15' 30"; and from its base the cor- responding angle is 10" 18' 15": find the height of the other tower, and the breadth of the river. 816. To determine the distance of a battery at A from a fort F, a dis- tance AB = 200 yards was measured in a direction at right angles to that of find the distance of the AF, Walking from J5 in a straight line towards F^ a pole was placed at a con- venient station C ; the distance AG was then measured and found to be 178 yards, and the angle BAG was 27" 50'; fort F from either station A or G. 817. In a besieged city two con- spicuous forts P and Q were visible from two batteries at A and B outside the city; the distance AB between the batteries was 1080 yards, and at A and B the following angles were ob- served, viz. BAP = 80" 10' 1 5", ABQ = 85" 23' 41", BAQ = 42" ir 29", ABP = 38" 51' 26"; determine the distance PQ between the forts, and if QP and BA were produced to meet, find at what angle they would intersect. 86 MATHEMATICAL EXAMINATION QUESTIONS. 818. From the extremities, A and -6, of a base-liue 3070 yards long, the horizontal angles subtended by two distant objects C and D were observed, viz. BAG = 41" 24' 30", CBD = 33' 47' 15", CAB = 48** 35' 30", DBA = 27" l6' 20" ; it is required to find the horizontal distance DC, 819. The distances between the points -4,-5, (7 in a fortress are known, and at another point P in the same plane as -i, -6, C, the angles AFC and BFC are observed : find the distance of P from each of the points A, B^ C ; supposing ^^ = 400 yards, ^C=600 yards, -BC = 300 yards, the angle ^ PC = 36" 40' and BPC= 11" 20', the points C and P being on the same side of the direct line from A to B. 820. The triangle RAB is in a horizontal plane, and T is an elevated object 103*517 feet vertically above R; the angle TAR =13'' 23' 20", TBR = 17" 41' 30", and ARB = 41" 19' 40" ; find AB. 821. From the extremities A and B of a. wall 500 yards long, running north and south, the distance between two objects C and D subtends equal angles of 30", and it is found that C is due east of A and north-east of B. Find the distance between C and Dj D being observed on the right of C from both ends of the wall. Witlwut the use of Tables. • 822. From the top C of a clifi" 600 feet high, the angle of elevation of a balloon B was observed to be 47" 22', and the angle of depression of its shadow S upon the sea was 6l" 10' ; find the height of the balloon, the altitude of the sun being Q5' 31', and By S and C being in the same vertical plane. 823. The angle of elevation of the top, C, of a tower on a hill observed at a point ^ is 13" 17' 20", and at a point B (not in the vei-tical plane passing through C and A)^ the angle of elevation is 22" 35' 15", the angles CBA and CAB are observed to be MENSURATION. 87 65° U' 30" and 47" 32' 10" respectively, and the difference of level of C and A is known to be 500 feet j find the distance AB, and the difference of level of A and B. 824. Two sides ^C, BG,of an equilateral triangle, subtend angles of 30" and 45" at a point J); find the distances DA, DB, DG^ supposing DC to fall between A and B, i> and (7 to be on opposite sides oi AB, and the side of the triangle to be 10. 825. The angle subtended by a diagonal of a square redoubt from a point in the prolongation of the other diagonal was found to be 21° 32' 10", and at a point on the same line, 200 yards nearer to the redoubt, the corresponding angle was 28" 1?' 15"; find the side of the redoubt,^ B)2Q. A circular reservoir subtends at a certain point an angle of 34" 22' 18", and advancing 150 yards directly towards its centre, I find it subtends an angle of 72" 18' 30"; find its dia- meter. 827. From the top of a tower 113-786 feet high the angles of depression of the top and bottom of a column standing on the same horizontal plane as the tower were found to be, 32" 15' 20" and 68" 54' SS" respectively ; find the height and distance of the column. 828. The distance between two horizontal parallel telegraph wires running N.W. and S.E., and vertically above one another, is 4 feet, and at noon the perpendicular distance between their shadows on a horizontal plane is 6 feet ; find the altitude {i. e, the angle of elevation) of the sun. MENSURATIOK 829. The lengths of the perpendiculars let fall from points in an irregularly curved line of fence upon a straight line of 5*86 chains, at equal distances from each other, are found to be 93, 84, 88 MATHEMATICAL EXAMINATION QUESTIONS. 72, 68, 43, 54, 37, 29, and 23 links; find the area included be- tween the extreme perpendiculars which fall upon the ends of the straight line. 830. The sides of a quadrilateral, taken consecutively, are 2416, 1712, 1948 and 2848 ; the angle between the first two is 30", and that between the last two 150°; find the area of the figure. 831. Find an expression for the area of a triangle whose sides are a, b and c. 832. Find the area of a triangular field whose sides are 7*32 chains, 4*57 chains, and 5*48 chains. 833. Find the number of acres in a triangular field whose sides are 10*42 chains, 874 chains, and 12 63 chains. 834. Three sides of a triangle are 6, 6 + J2, and 6- J9.; find its area. 835. The area of a right-angled triangle is 84-5 square feet, and one of its sides is 39 inches ; find its hypothenuse. 836. The area of a triangular field is 14 acres; find its sides, which are known to be in the ratio of 3, 5, and 7« 837. A quadrilateral field ABCD has its sides AB = 6 chains, BC=8 ch., CD = 8 ch., AI) = 9 ch., and its diagonal, £D = 12 ch. ; find its area in acres. 838. A mahogany plank 24 feet in length, is 18 inches wide at one end and 1 5 at the other ; the plank is cut across at a dis- tance of 3 feet 6 inches from the broader end : how many square feet are cut ofi* the plank, and how many are in the whole plank? 839. The floor of a room is a regular octagon, the distance between any parallel sides being 20 feet ; find the area in square yards. 840. The base of an isosceles triangle is 20, and its area is — ;- ; find its angles. *v^3 MENSURATION. 89 841. The parallel sides of a trapezoidal field of 25 acres are 1 chains and 1 5 chains respectively : find the perpendicular dis- tance across the field. 842. A trapezoidal field of which the parallel sides are 579 links and 854 links, and the perpendicular distance between them 723 links, is let at <£4. 10s. per acre; what income does it produce ? 843. The sides of a triangle are 2+^2, 2-^2, and 3; find its area. 844. Find the area of a regular polygon in terms of its side a 845. The sides of a right-angled triangle are 3, 4, and 5 ; find the area of the space contained by the segments of the sides 3 and 4, and the arc of the inscribed circle included between these 846. Express the area of a regular plane figure in terms of a the length of the side, and n the number of sides : and apply it to the determination of the area of the equilateral triangle, the square, and the regular pentagon, whose sides are each 10 units in length. 847. Deduce the expressions for the areas of the regular polygons of n sides circumscribing and inscribed within the circle whose radius is r; and thence show that they have to one another the ratio of 1 to cos^ . n 848. Show that the area of a regular polygon of n sides is equal to — . cot — - : where a is the length of its side. ^ 4 n 849. Find the area of an isosceles triangle, each of whose equal sides is 50, and each of whose equal angles is 75". 850. What must be the diameter of a carriage- wheel in order that it may make 500 turns in a milel 90 MATHEMATICAL EXAMINATION QUESTIONS. 851. A regiment, advancing in open column of companies, is wheeled by successive companies to the left; show that the dis- tance, in paces, lost by each company during the wheel, is two- fifths of the number of files in the company, supposing the ratio of files to paces to be 10 to 7, and that the right file of each com- pany takes exactly the full pace. 852. The driving-wheel of a locomotive engine being 6 feet in diameter, determine the number of strokes made per minute by each piston, when the train is running at the rate of 30 miles an hour; two strokes of each piston causing one revolution of the driving-wheel. 853. Prove that the area of the largest circle which can be cut from a regular hexagon is three times the ai*ea of the circle described on one of the sides as a diameter. 854. What is the area of the sector of a circle whose arc of 24" measures 10 feetl 855. Ten persons dine at a circular dining-table ; what is the area of the table-cloth, supposing each person to occupy 2 feet of the circumference of the table, and that the cloth overlaps 1 5 inches on each side 1 Find also the area unoccupied when a dinner plate of 10 inches diameter is placed before each person. 856. Find the area of the remaining portion of a circle whose radius is 20 inches, when a segment having an arc of 25", has been cut off from it. 857. Find the area of the segment of a circle whose height is one-half of the radius, when the radius of the circle is 1 foot. 858. Find the area of a segment of a cii'cle whose base is 54 and height 10. 859. A regular hexagon is inscribed in a circle whose radius is 10 inches, and another is circumscribed about it; find the area pf the latter, and show that the area between the boundaries of MENSURATION. 91 the hexagons is equal to one-third of the area of the inscribed figure. 860. If a regular hexagon, a square, and an equilateral triangle be inscribed in a circle, the square described upon the side of the triangle is equal to the sum of the squares described upon one side of each of the other two figures. 861. Show that the area of a regular polygon inscribed in a circle is a mean proportional between the areas of two polygons of half the number of sides inscribed within and circumscribed about the same circle. 862. The exterior diameter of the outer ring of a circular target is 5 feet, and it is divided into 6 concentric rings of equal breadth, alternately white and black, the outer ring being white ; find the number of square feet of white paint in the target. 863. Find the area of the sector of a circle whose arc of 20" is 18 inches long. 864. Find the area of a cii-cular segment whose height is 7 inches and its base 3 feet. 865. The angular points of an irregular pentagon are each at a distance of 100 yards from a certain point within it; and at this point the sides taken in order subtend angles of 45", 60°, 90", 45", and 120"; find the area of the pentagon and the length of each of its sides, without using tables. 866. If a regular hexagon be placed within an equilateral triangle, so that three of its sides are upon the sides of the tri- angle, show, analytically, that the areas of the figures will be as their perimeters, and that the areas of their circumscribing circles will be as 1 : 3. 867. A and B are two points in the circumference of a cir- cular pond of water, and a dyke from ^ to ^ is to be made so as to cut off the smaller part of the pond. If the circumference of 92 MATHEMATICAL EXAMINATION QUESTIONS. the whole pond be 100 yards, and the length of the dyke AB be 18 yards, how many square yards will the surface of the pond now occupy] 868. The paving of a circular court cost £50, at the rate of Ss. 4>d. per square yard; what is its circumference in feet? 869. Show that the ratio of a square to its circumscribing circle is 2 : -cr. 870. Prove that the areas of two sectors of circles are equal when their angles are inversely proportional to the squares of their radii. 871. Having a cord 20 yards long, and wishing to know the area of a circular reservoir, of which only a portion was accessi- ble, I found that when the cord was stretched between two points on its margin, the pei-pendicular distance from the point of bisec- tion of the cord to the nearest point of the circumference was 1*716 yard. What was the area of the reservoir? 872. If a be the length of an arc of a circle (radius r), and a the number of degrees in it, find the area of the corresponding segment. 873. A circular building is to be erected on a triangular plot of ground, the sides of which are 40, 30, and 50 yards; find the radius of the circle so that the unoccupied external area may be one-tenth of an acre. 874. If, in a wire rope, the diameter of each component wire be d inch, find the number of wires in each square inch of section of the rope; and show thence, that, if the circumference of the C rope be C inches, the bearing area of the section will be —j- . 875. Divide the surface of a sphere into four zones of equal surface ; and find the angular breadth of each zone. 876. Find the surface exposed to the force of the exploding power in a 13-inch shell whose thickness is 1*85 inch. MENSURATION. 93 877. The length and breadth of the rectangular base of a wedge are a and h inches respectively, and the length of its edge is c inches. If Y be its volume, and h the perpendicular distance of any point in the edge from the base of the wedge, prove that whether c be greater or less than a. 878. A cylinder and a hemisphere have equal circles for their bases ; find the form of the cylinder in order that its volume may be one-half of that of the hemisphere. 879. A brass gun was struck by a spherical shot : a section across the middle of the wound being a segment of a circle, the base of the segment was found to be 6 inches and its depth 1 inch ; find the weight of the shot which made the wound, the 9-lb. shot being 4 inches in diameter. 880. Show that if d be the diameter of a round shot which weighs 'p pounds, R and r the radii of the exterior and interior spherical surfaces of a shell, then the weight of the shell, in pounds, will be expressed by R^ d' .8^. 881. If a cone, the diameter of whose base is equal to its slant height, be enveloped in a sphere, and the sphere be enveloped in a cylinder, show that the respective volumes are as 9 : S2 : 48, and that their complete surfaces are as 1 8 : 32 : 48. 882. A sphere of lead, 2 inches in diameter, is j^laced within an inverted, hollow, regular hexagonal pyramid, 3 inches in the side and 5 inches deep; find the quantity of water which can be poured into the pyramid, and also the quantity when the sphere is just covered. The axis of the pyramid being, in both cases, vertical. 94 MATHEMATICAL EXAMINATION QUESTIONS. 883. It is required to construct a cylindrical gallon measure having a hemispherical bottom, the total depth being 12 inches: the imperial gallon being 277*274 cubic inches. 884. A cylindrical vessel 5 inches high and 10 inches in radius has three heavy spheres placed within it which rest at the bottom, touching each other, the bottom, and the curve surface of the cylinder; find the number of cubic inches of water which can be poured into the cylinder. 8S5. A sphere is placed within a triangular equilateral pyra- mid, or regular tetrahedron, and is found to touch all its sides; find the volume of the sphere, the side of the tetrahedron being 1 foot. 886. A tank or cistern for holding water is in the form of the frustum of a pyramid, the length and breadth of its rect- angular bottom are 10 and 6 feet, and it measures 15 feet by 9 at top ; how many gallons of water will it hold, supposing its depth to be 4 ft. 6in.? 887. Find the capacity of a cylindrical pontoon having hemi- spherical ends, its extreme length being 22 feet, and the length of the cylinder 19 feet. 888. Find the number of cubic feet of air in a conical tent 10 feet high and 14 feet in diameter. 889. A 13-inch shell weighs nearly 200 lbs., the thickness of the shell being 2 inches; find the thickness of the 36-inch shell which weighs 26 cwt. 890. Find the number of cubic feet of air in a rectangular hut, with a sloping roof; the sides of the base being 80 feet and 25 feet, the height to the eaves 10 feet, and to the ridge of the roof 1 5 feet. 891. Find the quantity of gas contained, after firing, in the bore of an 8 -in. gun at the instant the centre of the shot passes MENSURATION. 95 tlie plane of the muzzle, the total length of the bore, which is ter- minated by a hemisphere, being 8 ft. 7 in. 892. A hemisphere, radius 2r, has a cylindrical hole, radius r, bored through it perpendicularly to the plane of its base; the axis of the cylinder coinciding with a radius of the hemisphere. Find the remaining volume. 893. What quantity of paint would be required to paint the 8-inch shells in a square pile of 15 courses, supposing that 1 lb. of paint will cover 100 square feet 1 894. The mound on the field of Waterloo is 200 feet high, and is in the form of a cone, having the inclination of its sides to the horizon 30°; find its content in cubic yards, and its surface in acres. 895. The diameter of the 9 lb. solid shot being 4 inches; find the diameter of the 12 lb. and 24 lb. shot; and the thickness of the 13-inch shell which weighs I96 lbs. 896. A cylindrical pontoon 2 feet in diameter, having conical ends, is so far sunk under water that there is a mean pressure of 2 '3 lbs. upon each square inch of its surface; find the total pres- sure, the extreme length of the pontoon being 1 4 feet, and the height of each cone being equal to the diameter of its base. 897. The sides of a circular reservoir are inclined at an angle of 30° to the horizon, and the diameter of the horizontal bottom is 70 feet; find the number of gallons contained in it when the water is 10 feet deep. 898. Find the solidity of the frustum of a cone, the diameters of whose ends are 5 and 2, and the height between them 4. 899. What must be the thickness of an 8-inch shell in order that it may weigh 54 lbs. when unloaded; the 9 lb. shot being four inches in diameter 1 900. Find the weight in grains of a bullet in the form of a cylinder and cone upon a common base '6 inch in diameter; the 96 MATHEMATICAL EXAMINATION QUESTIONS. altitude of the cone being one-fourth of that of the cylinder, the total length of the bullet being 1'25 inch, and the weight of a cubic inch of lead 6 '57 oz. 901. Show that the capacity of a tent, considered as a prism of n sides surmounted by a pyramid, whose heights are respec- tively h and h^ and length of side a, is /o7 , ^ w.a" . 1800 (3h + L) . . cot . ^ ^' 12 n 902. Wliat is the content of a cylindrical pontoon having hemispherical ends, its extreme length being 20 feet and diameter 3 feet 1 And what is its weight, supposing its thickness to be one-sixteenth of an inch, and a cubic foot of its substance to weigh 4cwt] 903. Find the content of the bore of a gun 5 feet long and 4-5 inches in diameter, the end at the breech being a hemisphere of the same diameter. 904. The 9 lb. iron shot is 4 inches in diameter, and a cubic foot of water weighs 1000 ounces: find the weight of a 13-inch shell when filled with water, supposing its thickness to be 1'8 inch. 905. Find the weight of a S6-inch shell, 4 inches thick; sup- posing the 9 lb. shot to be accurately 4 inches in diameter. 906. The present 13-inch shell which is 1-85 inch thick, weighs 1 96 lbs. ; what would be its weight if the thickness were diminished to one inch 1 907. Show that the volumes of spherical shells of the same thickness are proportional to those of conic frusta, of equal heights, which have the diameters of their ends proportional to the external and internal diameters of the corresponding shells. 908. Show that if a gun or mortar have a conical chamber the sides of which are inclined to the axis of the gun at an angle MENSURATION. 97 of 30°, the available space for the charge of powder, when the shot is in contact with the sides of the chamber, is one-eighth of the volume of the shot. 909. A heavy sphere is just immersed in a conical glass fiiU of water; find the quantity of water in the glass, which is 4 inches deep, and 6 inches in diameter. 910. Find the volume remaining of a sphere of 16 inches radius, after a circular hole 8 inches in radius has been bored directly through its centre. 911. Assuming a cask, 41 inches long, (measured from head to head, externally, over the side) to be formed of two conic frus- tums having a common base; find the weight of water contained in it when the diameter of the head is 21 inches and the circum- ference at the bung is 7 feet, both measured externally, and the thickness of the wood is '5 inch, assuming the cubic foot of water to weigh 1000 oz. 912. Find the weight of a 14-inch shell made of lead 2 inches thick; the weight of the 13-in. iron shell which is 1*8 in. thick being 1 96 lbs., and a cubic foot of lead being to a cubic foot of iron, as 100 to 64. 913. The slant height of a frustum, cut from a pentagonal pyramid whose slant height is 10 inches, is found to be 7*5 inches, and each side of the base is 4 inches; find the content of the frustum. 914. What must be the thickness of a 36-in. shell, in order that it may weigh 1 ton; supposing a 13-in. shell to weigh 200 lbs. when 2 inches thick ? 915. Divide a cone into three equal parts by planes drawn parallel to its base. 98 MATHEMATICAL EXAMINATION QUESTIONS. SPHERICAL TRIGONOMETRY. « 916. State "Napier's Rules," and apply tliem to the deter- jnination of formulae for the solution of a right-angled spherical triangle in which A and h are given, and G is the right angle. 917. Prove that in every right-angled spherical triangle, whose right angle is c, cos c = cos a . cos 6, and cos A = tan b . cot c. 918. The side a and the hypothenuse c of a right-angled spherical triangle AJ3C being given, write down the formulae of solution for the angles and remaining side. 919. Prove that in every spherical triangle, cos c = cos a . cos h + sin a . sin b . cos C ; and show that, if tan B - tan a . cos C, this reduces to cos c = cos a . cos (b — 6). sec 0. 920. Show, that, in a quadi-antal triangle, cos (7= —cot a . cot b, when c is the quadrantal side. 921. Assumi n g that in an oblique-angled spherical triangle A,B,C: cos -(a^b) tani(^^5) = -f-— .cotf, cos ^{a + b) prove that 1 cos^(^-^) tan - {a + b) = ^ t^n- . cos^(^-f^) '^ 922. Find the area of a spherical triangle whose angles are 79" 10' 20", 57" 43' 2" and 43" 7' 18", on the surface of the eai-th whose radius is, approximately, 8000 miles. SPHERICAL TRIGONOMETET. 99 923. Prove, from first principles, that, in a right-angled spherical triangle ABC, having the right angle at C : cos c = cos a . cos b. 924. By the application of " Napier's Rules," write down the values of the sides and of the remaining angle of a right-angled spherical triangle in terms of the hypothenuse c, and one of the angles, B. 925. Prove that, in an oblique-angled spherical triangle, . cos b . cos G — cos a cos A = sin a . sinb and thence determine the value of sin - A in terms of the sides. 926. Show, from first principles, that in a right-angled spheri- cal triangle ABC, having the right angle at C, cos c = cot A . cot B, and sin a = sin A. sine. 927. The spherical triangle ABG has one of its angles ACB a. right angle, and the sides opposite to A, Bj G axe a, b, c respec- tively j prove the following properties : (1) tan 6 = tan j5 sin « j (2) cos ^ = cos 5 sin -4. 928. Show by " Napier's Rules," that the altitude of a regulai* tetrahedron whose side is 5, is - J6. 929. The sides and angles of a polar spherical triangle are the supplements of the angles and sides respectively of its primi- tive triangle. 7—2 100 MATHEMATICAL EXAMINATION QUESTIONS. 930. Show that in a right-angled spherical triangle sin {c — a). sin (c + a) = cos" a . sin' 6, c being the hypothenuse. 931. In a spherical triangle, given a= 148° 25' 34", h = 149" SV 48" and C= 109" 5T 57"; find the values of A and B. 932. Investigate Napier's "First Analogies," and show how the " Second Analogies" may be determined from them. State to what cases of solution of spherical triangles they are applicable. 933. Show that in a right-angled spherical triangle ABC, sin a . tan - ^ — sin 6 . tan - -5 = sin (a — b), C being the right angle. 934. Show, from first principles, that, in an oblique-angled spherical triangle. A /sm (s — h) sin (s - c) tan — = / — r^ '- — — i — — : . 2 \ sm.8 .sin{8 — a) 935. Show that in a spherical triangle cos c = cos a . cos b + sin a . sin 6 . cos C ; show what this becomes in the case of a right-angled triangle of which c is the hypothenuse ; and obtain the same result by the application of " Napier's Rules." 936. The angles at the vertex of a triangular pyramid are 37° 15' 45", 53' 22' 17" and 48° 33' 20" j find the inclinations of its planes to one another. 937. The plane angles at the vertex of a triangular pyramid are SS"" 14' 20", 42" 25' 50", and 50° 17' 10"; find the inclinations of its sides to one another. SPHERICAL TKIGONOMETBY. .... 101 938. A pair of compasse^^ , openeci so tliat the legs include an angle 2a, have the point? ,pUce'd ^i^ion a; table fe'othat the plane of the compasses is. inclined at an angle ^ to the plane of the table. Show, by " Napier's Kules," that, if 6 be the inclina- tion of each leg to the plane of the table, sin ^ = sin /? . cos a. 939. Solve completely the spherical triangle in which ^ = 34" 47' 40% a = 65° 22' 30", and 6 = 57° 32' 10". 940. The elevation of the top of a tower, seen from a point in a level straight road, is 10", and the horizontal angle between the tower and the road is 60"; find, hy Spherical Trigonoimtry, the greatest elevation the top of the tower can have from any point in the road. 941. A^ B, G are three points upon a plane inclined at an angle of 34" 16' 20" to the horizon j the line AB is horizontal, and AG is inclined 17^* 52' 5" to the horizon : find the angle GAB. 942. A, B, G being three places on the surface of the earth the distances of which from one another are known, and E the position of a balloon above the earth ; the angles AEB, BEG^ GEA were measured at the instant when the balloon was vertically over the line AB. Show how, by the solution of two spherical triangles and three plane triangles, the height of the balloon may be deter- mined, and give the necessary working formulae. 943. Two of the plane angles forming a trihedral solid angle are 30" and 60" degrees respectively, and their planes are perpen- dicular to one another ; find the remaining angle and the inclina- tion of its plane to those of the other angles. 944. The altitude of a regular pentagonal pyramid is to the slant height as Js : 2 ; find the inclination of its sides to one another, by the solution of a spherical triangle. 945. If a cube be cut by a plane which passes through the three diagonals of its adjacent faces, find the inclination of the 102 MATHEMATICAL EXAMINATION QUESTIONS. plane to those faces, by tte solution of a riglit-angled spherical triangle. - r . . 946. Two straight lines are drawn from a point in the inter- section of two planes, one in each plane, making angles of 30" and 60" respectively with the intersection ; find the angle between the lines when the planes are perpendicular to one another. 947. If A and B, two of the angles of a spherical tiiangle, be right angles, show that the area of the triangle is C — |— X (surface of sphere). 948. Two sides of a triangle on the surface of the earth are 20 miles and 40 miles respectively, and the angle included between them is 57" 18' 10"; find the remaining angles, and thence, the area of the triangle, the radius of the earth being assumed 4000 miles. 949. Show that if U" be the spherical excess in a triangle of which the area in feet is -4, and H be the radius of the earth .. -,„ 45A m miles, -o = . , , ^ , . ' 77 . (44jR)* 950. Find the area of the triangle ABC upon a sphere whose radius is 60 inches, where ^ =54° 13' 20", 6 = 32° 27' 14", and c = 45° 50' 30". 951. Find the angles, the spherical excess, and the area of an equilateral spherical triangle, described upon a sphere whose radius is 20 feet, each side of the triangle being 60°. 952. The angles of a spherical triangle upon a sphere whose radius is 4 inches, are 70°, 59°, and 81°; find its area. 953. The sides of a spherical triangle are 121° 15' 45", 57° 23' 15", and their included angle is 75° 25' 24"; find the area of the triangle upon the surface of a sphere of 16 ft. Sin. diameter. PRACTICAL ASTKONOMY. 103 PRACTICAL ASTRONOMY. 954. Define the terms "altitude," "azimuth," "right ascen- sion," " declination," " ecliptic," and " first point of Aries." 955. The meridional altitude of a star, observed on the south side of the zenith, is 57" 18' 6'\ and the latitude of the place 37° i4f' 6" N. ; find the north polar distance of the star. 956. The meridional altitudes of a circumpolar star in the southern hemisphere were observed to be 19" 32' 7" on the south- ern, and 72° 12' 4" on the northern side of the zenith; find the latitude of the place and the declination of the star. 957. By an observer situated on the equator of the earth, the meridional altitude of a star was observed to be 27" 13' 24". What was the declination of that star ? 958. The declination of a star is — 2 1" 37' 1 5", and the latitude of the place of observation 51" 28' 29'' N. ; find the meridional zenith distance. 959. The meridional altitudes of a southern circumpolar star were observed to be 81" 16' 4" and 13" 15' 10"; find its declina- tion and the latitude of the place. 960. At midnight, March 12, 1853, the R. A. of the moon was 27" 39' 19"'2, and her declination 6" 56' 59"7 N. ; find her lon- gitude and latitude, the obliquity being 23" 27' 32". 961. At midnight, February 6, 1853, the longitude of the moon was 302" 46' 29"'3, and her latitude was 2*' 56' 44"-8 S. ; find her R.A. and declination, the obliquity being 23" 27' 31". 962. At noon, January 18, 1853, the R.A. of the moon was 40" 54' 24"'75, and her declination 11" 53' 49"-7 N. ; find her longitude and latitude, the obliquity of the ecliptic being 23" 2/ 31". 104 MATHEMATICAL EXAMINATION QUESTIONS. 963. At noon, February 9, 1853, the R.A. of the Moon was 339" 47' Y'-G5, and her decimation 13" 39' 9"S. j find her longi- tude and latitude, with the same obliquity. 964. The Equation of Time being + 14"* 53% and the R.A. of the sun 15^ 27°" 34'; find the Sidereal Time corresponding to 7'' 33"^ 12' mean time. 965. The Mean Time being 2 1 »* 1 3"" 27', the Equation of Time -9" 47", and the Sidereal Time 16^ 53" 21" ; find the KA. of the sun. 966. The R.A. of the Sun being 5^ 14" 2', the Sidereal Time 15** 37" 10', and the Equation of Time + 12" 53' ; find the Mean Time. 967. Find the Sidereal Time of sunset at the autumnal equi- nox, and thence the Sidereal Time at the following mean noon; the Equation of Time being + 7™ 20' at sunset. 968. Find the Equation of Time when the Sidereal clock marks 15** 10" 4', and the Mean Time clock 4'' 7" 23', the R.A of the sun being lO** 58" 4' at that instant. 969. The Sidereal Time of sunrise being 20'' 14" 37', and the R. A. of the sun 3^ 5" 53' ; find the Equation of Time at the fol- lowing Apparent Noon, supposing the Mean Time of sunrise to be 17^ 0" 0*, and that the Equation of Time at the previous noon was + 8" 20'. 970. Find the mean solar interval corresponding to the side- real interval 8^ 43" 27'. 971. Find the sidereal interval corresponding to the mean solar interval 17*" 54" 30'. 972. The transit of a star on the equator was observed at 7^ 15" 43' Mean Time; find the Mean Time of its setting. 973. The Sidereal Time at Apparent Noon was 21'' 14" 33^ ; find the Sidereal Time at 6'' 0" 0" Apparent Time. PRACTICAL ASTRONOMY. 105 974. The Mean Time of the transit of the first point of Aries was 13^ 56"^ 4*; find the Mean Time of transit of a star whose KA. was 6^ 14°^ 37^ 975. The star Eegulus was observed to pass the meridian when a chronometer regulated to Greenwich Mean Time marked 15^ ST"" 32'; what is the longitude of the place of observation, the Equation of Time being +5°^ 8', and the R.A. of the sun 17^ 12"^ 3^? 976. At what time does the Sun set at Edinburgh, in lati- tude 55" 51' 0" K, when its declination is + 14" 46' 0"1 977. The azimuth of the Sun at setting was found to be N". 54° 15' 0'' W., when its declination was + 17" 32' 10"; find the latitude of the place and the Apparent Time of sunset. 978. The meridional altitudes of a .circumpolar star were ob- served at a place in the southern hemisphere to be 10° 17' 54" and 72° 25' 18"; find the latitude of the place and the declina- tion of the star. 979. The true zenith distance of a Pegasi was observed at 19^ 27^ 50' Sidereal Time to be 80° 14' 10"; find the latitude of the place, the R.A. and declination of the star being 22^ 57™ 26' and + 14° 24' 55'\ 980. The sidereal interval between the transits of Arcturus across the prime vei-tical was 2^ 1 7°" ^& ; find the latitude of the place, and the Sidereal Time of the star's setting, the R. A and de- clination of Arcturus being 14^ 8™ 58« and + 19" 5Q' 5&\ 981. The altitudes of a southern circumpolar star when on the meridian were 14° 27' 30" and 73° 14' 10"; find the decli- nation of the star, and the latitude of the place. 982. The interval which elapsed between the transits of Arc- turus across the prime vertical was observed to be S^ 4" 15'; find the latitude of the place, the declination of Arcturus being + 19° 57' 40". 106 MATHEMATICAL EXAMINATION QUESTIONS. 983. At a place in South America the altitude and azimuth of the Sun's centre at 23^ 30"* 12* Greenwich Mean Time were, AU. 21" 25' 40", Az. 69° 18' 20''. Find the latitude and longitude of the place ; the Equation of Time being 7"^ 42", to be added to appai'ent time, and the declination of the Sun's centre being -17"37'29''-5. 984. The R. A. and declination of a certain star being r and 8, and its angular distance from the moon when on the equator dy show that moon's R.A. =rdca, where a is derived from the equation cos a = cos J. sec 8. 985. At 13^ n'" 24', Greenwich Mean Time, the altitude of the Sun's centre was found to be 35^ 12' 4" and its azimuth S. 42" 38' 10" W. ; find the latitude and longitude of the place of observation ; the equation of time being 4™ 1 5' to be added to apparent time, and the declination of the Sun's centre 12" 13' 42" S. 986. The sidereal interval between the transits of a XJrsfe Majoris across the prime vertical was observed to be 2** 53"" 10'; what was the latitude of the place of observation, the declination of the star being + 62" 32' 3&''i 987. At 16*^ 52°^ 13', Greenwich Mean Tiihe, the star Sirius was observed to pass the meridian ; the R. A. of the sun was 21** 14™ 38'; the equation of time - 14™ 18': what was the longi- tude of the place ? 988. At 2^ 15"^ 21 '-5, Greenwich Mean Time, the tmnsit of the star a Lyrse was observed ; and it was found that the R. A. of the sun was, at that instant, 7*^23" l6'-2, and the equation of time - 5^ 8' -6 : what is the longitude of the place at which the observa- tion was made 1 989. The transit of the Sun's centre was obsei-ved at 12** 13"^ 4* -2, Greenwich Mean Time, when the equation of time was -f- 10™ 14'-7 ; find the longitude of the place. PRACTICAL ASTRONOMY. 107 990. The transit of y8 Leonis was observed at 22*" 17°" 53»-7, Greenwich Mean Time, when the Equation of Time was + 3™ 38'-8, and the R. A. of the sun 2^ 57°" 14.'-1 ; what is the longitude 1 991. At 18^ 42°^ 7', Local Mean Time, the First Satellite of Jupiter was observed to disappear in the shadow of the planet; and, upon reference to the almanac, this phenomenon was found to be predicted as taking place at 11'^ 51™ 26", Greenwich Mean Time : what was the longitude of the place of observation 1 992. The difference of the times of transit of 8 Piscium and the Moon's bright limb was observed to be 0'' 27™ 2r. At her Greenwich transit the almanac states that their K.A.'s were 8 Piscium 0^ 41"^ 2% Moon's l^right limb 1^4"^ BG'SS, and the change of the Moon's P. A. for one hour of longitude was 114 seconds; what is the longitude of the jolace, the star being to the westward of the Moon 1 993. The difference of the times of transit of Yirginis and the Moon's bright limb, was observed to be 12°" 15'; and the RA. at Greenwich transit were, 6 Yirginis 13^ 2°" 20^-97, the Moon's bright limb 13^ 32" 5^'39, and the change of the Moon's P. A. for one hour of longitude 136-6'9 seconds ; what was the longitude of the place, the star being to the westward ? 994. The star y Draconis was observed, when due east, to have an apparent altitude of 64° 31' 10''; find the latitude of the place of observation, and the Sidereal Time, the P. A. and decli- nation of the star being 1?^ 53"^ 12" and 5V 30' 29". 995. Pegulus was observed to be due west at 14'* 12'" 27', Sidereal Time ; find its altitude at that time, and the latitude of the place, the P. A. and declination of the star being 10^ 0^ 32' and 12'' 41' l". 996. The P.A. of Capella being 5^ 5"* 50", its altitude was observed at 23*^ 5=^50' Sidereal Time, to be 2?" 42' 10"; what was the azimuth of the star, and the latitude of the place ? the de- clination of the star being 45<* 50' 33'\ 997. At 18^ 0°" 0% Apparent Time, the altitude of the son's lower limb was observed to be l?" 16' 39" and the azimuthal 103 MATHEMATICAL EXAMINATION QUESTIONS. angle of his east (or left-hand) limb from a terrestrial object to the south 110° 15' 27"; and his declination and semidiameter were found in the Almanac to be 20* S2' 5" N. and 16' 12": what is the latitude of the place, and the azimuth of the object, east or west of south ] 998. Find the Mean Time of the true sunset in latitude 49" 13' 27" N. when the Sun's declination is + 18" 5& 39", and the Equation of Time + 3™ 54' ; and find the azimuth of his centre at setting. Find also these values in the same latitude south of the equator. 999. The sidereal interval which elapsed between the tran- sits of y Draconis over the prime vertical was observed to be jh jr-m i3».2^ find the latitude of the place, the declination of y Draconis being + 51° 30' 52". 1000. The mean solar interval between the observed equal altitudes of Antares was 4^ 17™ SS* ; find the latitude of the place, supposing the observed altitudes to be 54° 17' 27", the declination of Antares being 26" 6' 4". 1001. The mean solar interval between two observed equal altitudes of the Sun was S^ 27" 16' ; the Greenwich Mean Time of the first observation was 18** 7" 26"; the Equation of Time was — 7™ 43' ; the declination at first observ^ation was + 10° 15''7 ", and the approximate latitude 27° 4' N. ; find the difference between the time at Greenwich and the time at the place, supposing the diminution of declination in the interval between the observations to have been 2' 57"= 177" 1002. The apparent altitude of a star, whose declination was + SS° 14' 0", was observed to be 47° 22' 54", when its azimuth was 106° 15' 0"; find the time at which it would pass the meri- dian, the Mean Time of the observation being 14** 15™ 21*. 1003. The apparent altitude of a star, whose declination was + 15° 7' 43", was observed to be 62° 17' S&\ when its azimuth was 75° 15' 10"; what was the latitude of the place of obser- vation ? PEACTICAL ASTEONOMY. 109 1004. At 4^^ 23"^, Woolwich Mean Time, on May 19, 1852, the Sun was observed to cast no shadow in front or back of the Royal Militaiy Academy; in what direction does the building face, its latitude being 51" 28' 28'' K, the Sun's declination 19" 44' 21" N., and the Equation of Time + 3"^ 49* ? 1005. At a certain time, when the E.A. and declination of Sirius were 6^ 38"" 36' and - 16" 30' 49", and of Procyon 7^ 31'^ 3V and + 5" 36' 10", it was observed that the true setting of Sirius took place 3^ 42"^ 10* (sidereal) before that of Procyon. What was the latitude of the place ? 1006. The altitude of Eegulus was observed to be 16" 31' 5" at 9"" 15"^ 0^ and at 11^ 43"^ 8^ it was again observed and found to be 40" 11' 27"; what was the latitude of the place of observatiou, the declination of Regulus beiug + 12" 41' 30"? 1007. At 17^ 51"^ 6% Greenwich Mean Time, the altitude of the Sun's lower limb was observed, after it had passed the meridian, to be 62" 58' 54", and the azimuth of the east limb 112" 32' 0"; the declination of the Sun's centre was +15" 27' 40", its semi- diameter 15' 55'\ and the Equation of Time +3°" 7'; find the lati- tude and longitude of the place of observation. 1008. In south latitude 30" 17' 10", the altitude of /S Leonis, before coming to the meridian, was observed to be 17" 45' 9''? and its azimuthal distance from a distant terrestrial object to the west- ward of it was 60" 26' 23" ; what was the angular distance of that object from the meridian, the declination of /3 Leonis being + 15" 24' 14"? 1009. At a place in Canada, the altitude and azimuth of a Lyrse, before passing the meridian, were observed, at 5^ 14™ 10''6 Greenwich Mean Time, to be, altitude 15" 30' 59", azimuth 47"42'40". Find the latitude and longitude of the place of observation ; and thence determine its distance from Quebec (in latitude 46" 49' N., longitude 71" 16' W.), and also the bearing of Quebec. The R A. and declination of a Lyr« being 18^ 31"^ 55' and 38" 39' 14"; the 110 MATHEMATICAL EXAMINATION QUESTIONS. E,.A. of the sun 11^ 6"^ IS'-S, the Equation of Time +2°^ 24" ; and the radius of the earth 3956 miles. 1010. The observed altitude of the Moon's lower limb was 34° 17' 22'', when that of a Arietis was 42° 13' 4?" (to the east of the meridian), and their measured distance 21° 3' 18"; the Moon's bright limb being towards the star. The Moon's horizontal paral- lax was 54' 15" and her semidiameter 14' 49"; find the longitude of the place of observation, supposing the distances nearest to the above, tabulated in the almanac, to be, at 3^ Greenwich Mean Time, 20° 22' 21"; at 6** Greenwich mean time, 21° 27' 54"; and that the Local Mean Time of the observation was 15** 27™ 31". 1011. The apparent altitude of the Sun's lower limb was 34° 5' 44", and of the Moon's lower limb 56'' 56' 29", when the apparent distance between their nearest limbs was 35^ 16' 40". Find the true distance between their centres; the Moon's horizon- tal parallax being 54' 1 5", and that of the Sun 0' 8". 1012. The apparent altitude of a star was 20° 13' 26", and that of the Moon's upper limb 31° 32^ 16", when the apparent distance of the star from the Moon's bright limb (which was on the side next the star) was 72° 27' 22". The semidiameter of the Moon was 14' 54", and her horizontal parallax 54' 26". Find the true distance of the star from the Moon's centre. 1013. The apparent distance of the nearest limbs of the Sun and Moon was observed to be 80° 29' 40", when the altitude of the Sun's lower limb was 72° 9' 50", and that of the Moon's upper limb 18° 46' 38": find the true distance of their centres, the semi- diameter of the Sun being l6' 10", that of the Moon 16' 38", and her horizontal parallax 60' 58"; that of the Sun being as in ques- tion 1010. 1014. The measured distance of a star from the Moon's bright limb, which was away from the star, was 18° 37' 58", when the zenith distance of the star was 46° 33' 10", and that of the Moon's lower limb 58° 47' 23" : what was the true distance, the CO-ORDINATES. Ill semidiameter of the Moon being 16' 23'^, and her horizontal paral- lax 60' 21"? 1015. The distance between a certain star and the Moon's centre was observed to be 2?" 35' 0'', when the altitude of the star was 29" 47' 0'', and that of the Moon's centre 57° 22' 0"j what was the true distance, the horizontal parallax of the Moon being 60' 3"? 1016. The apparent altitude of the Moon's centre was ob- served to be 16'' 26' O", when that of Venus was 29" 41' O", and their measured distance 98'' 15' 3l"j the Moon's horizontal paral- lax was 60' 35" and that of Venus 20"; find the true distance. 1017. The measured distance of Jupiter from the Moon's centre was 120'' 18' 46", when the altitude of Jupiter was ob- served to be 8" 26' 0"; and that of the Moon's centre 19" 24' 0"; the Moon's horizontal parallax being 57' 14" and that of Jupiter l''-5 ; find the true distance. CO-OEDINATE GEOMETRY. 4 1018. Find the equation to the straight line which passes through the points (5 ; - 7) and (- 4 ; 3) ; find the tangent of the angle which it makes with the axis of x, and the points where it cuts the two rectangular axes. ^1019. Find the distance of the point (3, -5) from the line whose equation is 2x-8y + 7 = 0, and the angle which this line makes with the axis of x. N 1020. Find the equation of the line which joins the points F and Q j F being the intersection of the lines 7/ = 7x + 3 and 3y- 5x = 2 ; and Q being the intersection of the lines 5i/ + 4!X = 12 and 6?/ + 8a; = 4. 112 MATHEMATICAL EXAMINATION QUESTIONS. 1021. Find the area 'of the triangle whose angular points are (3; -2), (5; 4) and (-7 j 3). 1022. Find the equation to the straight line which passes through the points (5 ; - 7) and (- 1 j 4). 1023. The equations of the three sides of a triangle are 10a; + 5?/ = 4, 2y-3x = 6 and y = 0; find the co-ordinates of its angular points. 1024. The co-ordinates of the angular points of a triangle are for A, (3, -2), for B, (7, 1), and for (7, (5, 9); find the length of the pei-pendicular let fall upon AB from G, 1025. Find the equation of the straight line which passes through the point (5, - 3), and makes an angle of 30" with the axis of X. 1026. Find the perpendicular distance of the intersection of the lines 2?/ - 4a; = 10 and 3y + 9x -— 21 fix)m the line y- 3a; = 1 5. 1027. Find the intersection of the straight lines whose equa- tions are 2y + 5a;- 3 = and 7y-3x+ 10 = 0. 1028. Find the equation to the straight line which passes through the point (7, — 5) and makes an angle of 45" with the line whose equation is 2y-6x = 3. 1029. The equation of a straight line is 5x+10y — 7 = ; find its angles of inclination to the axes of x and y respectively ; and also the points where it cuts these axes. 1030. If y = ax + h he the equation to a straight line, show that the length of the perpendicular let fall upon it from the point (h, k) is h — ah — h n/(1+«')' and show the form which this takes when the equation to a line is of the form Ax-\-By + C=0. CO-ORDINATES. 113 1031. Find the equation of the straight line which passes through the point (3, -2), and which is perpendicular to the btraight line which passes through the points (-5, 7) and (2, 5). 1032. Find the length of the perpendicular drawn from the point (8 ; 4) to the line whose equation is 7/ = 2x—l6. 1033. Find the angle which the line Sy + 6x — 2 = makes with the line Sy - lOcc + 7 = 0. 1034. Find the angle included between the lines whose equa- tions are 2x+ 5 = Sy and 4^y + 3x = 0; and draw a figure indicating roughly the position of the lines, and mark, on the figure, the particular angle you have obtained. 1035. Show, by oblique co-ordinates, that the three straight lines drawn from the angles of a triangle to the points of bisec- tion of the opposite sides pass through the same point. 1036. Find the co-ordinates of the point of intersection of the three lines drawn perpendicular to the sides of a triangle at their middle points, and show that that intersection is equidistant from the angular points of the triangle. 1037. Show that the straight line which passes through the points (h, k), (7ij, k^) intersects the straight line which passes through the points (k, -h), (k^, -h) at right angles; and find the point of their intersection when h= 5,h = Sj A^ = — 4, and 1038. Find the equation to the line which passes through the intersection of the lines x-2y-a = 0, and £C + 3y - 2a = 0, and is parallel to the line Sa; -f 4y = 0. 1039. Find the equation of the straight line which cuts the straight line passing through the points (7,-1) and (-3, 5), at an angle of 45", and which also cuts the axis of a; at a distance 5 from the origin. 8 114 MATHEMATICAL EXAMINATION QUESTIONS. 1040. The equations of three straight lines are : y = 5x-i (1), y = lx-5 (2), ?/ = -4x + 2 (3); find the length of the portion of (3) intercepted between (l) and (2). 1041. Find the distance from (-3, 7) to (6, -5) when the axes of reference are at right angles to one another; and also when they are inclined at an angle of 60". 1042. Find the point of intersection of the line y = 5x — 7j with the line 5i/ + x-3 = ; and find the angle at which they intersect. 1043. A straight line whose equation is y + x — 5 = cuts two others whose equations are 59/ — 6x — 3 = and i/ — x + 3 = ; find the length of the intercepted portion of the fii*st line. 1044. Show that the measure of the angle contained by the two lines y = ax + h and y = a^x + b^ is expressed by tan ' :; ^ . 1 + aa. 5 1045. Find the angle included by the lines y = -x, and 3y + 6x — 8 = ; and determine the condition that the lines y = ax + b, and y = ax + 6' may be perpendicular to each other. 1046. Find the points at which the line 7y- 210? + 28 = cuts the co-ordinate axes, and the angle which it makes with the line which passes through the points (3, -2) and (-4, 7). 1047. The co-ordinates of the angular points of a triangle are (3, 0), (0, 3 J3) and (6, 3 J3) : find (l) the angles of the triangle, and (2) its area. 1048. Determine in inches the area of the triangle whose angles are at the points (7, 2), (4, - 5) and (- 3, 1), measured on the half-inch scale. CO-ORDINATES. 115 1049. Find the distance from tlie point (2, - 7) to the straight liDe which passes through the points (-4, 1) and (3, 2). 1050. Find the distance of the point of intersection of the two lines 7x — 5i/=l and 4a? + 2^/ = 20 from the origin. 1051. Find the area of the triangle of which (- 1, 2), (4, 4) and (6, -3) are the angular points. 1052. Find the general equation to the circle : (1) When the origin is at its centre. (2) When the origin is at its circumference. (3) When the origin is at any point within or without it. 1053. Construct the circle whose equation is and determine the equation of that diameter of it which passes through the origin of co-ordinates. 1054. A circle passes through the origin of rectangular co- ordinates, and through the point (3 ; 7) ', its centre is on the axis of X : find its equation. 1055. Find the radii and centres of the circles (1) 6x'-2y{7-3tj) = 0, (2) 3x'-6x + 39/ + 9l/-l^ = 0. 1056. Determine generally the points of intersection of a straight line and circle, when the origin is at the centre; and thence deduce the necessary relation between a, r and b when the straight line becomes a tangent to the circle. 1057. Find the radius and the co-ordinates of the centre of the circle whose equation is 7x' + 32/' - 42/ - (1 - 2xY = 0, and find the points in which it cuts the axis of x. 1058. Find the locus of the point to which if straight lines be drawn from the angular points of a given triangle, the sum of their squares will be a constant quantity s\ 8—2 116 J^IATHEMATICAL EXAMINATION QUESTIONS. 1059. Find the equation of tlie circle which passes through the three points (7, 5), (- 2, 4) and (3, - 3). 1060. Find the equation to the circle which passes through the points (7, 4), (- 5, 3) and (1, 7). 1061. Show that the general equation of the circle becomes x^ + y^- 9.ry = when the origin of the rectangular co-ordinates is a point in the circumference, and the centre is on the axis of y j r being the radius of the circle. 1062. The equation of a circle is (/+»') (1 + a')\ -2b{x + ay) = 0; find its radius 1063. Find the equation of a circle referred to oblique axes ; and find the centres and radii of the circles 2/' + a' + 2y - 6a; = 3, 6x — 1 and S'f/' + x — - — + 6(y-2) = 5. /^ 1064. Find the equation to the tangent to a circle at a point {h, k) in the circumference ; thence deduce the equation to the normal, and show that it always passes through the centre. 1065. The equations of three circles A, B and C are ^'^y'=^ {A\ x- + f^6x~10y + 25 = (B), x' + y^-4>{4^x + y)+52 = (C). Show that the line joining the centres of A and B is perpendicular to that which joins the centres of B and C, 1066. Find the equation to the common chord of the two circles x^ + y^=z25 and x' + y^ +6x^Sy = Q. CO-OEDINATES. 117 1067. The equation to a circle is x" + 9f-6x-l2y-¥A!l = 0', find its radius, and also the equation to the tangent which passes through the origin of co-ordinates. 1068. Find the angle contained by those diameters of the circles a;^ + 2/^ + 03 + 2?/ + 1 = 0, and a* + y^ + 2a; + 2^ + 1 = 0, which pass through the origin of co-ordinates. 1069. Determine the equation of the circle which has its centre at the point (1, — 3), and which touches the straight line 2/ - 2a; + 4 = 0. 1070. Find the points in which the line y = 5a; + 2 intersects the circle 2/Va;'-4.y-13a; = 9; and the length of the part within the circle. 1071. Determine the points in which the line y = 3a;-f-2 cuts the circle 2/^ + a;^ - 4a; + 4?/ = 7. 1072. Find the angle at which a tangent to the circle at the point whose abscissa is — 3, cuts the axis of x. 1073. ABCD is a rectangle; MK and NL are parallel to AD ; AN = BM, and AK\a joined ; find the locus of P. /' 1074. Investigate the general equation of the conic sections, and show how it may be reduced to the form which is the equation to the ellipse. 118 MATHEMATICAL EXAMINATION QUESTIONS. 1075. Define a parabola, and from the definition find its equation when the origin is a point in the curve and the axis of X is parallel to the directrix ; the abscissa of the focus being a, and the distance of the focus from the directrix being 2w. 1076. Investigate the equation to the elli|)se, considered as the locus of a point in a straight line of given length, which has its extremities in two fixed straight lines at right angles to one another. 1077. Give a general definition of the curves termed conic sections; deduce their general equation, and show under what conditions it represents each of the three particular curves. 1078. Determine the major and minor axes, and the eccen- tricity of the ellii^se x-5 = -y. 1079. A straight line moves so as to have one extremity in, and to be always perpendicular to another straight line whose length is Q,r ; find the locus of the other extremity, supposing the square of the first line to be always equal to the rectangle of the segments into which it divides the other. 1080. Find the locus of the point which is equidistant from the axis of y, and from the point (6, 0). 1081. The legs of a pair of compasses, standing vertically upon a horizontal table, are gradually extended so that the liinge descends vertically ; find the locus of the point of bisection of each leg. 1082. If (A, h), {h^, k^) be two points in a circle whose equa- tion is x^ + y^ = r^, show that the equation to the straight line joining those points may be reduced to the form lix + ky = r*, when h = h^ and k = k^; and apply this principle to the deter- mination of the equation of the tangent to a parabola. CO-ORDINATES. 119 1083. Prove that the locus of the intersection of two straight lines at right angles to one another, which pass through two fixed points, is a circle ; find its radius, and the position of its centre. 1084. If a straight line of given length move so that its extremities are always upon two straight lines at right angles to one another, show that any point in it will describe an ellipse whose semi-axes are the segments of the line; and explain the particular cases in which (1) the point is at the bisection of the line, and (2) at either extremity. 1085. The parabola being the locus of a point which is equi- distant from a given point and from a given straight line, find its equation when the given point is the origin of co-ordinates and the axis of y is parallel to the given line ; and show, from this equation, that the distance between the j^oints where the curve cuts that axis is equal to twice the distance from the given point to the given straight line. 1086. A tangent to an ellipse is inclined at an angle of 45° to the axes ; find its point of contact with the curve. 1 087. Find the point in which the line wliich joins the focus and the extremity of the axis minor of an ellipse meets the curve. 1088. Show, by means of its equation, y^ = 4<7nx, that the parabola is a curve which cuts the axis of x in but one point, and that every point in the curve is on the same side of the axis of y. 1089. In the parabola y^ = 4>ax, find the length of the normal which passes through the extremity of the latus-rectum, inter- cepted between the two branches of the curve. 1090. The major axis of an ellipse and the axis of a parabola are in one straight line, and the vertex of the parabola is at the centre of the ellipse j find the points of intersection of the curves, the parameter of the parabola being equal to the minor axis of the ellipse. 1091. Assuming the area of a parabola to be two-thirds of the rectangle of the same base and altitude, show that parabolic 120 MATHEMATICAL EXAMINATION QUESTIONS. areas of the same parabola and having parallel bases are as the cubes of their bases ; and bisect a given parabolic area by a line drawn parallel to its base. 1092. Show that the locus of the point whose distances from a given point on the axis of x, and from a given straight line parallel to the axis of y, are always in the ratio e : 1, is the equation 2/' + {l-e')a;'-2m(l +e)a;=0, where m is the distance of the given point from the origin : and investigate the several forms which this equation may be made to assume when the given ratio is greater than, equal to, or less than unity. 1093. A parabola and an ellipse have the same vertex ; and the axis of the parabola coincides with the major axis of the ellipse ; find the points in which the parabola cuts the ellipse when the parameter of the former is equal to half the minor axis of the latter. 1094. Find the equation of the parabola which passes through the points (0, 0), (3, 2) and (3, - 2). 1095. Prove that the perpendicular from the focus of a para- bola upon a tangent intersects that tangent in the tangent at the vertex. 1096. Find the locus of the point whose distances from two given points are always in a given ratio to one another. 1097. Show that a parabola having the same vertex and axis as an ellipse will cut the elKpse in two points if its latus-rectum 26* be less than — : and that it will not cut it at all if the latus- a rectum be greater than that quantity. 1098. Fiad the curve from any point of which two normals being drawn to a given parabola, they will be at right angles to one another. STATICS. 121 STATICS. 1099. Define " relative rest," " relative motion," and "equi- librium." 1100. Assuming that the resultant of two forces which are represented in magnitude and direction by the adjacent sides of a parallelogram is represented in direction by its diagonal; prove that such resultant is also represented in magnitude by that diagonal. 1101. Deduce the general expression for the resultant of two forces y, and/^ whose directions make an angle a with one another ; and also for the angle $ which the direction of this resultant makes with the direction of the force yj. 1102. Assuming the parallelogram offerees, show that, when three forces are in equilibrium, they will be to one another as the sides of a triangle formed by drawing straight lines parallel to their directions. 1103. A balloon which could just raise a weight of 10 cwt. is held to the ground by a rope which makes an angle of 75*^ with the horizon ; find the tension of the rope and the force of the wind upon the balloon. 1104. Two weights P and Q act by a string over two fixed pulleys at A and B, and support a third weight W, at (7, between them ; find the ratios of P, Q and IT, when AB is horizontal, the angles of the triangle ABG are in arithmetic progression, and C is a right angle. 1105. Two forces in the ratio of 2 : 5, and whose resultant is a mean proportional between them, make an angle 6 with one another : find cos 0, 1 1 06. Three forces re^' resented by the weights 50 lbs., 40 lbs. and 30 lbs., act, in one plane, upon a point; what angles does 122 MATHEMATICAL EXAMINATION QUESTIONS. the force 50 lbs. make with the other two when the three are in equilibrio ? 1107. What force must a man exert in a horizontal direc- tion to draw a weight of 3 cwt. four feet out of the perpendicular, supposing it to be suspended from a point twenty feet above that at which he applies his strength ? 1108. Two forces which act at two adjacent angles A and B of Si parallelogram ABGD are proportional to the coiTcspond- ing diagonals CEA^ DEB^ and act in those directions respec- tively ; find the direction, the relative magnitude, and the position of a third force which will keep the figure at rest. Show that the effect upon the figure will be the same whether it be acted on by forces proportional to, and in the direction of, EB and EA, or DB and BA, or | DA and i CB, — all the forces acting at E. 1109. A parallelogram ABC I) is acted upon by forces in the directions and proportional to AB^ BD^ and DC ', find the direc- tion and proportional magnitude of a fourth force which will produce equilibrium. 1110. Three forces, P, Q, and i?, all acting in the same plane upon a particle, keep it at rest; supposing the values of P and Q^ and the directions of Q and R to be given; find the value of R, and the direction of P. 1111. Four forces of 5 lbs., 6 lbs., 8 lbs. and 1 1 lbs. make angles of 30", 120", 225" and 300° respectively, with a fixed straight line : find the magnitude and direction of their resul- tant with reference to that line. 1112. Three tacks are fixed in a vertical wall and form an isosceles triangle ABC, in which AB = AC = 50, BC= SO and AB is horizontal : a string is passed over the tacks A and B, and under the tack (7; and, at its extremities, two equal weights of 20 lbs. are suspended : find the magnitude and direction of the pressure upon C STATICS. 123 1113. Two strings, AC and BD, are tied to two pegs, A and B, in a horizontal line, the string J3D passing through a smooth ring at C : find the position of C, and the tension of AC when a weight of 6 lbs. is suspended at D, and AC = ^AB. 1114. Two forces P and Q make an angle, a, with one ano- ther; find an expression for the magnitude of their resultant i?, and the angle which it makes with the force Q, 1115. Three forces, 5, 6, and 7, all in the same plane, and making equal angles with one another, act upon a point; what force, acting in the same plane, will keep the point at rest; and what angle must it make with the force 6 ? 1116. Three forces /j, Z^, /g make angles of 30" and 45" with one another taken in the above order; find the value oif^ in terms of/ and/^. 1117. A weight of 8 cwt. suspended by a rope from the top floor of a warehouse, on one side of a street 50 feet wide, is required to be drawn across by a rope from a floor at the same height on the other side of the street, attached to a point in the first rope 35 feet below the point of suspension; what will be the tension of the ropes when the weight is vertically over the centre of the street? 1118. A, Bj and C are three tacks at the angular points of a vertical equilateral triangle ; AB is inclined at an angle of 15" to the horizon and C is vertically below AB : a weight of 60 lbs. is suspended by a rope fixed at A, passing under C, over B, and then over C ; find the direction and amount of the pressure upon C 1119. A peg or tack A has four cords attached to it, at the ends of which four men pull, each with a force of 100 lbs., and in directions which are all in the same vertical plane and make equal angles of 30" with one another ; find the magnitude and direction of the strain upon A when a gun weighing 1 8 cwt. is hung upon it, and when the angles which the outer cords make with the horizon are equal. 124 MATHEMATICAL EXAMINATION QUESTIONS. 1120. Three tacks A, B and (7 in a vertical plane are so situated that AB is horizontal, the angle ABC is 60" and the angle BAG 45°; and a cord connecting two equal weights is passed over A and -S, under (7, and back again oyer B : find the pressures upon the tacks, and their directions. 1121. (a) If the angle between the directions of two forces be increased, theii* resultant will be diminished. And (jS) if three forces in a plane acting upon a point be proportional to the sides of a triangle constructed upon their directions taken in order, each of them will be equal and opposite to the resultant of the other two. But (y) if an angle of this triangle be in- creased, the opposite side will be increased, and therefore the resultant will be increased. Demonstrate the statements (a) and (/3), and reconcile the apparent anomaly involved in them as indicated in (y). 1122. Show that, when any number of forces f^^f^if^i : find the tensions of AC, CB, ED, and DB and the compression of CD, when the angle CED = 60". State what would take place if the cord ACEDB were continuous and allow- ed to run smoothly through eyes at C and D. 1137. A canal boat is drawn along the middle of a straight canal by two men hauling upon ropes 50 feet and 80 feet long respectively; find the ratio of the strains upon them, supposing the canal to be 30 feet wide. c 1138. A gun, to be lowered over the parapet of a fortress, is suspended from the extremity C of a spar AC, 10 feet long, held in its position by a stay BC 15 feet in length, the distance from ^ to jS being 10 feet; find the tension of BC and the thrust upon AG, when AB IB horizontal, and the weight of the gun S3 cwt. 1139. Define "the centre of parallel forces," and show that in the case of two such forces acting upon a rigid body, their "centre" divides the straight line which joins their points of application in the inverse ratio of the forces. 1140. Five equal bodies are placed so that their centres of gravity are at the angular points of a regular hexagon ; find the position of their common centre of gravity, when referred to the unoccupied angle. 1141. If three parallel forces be proportional respectively to a, h, c, the sides opposite to the angles of the triangle at which they act; show that the distance of their centre from the angle A is /hr. . (ft - a\ a + h+c V- (i-o). where s = 128 MATHEMATICAL EXAMINATION QUESTIONS. 1142. The triangle formed by joining the points of bisection of two of the sides of a triangle with its centre of gravity and with one another, is one-twelfth of the area of the triangle. 1143. Three parallel forces act at the circumference of a graduated circle, at the points marked 30®, 135" and 240", and their magnitudes are to one another as the cosines of these respec- tive arcs ; find the distance of the centre of these forces from the centre of the circle, and the angle which the line joining these centres makes with the first force. 1144. Five equal parallel forces act at five of the angles of a regular hexagon ; find the distance and direction of their centre from the remaining angle. 1145. The interior diameter of a 13-inch shell being 9*3 inches, find the position of the centre of gravity of one of un- equal thickness, in which the centres of the interior and exterior spherical surfaces are 1 inch apart; neglecting the fuze-hole. 1146. The opposite points of bisection of the sides of a parallelogi-am are joined, . and one of the parallelograms thus formed is cut from the figure : find the position of the centre of gravity of the remaining surface. 1147. If through the centre of gravity of a triangle two straight lines be drawn parallel to two adjacent sides and limited by the third side ; the triangle so produced will be equal to one- ninth of the original triangle. 1148. Three weights of 1, 2 and 3 lbs. are placed at the angles of an equilateral triangle ; find the distance of their com- mon centre of gravity from each of them when the side of the triangle is 2 feet. 1149. Find the centre of gravity of a solid composed of a cylinder and cone on the same base and of equal altitudes : 1st When their densities are equal; 2nd When the density of the cone is double that of the cylinder. STATICS. 129 And find their relative altitudes when the centre of gravity is in the centre of their common base. 1150. Find the centre of gravity of a cylinder having a cylindrical bore; the cylinder being 10 feet long and 10 inches in diameter; the bore 8 feet long and 4 inches in diameter. 1151. Find the position of the centre of gravity of a gun whose content is C and length of bore 6; the bore being cylindri- cal (circular section, radius r) with a conical chamber whose length is one-twelfth of h : supposing the distance of the centre of gravity of the unbored mass to be at a distance m from the muzzle. 1152. Prove that the centre of gravity of a pyramid is in the line joining its vertex and the centre of gravity of its base, and that it divides this line in the ratio of 1 to 3. 1153. An equilateral triangle is suspended at a point in one of its sides one-third of its length from the adjacent angle ; find the inclination of its other sides to the vertical. 1154. A beam whose centre of gravity divides its length in the ratio oi a : b, is placed in a smooth hemispherical bowl, and sub- tends an arc 2a at the centre of the sphere j find the inclination of the beam to the horizon. 1155. AB and BG are two beams united by a hinge at £, and resting on walls at A and C ; W is a weight suspended from B, and the weights of the uniform beams are F and Q. Find the horizon- tal thrusts at A and (7, supposing W=F = Q:= 100 lbs., ABW = 45" and CBW=30'. 1156. Prove that the centre of gravity of a triangle divides each of the lines drawn from the angular points of the triangle to the points of bisection of the opposite sides in the ratio of 2 : 1. 1157. A square lamina has one of its angles cut off by a straight line which bisects two of its adjacent sides; find the centre of gravity of the remainder. c. 9 130 MATHEMATICAL EXAMINATION QUESTIONS. 1158. Prove that the centre of gravity of a triangle remains unaltered, if the triangle formed by joining the points of bisec- tion of its sides be removed. 1159. A rectangle is divided into four triangles by its diago- nals, and one of these triangles is cut out from the rectangle ; find the position of the centre of gravity of the remaining sur- face. 1160. A square and an equilateral triangle having a com- mon side and of uniform thickness, are rigidly connected, and a weight equal to the weight of the triangle is suspended at the vertex of the triangle ; find the position in which the figure must be placed so that it may balance on a horizontal edge parallel to the common side. IIGI. Find the centre of gra^aty of the figure formed by describing an equilateral triangle upon one of the sides of a square. 11G2. A cylinder 10 feet long has a cone attached to one end of it by its base, which is of the same circular section as the cylin- der; find the height of the cone, so that the whole mass may balance about a point 8 feet from the other end of the cylinder. 1163. A 13-inch shell weighing 200 lbs., slung on a pole 8 feet long, has to be carried some distance by two men, one of whom, having just come out of hospital, is requii'ed to support only two-thirds the weight the other man carries; where must the shell be slung in order to produce this efiect 1 Find also the weight supported by the weak man when the shell is suspended 6 feet from his end of the pole. 1164. A cone and hemisphere of the same material and hav- ing the same base are fixed together, and it is found that they can be made to roll in a straight line from the top to the bottom of an inclined plane; find their relative dimensions. 1165. A straight uniform rod, 20 inches long, balances upon a fulcrum 8 inches from one end, when weights of 25 oz. and 10 oz. are suspended at its extremities; find the weight of the rod. STATICS* 131 1166. If from one point of the angular points of any regular plane figure straight lines be drawn to the other angles, show that the centres of gravity of the triangles thus formed lie in the cir- cumference of a circle, the radius of which is two-thirds of that of the circle inscribed within the figure. 1167. If at the angular points of a material plane triangle, weights A, B, and G be placed; find the position of the centre of gravity of the whole system, the weight of the triaugle being W; and show from the result that, when the weights A^ B, and G are' equal, the common centre of gravity is that of the triangle. 1168. Find the position of the centre of gravity of a hollow cylinder, the radii of whose interior and exterior surfaces are r and Ji, and their axes at a distance a from one another. 1169. Find the position in which a cylinder having a conical cavity (the axis and the base of which are coincident with those of the cylinder, and its altitude and the diameter of its base one- half of those of the cylinder) will rest when suspended freely at a point in the circumference of its base. 1170. If from a sphere 16 inches in diameter a cone having the diameter of its base equal to its slant height be turned out, find the position of the centre of gravity of the remaining solid. 1171. Find the centre of gravity of a solid composed of a cylinder and cone having a common base, the length of the cylin- der being 1 foot and the height of the cone 6 inches. 1172. A circular marble slab weighing 20 lbs. is supported as a table upon three vertical legs at its circumference, their dis- tances subtending angles of 75° and 135° at the centre; find the pressure on each leg. 1173. A cone and a hemisphere have a common base, and their common centre of gravity is in the centre of that base ; find the height of the cone, the centre of gravity of the hemisphere being three-eighths of its radius from the centre of its base; and show that the compound solid will rest in any position on the hemisphere when placed on a horizontal plane, but that there is- 9—2 132 MATHEMATICAL EXAMINATION QUESTIONS. no positioxi in wLicli it will rest on the hemisphere when the plane is inclined to the horizon ; assuming always that the plane and hemisphere are sufficiently rough just to prevent sliding. 1174. The centre of gravity of a hemisphere is at a point distant three-eighths of the radius from the centre of its base; find the position in which the solid will rest if suspended from a point in the circumference of its base; and find a point in its convex surface from which if it be suspended, the plane of its base will be inclined to the same amount as before on the other side of the vertical. 1175. A conical vessel full of water has mercury poured into it until the mercury rises to one-fourth of the side of the cone measured from the vertex. What is the position of the common centre of gravity of the mercury and water, the specific gravity of the mercury being m 1 « 1176. If the arm BF of a straight or bent lever be of such length and thickness as to cause the other arm FA to rest hori- zontal when a moveable weight w is suspended at II (between F and A); show that the distances from II at which the constant weight w must be placed in order to balance successive weights suspended at £, wiU be proportional to those weights. 1177. A weight is placed upon a horizontal table which has three vertical legs A, B, and C; the portion of the weight sup- ported by -4 is 8 lbs., that by -6 is 5 lbs., and that by C is 9 lbs. ; find the weight and its position upon the table, the distances be- tween the legs being 2 feet, 4 feet, and 5 feet respectively. 1178. A uniform bent lever, when supported at the angle, rests with the shorter arm horizontal; but if this arm were twice as long it would rest with the other horizontal ; find the lengths of the arms and the angle between them, when the whole length of the lever is 30 inches. 1179. A piece of timber, 25 feet long, balances upon an e^gQ at a distance of 10 feet from one end, and, when the edge is STATICS. 133 shifted to 12 feet, it requires 56 lbs. to be placed upon the other end to make it balance; what is the weight of the piece of timber ? 1180. Four weights, 2, 6, 14, and 10, are placed at equal distances on a straight lever. Determine the fulcrum when the lever is 21 inches long and the weights 2 and 10 are placed at its extremities; supposing the lever to be without weight. 1181. A uniform bar of iron 10 feet long projects 6 feet over the edge of a wharf, there being a weight placed upon the other end; and it is found that when this is diminished to 3 cwt. the bar is just on the point of falling over ; find its weight. 1182. A uniform leVer, whose weight is 8 lbs. and length 3 feet, has a weight of 20 lbs. suspended from one end, and 14 lbs. from the other; find the position of the fulcrum when there is equilibrium. 1183. The ratio of the arms of a bent uniform lever, inclined to one another at an angle of 120", is 5 to 4; find their inclina- tion to the horizon, when weights of 20 lbs. and l61bs. respec- tively are suspended from their extremities ; the fulcrum being at the angle. Find also the point on the longer arm upon which the lever will balance so as to have that arm horizontal; the lengths of the arms being 30 inches and 24 inches, and no weights being suspended. 1184. To the extremities of a smooth circular arc of yS degrees without weight, two weights P and Q are suspended ; find the position of the fulcrum over which the arc must be suspended, concavity downwards, to produce equilibrium. 1185. The arms of a bent uniform rod are as 3 to 5, and they are at right angles to one another; find the position in which it will rest when suspended by the angle. 1186. A uniform rod 2a feet long, and 2p pounds in weight, is balanced upon its middle point ; how much must it be length- ened in order that it may remain horizontal when a weight W 134 MATHEMATICAL EXAMINATION QUESTIONS. is suspended from one of its ends, the fulcrum remaining un- altered 1 1187. A uniform rod of unknown length, "weighing l^lb. per linear foot, rests on a fulci-um 4 feet from one end; find what weight suspended from that end wiU keep it at rest, the pressure on the fulcrum being 75 lbs. 1188. A weight of 10 lbs. suspended at one extremity of a horizontal lever, 3 feet long, at 9 inches from the fulcinim, is balanced by a force P acting at the other extremity and makin'^ an angle of 60° with the lever, which weighs 2 lbs. Find P when the centre of gravity of the lever is 7 inches from the point of application of the weight. 1189. Find the tension of each of two ropes which, passing round the circumference of a cylindrical pontoon and fastened to the upper extremity of a plane, support upon it the pontoon which weighs 4 cwt. ; the rojDes being parallel to the plane, which is inclined to the horizon at an angle of 30°. 1190. In the system of n pulleys in which all the strings are fastened to the weight, find the strain upon the axle of tlie pulley which is fixed; assuming the weights of the pulleys to be each equal to w. 1191. The weights of the several moveable pulleys in the system where each stiing is attached to the weight, are 1, 2, 3, and 4 oz. respectively, commencing at that over which the power first acts j find the weight supported by the force of 1 lb. 1192. A gun, in which the distance of the muzzle from the axis of the trunnions is t, is found to balance upon its trunnions when a weight w is suspended from the muzzle ; find the weight of the gun, supposing g to be the distance of its centre of gravity from the muzzle. 1193. A bent lever whose arms are a and h, inclined at an angle a to one another, has the weights P and Q respectively sus- pended at their extremities ; prove that the inclination of the arm P. a. sin a b to the horizon is cot" ^6 — Pa . cos a STATICS. 135 1194. A force / is required to lift the trail of a field-gun when a man whose weight is M rests his whole weight upon the muzzle of the gun; the length of the trail measured from the axle of the wheels being t, and the horizontal distance of the centre of gravity of the carriage g, that of the gun G (both behind the axle), and that of the man m, all measured from the axle ; find the weight of the gun, the weight of the carriage being (7, the angle which the line drawn from the end of the trail to the axle makes with the horizon a, and the gun horizontal. 1195. A gun is just raised from the ground at one end by a cord attached to the neck behind the breech and passing over a fixed pulley vertically over the neck, having a weight w sus- pended on the other side of the pulley ; and it is found that when the same arrangement is made at the muzzle, a weight w just raises that end; find the weight of the gun, and the distance of the C3ntre of gravity from the muzzle, the length from the muzzle to the neck being I feetj and the length from the muzzle to the base-ring l^ feet. 1196. A weight of 7 cwt. rests upon a plane inclined at an angle of 30° to the horizon ; what force, acting parallel to the hori- zon, will just prevent its sliding down the plane when the coeffi- cient of friction is -- j and what is the least force which, acting parallel to the plane, will draw it up ? 1197. Show that, if a be the inclination of a plane, i the angle which the supporting force P makes with the face of the plane, W the weight supported, and R the pressure upon the plane, the following relations subsist when there is equilibrium : cos I COS (a + 1) sm a 1198. In the single moveable pulley, show that, if the strings izo W make an angle a with the horizon. . cosec a. 136 MATHEMATICAL EXAMINATION QUESTIONS. 1199. Find the -weight which can be supported by a power of 1 5 lbs. by means of a system of pulleys, in which each string is attached to the weight, there being four moveable pulleys, each weighing 4 lbs. 1200. A force /*, making an angle i with the plane whose angle of inclination is a, acts upon a body which rests upon the plane, and whose weight is W; supposing that F is not sufficient to support the body upon the plane, find an expression for an additional force Q which, making an angle y with the plane, will just produce equilibrium. 1201. A gun, weighing with its carriage llcwt., is to be drawn up a plane inclined to the horizon at an angle of 20**; find the force which must be exerted to draw it up by ropes making an angle of 10" with the plane; and the pressure on the plane, supposing the amount of friction to be inconsiderable. 1202. What force is necessary to support a 13-inch shell (weight 196 lbs.) upon a plane inclined to the horizon at an angle of 30°, the force acting horizontally ? 1203. A shell is found to remain at rest on a plane inclined to the horizon at an angle of 15° when a horizontal force h is applied at its surface in a direction passing through its centre; find the weight of the shell. 1204. A guu and carriage weighing 12cwt. are to be drawn up a plane inclined to the horizon at an angle of SO'*, by four ropes fastened to the naves of the four wheels, and making angles of 15° with the face of the plane ; find the tension of each rope when the gun just begins to move up the plane. 1205. Find the power which must be applied to a system of pulleys in which each string is fixed to the weight, in order to raise a weight of 2 cwt., supposing that there are four strings, and that each pulley weighs 2 lbs. 1206. What weight will be supported by a power of 12 lbs. acting by means of a system of pulleys in which each cord is fastened to the weight, when there are five moveable pulleys, each weighing 1 ^ lb. 1 STATICS. 137 1207. Find the weights which must be suspended at Q and E to establish an equilibrium in the annex- ed system of pulleys when a force F acts vertically at P, the weights of the pulleys being neglected. 1208. Determine the ratio of the power to the weight in each of the following systems of pulleys : vy ^ >l' V P Q K ^ vy fh ^^ >'P "W 1209. Five turns of a screw working in a fixed collar have the efiect of raising the end of the screw one inch ; what would be the effective force at the end of the screw in the direction of its axis, if a force of 10 lbs. were applied at the extremity of a lever 5 feet long working in the head of the screw ? 1210. The length of each arm of the elevating screw of a gun is 6 inches, and it is found that one turn of the screw has the effect of elevating the piece 41'; what power is required to turn the screw, supposing the whole vertical resistance to be overcome to be 3 cwt, and the distance from the axis of the trunnions to the centre of the screw to be 3 ft. 6 in. 1 1211. Find the inclination to the horizon of the thread of a screw, which, with a force of 5 lbs. acting at an arm of 2 feet, can support a weight of 300 lbs. on a cylinder of 3 inches radius. 138 MATHEMATICAL EXAMINATION QUESTIONS. 1212. A locomotive having run off the line, it is proposed to raise one side of it by means of a machine consisting of a winch which turns an endless screw, which works in the teeth of a wheel having a pinion attached ; the teeth of this pinion are of the same dimensions as those of the wheel, and work in a vertical rack, at the extremity of which is the point of support : what effect would such a machine produce if a force of 56 lbs. were applied to the winch, the various dimensions being : Arm of the winch = 15 inches ; No. of teeth in wheel = 1 00 ; Distance between the threads = 1 inch; No. of teeth in pinion = 5 ? 1213. Ten men heave, with a force of 50 lb. each, at as many capstan-bars, 10 feet from the axis of the capstan; the cylindei" round which the hawser winds is 2 feet in diameter, and the hawser four inches ; find the tension of the hawser. 1214. Supposing that, in question 1201, the power is to be applied by a capstan and system of pulleys in which the same string passes round all the pulleys, all the strings being parallel to the face of the plane ; find the power to be applied with the fol- lowing data : Length of handspike 6 feet; diameter of axle 16 inches; number of sheaves in each block 4. 1215. An endless screw works in the teeth of a wheel, round the axle of which a rope is wound, which then passes continuously round a system of pulleys having four sheaves in the lower block ; find the weight which a power of 1 lbs., applied to the arm of the screw, which is 1 foot long, will suj^port, supposing the dis- tance between the threads of the screw to be 1 inch, the radius of the wheel 1 foot, and that of the axle 3 inches. 1216. A pinion of 12 teeth works in a wheel of ^6 teeth, upon the axle of which, 6 inches in diameter, a rope winds, the other end of which is coiled round a wheel 12 feet in diameter, and fastened to its circumference : upon the axle of this wheel, 18 in. in diameter, another rope is coiled which, passing down a STATICS. 139 plane inclined to the horizon at an angle of 45", is fastened to the truck to be drawn up the plane : what number expresses the mechanical advantage of this arrangement, if the first pinion be worked by a winch 2 feet long ? 1217. A cone whose height is h and the radius of its base r, rests with its curved surface upon an inclined plane, and is pre- vented from sliding ; find an expression for the tangent of the greatest inclination of the plane, so that the same surfaces may continue in contact. 1218. Find the strain upon a tent-peg fixed into the ground, when the efiective force of the wind upon the top of the pole in the direction of the horizontal line drawn in the plane of the pole and the peg is 120 lbs., and the angle which the tent-rope makes with the horizon is 45". 1219. An isosceles right-angled triangle is suspended by one of its acute angles ; find the angle which its hypothenuse makes with the horizon. 1220. Find the cylinder of greatest length, the radius of whose base is r, which can be made to rest with its base inclined to the horizon at an angle a. 1221. A uniform beam, whose length is 2c», rests with one end fixed upon a smooth horizontal plane by a hinge, and, at a distance c from this end, it rests upon a smooth sphere which is supported upon the same plane ; determine the horizontal force necessary to maintain the sphere in its position. 1222. A beam, loaded with a weight W at its middle point, rests upon two supports in the same horizontal line, and at a distance I from one another j find the moment of strain when the deflection is a. 1223. Two weights P and Q balance one another upon the surface of a sphere, being attached to one another by a string which passes over the highest point of the sphere. Find the posi- tion of equilibrium when P = ^Q. 140 MATHEMATICAL EXAMINATION QUESTIONS. 1224. A beam AB moveable, in a vertical plane, about A, has its other extremity B supported by a weight P acting over a pulley vertically above B ; find the weight of the beam wlien its centre of gravity divides its axis in the ratio of 5 : 7, from A towards B. 1225. A ladder AB resting with its foot against the base of a vertical wall AC, is to be partially raised by means of a rope pass- ing over a pulley at C, and attached to the ladder at a point i), which is a feet distant from A : what angle will the ladder make with the wall when a weight Q is attached to the rope, supposing the centre of gravity of the ladder to be b feet from yl, and the wall to be c feet high 1 1226. A mortar, whose weight is TT, being suspended from a uniform beam whose length is 2^ and weight w, at a distance b from its lower end, which rests on the ground at a distance d from a vertical wall, against which the other extremity of the beam rests : it is required to find the greatest tension of a rope, which, being fixed to the upper end of the beam, is hauled upon at a point whose horizontal distance from the foot of the wall is a. 1227. A rectangle whose sides are a and b, is suspended from a point in the side a, which is at a distance (, = tanA, and 7? = ^. ^(1 + /X,''). 1257. It is found that a force of 5 cwt., acting at an angle of 45°, is required to draw a block of stone over a rough horizontal surface ; find the weight of the block, supposing the coefficient of friction to be "62. 1258. A cone is placed with its base upon a rough horizontal plane, and a string is attached to its vertex ; find the dimensions of the cone when, a horizontal force being applied to the string, the cone will turn over when it is just upon the point of sliding along the plane. 1259. A post stands at a distance of 3 feet from a rough vertical wall, and a rough beam, 12 feet long, is placed over the post with one end resting against the wall ; find the equations of equilibrium necessary for determining the greatest inclination of the beam to the wall when the coefficient of friction for the post and wall is ft. c. 10 146 MATHEMATICAL EXAMINATION QUESTIONS. 12 GO. Find the force necessary to draw a block of stone up a plane, down which it would descend by the action of its o^^'^l weight, supposing the force to be inclined to the horizon at an angle equal to twice that of the plane. 1261. A cone, 4 inches in diameter and 20 inches high, stands symmetrically upon a cylinder 6 inches in diameter and 14 inches high which rests upon a plane: the coefficient of friction between the cone and cylinder is • 5, and between the cylinder and plane is -4. What will take place if the inclination of the plane be gi-adually increased 1 1262. In the inclined plane, show that the ratio of P to IF is that of sin (i =*= e) to cos (a =f e), where i is the inclination of the plane, a the inclination of the force P to the face of the plane, and e the limiting angle of friction. And show how this may be applied to the case of the screw, where h is the linear pitch of the screw, c the circumference of the cylinder, C the circumference desciibed by the point of application of the force P, and /x the coefficient of friction. 1263. Find the inclination of a plane down which a block of stone will just slide by the action of its own weight, the coefficient of friction being '63. 1264. What force, acting parallel to the plane, will be requi- site to draw a square block of timber up a plane inclined at an angle of 60", the coefficient of friction being -63^ and the weight of the block 5 cwt. 1 1265. Find the force (acting parallel to the plane) necessary to draw a 10 in. howitzer, weighing with carriage and limber 124 cwt., up a plane inclined to the horizon at an angle of 30"; 1st, assuming that there is no friction, and 2nd, that the friction is one-tenth of the normal pressure. 1266. Find the least force which, acting parallel to the plane, is necessary to drag a weight of 10 cwt. up a plane inclined to the horizon at an angle of 1 5°, the limiting angle of resistance being 30\ STATICS. 147 1267. A right cone whose height is equal to three times the diameter of its base, stands with its base upon an inclined plane; determine whether it will slide or fall over when the inclina- tion of the plane is gradually increased, if the coefficient of friction be -7. 1268. Find the least force necessary to draw a weight of 500 lbs. up a plane inclined at an angle of 15*^ to the horizon, the limiting angle being 30". 1269. The weight of a body standing upon a plane of 60^ in- clination is eleven-tenths of the force necessary to draw it up the plane by a cord parallel to the plane; find the coefficient of fric- tion between the body and the plane. 1270. A rough sphere is placed upon a rough inclined plane, the limiting angle of friction being e ; what is the inference if it neither roll nor slide ? State under what conditions it can be made to slide without rolling, and vice versa. 1271. Two forces, F acting parallel to the plane and Q making an angle a with the plane, just support a body upon a rough plane inclined to the horizon at an angle i, and when Q and F are interchanged, the body is on the point of moving up the plane ; find the values of F and Q. 1272. Two planes of equal slope, 60°, but of unequal height, are placed back to back, and two rough particles of equal weight, connected by a fine string which passes over a smooth pulley at the top of the highest plane, rest upon the planes, one upon each ; find the angle which the string makes with the lower plane, when the particle upon the other is on the point of moving up the plane, the limiting angle being 15^ 1273. If the force which, acting parallel to a plane, is neces- sary to draw a body up the plane be twice the force which, acting in the same direction, will allow it to slide down; what is the relation between the inclination of the plane and the coefficient of friction ? 10—2 148 MATHEMATICAL EXAMINATION QUESTIONS. 1274. If the force which, acting parallel to a plane inclined i to the horizon, is just sufficient to draw a body up, be n times the force which will just allow it to slide down the plane, show ?i + 1 that tan i . cot e = , c being the limiting angle. 71 — 1 1275. If F be the force which, acting parallel to a plane of known inclination, would just draw a body up the plane, and F' the force which, acting in the same direction, would just allow the same body to slide down the plane; find the coefficient of friction. 1276. Two rough planes, rigidly connected at their inter- section, are suspended upon a horizontal axle passing through it, and, upon them, two rough, heavy particles of equal weight, joined to one another by an inextensible string passing freely over the top are placed; find the angle through which the system may be made to move round the axle without displacing the weights from their position on the planes. 1277. A cylinder 6 in. in diameter and 14 in. high, begins to slide at the same time that it begins to topple over when the incli- nation of the plane on which it stands is gradually increased; find the coefficient of friction. 1278. A rectangular block of stone rests at an angle 6 with one of its edges on a horizontal plane and another against a vertical wall ; find the least value of ^ consistent with equilibrium, the sides of the vertical section of the block being a and h, and the coefficients of friction /w, and fi'. 1279. A heavy beam rests at an angle of 60", with one end against a smooth vertical plane and with the other upon a smooth horizontal plane ; find the value and direction of the oblique pres- sure of the beam against an obstacle in the horizontal plane which prevents its sliding, when the weight of the beam is W, and its centre of gravity is at a point one-thii-d of its length from the lower end. 1280. A beam 21 feet long rests with its ends upon a hori- zontal and a vertical plane respectively; find how far the lower end may be drawn out from the wall before the beam will slip ; STATICS. 14rd tlie coefficient of friction being ^ and J respectiNrely, and the centre of gravity, which is nearest to the lower end, dividing the beam in the ratio of 3 : 4. 1281. If a beam AJ3 rests with its ends upon rough horizontal and vertical planes respectively, and BE be drawn in the direction of the oblique resistance of the vertical plane to meet the ver- tical BGE drawn through the centre of gravity of the beam and intersecting the horizontal plane in i), show that the ratio of AD to DE is the coefficient of friction at A. 1282. A uniform rectangular board rests with one of its sidea upon a rough horizontal plane and parallel to the base of a rough vertical wall, against which the opposite side of the board rests ; find the least angle which the plane of the board can make with the horizon, the coefficients of friction being /x, and fxf ; and prove that this angle is the same whichever side of the rectangle be the base. State also the relation between the coefficients of friction when the inclination is 45". 1283. A beam is to be placed against a vertical wall, and it is required that its inclination to the liorizontal floor on which its lower end rests shall not be greater than 30"; find the position of its centre of gravity, so that it may not slip at that angle when the coefficient of friction at each end is '5. And show that although a beam may be made to stand at an inclination to the horizon when the floor is rough and the wall smooth, it cannot be made to stand in any but a vertical position when the floor is smooth and the wall rough. 1284. A hemisphere rests with a point in the circumference of its base upon a rough horizontal plane, and a point in its convex surface in contact with a rough vertical wall ; find the coefficient of friction when the body is just supported with its base in a plane parallel to the wall. 1285. A ladder which is divided by its centre of gravity into segments of 10 feet and 20 feet, is placed against the side of a house at an angle of SO"; find the highest round upon which a weight of 4 cwt. can be suspended; the weight of the ladder 150 MATHEMATICAL EXAMINATION QUESTIONS. being 3 cwt., and the coefficient of friction between the ladder and wall, and also between the ladder and horizontal pavement which supports its heavier end, being -j= . 1286. A uniform beam AB rests upon a rough horizontal plane AG and against a rough vertical wall CB ; show that it will stand when the coefficient of friction, being the same at both ends, is not less than BA-BG AG ' 1287. A rough body is supported upon a plane, inclined to the horizon at an angle a, by a string attached to a peg in the sur- face of the plane ; find the least angle which the string can make with the horizontal line drawn through the peg. 1288. Two uniform and equal rough rods, joined at an angle of 120°, are set astride over a rough horizontal cylinder in a plane perpendicular to its axis ; find the least angle which one of them can make with the vertical, the coefficient of friction being /a, the length of the rods 2a, the radius of the cylinder r, and the plaue of the rods being perpendicular to the axis of the cylinder. 1289. A uniform beam AB, 12 feet long, weighing 100 lbs., is placed upon a vertical post FB, the extremity A being fastened to the bottom of the post by a rope ^.4, and it is found that when the beam is just about to slide off the post in the direction AB, the angles FAB and FBA are 45" and 15" respectively, and AF=4! feet : find the friction at F. 1290. A flat bar weighing 10 lbs. is to be balanced at its middle point upon a horizontal edge, and its centre of gravity is 1 2 inches from one end and 1 8 inches from the other : find the greatest force which can be applied obliquely at the lighter end without causing the bar to slip ; the coefficient of friction being —t= . v3 STATICS. lol 1291. If a flat uuiform bar be supported upon an edge at its middle point, and a weight w be suspended at one end of it, find the greatest angle the direction of any force which will keep the bar horizontal can make with the vertical; /a being the coeffi- cient of friction between the bar and edge, and W the weight of the bar. 1292. A uniform beam length 2a rests against a smooth im- moveable hemisphere, of radius ?', placed, base downwards, upon a rough horizontal plane upon which one end of the beam is supported. Show that if B be the greatest inclination of the beam, 2r cot (2^ + e) = cos e sin e, ^ ' a the coefficient of friction being tan e. 1293. Show that, in the wheel and axle, when a force P, act- ing at the circumference of the wheel, su2)ports a weight Q upon the axle, F . {E=Fp sin e) = Q {r=i=p sin c) ± Wp sin e, where B, r and p are the radii of the wheel, the axle and their common axis respectively, and « is the limiting angle of re- sistance. 1294. A vertical water-wheel weigliing 5 cwt. has a cylindri- cal bearing of 3 inches radius ; find the pressure upon the circum- ference which will just overcome the statical ♦friction of the axle, the diameter of the wheel being 15 feet, and the coefficient of friction being "2. 1295. A balance having equal and uniform arms 10 inches long, turns upon a cylindrical axle 1 inch in diameter; find by how much a 20 lb. weight in one scale may be exceeded by the Aveight in the other, before the friction is overcome; the weight of each scale being 1 lb., of each arm 2 J lbs., and the coefficient of friction tan 30". 1^296. A uniform lever 2 feet long, weighing 1 lb., is balanced upon an axle whose radius is half an inch, when weights of 10 oz. and 14 oz. are suspended at its extremities, the former weight 152 MATHEMATICAL EXAMINATION QUESTIONS. being about to proponderate ; fiad the length of the arms, if the coefficient of friction be "4. 1297. Show that if P be the force which, acting at the cir- cumference of a horizontal wheel radius R, supported upon the plane end of its cylindrical axle, radius r, just overcoules the fric- OfxrW tion, P = ^ j where W is the normal pressure on the bearing. 1298. State Guldin's Theorem for determining the surface of a solid of revolution, and apply it to find the centre of gi'avity of an arc of a semicircle. 1299. The area of a parabola is two-thirds of that of its cir- cumscribing rectangle, and the volume of a paraboloid is .one-half of the volume of a cylinder of the same base and altitude, all the .sections of the solid through its axis being equal parabolas. Find tlie distance of the centre of gravity of any given semiparabola from its axis. 1300. The properties of Guldin being assumed, find the position of the centre of gravity of the arc of a circle in terms of the arc, its chord, and the radius of the circle. 1301. Find, by Guldin's Theorem, the volume generated by the revolution of a parabola about the tangent at its vertex; the curve being limited by its latus rectum. • 1302. Find the centre of gravity of the semi-ellipse cutoff by the major axis; and thence determine the volume of the solid generated by its revolution round that axis. 1303. A sphere is divided into eight equal parts by three planes perpendicular to one another; find, by integration, the position of the centre of gravity of one of these parts. 1304. Find the centre of gi-avity of the parabolic spandril ABC ; A being the vertex of the parabola, and its axis being parallel to BC. 1305. In a common isosceles trussed roof in which the two struts are parallel to the PRACTICAL MECHANICS. 153 opposite rafters, find the tension of the king-post, supposing the weight of the tie-beam to be 1 cwt., of the roof 10 cwt., and of the king post and of each strut 56 lbs.; and state what would be the effect upon the king-post if the tie-beam were to give way, supposing the parts of the truss to be rigidly connected. 1306. The radius of curvature of a deflected rectangular TT 7 3 beam being r^ , where E is the modulus of elasticity, h the depth in the plane of curvature, c the breadth and a the length of the beam ; show that the amount of deflection at its extremity is represented by . 1307. A beam of fir 8 inches square and 10 feet between its horizontal supports, has a |-inch iron wire securely fastened to its middle point, by means of which increased weights may be sus- pended until the beam or the wire give way; determine which will first take place, the tabular constant for the breaking strain of fir being 1100 lbs., and the tensile strength of iron wire ^Qt tons per square inch. 1308. Find the maximum pressure on a vertical revetment wall 20 feet high, the natural slope of the earth being 30", and a cubic foot of earth weighing 140 lbs.; supposing the friction against the wall to be the same as that of the earth. PRACTICAL MECHANICS. 1309. What allowance per mile must be made in the length of rails upon a railway for a change of temperature of 100", the modulus of expansion of wrought iron being -00000642 ? 1310. The length of a base line being measured with glass rods was found to be 23702 ft., but the temperature of the rods was observed to be 10° above the standard; what was the true length of the base; the modulus of expansion for glass being •00000431 % 154 MATHEMATICAL EXAMINATION QUESTIONS. 1311. Two steel six-foot rods are found to differ from each other by -0035 of an inch, when the difference of their tempera- tures is 8°'5 ; at what temperatures will they be accurately the same in length supposing the temperature of the longer one to be kept unaltered at 60°, the modulus being -00000636 ] 1312. The bore of the tube of a thermometer is '014 in. and the quantity of mercury in the thermometer is one-third of a cubic inch ; how many inches of scale would there be between 32" and 212°; the modulus of cubic expansion of mercury in glass beiug -00008696 1 1313. The weight of an empty thermometer tube is 250 grains, and when filled with mercury it is 1327 grains; find the diameter of the tube when the mercuiy rises 6 '2 inches for a change of 50". 1314. A steel bar 3 feet long is riveted at one end to a brass bar of the same weight, so that their free ends are towards the same parts. Find the length of the brass bar so that the distance of the centre of gravity of the mass from the free end of the steel bar may be unaltered by change of temj^erature, the modulus for brass being -00001052. 1315. Supposing the ends of the horizontal bar of an iron railing to be immoveably fixed in two vertical walls ; find the thrust exerted upon the walls when the temperature rises 30" F. ; the section of the bar being three square inches, the modulus of elasticity 29000000, and of expansion '0000642. 1316. The four iron wire stays of a mast are each one square inch in section and are inclined to the mast at an angle of 30°; find the increase of vertical pressui-e upon the mast due to a change of 40" F. 1317. The specific gravity of ice (water at 40" F. being the standard) is '91, find its modulus of expansion, supposing the expansion in volume to be three times the linear expansion ; and show this supposition to be very nearly correct. PRACTICAL MECHANICS. 155 1318. The strain upon a hempen cable being 10 tons, what must be its dimensions that it may bear this safelj, and what additional strain will be produced by its own weight if it be 50 fathoms long ? 1319. Show that if w be the weight of a unit of length of a bar of uniform section and length l^ the extension X produced by its own weight will be wT where m is the modulus of elasticity. 1320. The bearing area of a wire rope of which the circum- ference is C, being — j^ (see Qu. 874), find the dimensions of an iron wire rope which will bear with safety a strain of 1 tons. 1321. Find the length of copper wire rope which it will be just possible to suspend by one end. 1322. What must be the dimensions of a square cast-iron column to support safely, a mass of brick work 1 2 feet high, one foot thick and six feet long 1 1323. It is proposed to substitute sandstone columns for some of granite originally planned, what must be the change in dimensions % 1324. Define the *'unit of work," and find the number of units expended in raising a weight of 1 ton of building materials to a height of 40 feet. 1325. Find the time required to raise a block of granite 4 feet long, 3 feet broad, and 2| feet thick, to a height of 20 feet, by two men working at the winch of a traversing crane, the num- ber of units of work by each man per minute thus working being 2600, and the modulus of the machine "8. 1326. How many gallons of water per minute can be pumped up from the bottom of a shaft 200 fathoms deep by an engine working at a pressure of 15 lbs. to the square inch; the diameter 15G irATHEMATICAL EXAMINATION QUESTIONS. of the piston being 40 inches, the length of stroke six feet, the number of strokes 1 per minute, and the modulus 65 1 1327. Supposing the water in the last question to be carried over a wheel of 25 feet diameter which works a pump the water from which also passes over the wheel, find the whole quantity raised, the modulus of the wheel being -55. 1328. A stream of water 10 feet broad and 5 feet deep, flow- ing at the rate of 37 feet per minute, is couducted over a water- wheel 15 feet in diameter, which works a pump by which the water from the stream is raised to a height of 20 feet; what is the quantity of water raised per hour, the modulus being '65 ? 1329. Find the horse power of the wheel in question 1328, and compare it with that of the engine in question 1326. 1330. The diameter of the piston of an engine is 30 inches, the mean pressure of the steam is 10'5lbs. per square inch, the length of the stroke is 12 feet, and the number of strokes per minute is 10; find how many cubic feet of water per hour the engine will raise from a depth of 200 fathoms, the modulus of the engine being '6. 1331. In what time would two locomotives of 100 horse power each, convey a battery of field artillery from Woolwich to Brighton (60 miles), the engines, trucks, and carriages weighing W tons, and the men, guns, horses, and material A tons, sup- posing a tons to be conveyed in the trucks and carriages at each trip, and the resistances to be 10 lbs. per ton 1 1332. The piston of an engine is 3 feet in diameter, the length of stroke 5 feet, the number of strokes per minute 12, the pressure per square inch 15 lbs., and the modulus of tlie engine is '75 ; find the quantity of water which can be pumped from a depth of 30 fathoms. 1333. The piston of an engine is 48 in. in diameter; the pressure upon it 1 5 lbs. per sq. in. : the length of the stroke 5 feet, and the number of strokes 10 jjer minute; find the horse power of the engine. PRACTICAL MECHANICS. 157 1334. If the length of stroke be I, the number of strokes per minute n^ and the pressure p; what must be the diameter of the piston in order that the horse power of the engine may ha Kl 1335. What must be the horse power of an engine which will raise a tilt hammer weighing 1 1 ton 25 times a minute, the lift being: 2 feet ? "O 1336. A locomotive has to move a train of 12 carriages, each weighing, with the passengers, 4 tons, at the rate of 50 miles an hour; what must be its horse power if the resistances be 10 lbs. per ton and the weight of the engine and tender l6 tons? 1 337. If the weight of a train be 30 tons and that of the engine and tender 1 5 tons j find the radius of the driving wheel in order that with a pressure of 15 lbs., pistons of 14 in. diameter, a stroke of 3 feet, and a modulus of 75, the engine may work smoothly, and its full power be available, the resistances being estimated at 1 lbs. per ton. 1338. Find an expression for the diameter of the piston of a locomotive in terms of r the radius of the driving wheel, p the pressure per sq. in., I the length of stroke, ^ the total resistance in lbs. and m the modulus. 1339. Show that the length of stroke of a locomotive should be equal to the product of the total resistance, and the radius of the driving wheel, divided by the product of the modulus and the pressure on the piston, and multiplied by ir. 1340. A quantity of ballast is to be conveyed in trucks to a distance of 10 miles, by a train of 8 trucks, each weighing 2^ tons, and capable of carrying 7^ tons of ballast. The engine is of 20 horse power, and weighs with its tender 12 tons; and the resist- ances are 1 5 lbs. per ton. What quantity of ballast can be deli- vered in. 72 hours, allowing half an hour for discharge and loading at each trip ? 1341. How many bushels of coal will be consumed in 24 hours by an engine of 50 horse power, if its duty be 55 millions ? 158 MATHEMATICAL EXAMINATION QUESTIONS. 1342. How many bushels of coal will be consumed by an engine of 100 horse power and 70 millions duty, in raising 200000 cubic feet of water from a depth of 1 50 fathoms 1 1343. An engine of 65 millions duty has to pump water up three separate shafts of 70, 100, and 120 fathoms deep respec- tively, viz. 10 cubic feet per minute from the first, 20 cubic feet from the second, 50 cubic feet from the third. How many bushels of coal will the engine consume in 1 2 hours 1 1314. If the depths of a number of shafts of a mine be fi^A^fs^A ^^- ^^^^> ^^^ *^® number of cubic feet of water pump- ed from them per minute be t/?,, w^, w.^^ w^ tkc. respectively, find the horse power of the engine ; and its duty, supposing it to consume B bushels of coal in 24 hours. 1345. Find the number of units of work required to wind up a rope of 4 in. circumference 800 feet long 1 1346. What is the number of units of work required to wind the same rope, supposing it to be laid horizontally on the ground, the coefficient of friction being 70? 1347. Two boxes are worked up and down a shaft 150 feet deep by a connecting rope over a drum. The load each box will contain is 3 cwt. of coal. The engine has a piston of 8 inches diameter, the pressure of steam is 10 lbs., the length of the stroke 3 feet, the number of strokes 20 per minute, and the modu- lus "52. Find the quantity of coal raised per hour. 1348. Find the quantity of coal raised per hour by the same engine, supposing that only one box be used, that the time occupied in emptying the box added to the time of descent will be equal to the time of ascent of the loaded box, and that no time is lost in unhooking the empty box and hooking the full one at the bottom of the shaft; the rope being 3 inches in circumfer- ence and the weight of each boc 56 lbs. 1349. Find the number of units of work expended in dis- tributing 5000 tons of ballast equally over a line of railway, sup- DYNAMICS. 159 posing 18 tons to cover 10 yards length of the line, and tlie resistances to be 12 lbs. per ton. 1350. Find the number of units of work required to hoist the main-sail of a ship 50 feet by 30 feet to a height of 40 feet, supposing the sail-cloth to weigh 4 lbs, per square yard, and the yard to weigh 5 cwt. DYNAMICS. 1351. Define the terms "uniform motion,',' "variable mo- tion," " velocity," *' moving force," and " accelerating force." 1352. State the three "laws of motion," and give illustra- tions of their truth from familiar facts. 1353. State the second law of motion, and mention experi- mental facts which would lead to its assumption. What is the nature of the final evidence which is considered conclusive as to the truth of this law 1 1354. Define "variable velocity/' explain how it is pro- duced by the action of a constant force; state how this connexion is made use of in the dynamical measurement of such a force; and find the numerical representative of the force of gravity, suppos- ing the minute and the yard to be the units of time and space respectively. 1355. Find the unit of space when the accelerating force of gravity is 1 4, and the unit of time 5 seconds. 1356. Show that, when a body moves under the influence of a constant force yj the space s passed over in the time t will be re- presented by the formula 1357. An engine starts a train with a pressure which con- tinues uniform for 5 minutes, when it is found that the train 160 MATHEMATICAL EXAMINATION QUESTIONS. is moving at the rate of 23 miles an hour. At what velocity would the train be moving at the end of 15 minutes, if the pres- sure continued uniform during that time ? 1358. A balloon, rising with a uniform velocity of 30 feet per second, is carried by the wind over a horizontal distance of one mile in 1". 28'. ; find the angle which the line of its motion makes with the horizon, and the horizontal distance from an object at which a heavy body must be let fall from the balloon, when at a height of half a mile, so as to strike the object. Does the body fall in a straight line 1 1359. A wherry pulled at the rate of five miles an hour, is required to cross a river half a mile wide, to a point which is a quarter of a mile lower down the stream than the point of start- ing : what angle must the direction of the boat's head make with the bank, supposing the stream to run at the rate of 3 miles an hour; and how long will it be in crossing? 1360. A body weighing 322 lbs. is ui-ged forward on a per- fectly smooth horizontal plane by a constant pressure of 10 lbs.; and another body weighing 483 lbs. is moved forward in the oi:)po- site direction by a pressure of 75 lbs. Find the point at which they will meet, supposing them to have been 500 feet distant from one another at the commencement of their motion. 1361. A railway train and its locomotive of 40 horse power weigh 40 tons when empty, and 100 tons when loaded. Find the velocities in the two cases, supposing the engine to work with full power in each case, and that the resistance is 8 lbs. per ton. 13G2. Find the numerical value of the force of gravity when the minute and the foot are assumed as the units of time and distance. 1363. Two railway trains, running in the same direction on parallel lines of rails, at the rate of 20 miles an hour and SO miles an hour respectively, arrive, at the same instant, at a junc- tion where the lines diverge at an angle of 30"; find the relative velocity of the two trains after passing the junction. DYNAMICS. 161 1364. Find the pressure, in tons, which will bring each of the trains in the last question to rest, in three minutes, supposing them each to weigh 50 tons. 1365. A pressure of 2 tons acts upon a railway train weigh- ing 150 tons for 10 minutes; find the momentum the train has acquired. 1366. A railway train weighing 30 tons, detached from the engine, is moving against the wind at the rate of 20 miles an hour; how far will it run before coming to rest, supposing the resistance of the wind and the effect of friction to be equivalent to a pressure of 2 cwt. ? 1367. If a 68-lb. shot be propelled from the mouth of a gun with a velocity of 322 feet per second; find the statical pressure urging the shot forward while in the gun, supposing it to be uniform and the shot to take '05 second to traverse the bore. 1368. Distinguish between the statical and dynamical mea- sures of a force, and find the pressure, in ounces, of a force, whose dynamical measure is 10, upon a body which weighs 644 oz. at the surface of the earth : find also the weight of this body, sup- posing it to be removed to a distance from the surface equal to one- fourth of the radius of the earth. 1369. Show that the force of gravity is represented numeri- cally by 79000 very nearly when the units of distance and time are a mile and an hour respectively. 1370. Two bodies, A and B, of equal weights w, and a third body C of weight p, are attached to one another in alphabetical order by strings a inches in length, and C hangs freely over the edge of a table upon which A and B are placed in contact with one another; find the velocity with which A commences its motion, excluding friction. 1371. The steam from a locomotive makes an horizontal angle of 60° with the line of rails when the train is running at 10 miles an hour, and an angle of 30" when the train runs at 20 miles an hour; find the direction and velocity of the wind. c. 11 1G2 MATHEMATICAL EXAMINATION QUESTIONS. 1372. A splierical shot is rolling directly across the deck of a ship with a velocity of 10 feet per second; find the point at which it would strike the side, 6upi»osing the ship, which is steaming at 10 miles an hour, to be suddenly arrested in its course when the shot is 30 feet from the side; and find the velocity with which it would strike. 1373. A railway train 150 feet long passes a station in 5 seconds, and another train two-thirds of the length of the former passes the same point in 4 seconds; and it is observed that the line of steam from the former is half the length of that from the latter; find the velocity of the wind, which was blowing in the direction of the line of rails, 1374. A dragoon is galloping at the rate of 14 miles an hour past a sentry, who does not perceive him until he is at the nearest point to his post, 200 yards distant; what allowance must the sentry make in taking his aim, supposing the velocity of his liflc- bullet to be 1000 feet per second] 1375. A person travelling westward at the rate of 4 miles an hour, observes that the wind seems to blow directly from the south; and that, on doubling his speed, it appears to blow from the south-west : find the velocity of the wind and its direction. 1376. A body is thrown, with a velocity of 25 feet per second, horizontally from the window of a railway carriage moving at the rate of 30 miles an hour, and in a direction making an angle of 30" with the rear of the train; find the direction of the vertical plane in which it moves, and the perpendicular distance from the rail at which it strikes the ground, supposing the window to be 8 feet from the ground. 1377. A steamer whose course was N.E. had the smoke from its funnel passing off in a line to the ^vestward from the steamer, the wind being due S.; find the velocity of the wind, supposing that of the steamer to have been 1 2 miles an hour, and bearing in mind that the motion of the steamer has no effect upon the par- ticles of smoke after they have issued from the funnel. DYNAMICS. 163 1378. A boatman sees from the bank of a river, when at a distance of half a mile, a vessel which is sailicg through the water at the rate of 5 miles an hour, and coming up with the tide which is running at the rate of 3 miles an hour; and, as he wishes to board her, he immediately puts off. In what direction must he keep the head of his boat, supposing that he can pull at the rate of 5 miles an hour, and that the vessel maintains her dis- tance of one-fourth of a mile from the bank of the river? 1379. A boy attempts to drop a stone into the funnel of a locomotive as it passes under a railway-bridge at the rate of 40 miles an hour; where will it strike the train, supposing that it is dropped when the funnel is immediately underneath, and that the boy's hand is 48*3 feet above the roofs of the carriages'? 1380. The captain of a steam-frigate, steaming parallel to the coast at a distance of a quarter of a mile, wishes to fire a broadside at a certain point in an enemy's battery at the water's edge, when directly opposite to it; how many feet, and on which side of the object must the guns be laid, supposing the steamer to be running at the rate of 14 miles an hour, and the horizontal velocity of the shot to be 800 feet per second ? 1381. Two spheres A and B, of which the mass of ^ is double that of B, but the velocity of B double that of A, move in oppo- site directions; find their velocities after impact, e being the elas- ticity of the spheres. 1382. Two non-elastic bodies, moving in opposite directions, impinge directly upon one another with velocities of 20 feet, and 1 5 feet per second ; what will be their common velocity after impact, supposing the ratio of their masses to be 5 : 31 1383. A, B, and C are three ivory balls whose elasticity is |: ^ ( = 9 oz.) impinges directly on ^ ( = 6 oz.) which is at rest, with a velocity of 10 feet per second; and then B impinges directly upon C, at rest ; find the weight of C, in order that it may move on with a velocity of 1 5 feet per second. 11—2 164 MATHEMATICAL EXAMINATION QUESTIONS. 1384. Two spheres, A and B, moving with equal velocities of 9 feet per second in directions inclined 60° to one another, impinge when they are at equal distances from their respective starting-points. Determine their motions after impact; their common elasticity being §, their volumes equal, but the mass of 4 double the mass of B. 1385. Two spheres, A and B, moving in opposite directions, impinge directly upon one another with velocities of 5 and 2 respectively; find the ratio of their masses in order that A may be reduced to rest after impact, and find the velocity of B after impact, their common elasticity being §. 1386. Two bodies, A = \6 and ^ = 24, moving in lines which are inclined 30° to the common tangent at impact, with velocities of 20 and 1 6 respectively, strike one another obliquely : find their distance from one another 4 seconds after impact, their common elasticity being -J. 1387. Two spheres, A and J?, impinge upon one another with velocities of 10 feet and 8 feet respectively, in directions which make angles of 60° and 1 50° with the line of centres at impact. Find their velocities and the directions of their motions after impact, the mass of A being double that of B, and their modulus of elasticity being J. 1388. Two perfectly elastic spheres, of masses m and m', ap- proach each other obliquely with velocities . V and F', and in directions which make angles a and /8 with the line of centres at the time of impact ; determine the motion after impact, when V : y : : B.cos/i : 4 . cos a. 1389. Two spheres, A and B, having equal velocities, impinge directly upon one another; find the ratio of their masses, so that, after impact, the motion of B may be reversed, and its velo- city doubled. 1390. If two planes inclined at an angle a to one another be placed vertically upon a horizontal board, find the angle at which DYNAMICS. - 1G5 an elastic sphere rolling along the board must strike one of the planes in order that after three successive impacts it may retrace a poiiiion of its course. 1391. Show that if a body be projected from the angle A of a plane triangle ABC so as to strike the side CB at a point i5, then, if its course after reflexion at D be parallel to AB, 4. n»i (l+€)cot.5 tan DBA=\ . 1 - € cot B 1392. Three equal spheres, A^ B, and C, are at the angular points of a triangle ABC (the angle C beiing greater than 60°) t what will be the resulting velocity of (7, and the direction of its motion, if A and B move uniformly along the sides AC and BC with velocities proportional to those si feet long, inclined at an angle of 30", at the same instant that another sphere of 10 oz. is allowed to roll from the top; with what velo- city must the former be projected so that, after collision, the latter may rebound to the top of the plane; the coefficient of elacticity being '751 1426. Two bodies, weighing 10 and 5 ounces respectively, are 170 MATHEMATICAL EXAMINATION QUESTIONS. connected by a string passing over a pulley at the top of a double inclined plane; find the tension of the string when the inclina- tions of the planes are 50° and 45" respectively. 1427. A sack of com weighing 2 cwt. is lowered from a height of 50 feet by a rope, which, passing over a fixed pulley, has a bundle of empty sacks weighing 1^ cwt. attached to the other end, and these are drawn up by the descent of the full sack ; how long will the sack take to descend, and with what velocity will it strike the ground i 1428. Two weights hang freely over a single fixed pulley; prove that, if friction and the inertia of the pulley be neglected, the force of gravity at different places on the earth will be propor- tional to the S2)aces descended from rest in the same time by the heavier body. 1429. A weight of 16 ounces draws another of 10 ounces down a plane inclined at an angle of 45° to the horizon; find the vertical velocity of the latter at the end of three seconds, and the tension of the string. 1430. A weight of 5 oz. hanging freely, pulls a weight of 20 ounces along a perfectly smooth horizontal table, by means of a string to which both are attached; how long will the latter take to reach the edge of the table, from which it was originally 10 feet distant, and at what horizontal distance from the edge of the table will it strike the ground, supposing the length of the string and the height of the table to be each 1 feet 1 1431. The velocity acquired by a body in falling from rest, down a plane, is 25; find the velocity it would acquire if pro- jected from the top, down the plane, with a velocity of 25. 1432. A railway train weighing 100 tons, moving at the rate of 40 miles per hour, comes to an incline of 1 in 50; what addi- tional pull, in tons, will be required from the engine to keep the speed at the same amount? and how far would it run up if the engine were to shut off" the steam at the base of the incline ? 1433. The " ways" along which a ship was launched were DYNAMICS. 171 found to have an inclination of 1 in 20 : find tlie velocity with -which she was propelled along the surface of the water, supposing the time she took to slide off the ways to be 10 seconds. 1434. What velocity will a railway train acquire by its descent of an incline of 1 in 70, one mile in length 1 1435. Find the velocity acquired by a railway train running down an incline of 1 in 100, one mile long. 1436. Find the line of quickest descent from a given point to a given plane. 1437. Find the length of the line of quickest descent from a given straight line inclined at an angle a to the vertical, to a point in the same vertical plane at a perpendicular distance p from the given line. 1438. If a be the shortest horizontal distance of a given point from a given plane whose inclination is i, show that the length of the line of quickest descent from the point to the plane is 2a . sm - . 1439. Find the plane of quickest descent from a given plane to a given point. 1440. Find the straight line of quickest descent from a given point to a given circle. 1441. Find the line of quickest descent from a given circle to a given circle within the former. 1442. A balloon with its car, &c. weighs 750 lbs., and is capable of just lifting 1200 lbs.; find the height to which it will rise with the first-named weight in twenty seconds, supposing the resistance of the air to be equivalent to a constant downward pressure of 50 lbs. Find also the time in which a body^ released from the car at that point, will reach the earth. 1443. Investigate from first pri^ciples the expression sin (2(5 — ij = 1^ cos** i + sin i, 172 IMATHEMATICAL EXAMINATION QUESTIONS. where e is the elevation which, with the velocity V, will give the range r upon a plane inclined at an angle i to the horizon. 1444. A body projected obliquely, and acted on by gravity, strikes a plane inclined to the horizon at an angle /3; find the range on this plane, and the time of flight, the angle and velocity of projection being respectively a and V. 1445. If a body be projected at an angle of 60°, and strike a plane inclined at an angle of 30" ; show that the range in feet will be sixteen times the square of the time of flight in seconds, if (J = 32. 1446. Find the elevation and velocity with which a body must be projected, in vacuo, in order that it may pass through two j)oint8 distant h and h' from the point of projection, and k and k' above its level ; find also the time of flight to the latter point. 1447. If any number of bodies be projected from the same point, at the same time, with the same velocity in the same verti- cal plane, but at various angles, they will all, at any subsequent instant, be in the circumference of the same circle. 1448. What would be the range, m vacuo, of a 32-lb. shot fired at an elevation of 10°, wit"h a 6-\h. charge, upon a plane hav- ing an upward slope of 4°, supposing the initial velocity to be expressed by where C is the weight of the charge, and W that of the shot ? 1449. Investigate an expression for the determination of c, the elevation of a gim, which will c&,u&e a shot fired with a velo- city V to strike a platie, inclined to the horisson at an angle i, at a distance r from the guli. 1450. Find the range of a shell, fired at an angle of 45° with a velocity of 500 feet, upon a horizontal plane, and the length of fuse requisite to cause it to explode upon reaching the ground, supposing the fuse to burn at the rate of 1 indh in six seconds. DYNAMICS. 17a 1451. Show that if a body be projected from the bottom of an inclined plane and strike the plane perpendicularly, the time of flight will be, in the usual notation, — '- r^-: — - . g.sin.1 1452. Find the i*ange of a shot, fired at an elevation of 30" with a velocity of 400 feet per second, upon a descending plane of 15". 1453. Prove that the maximum range upon a plane inclined to the horizon is obtained when the body is projected in the line of direction which bisects the angle between the vertical and the plane ; and that equal differences of elevation above and be- low this line will give corresponding equal ranges. 1454. A sphere is rolled with a velocity of 20 up a smooth board 8 feet in length, inclined at an angle of 30° : find the horizontal distance from the foot of the board at which it will strike the ground. 1455. A shell is to be fired from the top of a cliff 300 feet high with a velocity of 600 feet per second, to strike a ship at anchor 600 yards from the base of the cliff; what must be the ele- vation of the gun and the length of the fuse, supposing it to burn at the rate of '167 inch per second? 1456. Let ABC be an isosceles triangle, resting with its base AB upon a horizontal plane, and let its surface be inclined to the horizon at an angle of 60°; it is required to find the velocity with which a ball must be projected in the direction ACj so as to roll off the surface of the triangle at the point of bisection of BC. 1457. A particle is projected at an angle a, and strikes a plane inclined at an angle i ; find the time at which it is a feet distant from the plane. 1458. A perfectly elastic particle is projected with a velocity V, at an elevation a, directly up a plane inclined at an angle i to the horizon. Find the greatest perpendicular distance of the par- 174 MATHEMATICAL EXAMINATION QUESTIONS. tide from the plane; show that the velocities parallel to the plane, at the successive points of impact, are in arithmetical progression j and find the number of bounds the particle will make before it commences its descent down the plane. 1459. If X and y be the rectangular co-ordinates of a point re- ferred to horizontal and vertical axes, whose origin is the point of 2>rojection of a body, show that 9^' y = a;tane-^ wliere e is the inclination of the line of projection to the horizon and V the velocity of projection. 1460. Two particles are projected with the same velocity so as to have the same range on the same horizontal plane ; com- pare their times of flight. 1461. If the wind were blowing at the rate of 40 miles an hour, what would be the equation to the line of motion of the balloon in Question 1442? And what would be its position and the direction of its motion at the end of ten seconds from the time of starting? 1462. Find the length which a fuse must be cut in order that it may bui-st a shell when it strikes the ground at a range of 1200 yards upon a horizontal plane, if the gun be fired at an elevation of 4"; the fuse burning at the rate of a fifth of an inch per second. 1463. What is the charge of powder which will give a range of 1650 yards on a horizontal plane to a 13-inch shell fired at an elevation of 45** ? And what would be its range with that chai'ge on a plane inclined to the horizon at an angle of 2" 40' ] 1464. Find the range of a shot fired at an elevation of 30" with a velocity of l6l feet per second, upon a plane which rises 1 in 15. 1465. If a triangle ABC have its base AB horizontal, and a body be projected from A at an elevation e, so as to pass through DYNAMICS. 175 C and B; show that tan e is equal to twice the area of the triangle divided by the rectangle of the segments into which the perpen- dicular from G divides the base. 1466. A perfectly elastic body is projected from a point A at the base of a plane inclined to the horizon, and strikes the plane perpendicularly: what is the distance from A at which it will again strike the plane ? 1467. Two elastic spheres discharged at the same instant from points in a horizontal plane, describe the same parabola in contrary directions; after impact one of them retraces its path and the other falls vertically; find their elasticity of recoil and the ratio of their masses. 1468. A sphere whose elasticity is J, is propelled, with a velocity 10, down a plane inclined at an angle of 60" to the hori- zon, and at a distance of 15 feet from its starting-point strikes a hard and immoveable obstacle, which presents a horizontal surface to its impact; find the point where the sphere will again strike the plane. 1469. Two equal inclined planes are placed back to back, and a body projected up one of them flies over the top and strikes the ground just at the foot of the other; find the velocity of pro- jection, the inclination of each plane being 30", and their common altitude h feet. 1470. A perfectly elastic sphere is let fall from a height of 16'1 feet above the point where it strikes a plane inclined 45° to the horizon ; find the position of the body at the end of 2 seconds. 1471. A body is projected with a velocity of 800 at an eleva- tion of SO*' upon a plane inclined 15° to the horizon, and strikes the plane at an angle of 40° ; find the range. 1472. If an elastic body be projected on a horizontal plane, show that the successive ranges will be in a geometric progression, having the common ratio e. 1473. The "sighting" of a rifle fixes the angle between the 176 MATHEMATICAL EXAMINATION QUESTIONS. direction of the object aimed at and the line of fire : show whe- ther, if aim be taken at the top of a vertical wall with the sight adapted to the horizontal distance of its base, the bullet will pass above or below the top of the wall; and find the error in the oblique range. 1474. Three points, B^ C, and i), in the same horizontal line, are equidistant from one another, and from the horizontal line AH, vertically below them ; prove that if any line CA be drawn to the horizontal line, the angle BAH is less than 9.CAU, and that if a body be projected from A at an angle a = 9.CAH so as to pass through B ascending, it will also pass through D; except in the case when A is vertically below B. 1475. A shell is projected with a velocity of 805 feet per second ; find its greatest range upon a plane inclined upwards at an angle of SO". 1476. A perfectly elastic ball is suspended from the centre of the ceiling of a room h feet high, and 2a feet wide, by a string ; find the length I of the string, so that the ball, being struck ho- rizontally by an equal and perfectly elastic ball projected from the middle of the side of the floor, may just touch the ceiling. 1477. A shot is fired with a velocity of 400 at an elevation of 30", and is observed to strike an object at the end of 4 seconds; find the inclination of the line which joins the object and the gun. 1478. A pendulum oscillating in 2 seconds at the sea-level, has one inch cut from its length, and is then cari'ied up a moun- tain 17600 feet high; find the time of oscillation in that position, the radius of the earth being S^60 miles. 1479. A seconds'-pendulum is carried to the top of a moun- tain 3000 feet high ; assuming that the force of gravity varies in- versely as the square of the distance from the earth's centre, and that the radius of the earth is 4000 miles, find the number of os- cillations lost in a day. DYNAMICS. 177 1480. If two clocks, having pendulums of different lengths, gain or lose to the same amount, show that the corrections of their pendulums will be nearly in the ratio of their lengths. 1481. Find the velocity with which the earth must revolve in order that bodies at the surface, in latitude 60°, may lose all their weight. 1482. Show that if the vertical radius of a quadrant of a circle be divided into n segments, which are to one another as the odd numbers 1, 3, 5, 7, &c., and horizontals be drawn to meet the arc, the points of intersection will be such that a body falling down the arc from rest at the extremity of the horizontal radius, will have, at those successive points, velocities which are as the natural numbers 1, 2, 3, 4, &c. 1483. A pendulum which makes n oscillations in a certain interval, at the surface of the earth, makes n — S oscillations at the bottom of one mine, and ti - 5 at the bottom of another, in the same absolute interval as before ; compare the depths of the mines. 1484. A heavy particle is suspended from the centre of the ceiling of a room 15 feet wide by an inextensible string 6 feet long, and being drawn aside, in a plane perpendicular to the side of the room, until it touches the ceiling, is allowed to swing ; find the point at which it will strike the opposite wall if the string be cut when the particle has made half an oscillation. 1485. A simple seconds' pendulum is found to make 10560 oscillations at the level of the sea, and 10558 in the same time at the summit of a mountain ; find the height of the mountain, the radius of the earth being 7960 miles : find also how far a body let fall at the top of the mountain would fall in 3 seconds. 1486. The number of vibrations made in 3 hours by a pen- dulum at the surface of the sea was 20000; and the number made in the same time, by the same pendulum, at a height of 10560 feet was 19990: find the radius of the earth. c. 12 l78 MATHEMATICAL EXAMINATION QUESTIONS. 1487. If L be the length of the seconds' pendulum at the level of the sea, and I the length of the seconds' pendulum at a height A, show that the radius of the earth is 1488. Find the change in the daily rate of a clock having a brass pendulum beating seconds, due to a rise of temperature of 20"; the expansion of brass being -00001 of its linear dimensions for each degree, 1489. Find the length of the simple pendulum which oscil- lates half-seconds; the force of gi'avity being expressed by 32 "2. 1490. Find the daily rate of a clock the pendulum of which is 38*64 inches long, supposing each vibration to mark one second upon the dial plate. 1491. Find the length of the seconds* pendulum when the force of gravity is 32*2; and determine the daily rate of a clock having such a pendulum made of brass, due to an increase of temperature of 10 degrees, supposing brass to expand "00001 of its length for each degree. 1492. A body is suspended freely by a fine thread from the ceiling of a railway carriage, and it is observed to attain a devia- tion of 40' from the vertical; find the radius of curvature of the line of rails at that point, supposing the train to be moving at the i-ate of 15 miles an hour. 1493. A horseman is galloping in a horizontal circle of 20 feet radius at the rate of 12 miles an hour; find the natural incli- nation of the plane which passes through the axes of the bodies of the man and horse, to the vertical. 1494. How much must the velocity of the earth's rotation be increased in order that bodies at its equator may lose all their weight ; the radius being assumed as 4000 miles ? 1495. A curve on a railway has a radius of a quarter of a mile; find the difference in level of the lines of rails in order that DYNAMICS. 179 tlie pressure on the wlieels may be equal, wlien a train runs rotlnd the curve at the rate of 20 miles an hour; the "gauge" or dis- tance between the rails being 4 feet. 1496. A weight W is suspended from the ceiling of a railway carriage by a string a feet long; find the tension of the string and its inclination to the vertical when the carriage is running round a curve of 1 mile radius at the rate of m miles an hour; find also the number of small oscillations which it would then make in one minute. 1497. A 24- lb. and a 124b. shot are successively whirled round on a horizontal table with a given length of string of the same strength attached to them, the velocity in each case being in- creased until the string breaks; find the ratio of these breaking velocities, 1498. A boy whirls round a stone in a sling 3 feet long at the rate of six turns in one second, and the stone is discharged at an angle of 30" to the horizontal plane; find the distance at which the stone will fall and the strain upon the sling, supposing the velocity to be doubled by the action of the boy's arm at the instant of dis- charge. 1499. Find the centrifugal force at the surface of the earth in latitude 60^, estimated in the direction of the radius of the earth at that point, supposing the earth to be a sphere whose radius is 4000 miles. 1500. A particle suspended in the car of a balloon revolves as a conical pendulum in a plane 21 inches below its point of sup- port, and it is observed to make 7 revolutions in 10 seconds. Find how high the balloon will have risen above the earth in 2 minutes, supposing the force of gravity and the acceleration of the balloon to have remained constant. 1501. A body being suspended by a string 2 feet long, and made to revolve as a conical pendulum, it is found that the tension of the string is twice the weight of the body; compare the time of revolution with the time of oscillation of the body as a simple pendulum. ^ 12—2 18d MATHEMATICAL EXAMINATION QUESTIONS. 1502. At what angle is the arm of a conical pendulum in- clined when it revolves in the time of its oscillation as a simple pendulum 1 HYDROSTATICS. 1503. The volumes of three fluids and their respective spe- cific gravities being given ; find the specific gravity of their mix- ture, supposing a loss of one-tenth in volume. 1504. A cylindrical tube 1 foot long and 1 square inch in section is found to weigh wlhs. when empty, and IF lbs. when filled with a certain fluid ; find the specific gi-avity of the fluid, water being the standard. 1505. Explain the experimental process by which the equal transmission of fluid pressure in all directions is proved; and define the meaning of the term " pressure at a point." 1506. The volumes of two fluids are as n to m, and their specific gravities are p and p ; when mixed together, they lose one pth of their volume : find the specific gi^avity of the mixture. 1507. A coin, known to be composed of platinum and silver, is found to be of exactly the same size and weight as a sovereign; find the relative weights of the two metals in it, the specific gravities of platinum, silver, and gold being 21, 10*5, and 17 '5 respectively. 1508. The specific gravity of silver being 10*5 and of copper 8*9; find the relative weights of the two which must be mixed in order to form a compound which shall weigh one-ninth more in air than in water. 1509. A cubic foot of teak weighs 100 times as much as a cubic inch of lead; compare their specific gravities. 1510. A certain mass of metal weighs SO oz. in one fluid, and 35 oz. in another; what will be its weight when . immersed HYDROSTATICS. 181 in a mixture of equal volumes of the two fluids, if it weigh 40 oa, in air? 1511. In the hydrostatic bellows, when the weight is con- stant, and water is poured into the pipe, find the rise in the surface of the water in terms of the volume poured in and of the sectional areas of the pipe and of the bellows. 1512. The radii of two spheres are 2 inches and 3 inches, and their weights are 8 lbs. and 10 lbs. respectively; fiiid the ratio of their specific gravities. 1513. The specific gravity of copper being 8*8 and of tin J- 3, find the weights of each of these metals in a mass of gun-metal weighing 500 lbs., its specific gravity being 8*6. 1514. A rectangle is immersed vertically to a depth a below the surface of a fluid, and has two of its sides horizontal ; find the position of the horizontal line which divides the surface into por- tions the pressures upon which are equal. 1515. A fluid A has a specific gravity 1-25, and another fluid B has a specific gravity -85 ; 5 fluid ounces of A are mixed with 7 fluid ounces of B, and 3 fluid ounces of water are added : find how many grains a mass of lead weighing 745 grains, will weigh when immersed in the mixture ; the specific gravity of lead being 11-4. 1516. A piece of gun-metal composed of copper and tin, is found to weigh W ounces in air, and to ounces in water; show that the ratio of the weights of copper and tin is expressed by the fraction c W-{W-w)t t "" c{\V-w)-W\ where c is the specific gravity of the copper and t that of the tin. 1517. A vertical cylinder contains a quantity of water of which the depth is equal to twice the diameter of its ciitjular base, and a right cone of a density equal to five times the density 182 MATHEMATICAL EXA3IINATI0N QUESTIONS. of the fluid, having its base exactly fitting the cylinder, rests (vertex downwards) upon the fluid; find the total pressure upon the curved surface of the cylinder in terms of the weight of the water, supposing the axis of the cone to be equal to the diameter of its base. 1518. Find the ratio of the pressures upon the upper and lower halves of a regular hexagon immersed vertically in two fluids, the one of double the density of the other, one of the sides of the hexagon being in the surface of the upper fluid, and its centre of gravity in the surface of the low er. 1519. A vessel in the form of an inverted frustum of a cone is filled with water: compare the pressure upon the bottom or smaller end with the weight of the water, the frustum being 12 inches deep, and the radii of its ends 4 inches and 1 inch. 1520. A watch-chain which weighs 200 grs. in air, weighs only 184'7 grs. in water: find the ratio of the volumes of brass and gold in it; the specific gravity of brass being 7 '8, and of gold 19-3. 1521. A hollow equilateral triangular prism, the sides of which are squares, is placed on end and half-filled ^vith water; it is then placed horizontally upon one of its sides. Compare the pressures on the end in the two cases. 1522. If a hollow right cone, standing with its base down- wards upon a horizontal plane, be completely filled with water, show that the weight of the cone must be equal to twice the weight of the water in order to prevent the cone from rising; and thence deduce the amount of the whole normal pressure upon the curved surface, and the consequent position of the centre of gravity of that surface, 1523. A cylinder is completely filled with water; compare the pressure sustained by its curved surface when the axis is ver- tical and when it is horizontaL 1524. Compare the pressures upon the upper and lower por- tions of a circle immersed vertically in a fluid, the circle being HYDROSTATICS. 183 divided by a horizontal diameter, and the centre of gravity of 4 a semicircle being at a distance equal to — of the radius from the centre; and find tlie depth to which the circle must be immersed in order that the pressure upon the lower half may be the double of that upon the upper. 1525. Compare the pressures upon the three upper and upon the three lower faces of a cube suspended by one of its angles in a homogeneous fluid, at a depth equal to the side of the cube. 1526. Compare the pressure on the base and on the three sides of a regular tetrahedron filled with water, and having its base horizontal. 1527. A cylinder is divided by a plane which cuts its axis perpendicularly at a distance from the lower end equal to one- fourth of its length ; find the depth to which it must be vertically immersed in a fluid, that the total fluid pressures upon the whole surfaces into which it is divided may be equal. 1528. A hollow thin square prism, filled with water, stands upon one of its rectangular faces, and one of its ends is loose, though water-tight ; find the magnitude and point of appli- cation of a single force which would hold that end ia its po- sition. 1529. Supposing both ends of the prism in the last question to be fixed, and the prism to be filled with equal volumes of two fluids of densities p and 3p which do not mix; find the pressure produced upon that portion of each end in contact with the latter fluid. 1530. An isosceles triangular lamina floats vei-tically in a fluid with one of its equal sides parallel to the surface of the fluid, and with its centre of gravity in that surface, the vertex of the triangle being supported by a string : find the ratio of the den- sities of the fluid and lamina, and the tension of the string. 1531. A solid body is floating between two fluids of specific gravities s and /, and the part immersed in the denser fluid 184 MATHEMATICAL EXAMINATION QUESTIONS. is observed to be the same as if it were floating in a mixture of equal volumes of the two fluids; find the specific gravity of the solid. 1532. A hollow water-tight 8-inch cube has one of its verti- cal sides loose, and capable of revolving about its upper edge; find the pressure at the lower edge necessary to keep the side closed when the cube is filled with wat«r. 1533. An isosceles triangle is immersed in a fluid so that its vertex is at the surface and its base horizontal; find the locus of the centres of pressure of the straight lines drawn in the triangle perpendicular to its base. 1534. A rectangular sluice gate ABCD has its two sides AD, BC and the axis EF upon which it turns, horizontal; find the height to which the water must rise above EF before the gate turns, when AB = 7 feet, and AE=4! feet. 1535. A uniform cylinder of length 2b inches, and density p, floats in a fluid, with its axis vertical and half immersed ; another fluid is then poured upon the top of the former (with which it does not mix) to the depth a inches, which causes the cylinder to rise b inches in the lower fluid. Find the density of the upper fluid. 1536. A cylinder rounded off* at one end in the form of a hemisphere, is found to float in water with one-half of the radius of the hemisphere immersed, and the equilibrium is found to be neutral ; find the density of the body. 1537. A heavy cone, having its base 6 inches in radius, is suspended in water by a point in the circumference of its base so as to be completely immersed ; find the magnitude and direction of the resultant pressure upon its curved surface, and compare the former with the total pressure upon the same surface, the point of suspension being 2 feet below the surface of the water, and the height of the cone 12 inches. 1538. A hollow cube having a weight suspended at one of its angles, floats with the three nearest angles at the surface of the HYDROSTATICS. 185 water; find the additional weight which will just bring the high- est angle down to the surface, the edges of the cube being two feet long. 1539. Find the thickness of an iron spherical shell 20 inches in diameter which will just float in water, the specific gravities of iron, water, and air being 7*5, 1, and -00125 respectively. 1540. A ship in dock is observed to have risen 3 inches out of the water, owing to the discharge of 50 tons of her cargo; find the area of her section at the water-line. 154-1. A body in the form of an equilateral cone and a hemi- sphere on the same base, floats in a fluid; find the position and character of the equilibrium when the density of the fluid is double of that of the body. 1542. An empty hollow sphere of 4 inches radius, and weigh- ing 3 lbs., is fastened to the bottom of a vessel filled with water by a fine string; find the tension of the string when the sphere is wholly immersed in the water, and also when the water is replaced by the same bulk of oil, the specific gravity of which is '92. 1543. A thin hollow cone 8 inches in diameter and 12 inches deep has a sphere of lead 3 inches in diameter placed within it; find the depth to which it will sink in a fluid the specific gravity of which is 1*135, that of lead being 11*35. 1544. Four cylindrical pontoons 20 feet long, having hemi- spherical ends of 2 feet radius, are lashed together and a platform is laid upon them, and it is found that they are then half im- mersed; find the additional weight they will carry. 1545. Four cylindrical barrels 4 feet long by 2 feet 6 inches diameter are lashed together to form a raft, and it is found that, when floating in the water, one-third of their diameters is im- mersed; find the additional weight which the raft will support when the barrels are completely immersed. 1546. A cylindrical pontoon, having hemispherical ends, is 25 feet long and 2 feet in diameter, and is immersed to half its 186 MATUEMATICAL EXAMINATIOJ^ QUESTIONS. depth ; find the additional weight it will bear before it is wholly immersed: fiud also the total pressure upon its surface in the latter case. 1547. A diving-bell is sunk until the depth from surface to surface is 11 fathoms; find the volume of air which must be pumped in, in order to fill the bell, the temperature of the air in the bell being f lower than that of the external air. 1548. A cylindrical diving-bell, 9 feet high, is to be sunk to the bed of a river 40 feet deep; find the height to which the water will rise within it. 1549. The water in a conical diving-bell has risen one-third of its height inside the bell, where the thermometer stands at 6*0", the atmospheric temperature being 40". Find the depth from surface to surface. 1550. A balloon filled with gas at a temperature of 60" and pressure of 30 inches, ascends to a height where the thermometer indicates 35^ and the barometer 25 inches; find the change of volume of the gas, and the height to which the balloon has risen. 1551. Calculate the height of a point on Shooter's Hill, above the wharf in the Arsenal, from the following observations : — Barometer. Temp, of Mer. Temp, of Air. Arsenal 29475 37 34 Shooter's Hill 29*022 35 32 usinfj the formula II (in feet) = 60000 log - , in which the values of h and h' are supposed to be observed at temperature 32". 1552. A sphere, sp. gr. 1*25, is placed in a reservoir of water 20 feet deep ; find the time in which it would reach the bottom, neglecting the resistance of the water. 1553. An india-rubber hollow sphere, which contained air of three times the external density, was suspended within a diving- DIFFERENTIAL CALCULUS. 187 bell, wliicli was then sunk to a depth x, when the sphere was found to be one-tenth less in circumference than v/hen the bell was at the surface. Find x, on the supposition that the force of compression by the india-rubber remains practically unaltered. 1554. Describe the " Air-Pump; " and determine the increase of temperature required to restore the elastic force of the air in the receiver after two strokes of the piston, supposing the tempe- rature of the air to be 50'^, and the capacity of the receiver to be ten times that of the pump. 1555. Describe the action of the air-pump; find the density of the air in the receiver after 5 strokes of each piston, the ca- pacity of the receiver being to that of each barrel as 9 : 1 ; and calculate the increase of temperature from 40" Fahrenheit which, under a constant pressure, would have produced the same amount of rarefaction in the air. 1556. Describe the common suction-pump; and determine the force necessary to move the handle downwards, when the length of the pipe from the piston to the surface of the water is 15 feet, the diameter of the piston 6 inches, the length of the handle 5 feet, that of the arm at right angles to it and connected with the piston-rod, 1 foot, and the play of the handle 60?. DIFFERENTIAL CALCULUS. 7x + 2 1557. Find the limiting value of , when x approaches oX — D infinity. 1558. Define the *' differential coefficient" of a function ; and find the differential coefficient of l+xj2 + x' ^, _yxj2 u = los—- — —^ .-I- 2 tan l-xj^ + x"" 1-x^ 188 MATHEMATICAL EXAMINATION QUESTIONS. 1559. Show that d . xyz - ocydz + xzdy + yzdxy and apply this form to the differentiation of the function u = a?. e*. tana;. 1560. Prove that d.a =a. log, adoc 1561. Investigate the following formulae : (a) c?. (a;") = ?ix"~\cfo;; (6) d {xy) -x.dy + y.dx; (c) ^ /^\ y-^-«^-^2/. y {d) c?. tan a; 3= sec' a;, do:. Differentiate the following functions : 1562. ^=Vj^- 1563. w=--^-V(l+a:'). J{x' + a')-J{x* + h') 1565. w= — -. 1566. ^^ (^'-^^) a: + ^(l-ar) (a'-ar'f 1567. "-^,^"^t 1568. ,. = 4^:^. a3 + .y(a;+l) x^ + x-\ 1569. u= . ^\, , 1570. y "^^ (a + ^r' ' a'ia' + x^)^' 1572. u = log {^{1 +x')+J{l- x')}. 1573. w = a:='-3log{l+a;')i 1574. « = log ^^^^^^^^ ^^^^ a;, log 05 , ,, V 1575. u = — — 5_ + log (1 - x), 1 — X 1576. log (a;. e*=^*) and tan~^ x /-- . DIFFERENTIAL CALCULUS. 189 1577. w = «". e^''^*. 1578, u = tsLn-' ^^ . 1 —x' 1579. u = &m(nx).siii''x. 1580. w = cos-'^— ^. ^ ^ x"'+l 1581. w-eMogsina-. 1582. w= -^f- - i Wtan-. tan - 1583. u= .^ : ,... .tan-^ 1584. w = tan~'a; + tan~'a;^ 1585. u=J{d-D.{x+a).Bm.{x-a)]. I 1 9^^ — 1 1588. u = \og.tdiYie'^\ 1589. w= -y-*^^~'— T" j:" 1590. y = a; . J{a' - x') + a\ shr' - + log "^f^+J^ _ 3^ Ct Oj — X 1591. v = 7Tlo2-g — ', 7-. tan — ^ — . iC 1 1592. y — \oQ^ ,— -tan~^a;. 1593. Investigate Maclaurin's Theorem; and, by means of it, expand sin~^ic in series arranged according to the ascending powers of X : 1594. u = log . cos X, 1595. Yers x. 1596. w = cos~*x. 1597. e''. cos (a; ;y3). 1598. e'.tana;. 1599. zt = tan-'a?. 1600. If u=f{x + y), show that the first differential coeffi- cient will be the same whether y be considered constant and x variable or vice versd; and show how this theorem may be em- ployed in the investigation of Taylor's Theorem. 190 MATHEMATICAL EXAMINATION QUESTIONS. Expand by Taylor's Theorem in series arranged according to the ascending powers of h : 1601. sin-^aJ + A)- 1^02. co^-'{x + h). 1G03. log (a; + 70". 1604. tan (a; + 70. 1605. tan-'(x + 7/). 1606. Show, by Taylor's Theorem, that if u=f{x) have a maximum or a minimum value, depending upon the variable x \ then, for that value,'— =0; and that if, for that value, -y-r do dx dxr not also vanish, it must be negative in the former case and posi- tive in the latter. 1607. Define the meaning of "maxima and minima values" of the function of a variable; and determine the form of a thin lioUow cone which, with a given capacity, shall be constructed with the least expenditure of material. 1608. Explain how the maxima and minima values of a function of one variable are determined; and find the shortest line to a parabola from a given point in its axis. Explain the result when the distance of the point from the vertex is less than half the latus rectum. 1609. A boatman, S miles out at sea, wishes to reach a point on the beach 5 miles from the nearest point of the coast; he can pull at the rate of 4 miles an hour, but he can walk at the rate of 5 miles an hour: find the point at which he must land. 1610. Find the greatest cone B which can be described within a given cone A ; the vertex of B being the centre of the base of A. 1611. Insciibe the greatest rectangle in a given parabolic area terminated by a double ordinate perpendicular to the axis. 1612. Find the dimensions of a cylinder of a given volume F, such that its surface shall be the least possible. DIFFERENTIAL CALCULUS. 191 1613. Find the greatest right cone which can be cut from a given sphere. 1614. A steamer w^hose course is due west and speed 10 knots is sighted by another steamer going at 8 knots ; what course must the latter steer, so as to cross the track of the former at the least possible distance from her 1 1615. If from a circular piece of paper a sector be cut out, and the straight edges of the remaining sector be joined, a cone will be formed ; find the arc of the first sector, so that the cone may have the greatest volume possible. 1616. Jjet ABC D be a parallelogram of which AC is a dia- meter; it is required to find in AC a point F, such that I>F being drawn and produced to meet CD in F, the sum of the triangles AFB and FFC may be a minimum. 1617. "Within the angle ACB made by two straight lines CA^ CB, a point F being given ; it is required to draw a straight line through F, so that the sum of the segments of the other lines intercepted between it and the angular point C may be the least possible. 1618. Let i) be a point in the diameter AB of a semicircle APB'y it is required to find the position of the point P, so that the sum of the distances AF and FD may be a maximum. 1619. A square piece of sheet-lead is to have its edges turned up perpendicularly, so that the vessel thus formed may contain the greatest possible quantity of fluid. Find the depth of the vessel. 1620. A weight P draws another ^ up by a string passing vertically over a pulley; find Q so that the momentum acquired by ^ in a given time may be a maximum. 1621. Find the subtangent at the point, x = a,of the curve y^ = x^ . (2a — x); and also its asymptote. 192 MATHEMATICAL EXAMINATION QUESTIONS. 1622. Find the equation to tlie normal in the " Witch," xif = a^ {a- x). 1623. Show that \iy=fx be the equation to a curve having a rectilinear asymptote, the intercept of the asymptote on the axis dx of a; is the limit towards which x — y-j- approaches as x approaches infinity. And find the asymptote of the curve y' = a;' + ax'. 1624. Find the subtangent and the radius of curvature of an ellipse, at the point whose abscissa, measured from the centre, is one-fourth of the major axis of the curve. 1625. Find the radius of curvature of the parabola at the vertex and at the extremity of the latus rectum. 1626. Find the radius of curvature of a cycloid at the ex- tremities of the axis and at the points corresponding to a; = a, the equation of the curve being x = a, vers"' - - (2ay - y^) -. 1627. Find the radius of curvature at the point where x = in the curve xy — a', and show that the equation to the evolute is 1628. Find the radius of curvature at the vertex of the curve y^ = a' {x^ — 6"), and if it have asymptotes determine them. 1629. Trace the curve y^-a^ — x^\ finding its asymptotes and points of inflexion. 1630. Trace the cui've y- — x*=^ax^ and investigate its jn'o- perties with regard to asymptotes and points of inflexion. Find also its radius of curvature at the point a; = 0. INTEGRAL CALCULUS. 193 1631. Trace the curve y (a* + cc') = aa;', and determine its pro- perties with respect to asymptotes and points of contrary flexure. 1632. Trace the curve y = x — x^; finding its points of in- flexion and asymptotes, if any. Find also its radius of curvature at the extremity of its greatest ordinate. INTEGEAL CALCULUS. Find the values of the integrals : 1633. f-^. 1634. f-^:^^,efe. 1635 (a + 3xy J{2rx-xy dx ,«„« f dx 5. f f , . 1636. ( ^ -.. Sx+\ , ,^„„ 30dx f 303+1 }a?-\-^x^-¥x 1637. , ' dx. 1638 X —X 6x _-,_ x'dx ^n.. f dx 1645. jb^,ia^-xi,dx. 1646. Jg--^^}c/a:. 1647. r,^!^. 1648. kx'iS-^x'f.dx. J{a'-x'f J .4 1649. du = ^^^^-^,.dx. 1650. (mnhx'^\{a+hx'')'^\dx. (a + xy J x + 1 J J 1651. • '"'^" ' --^^ ^ 13 (0.-2) C. ]94 MATHEMATICAL EXAMINATION QUESTIONS. 1653. [ ^'^ , . 1654. [(2ax-x')Kdx. J {a+ bxy J 1655. / ^ , . 1656. (?^±^. 1657. f_^^. 1658. f ^— ,. J {2ry-yy J{a + bx + caf)^ 1659. f-^. 1660. f-^-^,. 1661. j-^^,dx, 1662. jx'logx.da;. 1663. L-logaj.tia;. 1664.' f— 7^ ). ix^e'dx. 1665. UVc/a;. 1666. du xy dx X —X 1667. {x.e^\dx. 1668. f^^e^a:. j ycos x 1669. .[!i^l^. 1670. lt^tJx,dx. J cos ic y 1671. i^csin-'x.dxr 1672. f"^''' ^^"^^ dx. J J (l-xf 1673. / sin* a; . cos^ re . c?a;. 1674. cos^£c. c?a;. 1675. ~. 1676. f-^. y cos X cos* a; adx 1677. {^. 1678. [t9.n'x.dx. J sm aj y 1G79. Find the value of n . l .. - Jo ^/(a -6a;') INTEGRAL CALCULUS. 195 1680. The equation to the cycloid (origin at the vertex) being y = a Yers"' - + (2aic - a;')^, show that the length of a cycloidal arc measured from the origin, is twice that of the chord of the corresponding arc of its gene- rating circle.. 1681. Find the area of the portion of a parabola cut off by the latus-rectum ; and the volume of a conic frustum generated by the revolution of a rectangular trapezoid round its perpendicular side. 1 682. Find the area of the curve y^ = a^. {x^ — h^ between the abscissas a and h. 1683. Find the area of the curve xif = 9.a — x between the limits £c = and x = a. 1684. Find the area of the curve a^ y — a from a? = to a; = a y a 1685. Find the area of the curve y^ = x^ .{x- of between the limits x = and x^a. 1686. Find the area of the "Witch of Agnesi" bounded by the curve whose equation is xy^ = 4r^ (2r — x), and by a straight line perpendicular to its axis and passing through the centre of the generating circle. 1687. Find the area of the curve y = x-x^ intercepted be- tween the axes. 1688. Find the area of the curve 2/ . (4a^ + £C*) = 2 (a; + a) a" between the limits a; = and x = 9,a. 1689. Find the area of the curve expressed by the equation xf = {a-x)% between the limits x-0 and x-a. 13—2 196 MATHEMATICAL EXAMINATION QUESTIONS. 1690. Find the volume of the solid generated by the revolu- tion of the curve xi/ = (a + x} (b' - a^^ round the axis of y. 1691. An ellipse revolves round a tangent at the extremity of the major axis ; find the volume of the ring generated by the area of the semi-ellipse furthest from the tangent. 1692. Find the volume of the solid formed by the revolution of the curve xy' = (a - x)' round the axis of y, between the limits x = and x = a. 1693. Let DC and CA be the semi-axes, minor and major, of an ellipse, and from any point E in the arc DAy draw £F parallel to AC and meeting DC in F; and let FC=h: it is required to find the area DFF, and the volume of a conoidal bullet generated by that area about FF. 1694. Find the volume of the solid generated by the revolu- tion of the curve y'(a' + x') + a^x = a* about the axis of x ; from a: = to a? = a. 1695. Find the volume aud curved surface of a paraboloid between the limits x = a and a; = 6, the equation to the generating curve being 3/* = ^Tnx, 1696. Find the centre of gravity of a quadrant of a circle. 1697. Find the centre of gravity of a circular sector of which the arc is 2a; and thence deduce that of a semicircular area. 1698. Find the centre of gravity of a segment of a circle in terms of the radius r of the circle, the semi- arc a of the segment, and the radius p of the base of the segment : and show what this becomes in the case of the semicircle. 1699. Find the centre of gravity of a semi-parabola of which the abscissa is a and the ordinate b. INTEGRAL CALCULUS. 107 1700. Find the centre of gravity of a circular anSi a, in terms of the arc, its chord, and the radius of the circle. 1701. Find the centre of gravity of the solid generated by the revolution of the figure formed by two straight lines VAy AB Sit right angles to one another, and the parabolic arc VJB about VA ; when V is the vertex of the parabola of which the axis is parallel to^^. 1702. Find the centre of gravity of a material line, the den- sity of which varies directly as the distance from one of its ends. 1703. Let the density of the sections of a right cone parallel to its base vary inversely as their distances from the vertex ; find the centre of gravity. 1704. Find the centre of gravity of a cone, the density of every point of which varies inversely as the nth. power of its dis- tance from the plane of the base. 1705. Find the centre of gravity of the solid formed by the revolution of the area of the curve about the axis of X'j between the limits a; — 0, and x = a. 1706. At a point D, in an ellipse, the ordinate BH is equal to the abscissa HC, C being the centre. Find the centre of gravity of the segment cut off by the double ordinate DHE. 1707. Let the density of a triangle vary as the wth power of the distance of any point in it from a straight line drawn through the vertex parallel to its base; find its centre of gravity. 1708. Let the density of a quadrant of a circle of uniform thickness vary as the n\h power of the distance of any point in it from the centre of the circle; find its centre of gravity* 198 MATHEMATICAL EXAMINATION QUESTIONS. MOMENTS OF INERTIA. Find the moment of inertia of 1709. A uniform rod about an axis through its centre of gravity and perpendicular to its length. 1710. A circle about an axis passing through its centre per- pendicular to its plane. 1711. An equilateral triangle about one of its perpendiculars. 1712. A circular arc about the diameter which bisects the arc. 1713. A circular arc about an axis passing through its vertex and perpendicular to its piane. 1714. The circumference of a circle about any tangent. 1715. A circular area about any diameter. 1716. A circular ring about an axis perpendicular to its plane passing through its centre. 1717. A cylinder about its axis. 1718. A sphere about any diameter, 1719. A right cone about its axis. 1720. A spherical shell about a diameter, 1721. A hollow cylinder about its axis. 1722. A spherical lamina about a diameter, 1723. A cylindrical lamina about its axis. 1724. A parallelogram about an axis perpendiculai- to its plane and passing tlirough the intersection of its diagonals. 1725. A parabolic area about an axis perpendicular to its plane and passing through its vertex. 1726. A cube about its diagonal. CENTRE OF OSCILLATION. 199 1727. A cube about the diagonal of one of its faces. 1728. A cube about one of its edges. 1729. A cone about its slant side. 1730. A spheroid about its axis of generation. 1731. An ellipsoid about one of its axes. CENTRE OF OSCILLATIOK Find the time of a small oscillation of 1732. An equilateral triangle about an axis perpendiculai* to its plane, through one of its angles. 1733. A cube about one of its edges. 1734. A sphere about an axis touching its surface. 1730. A right cone about an axis touching the circumference of its base. 1736. Show that an arc of a circle will oscillate about an axis through its middle point perpendicular to its plane in the same time as if its mass were collected at the opposite extremity of the diameter of the complete circle. 1737. Show that a cylinder of 8 inches radius will oscillate about an axis on its sur£a,ce parallel to its geometrical axis, in the same time as if its mass were collected at a point one foot distant from the axis. 1738. Show that a hemispherical surface will oscillate about a diameter of its base in the same time as a simple pendulum, the length of which is two-thirds of the diameter. 200 MATHEMATICAL EXAMINATION QUESTIONS. MOTION IN A RESISTING MEDIUM. 1739. Find the time in which a body of given "weight falling from rest through the air will acquire a velocity v ; assuming that the resistance varies as the square of the velocity. 1740. Find the space through which the body will have fallen when it has acquired the velocity v. 1741. If a particle be projected in a medium the resistance of which varies as the velocity, find the space described in the time tf supposing no other forces to act. 1742. A body is projected vertically upwards with a velocity F; find the height to which it will rise, the resistance of the air vaiying as the square of the velocity. 1743. If in 1741 the resistance vary as the square of the velocity, find the space described in the time ^ by a body whose weight is w. 1744. A body is projected with a velocity V obliquely into the air at a small angle of elevation a ; show that, if the vertical resistance of the air be neglected, the range on a horizontal plane is w , (JcV . ^ ,) =- . log { sin 2a + 1 > , kg ^\w J' where w is the weight of the body, and k the resistance due to a unit of velocity. EXAMINATION PAPERS. 203 A. I. Geometry. 1. Deduction. If two sides of a triangle be bisected, show (from the First Booh of Euclid) that the line joining the points of bisection is parallel to the third side, and equal to half that side ; and thence show that if all the sides of a quadrilateral figure be bisected, and the adjacent points of bisection be joined, the figure so formed will be a parallelogram equal to half the given quadrilateral. 2. In every triangle the square of the side opposite any of the acute angles is less than the squares of the sides containing that angle, by, &c. 3. If two straight lines within a circle cut one another, the rectangle contained by the segments, &c. 4. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal. 5. The circumferences of circles are to one another as their diameters. 6. If two straight lines meeting one another be parallel to two other straight lines which meet one another, but are not in the same plane with the first two; the plane which passes through them is parallel to the plane passing through the other. Coordinate Geometry. 7. Investigate the relation between a and a\ so that the lines y = ax + (3, y=za'x + ^', may be perpendicular to one another. The line y = ax^h passes through the point (1,-2), and is per- pendicular to the line by — \Qx+ 12 = 0; find the values of a and h. 8. Construct the circle denoted by the equation a;* + 2/^ - 6a; + lOy - 15 = ; and find the position of that diameter of it which passes through the origin of coordinates. 201 A. II. Arithmetic and Algebra. 1. Define the terms "fraction," "power," "root," "index," "logarithm" and "modulus." Explain also the reasoning by which it is shown that a — {b — c) = a — b + c. ^J- i A pei-son owes £800 bearing interest at 5 per cent, per anniyb. At the end of each year he pays £120 for interest and in pq,rtL payment of the principal Find the amount of his debt at the end of the second year. 3. Reduce ] 4 — Ji ki to its most simple form : and find (a+cy-(b + dy ^ ' a* + 6* - c' + 9.ah a + h + c the value of 6* - c' + 26c a-h + c 4. Solve the following equations : 5x + () = -x->r ... (l); sl(x + S)-J{x + 5) = Jx (2); (x-T/) {a'x' + by) = a' (x' -/), xr/ = c' (3). 5. A and B agree to pay their expenses for a certain time in the proportion of the numbers 4 and 7- At the end of this period it was found that A had paid the sum of £102, and B £73. What has the one to pay and the other to receive in order to settle the account 1 C. The equation x^-x'- 33x — 63 = has two equal roots : find them by means of the derived equation; find also the third root. 7. Define a geometrical progression, and show that if each term be subtracted from the preceding, the successive difierences constitute also a geometrical progression. Sum the latter series to n terms, when the first term of the original series is 2 and ratio - . o 8. Investigate the formula for the number of shot in a square pile; and show that if the number in a square pile be to the number in a triangular one of the same number of courses as p to q, then the number of courses in each is - — - . 2q-p 205 A. III. Plane Trigonometry. 1. Explain the law of the "algebraic sign" in each of the four quadrants of the circle, in reference to the " tangent" of an arc or angle; and prove the relations tan (90" + A) = - cot A, and tan (1 80" - ^) = - tan .4 . 2. If sine a = t > what are the surd expressions for the cosine, secant, tangent and cotangent of a ? 3. The side a and its opposite angle J, of a right-angled triangle ABC (right-angled at C), are 152 feet and 18" 25' l6"; find the other parts of the triangle. 4. Prove the following relations : — sin (A + £) = sin Acos£ + sin B cos A ; tan (A-B) = , — ^ ; cos*-4 - sin**^ = cos (A + J5) cos (A-B). ^ ' 1+tan^tani? \ / \ / 5. If a, 5, c be the sides, and Ay By C the opposite angles of a ^ = - (a + 6 + c) ; show 1^^ / (8-h){8-c) 2 V s{s-a) * 6. Given a =126 feet, 6 = 132 feet, and c= 140 feet; find J, B and C. 7. An object is seen from two ships A and ^ in a river, 480 feet asunder. At A the elevation of the object above the level of the river is found to be 28° 20', and the angle subtended by it and the ship Bj 48" 25'. Also at B the angle subtended bj the object and the ship A is found to be 20" 15". Find the height of the object above the level of the river. 8. Prove that if the sides a and 6 of a plane triangle ABC include an angle of 60", cos(60"-5) = ^*. plane triangle, and also s = -{a + b +c); show that tan 206 IV. Spherical Trigonometry. 1. State Napier's Rules for the solution of right-angled sphe- rical triangles; and exemplify them : 1st, when a side is taken as middle part; 2nd, when the complement of an angle is taken as middle part; 3rd, when the complement of the hypothenuse is taken as middle part. 2. \i a,h, c are the sides opposite to the angles A^ B, C of & spherical triangle, show that cos a = cos 6 cos c + sin 6 sin c cos ^. 3. Show that, if 8=-(a + h + c\ cos -o) 1 . /sin 8 sin (s — a 2 \/ sin 6 sin c 4. The three sides of a spherical triangle are 50° 37', 83° 19', and 40° 12'; find its three angles. Astronomy. 5. Define the following terms : — " pole" and " equator" of the heavens; "meridian"; "ecliptic"; "the obliquity of the eclip- tic"; "the declination and right-ascension"; "latitude and lon- gitude of a heavenly body;" "the latitude and longitude of a j^lace on the earth." 6. Point out, by means of a figure, how the time of rising of the sun, and its azimuth when rising on a given day and at a given place, are determined. 7. What angle or arc in the heavens is the measure of the latitude of a place on the earth ? What other angle or arc is it equal to? and how is the latitude of a place determined by obser- vations on a circumpolar star ? 8. Point out, by means of a figure, how the latitude and lon- gitude of a star are determined from its observed right-ascension and declination. 2or V. Statics. 1. A straight lever is inclined at an angle of 60" to the hori- zon, and a weight of 360 lbs. hung freely at the distance of 2 inches from the fulcrum is supported by a power acting at an angle of 60" with the lever, at the distance of 2 feet on the other side of the fulcrum : find the power. 2. The arms of a lever are in the ratio of 2 to 1, and are at right angles to each other. A weight of 20 lbs. is suspended freely from the shorter arm, and 5 lbs. from the longer: find the angle which the longer arm makes with the horizon when the lever is in equilibrium. 3. A point is kept at rest by three forces represented by 5 lbs., 6 lbs., 7 lbs. : determine the angles which the directions of these forces make with each other. 4. A weight of 400 lbs. is supported on two props, 4 feet and 8 feet long respectively, resting at their other ends, 10 feet apart, on a horizontal plane : find the pressure on each prop, and the ho- rizontal thrust of each on the plane. 5. Determine the weight that will be supported by a power of 40 lbs. applied to a system of pulleys of the third kind, where each string is attached to the weight, in which there are three moveable pulleys, each weighing 5 lbs. ; giving the whole investi- gation, on the principle of the tension of the string applied to this particular case. 6. A power of 10 lbs. supports a weight of 19 lbs. on a plane inclined 30" to the horizon : find the angle which the direction of the power makes with the plane. 7. At what distance from each other must the threads of a screw be cut, that a power of 28 lbs., acting at the extremity of an arm 25 inches long, may press by means of the screw with the weight of 5 tons? 208 YI. Dynamics. 1. A weight of 84 oz. is connected with another of 77oz. by a string hanging over a fixed pulley: how far will the heavier descend, and what velocity will it acquire in 5 seconds? 2. A body is projected with a velocity of 40 feet per second down a plane inclined to the horizon at an angle of 30** : in what time will it describe 200 feet on the plane, and what will be its velocity at the end of that time? 3. Show that the times of descent down all chords drawn through either extremity of a vertical diameter of a circle are equal, and that the velocities acquired at the lowest point of these chords are proportional to their lengths. 4. Find the straight line of quickest descent : (1) From the circumference of a given circle to a given point within it : (2) From a given point without a circle to the circumference of the circle. 5. Deduce an expi*ession for the time of flight of a projectile, in terms of the range, the angle of elevation of the projectile and the angle of inclination of the range plane; and thence determine the length of fuse for a range of 1200 yards on a plane rising at an angle of 5** 30', the shell being fired at an elevation of 15**, and the fuse burning at the rate of an inch in three seconds. 6. A = 3 oz., B = 7 oz., (7=5 oz., are three perfectly elastic balls : after A has impinged directly with a velocity of 1 feet per second upon By at rest, C impinges directly upon B, in the direc- tion opposite to -4's first motion, with a velocity of 12 feet per second: find the ultimate velocities of A, B, C, considering the original velocity of A as positive. 7. Find the amount by which a seconds pendulum must be shortened in order that it may keep true time at an elevation of 15000 feet. 209 A. YII. Hydrostatics. 1. State the principle on which the pressure of a fluid upon any plane surface is determined : and the altitude of a triangular prism and each side of its base being one foot, find the pressures on its sides and ends when filled with water: (1) when the prism is placed upright on one of its ends; (2) when a rectangular face is horizontal with the opposite dihedral angle upwards; a cubic foot of water weighing 1000 oz. 2. The specific gravity of gold being 19-25, and of copper 8-9, what are the weights of copper and gold respectively in a compound of these metals which weighs 800 grains in air, and 750 grains in water? 3. A pontoon in the form of a right hexagonal prism floats with one of the rectangular faces horizontal j its length is 8 feet ; each side of the hexagonal ends is 1 ft. 6 in. ; and its weight with the portion of bridge which it supports is 1461 '4 lbs. : to what depth will it be immersed in the water, a cubic foot of which weighs 62*5 lbs.? Find also the additional weight which it bears when sunk to the depth of 2 feet. 4. Describe the Mercurial Barometer ; state on what the height of the column of mercury in it depends; what it is a mea- sure of; and why it is difierent at difi'erent altitudes above the earth's surface. 5. Describe the Condenser: R being the volume of the re- ceiver, and h that of the barrel of the condenser : find the density of the air in the receiver after n strokes of the piston, the density of the external air being taken as the unit. 6. A Diving-bell in the form of a cone, the diameter of the base of which is 8 feet, and the axis 10 feet, is let down into the sea until the water rises 5 feet within it : find the depth to which it was let down, and the density of the contained air. c. 14 210^ A. YIII. Differential Calculus. 1. Find -J- in the following functions : u = (a'- x') (a' + x')\ u = log,{J{a' + x') - J(a' - x% u = e sin x. 2. If w = tan Xy find - -s ; u = x log x, find -^ ; u = oce', find 3. Of all cylinders inscribed in a given hemisphere whose ladiiis is a, find that which has the greatest convex surface. 4. Find the subtangent to a curve whose equation is {l/-h)J{a'-x') = a\ Integral Calculus. 5. Find the following integrals : C a^x'dx C dx C x' dx 6. Find the area of a curve whose equation is y' {a' - xy = a' x\ from a; = to a. 7. Find the volume of a segment of an ellipsoid generated by the revolution of an ellipse about its major axis; and show that the volume of the whole ellipsoid is two-thii*ds of that of its cir- cumscribing cylinder. 8. Show that the distance of the centre of gravity of a para- bola from its vertex is | x. 9. By the property of Guldin, find the volume of a solid generated by the revolution of a parabola about a double ordinate — 26, the corresponding abscissa or height of the parabola being a. 211 B. I. Geometry. 1. Give definitions of the following: — 1. "a diameter of a circle;" 2. "a segment of a circle;" 3. "a square;" 4. "parallel straight lines;" 5. "the rectangle contained by two straight lines;" 6. "an angle in a segment of a circle;" 7- "a rectilineal figure described about a circle;" 8. "duplicate ratio," and "tri- plicate ratio;" Q. "reciprocal figures;" 10. "a dihedral angle;" 11. " a solid or polyhedral angle." 2. Parallelograms upon the same base, and between the same parallels, are &c. 3. If a straight line be divided into any two parts, the square of the whole line is equal to &c. 4. The opposite sides of a quadrilateral described about a circle are together equal to the other two opposite sides. 5. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles propor- tionals, the triangles shall be equiangular, and &c. 6. If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane. Co-ordinate Geometry. 7. The line y = ax + h passes through the points (5, — 7) and ( - 3, 2) ; find the values of a and K Construct also this line. 8. Determine the radius of the circle which passes through the points (0, 0), (1, 2) and (1,-2); and show also that (^, f ) is a point in the same circle. 9. Investigate the equation in general of the conic sections, and show that such equation represents three distinct curves. 14-2 212 B. II. Arithmetic and Algebra. 1. If in quick marching, 110 steps are performed every minute, and each step is 30 J inches in length, what is the rate, in miles, of marchiug per hourl 2. Find the value of the expression : 1 1 1 4- + ^ x(x-a)(x — b) a (a - x) (a — b) b(b — x) {b — a) 3. Solve the following equations : a{x-y)^h{x+7j), o(?-7f = c\ J'{3 + Jx) + J{4^ - Jx)- ^{7 + 2 Jx) = 0. 4. Two railway trains are dispatched from a station, along two different lines, the one starting an hour before the other. As the rate of motion of the later train is 5 miles an hour more than that of the other, it arrives at a distance of 100 miles from the station at the same hour of the day in which the earlier one completes an equal distance. Find their rates of motion. 5. Find the least root of the equation a'- 13-572a; + 24 7148 = by the method of continuous approximation, and also by the com- mon method. 6. Investigate a formula for the sum of n terms of an arith- metical series, a being the first term, and d the common difference 33 of the terms. Sum also to 25 terms, the series 18 + — + 15 + (fee. 7. Find an expression for the number of shot in a rectan- gular pile of n courses ; m+1 being the number of shot in the top row. Determine also the number of balls in the length and breadth of the lowest course of an incomplete rectangular pile of 25 courses; the number of balls in the length and breadth of the top coui-se being, respectively, 24 and 1 8 balls. 8. Develope a' in a series of ascending powers of x; and thence deduce the base of the Napierian system of logarithms. 213 B. III. Tkigonometry and Mensuration, 1. Trace the changes of algebraic sign of the trigonometrical functions sine and cosine of an arc, for the first four quadrants of the circle; and explain why sec^ and sec (180 + -4), which coin- cidej should be one positive and the other Tiegative. Prove also that sec^ = — —7, tan (180 -^) = - tan ^. cos ^ ^ ' 2. Prove the relations j cos {A +3) = cos A cos B — sin A sin i>, cos 9.A = 9, cos^-4 -1 = 1-2 sin^J, 2 sin ^ cos 5 = sin (^ + ^) + sin (^ - ^). 3. Given two sides of a triangle, 200 feet and 2.50 feet, and their included angle 52" 23' : find the third side and the remaining angles. 4. An object B on the summit of a hill, and two stations C and D are in the same vertical plane : at G the angle of eleva- tion of B above the horizontal plane is 50* 1 8' 20'^ ; at D the angle of elevation of B is SO** 16' 12" ; the distance Ci) = 124 yards ; and the line DC rises from D towards C at an angle of 4" 12': find the distance of B from i>, and its vertical height above that station. 5. Deduce an expression for the area of a triangle in terms of two of its sides a, b and their included angle G, 6. The sides of a plane triangle are 6, 6 + ^'(2) and 6 - ^2 ; find its area. 7. The area of a regular polygon of n sides, whose side = a, is a? , 180 n, — .cot . 4 n 8. The convex surface of a cylinder inscribed in a sphere is one-third the surface of the sphere : find the radius and altitude of the cylinder in terms of r, the radius of the sphere ; and show generally, that if the convex surface of the cylinder is one n^"^ of that of the sphere, the radius of the cylinder, is "^ oTln ' ^* 214 B. TV. Spherical Trigonometry. 1. Define the following : — " a great circle of the sphere;" " a small circle;" "the poles of a circle;" "a spherical angle;" "u spherical triangle;" " the polar triangle." 2. Show that the sides of the polar triangle are the supple- ments of the angles of the primary triangle ; and the angles of the polar triangle are the supplements of the sides of the primary triangle. 3. If a, h, c are the sides opposite to the angles A, By C of a spherical triangle, show that cos c = cos a cos 6 + sin a sin b cos C. 4. Show that, if 8 = -{a + b + c), tan - A jit ji) _ /sin {s — h) sin {s - c) ^ sin 8 sin (s — a) and thence that, tan ^(A +B) = 14 ft cot ^ C. ' -^ ^ ^ cos i (a + 6) ^ 5. The base of a right-angled spherical triangle is 57" 19'> and the adjacent acute angle 31** 53': find the remaining parts of the triangle by Napier's Rules. Astronomy. 6. Define the following terms :—" Zenith;" "Meridian;" "Prime vertical;" "Azimuth;" "Hour cii'cles;" " Right Ascension and Declination;" "Latitude and Longitude of a heavenly body;" " Latitude and Longitude of a place on the earth." 7. Point out by means of a figure how the time from appa- rent noon is determined from the observed altitude of the sun on a given day, knowing the latitude of the place of observation. 8. Find the latitude of a place in terms of h, the observed time of the sun being on the prime vertical, and 8 the north polar distance of the sun. 215 B. V. Statics. 1. Two forces represented by 5 lbs. and 7 lbs. act at a point, in directions making an angle of 50" with each other : what weight will represent their resultant, and what angle will its direction make with that of the greater force? 2. Suppose that in the common gyn, the windlass or axle, round which a rope passing over a pulley fixed at the top of the gyn is coiled, is 10 inches; and the lower block in the system of pulleys (of the second kind, where the same string passes round all the sheaves) contains three sheaves, the rope being fixed to the upper block : what weight will a man support on the hook of the lower block, when applying a force of 100 lbs. on the end of a handspike, 7 feet long, placed in the windlass ? 3. If in a system consisting of any number of particles a. point be taken, and if each particle be multiplied by the square of its distance from the point, show that the sum of these products will be the least when the point is the centre of gravity of the system. 4. A beam, 8 feet long and weighing 100 lbs., is suspended from a hook, by two cords, 6 feet and 10 feet long respectively, attached to its ends : find the angle which the beam makes with the vertical when in the position of equilibrium, and also the tension of each cord. 5. ^jB is a beam, which with a weight W, suspended from B, is to be suppoi-ted against a vertical wall VL, by a chain CD fixed to the wall: given the weight of the beam = 100 lbs., the weight Tr=300lbs., the length of the beam ^^ = 8 feet, the angle £AL, which the beam makes with the vertical, = 30" ; find the distances AC, AD of the points of attachment G, I), of the chain to the wall and to the beam, from the point A, where the beam rests against the wall, so that, CD being perpendicular to A£, the whole may be in equilibrium; and determine the tension of the chain CD, and the pressure against the wall VL, 216 B. VI. Dynamics. 1. A = 2 and B—3 are two perfectly elastic bodies : A, moving with a velocity of 20 feet per second, impinges directly on B moving in the opposite direction with a velocity of 12 feet per second : find their directions and velocities after impact. 2. With what velocity must a body be projected vertically upwards from the base of the Monument to reach the top, 210 feet above, in 3 seconds? and what will be its velocity when it reaches the top] 3. P and Q are two equal weights, each 16 oz., connected by a string passing over a fixed pulley : what weight must be added to F that it may descend through 1 foot in 2 seconds, supposing no inertia in the string or pulley, and that g = 32 feet 1 4. Find geometrically, the straight line of quickest descent, (1) From a given straight line to a given point below it; (2) From the circumference of a given circle to a given point without it, and below the highest point of the circle. 5. A body weighing 3 lbs. revolves as a conical pendulum and makes three revolutions in seven seconds ; find the tension of the string, 10 feet long, by which it is held. 6. Given the velocity of projection v, the angle of elevation e; investigate the expression for the range on a plane passing through the point of projection and making a given angle i with the horizon. 7. Investigate the expression for the time of flight in terms of e, i and r the range. 8. If 1 inch of fuze burn 4-9 seconds, find the length that a fuze must be cut, in order that a shell may explode on reaching the ground at a range of 800 yards, on a horizontal plane, when fired at an angle of elevation of 45". 9. With what velocity must a shell be fired, at an angle of elevation of 30", to strike an object at the distance of 900 yai-ds on a plane which rises at an angle of 5" ? 217 B. VII. Hydrostatics. 1. A hexagonal prism being filled with water is placed with one of its rectangular faces horizontal; find the pressure on each end, and also on each of the rectangular faces, each side of the hexagonal ends being 1 foot, and the length of the prism 2 feet. 2. The specific gravities of platinum, gold and silver being respectively 21, 17'5 and 10*5, and the value of an ounce of each 305., 805. and 5s. respectively, it is required to find the value of a coin composed of platinum and silver which is equal both in weight and in magnitude to a sovereign. 3. A cylindrical pontoon with hemispherical ends, 3 feet in diameter and 20 feet long in the cylindrical part, when floating has a fourth of the diameter immersed ; what is the weight of the pontoon; and what additional weight does it bear when only a fourth of the diameter is above the water, a cubic foot of water weighing 6*2^ lbs. ? 4. A diving-bell in the form of a paraboloid, the diameter of its base being 9 feet and its height 12 feet, is let down into the sea until the water rises 4 feet within it; find the depth to which it was let down, and the density of the air within it, the pressure at the surface of the sea being equal to 33 feet of sea-water. 5. Describe the air-pump, and show the degree of rarefaction of the air in the receiver after n turns of the handle, the content of each barrel being an m*^ part of the content of the receiver. 6. A cylinder 1 feet high being filled with water, it is re- quired to find where a small orifice must be made in its side so that the water issuing from it may strike the horizontal plane on which the cylinder stands at the distance of 8 feet from the base. 7. An oaken sphere is let fall from the surface of the sea; in what time will it strike the bottom at the depth of 20 fathoms, neglecting the resistance of the water, the specific gravity of the oak being 1-17, that of the sea- water 1*03, and (/= 32 1 218 • B. YIII. Differential Calculus. 1. Find the differentials of the following functions of x : u = (a-x)J{a + x), «=-,-^-^, . X + a • ? 9 u = log, , u = sm X cos X. 2. If u is a function of x, and w, represents u when, in it, x becomes x + hj what is the value of u^ (Taylor's Theorem) 1 3. u being a function of x, show that when u is either a . . die , . . maximum or a mimmum, —- — ; that it is a maximum when ax d'u . T . . . dhi . j-j IS negative ; and a mimmum when j-j is positive. 4. Of all cylinders which can be inscribed in a given cone whose altitude is a and the radius of whose base is b, find (1) That which has the greatest convex surface; (2) That whose whole surface is the greatest. 4. Find the subtangent to a curve whose equation is aj^?/' =a*- X*. 6. Find the expression for the radius of curvature in the parabola and in the ellipse ; and find its value at the vertex in the parabola, and at the extremities of the axes in the ellipse. 219 B. Integral Calculus. 1. Find the following Integrals : [ x^dx r(2x-5)dx r x^dx C x'^dx J {a'-xf' Jx^-5x+6' Jj{a' + x')' ]j{l-x')' 2. Find the area of the curve whose equation is y{a-x) = x{a-\- x), from x = to x = ^a. 3. Find the length of the arc of a cycloid whose equation is X y = J{9,ax — x^ + a vers"' - , from x^O to x = a, and also from x = ato x = 2a. 4. Find the volume of a solid formed by the revolution of a curve whose equation is ay^ = {a-x) {a^ + x^), about the axis x, from X = to x = a. 5. Give the expression for determining the distance of the centre of gravity in a solid of revolution ; and determine the posi- tion of the centre of gravity of a spherical segment. G. State Guldin's properties of the centre of gravity; and from these properties find the volume and the surface of the solid generated by the revolution of a semicircle, diameter = 2r, about an axis parallel to its diameter, at the distance c from the centre. 7. Assuming that when a pendulous body vibrates in a cycloid, the force varies directly as the arc from the lowest point, find the time of vibration of such a pendulum whose length is I. 8. The length of the seconds' pendulum in the latitude of London being 39-1386 inches, determine from this the value of g, the number representing the force of gravity. 220 I. Geometry. 1. Give definitions of the following: "a right angle;" "pa- rallel straight lines;" "an angle in a segment;" "ex sequali;" "a plane perpendicular to a plane;" "a dihedral angle;" "similar solid figures." 2. The difierence of any two sides of a triangle is less than the third side. 3. The straight line drawn at right angles to the diameter of a circle from the extremity of it, falls without the circle ; and no sti*aight line can be drawn between that straight line and the cir- cumference so as, (fee. L The sides about the equal angles of equiangular triangles are proportionals; and those which are opposite to the equal angles are, &c 5. If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane. 6. Through two given straight lines to draw planes parallel to each other. Co-ORDiifATE Geometry. 7. State clearly what is meant by the " equation of a straight line or curve;" and find the equation of the line which passes through the points {1,4) and (- 3, 7)* 8. The equations of the sides of a triangle are y = a; - 2, y=^x + 3y and y = 3a; + 1 ; construct this triangle, and find the co-ordinates of its angular points. 9. Investigate the equation of the circle from its geometrical definition ; and find the radius of the circle denoted by the equa- tion x^ + if - 6aj + 4?/ = 12. 221 C. II. Arithmetic and Algebra. 1. If 30 men can dig a certain trench in 11 days, how many men can dig a trench of the same sectional area, but 4 times as long, in 2 days? 2. Keduce to its simplest form the expression and extract the square root of 23 — 8 J 7 in a binomial surd. 4 3. Solve the equations : Jx + J (2 + x) v/(2 + x) '1 X 1 + - y = a, 1 X 1 + - z = h, 1 1 + - z = c. 8/ ^^ 4. If 962 men were drawn np in two solid squares, and it were found that one square had 18 ranks more than the other, what would be the strength of each square ? 5. Find the position of all the roots of the equation x^-l6x^+86x'-l68x + 69 = 0, and determine the least root to four places of decimals. G. The number of shot in a rectangular pile consisting of ten courses is equal to six times the sum of the number of shot in a triangular pile and square pile, each of the same number of courses as the rectangular pile; find the number of shot in the ridge of the rectangular pile. 7. Show that log^„ n = j-^— . log, n ; and if log^^ 3 = -477 1 2 1 3 OS and log^„ 5 = '6989700, find log^„ 6. 8. The hypothenuse of a right-angled triangle is 10, and the excess of the perpendicular above the base is 2 ; find the sides of the triangle. 222 C. III. Trigonometry and Mensuration. 1. Define the tangent, cotangent, secant, cosecant of an angle; and express each of these functions in terms of the sine. Explain also the double sign in the results; and prefix the proper sign when the angle is between 270** and 360". 2. The sides a and 6 of a plane triangle, right-angled at C, are 183 and 197; find the angles and the remaining side. 3. Prove the following formulae : -^ 7 = tani(^ +B)iBjiXlA-B)\ cot^-tan^ = 2cot2X 4. State the three cases for the solution of plane ti'iangles; ijrove the formula =- = ;— -^ -. ; state to what case this ^ a-h tan ^{A-B)' applies, and how. o. Prove that in a plane triangle right-angled at C, sin 2 A = j^ 5 ; cos 2A = , , ^ . 6. A person on a level plain, on which stands a tower AB surmounted by a spire BC, observes that when he is 100 feet dis- tant from the base A of the tower, ^ is in a line with the top of a hill, and the angle of elevation of B above the horizontal plane is 43**; but when he is 80 feet further from A, (7 is in a line with the top of tlie hill, and the elevation of C is 34**; find the height of the spire and of the hill. 7. Investigate a formula for the area of a triangle ABC in terms of the base b and the angles at the base A and C. Deter- mine also the area of the triangle of which the base is 410 feet, and the angles at the base 30*> 17' and 50° 1 1'. 8. Prove that the areas of an equilateral triangle and hexa- gon, of equal perimeters, are to one another as 2 to 3. Find also the absolute values of these areas when the perimeter of each is 15 inches. 223 C. TV. Statics. 1. Define the terms: — "absolutely at rest;" "relatively at rest;" "force;" "equilibrium." 2. Tliree ropes are fastened at one of their extremities to a ring round a post, and three men A, B^ G pull horizontally with forces of 80 lbs., 90 lbs., 100 lbs. at the other extremities of these ropes; find the directions in which A, B, C must pull, that is, the angles which the ropes must make with each other, that the post may be undisturbed. 3. 'Forces /,/', f'\ f\ (fee, make angles a. a, a', a", (fee. with the axis Ax: find the equivalent forces in the directions of the rectangular axes Ax, Ay; find also the resultant of these forces, and the angle which it makes with the axis Ax : and apply this to the case where the forces are represented by 5, 7> 2 and 9 lbs., and the augles which they make with Ax are 0', 30^ 60'^, 90\ 4. Three forces represented by 40 lbs., 60 lbs., 80 lbs. act in a vertical plane upon a point, and their respective directions make angles of 30**, 60°, 120° with the horizon; find the magnitude and direction of a fourth force that shall counterbalance their effect upon the point. 5. A power of 50 lbs. is applied by means of a single moveable pulley to the arm of a screw : what will be the pressure of the screw, the distance between the threads being one-third of an inch, and the length of the arm 1 2 inches % 6. If the axis of the trunnions of a gun be 2 inches below the axis of the bore, and the centre of gravity of the gun be 5 inches behind the intersection of the perpendicular from the mid- dle of the axis of the trunnion upon the axis of the bore, with this axis ; what is the greatest depression that can be given to the gun without its turning over? 224 C. V. Dynamics. 1 . State the three " Laws of Motion." 2. A shot was let fall from the top of a cliff; one second afterwards another shot was projected vertically downwards, from the same point, with a velocity of 35^ feet; and the two were heard to strike the sea at the base of tlie cliff, at the same instant, ()J seconds after the first was let fall; find the height of the cliff; and, from this, the velocity with which the sound travelled. 3. A weight of 100 oz. is placed on a perfectly smooth plane inclined at an angle of SO" to the horizon, and is attached by a string passing over a pulley at the uj)per edge of the plane to an- other weight hanging freely : find what this weight must be that it may descend 27 '5 feet in 5 seconds. Find also how far this weight would descend in the same time if the string passed over a pulley at the lower edge of the plane ; and find the tension of the string in each case. 4. "What is the length of a pendulum which vibrates three times during the time that a body descends 64-4 feet on a plane inclined 30° to the horizon? 5. Determine the equation between v the velocity of projec- tion, e the angle of elevation, and r the range of the shot on a ])laue which passes through a point at the distance h feet vertically below the point of projection, and which rises at an angle i from the horizon. G. A shot is to be fired from a battery to clear (1 foot above) a parapet, at the horizontal distance of 600 yards from, and 11 feet above the level of the battery, and then strike a point 60 feet beyond, and 5 feet below the top of the parapet: at what elevation and with what velocity must the shot be fired 1 7. J[=4oz., £ = 3oz., C=5oz., are three perfectly elastic spheres; B being at rest, A impinges directly upon it with a ve- locity of 14 feet; and, immediately after this impact, C impinges directly upon B with a velocity of 12 feet in the direction opposite to that in which A had impinged; find the velocities of ^, B, (7, after all the impacts. 225 c: YI. Hydrostatics. . 1. A prismatic vessel, whose base is an equilateral triangle, and altitude is equal to one side of its base, is filled with fluid : compare the pressure on one of the triangular ends with that on one of the rectangular sides, when it stands upright on a triangu- lar end. Also, when a rectangular side is horizontal, compare the pressure on one end with that on one of the inclined sides, (1) When the horizontal side is upwards; (2) When the horizontal side is downwards. 2. The specific gravities of platinum, gold and silver being respectively 21, 17*5 and 10-5, and the values of an ounce of each 305., 80s., and 5s. respectively, it is required to find the value of a coin composed of platinum and silver which is equal in weight and magnitude to a sovereign. 3. A cylindrical pontoon, 20 feet long and 3 feet in diameter, has a fourth of the diameter immersed when floating; what is the weight of the pontoon, and what additional weight will it just bear, a cubic foot of water weighing 62 J lbs. ? 4. Find the altitude of the roof of Severndroog Castle, Shooter's Hill, above the wharf in the Arsenal, from the follov/- ing observations : — Thermometer. ; Barometer. Attached. Detached. Arsenal 29-475 in 37 34 Severndroog Castle... 29-022 in 35 32 5. A diving-bell, in the form of a rectangular prism, and whose height is 8 feet, has descended until its upper surface is 40 feet below the surface of the water : to what height will the water rise within it, and what will be the density of the contained air, that of the external air being 1, the height of the barometer 30 inches, and the density of mercury to that of water 14 to 1 nearly? 6. Describe the common suction-pump and explain its mode of action by means of a figure. c. 15 226 ^^ VII. Differential Calculus. 1. u and V being functions of x, show that d(uv) du dv dx dx dx* and thence that, w, *, t being functions of a;, d{uls) du dt da dx dx dx dx ut8 u t s ' 2. Find the differentials of the following functions of x : s u = (2a' + Sx') (a'-x')K u = -^p—^ . Cb ~" X^ ,«• + «• . . 3. Find the sides of the greatest rectangle that can be in- scribed in a given regular hexagon, a side of the rectangle being parallel to a side of the hexagon. 4. Of all cylinders inscribed in a given cone whose altitude is a, and radius of its base 6, find, 1st. That which has the greatest convex surface; 2nd. That whose whole surface is a maximum. 5. Find the subtangent to a curve whose equation is a,"* a^ — x^ — = J- ; and give the values of the subtangent corresponding to x = ^a and x = a, 6. From the variable points C and P, equidistant from a given point A in the same straight line as C and P, perpendiculars, CBy PJ/, to CF are di-awn, of which CI) = AC, and FM is inde- Unite. From a given point J3, in FC produced, the straight line CDM is drawn cutting FM in M. Find the equation to the locus of the point if, and the expression for the subtangent to the curve. 227 C. VIII. Iktegral Calculus. 1, Find tlie following integi-als : /xdx 3+x-x'* a^dx /x" ^dx {a + hxy f x'dx f a^ 2. Find the area of the curve whose equation is a*y^ = (ic* + y') x*, from x^O ix> x = ^a. 3. From the general expression for the volume of a solid of revolution, find an expression, 1st. For the volume of a paraboloid j 2nd. For the volume of a spherical segment, and of the whole sphere, ^ 4r. Give the differential equatious of rectilinear motion when a body is acted on by any force ^ 5. If a meteorolite were to fall to the earth from a height equal to ten times the radius, with what velocity would it strike the earth, the force varying inversely as the square of the distance from the centre, abstracting the resistance of the atmosphere near the surface 1 6. Give the expression for determining the distance of the centre of gravity in a solid of revolution ; and determine the po- sition of the centre of gravity of a spherical segment, 7. Prove Guldin's properties of the centre of gravity; and, from these properties, find the volume and the surface of the solid generated by the revolution of a semicircle, diameter = 2r, about an axis parallel to its diameter, at the distance c from the centre. 15—2 228 D. I. Geosietrt. 1. Give Euclid's definitions of the following : — (1) A rectilineal figure described about a circle; (2) the same ratio, or equal ratios; (3) duplicate ratio and triplicate ratio; (4) similar rectilineal figures. 2. The difference of the angles at the base of any triangle is double the angle contained by a straight line di-awn from the vertex perpendicular to the base, and another bisecting the angle at the vertex. 3. In a circle, the angle in a semicircle is a right angle;* but the angle in a segment greater than a semicircle is 4 was detained 20 minutes on the road before he met £. 2. Simplify the expression ^ ; and show that f — ^-jrf= (2 - V^)^ 3. Solve the following equations : J{5x + 10) = J5x + 2, 2 2 m n _ n m ^ ~ x + J{2-x') ^ x-J{2-x') ' x'^y^^'x'^lj^^' 4. In the front of a detachment from an army were 175 more men than in the depth; and by increasing the front by 50 men, the detachment was drawn up in 20 lines. Find the num- ber of men in the detachment, 5. The first term of an arithmetic series is cl, the common difference d, and the sum of n terms s; find an expression for s. Find also the first term, the common difference, and the sum of n terms of the arithmetic series, of which the general form of the n^ term is -^{2n - 1). 6. The first term of a geometric series is 5, and the ratio 2 : how many terms of this series must be taken, that their sum may be equal to 33 times the sum of half that number of terms 1 7. Find the value of (■0^^^^ ^ (-^^^J^ by logarithms. (•024)^ III. Trigonometry and Mensuration. 1. Show by a figure that tan A = j , and cotan A cos A tan A * 2. Shew that sin 60=-rj3; and thence find coa 60", tan 60^ cot60^ sec 70", vers 60". 3. A and -5 being any two angles, show that / J 7i\ cot A cot i? T 1 cotan (A * ^) = — —^ — -j- . ^ ' cot ^ ± cot -4 4. O) 5, c being the sides of a triangle, respectively opposite tani(.4+^) the angles A, B, C, prove that 7 ^ — — — ; state to "-* Unl{A-S) -what case in the solution of plane triangles this is applicable, and point out how it is applied. 5. The three sides of a triangle are 300, 500, 700; find each of its angles without employing logarithms. 6. Wanting to know the distance between two church towers B and H, standing on a horizontal plane, I measured a base of 880 yards on the same plane, from the extremities -4, C of which, I took the following horizontal angles, viz. from the station Aj the angles between the other station C and the towers B and // were respectively 9l"13' and 67*'l7'j fronx the station C the angles between the other station A and the towers B and H were respectively 79* 24' and 99° 4-7' : find the distance between the two churches B and if, 7. The three sides of a triangle are 6, 6 + a/2, 6 - ^2 ; find its area. 8. Show that the area of a regular hexagon of which a is the 3 iJS side is —^— «* : and the distance between two parallel sides of a regular hexagon being 20 yards, find its area and also that of its circumscribing circle. 231 D. TV, Spherical Trigonometry. 1. Define the following teims: — "diametral plane/' and "tangent plane, of the sphere;" "side of a spherical triangle;" the "spherical excess;" "quadrantal triangle." 2. State Napier's Kules for the solution of a right-angled spherical triangle; and prove that if a, 6, c be the sides of a spherical triangle, right-angled at B^ cos b = cos a cos c. 3. Given the side a = 38° 1?', the angle A = 50M6', and the angle B = 90°, of a spherical triangle ABC, to find the remaining parts by Napier's Rules. 4. Show that in any spherical triangle, if a, 5, c be the sides, Ay B, G their opposite angles, and « = - (a + 5 + c), . - 1 . sin (« - 6) sin (s — c) „ 1 . sin s sin is — a) sm'-.l = : , . ^^ -y cos^-^ = . , . -, 2 sm 6 sm c 2 sm 6 sin c 5. The two sides a and 6 of a spherical triangle ABC are 108" and 99" 21', and the angle ^ = 120"; find the angle B, Give also the equations in a logarithmic form, for the determination of the remaining parts of the triangle. Astronomy. 6. Define the following terms : " The Poles and Equator of the Earth," " The Poles and Equator of the Heavens," " The Declination, and Eight Ascension of a heavenly body," " The Lati- tude and Longitude of a place on the Earth." 7. Show by a figure that the Latitude of a place on the earth is equal to the altitude of the pole above the horizon. 8. State how the Latitude of a place is determined by obser- vations of a Circumpolar star. 9. State the principles upon which the determination of the longitude of a place depends. 232 V. Statics. 1. Give definitions of the following: 1. the "resultant" of two or more forces; 2. the "components" of a force; 3. the "centre of gravity;" 4. the "lever;" 5. the "inclined plane;" 6. the "pulley." 2. A force of 30 lbs. in a direction making an angle of 60" with the axis of x is the resultant of two forces which make angles of 45" and 120" respectively with the same axis; find the magni- tudes of these forces. 3. A plane figure, consisting of a square with an equilateral triangle described upon one of its sides, is placed with its plane vertical and an angle of the square resting upon a horizontal plane; find the angle which a side of the square makes with that plane when the figure just balances on the point of support. 4. Spheres of which the weights are 4 oz., 6 oz., 5 oz., 7 oz., are placed with their centres at the angular points -4, 2?, C, i) of a trapezoid of which AB and DC are the parallel sides; AJB = 10 inches, BC= 4 inches, CD = 7 inches, and DA = 5 inches : find the jDOsition of their centre of gravity — 1st. When the bodies are supposed to be connected by inflexi- ble rods without weight ; 2nd. When the connecting rods are of uniform density, and weigh I oz. per inch. 5. If two forces acting upon the arms of a lever keep it at rest, they are to each other inversely as the perpendiculars drawn from the fulcrum upon their directions : prove this. 6. A circular plate of metal, weighing 10 lbs., supported by a hook at the point A in its circumference, has a weight of 50 lbs. suspended from the point ^ diametrically opposite to ^: with what force must the point JD at the upper extremity of the diameter JDU at right angles to AB be pressed vertically upwards, that the dia- meter AB shall incline downwards at an angle of 30" to the horizon ? 233 D. YI. Dynamics. « 1. Define the terms "inertia;" "velocity;" "momentum;" " accelerating force." 2. A ship's apparent course is N.W. by W., 8 knots an hour, and the tide sets her "W.S.W. at the rate of 3 knots an hour: what is her true course ? and what her rate of progress in that course 1 3. A shot is fired from the ground vertically upwards with a velocity of 322 feet per second ; and 6 seconds after, another shot is fired, with a velocity of 1288 feet per second, vertically upwards from the same spot : at what distance from the ground will the second shot pass the first ? 4. A weight of 120 oz. is placed on a smooth horizontal plane, and is attached by a string to a weight of 41 oz. hanging verti- cally over the edge of the plane; find how far this weight will descend in 5 seconds, the velocity it will acquire in that time, and the tension of the string, 1st. When the friction is neglected; 2nd. "When the friction is equal to one-eighth of the weight on the plane. 5. From what height must the ram of a pile-driver, weighing 16 cwt., descend upon the head of a pile, that it may strike it with a momentum equal to that of a 42 lb. shot fired with a velocity 1610 feet per second? 6. A railway train, without locomotive engine, descends, from rest, a mile or I76O yards along an inclined plane which falls 1 foot in 400 feet, and then ascends an inclined plane of 1 foot in 600 feet : how far will it ascend along the latter plane, supposing no motion to be lost by friction or in the transfer from one plane to the other 1 7. With what velocity must a shell be fired, at an angle of elevation of 45", from a battery on a clifi" 400 feet above the level of the sea, to strike a ship at the horizontal distance of 2 miles from the base of the 0119*? 234 VII. DiFFEREXTIAL ANB INTEGRAL CaLCULUS. 1. u and V being functions of «, show that , w du dv , u vdu — udv ' V dx dx a - = 5 or — r— = j . V V dx V du 2. u being a function of x, find -j~ , when u-3 sin"' a; - (2a^ + 3x) J{1 - a;"). 3. Of all cylinders which can be inscribed in a given cone whose altitude is a and the radius of whose base is 6, find 1st. That which has the greatest convex surface; 2nd. That whose whole surface is the greatest. 4. Find the subtangent to a curve whose equation is x'y' = d*- X*, 5. Find the following integrals : r x'^'^dx C xdx C x*dx ( x^dx •'^^^' i^^^^' }W-^' J7(y^r 6. Show, by means of Taylor's Theorem, that the area of a curve is jydx, 7. Find the area of the curve whose equation is a*y* = (a;' + y') a;*, from aj = Otoa; = ^a. 8. From the genei*al expression for the volume of a solid of revolution, find an expression, 1st. For the volume of a paraboloid; 2nd. For the volume of a spherical segment, and of the whole sphere. 235 E. I. Geometry. 1. If a straight line faXia upon two parallel straight lines, it makes, 8" =5.8 — Vm x^ + y' = h'xY) 5. Find all the roots of the equation cb'- 2a;*- ^'99x + 5'6l = 0. 6. The number of shot in a square pile of n courses being -^^ ^ ^- — , show how the formula for the number of shot 3.2.1 ' in a rectangular pile of n courses, and having m + l shot in the top row, is obtained : and compute the number of shot in an in- complete pile of 20 coui-ses, having 30 shot in the longer, and 1 8 shot in the shorter side of its upper course. 7. Prove that in the expansion of (l+z)" the second tenn is nz whether n be an integer or a fraction, positive or negative. 8. From the equation between a number and its logarithm to the base a, show that log« (iV^ x iV,) = log« iV + log, iTj, ^0Sa~y- = l0g,^-l0g,]^^, l0g,iV"' = wl0g,ir. 9. Logj„ 731 being 2-8639174, prove that log,„ -00731 = 3-8639174, and log,„ 731000= 5-8639174. 245 F. III. Trigonometry and Mensuration. 1. Define the terms " complemeat," " supplement," " secant," and " cosecant" of an arc or angle; and prove that cota = -^ , sin (90° + a) = cos a, sin 30" = - . sm a 2 2. Solve the right-angled triangle ABC, of which B is the right angle, and a = 113-4, feet, G= 2(f 17' 12". 3. Prove that sin ^ - sin -B = 2 cos - (^ + B) sin - {A - B). 4. Show that the sides of a plane triangle are to one another as the sines of their opposite angles. Also solve the triangle of w^hich a = 1 56, 6 = 127, and 5 = 50". 5. Two objects A and B subtend an angle of 50" 18' 20" at a station S. Given the distances SA = 1784 yards, SB = I692 yards, to find the horizontal distance between the objects: (1) when A, B and the eye (>S') of the observer are in the same horizontal plane; (2) when A and ^S' are in a horizontal plane and ^120 feet above it. 6. In an isosceles triangle in which a = b, prove that cos ^ = — - , and vers C = 7— j . 2a 2a 7. Investigate the formula for the area of a plane triangle when two sides and the included angle are given; and thence find the area of the triangle in which 6 = 124 feet, c = l6l-4feet, and ^ = 71" 12'. 8. Assuming that the area of a circle is equal to half the product of the radius and circumference, it is required to deduce the particular form, (l) in terms of the radius, (2) in terms of the diameter, (3) in terms of the circumference. State also, gene- rally, how the expression for the area enunciated above, is obtained by means of the inscribed and circumscribed polygons. 246 F. lY. Spherical Trigonometry. 1. State Napier's niles for the solution of right-angled sphe- rical triangles, and point out how they are applicable to each case. 2. The angles of a spherical triangle being A, B, (7, and the sides opposite to them a, )8, y, show from the equations . - 1 , sin (o- — ^) sin (o- — y) . - 1 . sin (t sin (a - o) sin* -A = ' n ' — ^^ and cos''- A = ; — .r-i , 2 sin /j sin y 2 sin fS sm y cos -(a-B) 1 2 ^ '^'^ 1 I ^that sin -(A + B)= ■ . cos - C and sin - (A - B) tit ^ tit ^ - cos-y sinl(a-^) = ; COB-C. 3. Deduce Napier's first and second analogies, and state to what cases in the solution of spherical triangles they are applicable. 4. State what is underetood by the spherical excess; and JS" being the spherical excess in a triangle of which the area in feet is n, and r the radius of the Earth in feet, show that 180.60.607i £" Tir 5. The radius of the Earth being 3954 miles nearly, find the spherical excess in a triangle whose sides are respectively 40, 50, 60 miles. Astronomy. 6. Define the following terms: pole of the heavens, meri- dian, horizon, prime vertical, right ascension, declination, parallax, sidereal year, tropical year. 7. The latitude of the place of observation being known, show how the time of the Sun's rising and setting on a given day, and also his azimuth at that time may be determined. 247. F. V. Statics. 1. Forces f^ f\ f'\ f"\ &c. make angles a, a', a", a ", (fee. with the axis Ax\ find the equivalent forces in the directions of the rectangular axes Ax^ Ay^ find also the resultant of these forces, and the angle which it makes with the axis Ax : and apply this to the case where the forces are represented by 5, 7, 2 and 9 lbs., and the angles which they make with -4a; are 0**, 30\ 60°, 90°. 2. A power of 50 lbs. is applied by means of a single move- able pulley to the arm of a screw : what will be the pressure of the screw, the distance between the threads being one-third of an inch, and the length of the arm 12 inches i 3. If the weight of a gun and carriage exclusive of the wheels be 1 8 cwt., and the centre of gravity of this mass be at the hori- zontal distance of 10 inches from the axletree; what weight will each of two men support when lifting the ti-ail at a horizontal distance of 8 feet from the axletree 1 4. A uniform beam AB, 8 feet long and weighing 100 lbs,, is hooked at J. to a vertical wall FZ, and is supported by a chain fixed to it at G and to the wall at JD: AG — 6 feet, and each of the angles AG By GAB is 60^ ; find the strain on GB, and the direc- tion and amount of the reaction at A when a weight of 1000 lbs- is suspended from B. 5. A heavy six-pounder gun weighing 36 cwt. 3 qrs. is to be drawn, up a slope of 30° inclination to the horizon by means of drag-ropes parallel to the slope : the friction being one-ninth of the perpendicular pressure on the slope ; what is the least num- ber of men that will be able to move the gun, supposing each man to draw with a power of 1^ cwt. ? 6. Two uniform beams, each 20 feet long and weighing 1 00 lbs., rest against each other in the form of a roof, and are supported on the top of two vertical walls, 30 feet apart, to which they are at- tached : find the direction and the amount of the reaction at the top of each wall, and the amount of the horizontal force tending to overturn the wall. 248 YI. Dynamics. 1. A and B are two ivory balls whose elasticity is represented Ly seven-eigliths : A'& diameter is 1 inch, and j5's is 1 J inch, and J, moving with a velocity of 5 feet per second, impinges directly upon By moving in the opiK)site direction with a velocity of 2 feet per second : in what direction and with what velocities will they move after impact ? 2. From what height must the ram of a pile-driver weighing 1 4 cwt. descend upon the head of a pile, that it may drive it into the earth six inches; supposing the resistance of the pile to be represented by a pressure of l6 tons. 3. A shot is fired vertically upwards with a velocity of 1610 feet; find to what height it would rise if the air caused no resist- ance to its motion ; in what time it would rise through the fii-st mile of its ascent; and in what time it would again reach the ground. 4. Two weights, -4 = 125oz., i?=S6 oz., are attached to the two ends of a string ; A being i)laced on a horizontal plane, the string passes over a pulley fixed at the edge of the plane so that the string may be parallel to it, and B hangs freely ; find the space described by -4 or ^ in 4 seconds, — 1st, supposing no fric- tion on the plane ; 2nd, supposing the friction to be a fifth of the weight A. 5. Find geometrically the plane of quickest descent : (1) from a given plane to a given point below the plane ; (2) from a given • point without a circle to the circumference of the circle. 6. Deduce expressions for finding the angle of elevation at which a shot is to be fired, and its velocity of projection, so that the shot may pass through two given points of which the co- ordinates are p, q, p^, q^. 7. An invariable pendulum which made 86380 vibrations in a day at one station was found to make 86423 at another. Com- pare the force of gravity at the two stations. 249 F. VII. Differential and Integral Calculus. 1. If V =/{x), z=cfi(x) and u = vz, show that du dv dz dx fix dx' 2. Find -^ when dx 1-x ' ^^ X ' ^(l+a;")' 3. Investigate Maclaurin's Theorem ; and expand sec x, in a series terminating at the term which involves the fifth power of X. 4. Find the greatest cylinder that can be inscribed in a given cone. 5. Show that in any curve, the value of the subtangent is dy^ dx 6. If R be the area of a curve referred to rectangular co- ordinates X and 2/j show that -— = y. clx 7. Deduce the differential expression for the radius of curva- ture of a curve ; and find the co-ordinates of the centre of the circle of curvature. 8. Find the value of u in the following differential equations : du 5ax du hx^ du 2a? — 1 7 Hx ~ J{a^ - ic*) ' dx~~ {a - xf ' dx ~ 3x^-7x-6 ' 9. Find the volume, from x = to x = a, of a solid generated by the revolution, about the axis x, of a curve whose equation is x^+(a + x)y^-a' = 0. 250 a I. Geometry. 1. If a side of a triangle be produced, the exterior angle is equal, show that siD.^A = j-.{s{s~a){s-h)(s- c)}^, s being equal to . 5. The two sides of a triangle being 20 and 30, and their included angle 60", solve the triangle. 6. The angles of elevation of the top of a tower from two stations in the same vertical plane with it are 60" and 30", and the distance between the stations is 120 feet : find the height of the top of the tower above each station, (1) When both stations are in the same horizontal plane; (2) When the station nearest to the tower is 10 feet above the level of the more remote. 7. Investigate the formulae for the areas and jDerimeters of regular polygons of n sides, described in, and about a circle of given radius r ; and apply them to find the perimeters and areas of the regular polygons of 900 sides described in, and about the circle of which the radius is 100 feet. 8. If the circumference of the dial of a clock be 15 feet, what will be the area of the sector passed over by the minute-hand in 12 minutes ? 9. The curved surface of a hemisphere being 157*08 feet, find the area and circumference of its base. 260 IV. Spherical Trigonometry and Astronomy. 1. Show that in a right-angled spherical triangle, the tangent of the angle at the base is equal to the tangent of the perpendicu- lar divided by the sine of the base. And give Napier's rules for the solution of right-angled spherical triangles. 2. State Napier's four analogies, and point out the cases in spherical triangles to the solution of which they are applicable. 3. Show that in any spherical triangle , cos a — cos B cos v cos A = ; — ^— . sm p sin y 4. The three sides of a spherical triangle are 50" 3?', 83° 19', and40M2'j find its three angles. 5. If n = number of square feet in the area of a spherical triangle whose sides are small compared with the radius of the sphere, and B = number of feet in a degree on the surface of the sphere, show that the number of seconds in the spherical excess is SOtt . jr^ I and point out how this may be applied to determine the sum of the errors of the observed angles in such a triangle. 6. Define the following terms : equator of the heavens ; eclip- tic; tropics; oblique ascension; zenith; azimuth; parallax; sidereal day ; solar day ; equation of time. 7. The right ascension and declination of a heavenly body being determined by observation, show how its latitude and longi- tude may be computed. 8. Show how, by observing the Sun's altitude on a given day, at a given hour, the latitude of a place on the Earth is deter- mined ; and also the Sun's azimuth at the time of observation. 9. An eclipse of one of Jupiter's satellites, which occuiTcd at 19^ 52°" 19'-3, Greenwich time, was observed at 11'' 17°" 25'-8: what is the longitude of the place of observation ? 261 H. V. Statics. 1. Define the terms : "force," "composition offerees," "equi- librium," "centre of gravity," "fulcrum," ** moment of a force about a point." 2. If the men on one drag-rope of a gun pull with a force of 400 lbs., those on the other with a force of only 300 lbs., and the directions of the drag-ropes make an angle of SO** with each other ; with what force will the gun be urged forward, and what angles will the direction in which it is urged make with the drag-ropes^ Find also the force that would be gained by the forces on the ropes being equally distributed, and the ropes being parallel. 3. Two weights of 100 lbs. each are suspended by a string passing over two fixed smooth points A, B : find by how much the pressure on the upper point exceeds that on the lower when the line AB makes an angle of 45*' with the vertical. 4. Find the magnitude and position of the resultant of two parallel forces acting on a rigid body. 5. The diameters of the ends of a frustum of a cone being a and 6, and the height of the frustum h ; show that the distance of its centre of gravity from the end of which the diameter = a is h a'+^ab + Sh^ 4' a'+2ab + b' ' 6. A system of pulleys of the second kind, in which each pulley hangs by a separate string, and consisting of four move- able pulleys, has the last pulley hooked to the arm of a screw at the distance of 10 inches from its axis: the distance between the threads of the screw being half an inch, and a power of 30 lbs. being applied at the free end of the string passing round the first pulley, so that the strings being parallel, the force on the arm acts at right angles to it : it is required to find the pressure pro- duced on the head of the screw. 262 H. VI. Dynamics. 1. A body being let fall from the top of a cliff 322 feet high, it is required to find with what velocity a body must be projected vertically upwards from the base of the cliff, that it may meet the falling body half way in its descent. 2. A weight of 84 oz. is connected with another of 77 oz. by a sti-ing hanging over a fixed pulley : how far will the heavier descend, and what velocity will it acquire in 5 seconds, (I) Sup- posing no inertia in the pulley; (2) When the inertia of the pulley is represented by lO^^^oz,? 3. Investigate an expression for the velocity with which a shot must be fired at a given angle of elevation e, that it may strike an object at a given distance r fix)m the point of projection, on a plane passing through that point and making a given angle i with the hoi'izon. 4. With what velocity and at what elevation must a shot V)e fired, that, clearing (1 foot above) a parapet 14 feet above the point of projection and at the horizontal distance of 1800 feet, it may dismount a gun 9 feet below the pai-apet, and at the hori- zontal distance of 1 50 feet beyond it 1 5. A stone being let fall from the top of a cliff, a pendulum 34^ inches in length was observed to make 6 vibrations during the descent of the stone to the sea : determine the cliff's height, the length of the seconds' pendulum being 39\ inches nearly. G. A particle suspended by a string 3 feet long is struck horizontally so as to produce in it a velocity of 12 feet per second; find the arc which the particle will have described when the string begins to slacken, and the highest point to which the I^article will rise. 263 H. YII. Differential Calculus, bhow that — = cos x, and ; = _ sin x. ax ax u being a function of x, find --, when OjX (1 -xy ""= j(x^li)+x ' «=-'«;•»«'■ log.*- x—\ 3. Find the value of x which renders -7, a maximum or a minimum. 4. Find the altitude and diameter of the base of the greatest cone that can be inscribed in a given sphere. 5. Given the base of an inclined plane = 5, to find its height so that the time of a body falling down the plane may be a minimum. 6. ADB is a semicircle of which the diameter AB = '2a', the chord DB is bisected in E and AE joined, intersecting DP, per- pendicular to AB, in M : show that the equation to the locus of the point M is y^(2a + xy = x^ (2a — x), where x — AP, and y = PM, and from this equation determine the value of the subtangent and of the subnormal to a point in the curve which is the locus of the point M. 7. The equation of the cycloid being 2/ = J{2ax — x^)-ira vers"* - , a show that the length of a cycloidal arc is twice the corresponding chord of the generating circle. 8. Find the area, from a? = to ic, of a curve whose equation x^ y-a IS -2 = . or y 264 H. YIII. Integral Calculus. 1. JFind the following integrals : x'dx 2. Find the volume of the solid generated by the revolution of the curve whose equation is y* (a* + x^ + a^x = a*, about the axis Xf between the values a; = and x = a. 3. Find the distance of the centre of gravity of the segment of a sphere, from the extremity of the diameter; and state what this distance is in a hemisphere. Find also the distance of the centre of gravity of the segment of a spherical shell of very small thickness, and deduce the dis- tance for the hemispherical shell, including the plane circular base. 4. Investigate the 1st of Guldin's properties of the centre of gravity. 5. Show by Taylor's theorem, that in all variable motions 6. A body attracted to a centre by a force varying inversely in the sesquiplicate ratio of the distance, begins to fall at the dis- tance a ; what will be its velocity at the distance ^ a, supposing the force at that distance to be equal to g, the force of gravity ? 7. Show that, in a system of material points, the Moment of Inertia with respect to any given axis is equal to the moment about an axis parallel to this and passing through the centre of gravity, plus the moment of the whole system collected in its centre of gravity, about the given axis. 8. The Moment of Inertia of a circle, radius r, about an axis TTT* passing through its centre and perpendicular to its plane being -^- , find the Moment of Inertia of a sphere, radius a, about its dia- meter. 265 J. I. Geometry. 1. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes, &c. 2. The sides about the equal angles of equiangular triangles are proportionals ; and those which are opposite to the equal angles are, &c. 3. If two straight lines be cut by parallel planes, they shall be cut in the same ratio. 4. The triangles formed by joining the corresponding extre- mities of three equal and parallel straight lines, not all in the same plane, are equal, and their planes are parallel. 5. The traces of two planes being given, find the projec- tions of their intersection, when the traces of the given planes intersect in two points. 6. Draw a plane that shall pass through a given point and through a given straight line. Co-ordinate Geometry. 7. Determine the equation to a straight line which is drawn through a point whose co-ordinates are £c^ = 5, y^ = — 2, perpendi- cular to the straight line whose equation is y = 2x + 5, and construct the line by scale. 8. The co-ordinates to the centre of a circle referred to rect- angular axes are a and yS : find the equation to the circle. Also determine the position and magnitude of the circle which is the locus of the equation 2/^ 4- cc'' - 4y + 6x - 1 2 = 0. 9. Give the definition of a " conic section," and state when the conic section is a ^parabola," an "ellipse," or an "hyperbola." 2m II. Arithmetic and Algebra. 1. What sum of money lent at £3. 4«. per cent, per annum, simple interest, will amount to £10000 in 7 J years? 2 5 2. A father left ^ of his property to his eldest son, - of the remainder to his second, and the rest to his third : the difference between the shares of the second and third was c£l800 : what was the share of each 'i 3. Given the sum of three numbers equal to 1 3 ; the sum of the products of every two equal to 47; and the squares of the numbers in arithmetic progression : find the numbers. 4. When the price, per cwt., of tin to that of copper was as 3 : 2, 11 cwt. of gun-metal cost £64. 85., but when the prices of tin and copper were reversed, 11 cwt. of gun-metal cost £S9. 128. : what was the price of copper in the first case, and what the weight of copper and tin in the metal 1 5. Solve the following equations : (1) 1 __i .1 ^ ^ x-J{2-x') x + J{2-x') '• (2) v'« + vz^ = 84 and (v^ - z^) {v-z) = 7. 6. The equation a;' - 11 a;'' -f 7a; + 147 = has two equal roots : find them by means of the derived equation, and the third root by depressing the equation. 7. Decompose the fraction -^ ; into three fractions ^ a;^-2a;--a; + 2 having denominators of the first degree. 8. Investigate an expression for the number of shot in a square pile of n courses, that is, an expression for the sum of n terms of the series 1^ + 2^ 4- 3* n'. 9. By means of a table of logarithms find the value of the 3-14159r4/(-08357 ) (•4753)^ V(5-37462) ' fraction {I'l^^T ^^^51) 267 J. III. Trigonometry and Mensuration. 1. A and JB being any two angles, show that X / ^ T>\ t^n A. =fc tan B tan (A^B)= _ , ^ ^ l=Ftanyl.tan^' «,..! X /A T}\ cotan J cotan J5 T 1 and cotan (^ ± 5) = — — . cotan £ ± cotan A 2. From sin SO" = - and sin 45" = - J2, show that tan 1 5" = 2 - ^3, and tan 75" = 2 + ^^3. 3. Show that, in any triangle, the sides are to each other as the sines of their opposite angles; and apply this to finding the remaining side and angles of a triangle of which two sides are 315*753, 238 '825, and the angle opposite the greater of these two sides is 97" 1 3'. 4. From the top of a hill I observed two church spires exactly in the same direction on the horizontal plane below the hill : the angle of depression of the base of the nearer spire was observed 15" 23', and of the further spire 4" 12': the top of the hill being known to be 315 feet above the horizontal plane on which the churches stand, it is required to find, from these obser- vations, the distance between the two spires. 5. Two sides of a court-yard in the form of a parallelogram are 40 feet and 70 feet, and its shorter diagonal is 50 feet; what will it cost paving at 3s. Qd. a square yard 1 6. The side of an equilateral triangle, of a square and of a regular hexagon are in arithmetic progression; the sum of this progression is 36 feet ; and the perimeters of the three figures are equal; find their areas. 7. The diameter of the horizontal bottom of a circular reser- voir is 200 feet; the sides of the reservoir being inclined all round at an angle of 45", find the number of cubic feet of water contained in it when the depth of the water is 10 feet 268 J. IV. Statics. 1. Define the terras (1) "force," (2) "gravity," (3) "weight," (4) " resultant," and (5) state what is meant by the " parallelo- gram of forces." 2. Assuming the parallelogram of forces, show that if/ and / represent two forces acting at a point, the angle which their directions make with each other, and r their resultant, ^' =/'+// +2#, cos ^: find the resultant of two pressures acting at a point in direc- tions making an angle of 60" with each other, and find also the angle which the resultant makes with the direction of the pressure 7 lbs. 3. The extremities of a string 14 inches long are fixed to two points 10 inches apart, in the same horizontal line, and a weight of 8 lbs. is suspended, 1st, from a knot in the string 6 inches from one extremity ; 2nd, from a smooth ring which slides freely along the string : find in both cases the tensions of each of the parts into which the string is divided, and state clearly the mechanical principles in- volved in each step of the investigation. 4. Define the centre of gravity of a body, or system of bodies; and show that if m, m^, m^, m^ are bodies in a straight line, and a, a^, a^, a.^ are their distances from a fixed point in that line, the distance of their centre of gravity from the same point is 5. If a and h be the two parallel sides of a trapezoid, and h the line which bisects these sides, show that the centre of gravity of the trapezoid will be in this line, and its distance from a along it will be „ . o a + 269 V. Dynamics. 1. A body projected vertically upwards from the bottom of a tower with a velocity of 60 feet per second reaches the top in 2 seconds : what is the height of the tower 1 and how much above the top does the body rise ? 2. A weight of 3 oz. draws a weight of 12 oz. down a plane inclined 30" to the horizon, by means of a string passing over a pulley at the bottom of the plane: find the vertical descent of each of the weights in 3 seconds, the friction and inertia of the pulley being 1 oz., and g=32 feet. 3. Investigate the expression for the velocity of a shot which, being fired at an angle of elevation e, shall strike a mark at the distance r, on a plane passing through the point of projection and making an angle i with the horizon. 4. Show how the value of e is determined from the equation 2v^ cos e sin (e — i) r=- — . Y-' g cos I in terms of r, v, i ; and prove, that the range on a given plane is a maximum when the direction of projection bisects the angle between the plane and the vertical. 5. At what elevation must a shot be fired with a velocity of 400 feet that it may range 2500 yards on a plane which descends at an angle of 30" 1 6. The length of the seconds' pendulum being 39*139 inches, find (1) the value of g the force of gravity, (2) the length of the pendulum which vibrates 80 times in a minute. 7. A clock intended to beat seconds gains 7 '4 seconds a day : how must its pendulum be altered that it may go right 1 270 YI. Hydrostatics. 1. A cylindrical vessel, of which the diameter is 6 inches, and the length of the axis is 1 foot, being filled with water is closed at both ends : find the pressure on the concave surface, and also on one end of the cylinder, (1) When the axis is vertical ; (2) When the axis is horizontal; and compare the whole pressures in the two cases. 2. Define the " centre of pressure ;" and deduce the differen- tial expression for the distance of the centre of pressure of a plane surface immersed in a homogeneous fluid from the surface of the fluid. 3. A sluice-gate in the form of an isosceles triangle, of which the altitude is 4 feet and base 5 feet, the vertical angle being downwards, turns upon a horizontal axis parallel to the base, the part below the axis being prevented turning in the direction of the pressure, but the upper part being free to turn : at what dis- tance from the top of the gate must the axis be placed, that the gate may open, when the water rises more than 8 feet above its top? 4. Some silver being alloyed with 100 grains of copper of the specific gravity 8*79, the alloy was found to weigh 1150 grains in air and 1034*5 grains in water : what was the specific gravity of the silver employed ? 5. The whole length of a cylindrical pontoon with hemi- spherical ends is 23 feet; its diameter is 3 feet; and its weight, with the portion of the bridge it supports, is l660 : what additional weight does it bear when the axis of the cylinder is on the level of the surface of the water ? 6. Describe Nicholson's hydrometer, and point out the man- ner in which it is applied for the determination of the specific gravities both of solids and of fluids. 271 J. YII. Differential Calculus. 1. u beinsf a function of x, find -7- , when ° ax 2. Show tliat when w, a function of x, is a maximum, ^=0, and^xsnegaUre. 3. Find the value of x which renders — — 5 a maximum or a 1 +a;'' minimum. 4. Find the altitude and the radius of the base of the least cone that can circumscribe a given sphere whose radius is r. 5. Find the subtangent to a point in the curve whose equa- tion is ai/^ ={a + xy{2a - x). Integral Calculus. 6. Find the following integrals : r 7xdx C{3x + 9)dx f x^dx f dx J 3{a' + x'f' J af + x-2 ' J J{a'-of)' J (^a'-x^' 7. Show that the area of a curve between the curve and the ydx. 8. Find the volume, from x = a to a; = 2a, of the solid gene- rated by the revolution about the axis a;, of the curve whose equation is xy^ = {a + xy{2a - x). 272 K. I. Geometry. 1. In a circle, the angle in a semicircle is ^ = T^ Tn^ w = log,— — ... 3. Find the values of x which render x* - 12x^ + 52x^ - g6x a maximum or a minimum. 4. Find the following integrals : j 10x\ XI {a^ - xjdx ; { x\ J (a' + x') dx ; f {5x + 3)dx ^ r J a;' + 2x -3 ' J \ dx 5. Find the area of the Cissoid of Diodes, between the values x = and x = -a. 6. Show that if V be the volume of a solid of revolution dV dTx^'^y- 7. Give the general expression for the distance of the centre of gravity of a mass from the axis of y ; and apply this to finding the distance of the centre of gravity of any segment of a si^here, and also of a hemisphere, from the centre of the sphere. 8. State Guldin's properties of the centre of gravity : and, by this method, find the volume of the solid formed by the revolu- tion of a parabola whose altitude is a and base 2b; 1st about a tangent at its vertex; 2nd about its base. 286 M. I. Geometry. 1. Show that in any triangle, if straight lines be drawn from each of the angles to the middle of the opposite side, four times the sum of the squares of these lines is equal to three times the sum of the squai-es of the sides of the triangle. 2. ADB is a semicircle of which the centre is C, and A EG is another semicircle on the diameter AC ; AT is 2i, common tan- gent to the two semicircles at the point A : show that if from any point Fj in the circumference of the first, a straight line FG is drawn to C, the part FG, cut off by the second semicircle, is equal to the perpendicular FH on the tangent. 3. Equal triangles, which have an angle in the one equal to an angle in the other, have their sides about the equal angles, s, jd per acre : how- much per cent, did he gain on the purchase-money? 2. Divide £16815 among three persons A, B, C, so that ^*s share shall be to ^'s as 3 to 7, and £'s to Cs as 6 to 5 ; without using Algebra. 3. Given Jx- ,Jy = 9. and {x + y) Jxy = 510; determine all the values of a; and y which satisfy these equations. 4. The number of courses in an incomplete square pile is equal to the number of courses wanting to complete the pile, and the number of shot in the incomj^lete pile is equal to 6 times the number of shot wanting to complete it : find the number of shot in the incomplete pile. 5. Find the greatest root of the equation, x^ — 3x^ — 2a; + 1 =0, and the integer limits of the other two roots. 6. The sum of three numbers in geometric progression is 21, 7 and the sum of their reciprocals is — : find the numbers. 30 7. Decompose the fraction -3 ^ — 77- into three fractions X — X — ox having denominators of the first degree. 8. Show that a' =1 + Ax + — — + — '— + &c. 1.2 1.2.3 where A=a — l--(a-iy + --(a-iy~&c. 9. Show that, if y and y^ are any two numbers, ^oga {yy) = loga2/ + log« 2/^, and log„ ^ = log„ y - log„y^ ; also, n being any number whole or fractional, log,, y'' = n log^ y. 288 M. III. Trigonometry and Mensuration. 1. Sin 30" being |, determine the values of sin 15°, cos 15". 2. Show than tan (^ *^)== , ^"^"^"^ „ ■ * ^ 'It tan A . tan B .1 X sin ^ + sin 5 tan X(A-B) 3. Show that ^- ^-j^ = i fyi »;. sin ^ - sin jtf tan ^(A + £) 4. In a plane triangle, show that a sin A a sin -4 6 sin i5 b sin B* c sin (7 ' c~ sin 0* * state to what case in the solution of plane triangles this is appli- cable, and point out when the solution is ambiguous and when not. 5. From the top of a tower, 60 feet high, standing near the edge of a cliff, the angle of depression of a ship's hull was observed 15" 19' 36", and from the base of the tower, the angle of depression of the ship's hull was 13" 5' 20": find the horizontal distance of the ship from the tower, and the height of the tower's base above the level of the sea. 6. Simultaneous obsei-vations at two stations, A and B, in the same vertical plane as a balloon, gave the angles of elevation of the bottom of the car 13" 59' 40", and 21" 00' 12" : the station A was 6810 feet further from the balloon, in horizontal distance, than B, and 300 feet above the level of B : find the vertical height of the car above the level of B and its horizontal distance from that station. 7. The three sides of a triangle are 6, 6 + J2, 6- J2; find its area. 8. Show that the volume of the frustum of a cone whose altitude is a, and the radii of whose base and upper surface are B and r, is ^iraiE' + Er + ry 9. ABC is a cone whose altitude AJD = 4 inches, and dia- meter of its base BG = 6 inches; and EFGK is a cylinder in- scribed in it. The convex surface of the cylinder is one-fourth that of the cone : find the dimensions of the cylinder. 289 M. TV. Statics. 1. Show that, if from any point Z, in the direction of the re- sultant Ji, of two forces F and Q, perpendiculars, LM, ZxV, be let fall upon the directions of the forces F and Q, then F : Q :: LN \ LM. 2. The ends of a thread, \Q inches long, are fixed to pins at the points A and i?, 14 inches apart, and at the point G in it, at the distance of 6 inches from -4, a force of 30 lbs. is applied by- means of another thread attached there, and so that, when the threads are stretched, the angle AGD is a right angle : find the strain upon the pins A and B^ that is, the tensions of the strings CA and GB. 3. An equilateral triangle has a square described on one of its sides : find the position of the centre of gravity of the whole figure. 4. Show how the resultant of any number of forces, acting in the same plane, is determined, by resolving the forces in the direc- tions of rectangular axes ; and apply this to finding the resultant of three forces, A = 3, B = 4!, G = 2, J) = 6, and the angle which its direction makes with that of the force -4, the angles which the directions of the forces By G, D make with that of A being 30", 45*, 60° respectively. 5. Suppose that the vertical pressure of a mortar on a prop placed under a particular part of its bed would be 9 cwt. ; that a 6-feet handspike is placed under the bed so that this point rests upon the handspike at the distance of 6 inches from its lower end, which rests on the platform ; and that the handspike is inclined to the horizon at an angle of 30* : with what force must a man press perpendicularly to the handspike, at its upper extremity, so as just to sustain the vertical pressure of the mortar, the coefficient of friction being '5. a 19 290 M. V. Dynamics. 1. Two inelastic bodies, A and J?, move in the same direction with uniform velocities a and h (a greater than b), show that after the impact of -4 on jB they will move with a common velo- Aa + Bh city equal to — — ^ , 2. Two ivory spheres, 3 inches and 4 inches in diameter, move in opposite directions with uniform velocities, 25 feet and 10 feet per second; find their motions after impact, the elasticity of ivory being represented by the fraction f. 3. To what height would a shot rise if fired vertically with a velocity of 1200 feet per second, and in what time would it again reach the ground, abstracting the resistance of the air 1 4. A body, A^ resting upon a plane inclined at an angle of 30" to the horizon, is connected with an equal body B by means of a string passing over a fixed pulley at the top of the plane, B hang- ing freely : in what time will A describe 100 feet on the plane ? (1) Supposing no friction on the plane. (2) Supposing the friction to be one-tenth of the pressure on the plane. 5. Find the straight line of quickest descent from a given point without a given circle to the circle. 6. The length of the pendulum vibrating seconds (in a vacuum) in the latitude of London having been found to be 39*139 inches, determine the value of g which represents the force of gravity. 7. If a clock constructed to beat seconds gains 1 minute in a day, how, and to what extent, must its pendulum be altered ? 8. Show how the velocity and angle of elevation may be determined, that a shot may pass through two given points of which the co-ordinates are 2?, q and^^, q^. 291 M. VI. Differential Calculus. 1. u being a function of a?, find -j- when 1-a; ' _ ja' + x')^ 2. Find the values of x which render («'' + lOo; + 1 1) (7 - x)" a maximum or a minimum. 3. Of all cylindrical pontoons terminated by equilateral cones, having the same given surface = a', to find that which has the greatest volume. 4. Find the value of the subtangent in a curve whose equa- tion is y^x' = h{a'-xy. Integral Calculus. 5. Find the following integrals : f 6x*dx f 3dx ' [ 2 f/-, 2\j 6. Find the area of the cur^'^e whose equation is a V = y^ (a^ + x^), from x = to x=-a, 7. In what time would a body let fall from a point at a dis- tance from the earth's surface equal to its radius, reach the earth, and with what velocity would it strike ; the force of gravity varying inversely as the square of the distance from the earth's centre? 8. Give the differential expression for the distance of the centre of gravity of any body from the axis 2/ ; and point out what this becomes for an area, and also for a solid of revolution about jbhe axis x. 9. Find the distance of the centre of gravity of a parabola, and also of a paraboloid, from the vertex. 19—2 292 I. Geometry. 1. Bisect a triangle by a line drawn from a giren point in one of its sides. 2. In a circle, tlie angle in a semicircle is a right angle; the angle in a segment gi-eater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle. 3. Triangles, and also parallelograms, having the same alti- tude, are to one another, &c. 4. If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall be at right angles to the plane in which they are. CO-ORDIKATE GEOMETRY. 5. Find the equation to a straight line passing through two given points, the rectangular coordinates to which are respec- tively x' = 3, y = 5, x" = 7, y" = ^ } and find the distance from the origin at which this line cuts the axes, and also the angle which it makes with the axis of a;. ' 6. The equation to a given straight line being y = 3x — 4f, find the length of the perpendicular let fall on it from a point whose co-ordinates are ic' = 2, y=5; and represent correctly the positions of the given line and the given point. 7. Find the equation to the circle whose radius is r, and the co-ordinates to its centre a/, y ; and thence deduce the values of y corresponding to a; =: 7, in the circle whose radius is 5 and co-ordi- nates to its centre a/ = 9, y' = 2. 8. In the parabola, if a perpendicular on the tangent at any point be drawn from the focus, the tangent at the vertex will pass through the point of intersection. 9. In an ellipse show that the square of the transverse axis is to the square of the conjugate, as the rectangle of the abscissa; is to the square of the ordinate. 293 N. ' II. Arithmetic and Algebra. 1. A father left f of his property to one son, f of the re- mainder to a second, and the rest to a third : the difference between the shares of the second and third was £1200: what was the share of each ? {Arithmetically.) 2. Solve the following equations.: (1) x^ + 2 J{x''-ax) = 2{ax+J(ax)}. (2) x + 2/ = ax^f and x^ + 9/^ = Ifx^'if, (3) 7'''' =r 2 + Ti applying logarithms. 3. Form an equation of which the roots are 5 + ^S^ 5 — JS, 2 + ^(-l), 2-V(-l)and-S. 4. Find the least positive root of the equation a' - lOcc* + SSx - 20 = 0, by Homer's method, and state the character of the other roots. 5. The first term of a geometric series is 5 and the ratio 2 : how many terms of this series must be taken, that their sum may be equal to 33 times the sum of half that number of terms ] 6. Show, by the method of indeterminate coefficients, that the number of shot in a square pile, of which n is the number of courses, is (2n+l).(n+ l).w 3.2.1 * 7. Assuming that the coefficient of the second term of the expansion of (1 + x)" is n, whatever n may be, prove the Binomial Theorem. 8. Given the sum of two sides of a triangle = 5, the perpendi- cular from the vertical angle on the base = «, and the diameter of the circumscribing circle = d : find all the sides of the triangle. 9. In a triangle, given the perpendicular from the vertical angle on the base = o^, the base = 2&, and the difference of the other two sides = 2c?; to find those sides. 294 N. III. Trigonometry and Mensuration. 1. Find the value of cot (A * B) in terms of cot A and cot B, 2. Given tan d = 2 sin ^ ; find 9, 3. In any triangle, a, 6, c being the sides opposite to the angles A, B, G, show that a + h tsin^{A+B) a-%~tB,ni(A--B)' and point out how this is applied to the solution of the case where two sides and the included angle are given. 4. a, 6, c being the sides of a triangle opposite to the angles A, B, C, show that cos B = — ; and deduce from this the value of cos ^ B, 5. If the exterior face of a front of fortification is 360 yards, the face of each bastion 100 yards, the angles which the lines of defence make with the exterior face each 16°, and the fianks are perpendicular to the lines of defence, — what is the length of the flank and of the curtain 1 6. Investigate the formula for the area of a triangle when its three sides are given ; and find the area of a triangle of which the three sides are 4/, 4> + JS and 4 — J3. 7. How many square feet are in the floor of a room, which is a regular octagon having each diagonal l6 feet ] 8. Deduce an expression for the surface of a square pyramid of which the altitude is a, and a side of the base is b. 9. If the depth of the ditch of a field-work is 10 feet, and its width at the bottom 8 feet less than at the top, what must be the width at the top that the ditch may furnish earth for tlie work the area of whose vertical section is 143 feet, allowing an increase in area of a tenth in the earth thrown out, when thus applied i 295 N. IV. Spherical Trigonometry. 1. Show that in a spherical triangle ABCy right-angled at B^ a, ft 7 being the sides opposite the angles AyByC^ . . sin a tan a sin ^ = — — J. , and tan A - ~ ; sin yS sm y ^ and state Napier's Rules for the solution of right-angled spherical triangles. 2. Show that, in any spherical triangle, . cos a — cos i8 cos V cos^ = . Q . '-, sm p sm y 3. Show that, in any spherical triangle, if cr = - (a + ^ + y), At «;n« ^ >< _ sin(o— y)sin((r-ff) ^ 1 _ sin o- sin (cr - a) sm -—JL -. j^—. — — . cos —-a.— : 5 — ; . 2 sm/jsmy 2 sinpsmy Astronomy. 4. Define the following terms : Meridian, Prime Vertical ; Right Ascension, Declination, Latitude, Longitude of a heavenly body; Equator of the Earth, Latitude and Longitude of a Place on the Earth. 5. Show, by a figure, how the obliquity of the ecliptic is determined from the Sun's meridian altitudes at the summer and winter solstices. 6. Given the latitude of the place and the Sun's declination, show by a figure how the length of the day, and the Sun's azimuth at rising or setting, may be determined. 7. Show how the hour lines are determined in a vertical south dial (in north latitude), and also in a vertical east or west dial. 8. Show how the latitude of a place is determined from two observed altitudes of the Sun on a given day, and the observed in- terval of time between the observations. 296 N. V. Statics. 1. A force represented by 100 lbs. acting in a direction which makes an angle of 60" with the horizon, is counteracted by two forces making angles of 30° and 120" with the horizon : find the weights which represent these forces. 2. Four equal bodies are placed in the angular points of the four solid angles of a triangular pyramid, show that their centre of gi'avity coincides with that of the pyramid. 3. A handspike is placed under the breach of a heavy gun, and rests on its carriage to support it for elevating : what force must be applied at the further end of the handspike, supposing its whole length to be 5 feet 6 inches ; the distance of the line of support on the carriage cheek from the end on which the gun rests, 9 inches; the distance of the part of the breech of the gun resting on the handspike from the axis of the trunnions 3 feet ; the distance of the centre of gravity of the gun from the same axis 6 inches ; and the weight of the gun 28 cwt. 2 qrs. 1 4. In a system of pulleys in which the stiings are parallel, and each string is attached to the weight, there being 4 pulleys in the system : show that, when the power P is in equilibrium with the weight TT, (1) Neglecting the weight of the pulleys, W= (2* - 1) P, (2) If A is the weight of each pulley, W= (2" - 1) (P + ^) - 4^. 5. A power of 50 lbs. is applied at the extremity of an arm, 15 inches long, to a screw in which the distance between the threads is a third of an inch : what is the pressure exerted at the head of the screw, in the direction of the axis ? 6. A 13-inch shell weighing 200 lbs. rests between two planes AB, BC inclined at angles of 30° and 45° to the horizon : find the pressure on each plane. 297 N. yi. Dynamics. 1. A glass sphere half an inch in diameter, moving with a velocity of 24 feet per second, impinges directly on another sphere of glass 1 inch in diameter and moving in the same direction with a velocity of 6 feet per second : in what directions and with what velocities will the bodies move after impact, the elasticity of glass being ^? 2. A body is projected vertically upwards with a velocity of 483 feet per second, in what time will it rise through 1610 feet? 3. Two bodies A and B, each weighing 5 lbs. avoirdupois, are connected by a string passing over a fixed pulley : what space will they describe in 10 seconds when an ounce weight is added to A, and what velocity will they have at the end of that time 1 4. Find geometrically the straight line of quickest descent ; (1) From a given straight line to a given point. (2) From a given point within a circle to the circumference. (3) Between the circumferences of two given circles, the one being wholly within the other. 5. Investigate the expression for the range of a projectile on a given plane passing through the point of projection, in terms of V, the velocity of projection ; e, the angle of elevation of the pro- jectile ; and i, the inclination of the plane of the horizon. 6. In Atwood's machine, a weight of 22 oz. being placed on one side, and 23 oz. on the other, in what time will the heavier descend 64 inches, and what velocity will it have acquired, sup- posing the inertia of the wheels to be 3 oz. and ^ = 32 feet 1 7. A particle is allowed to roll down the exterior of a para- bola whose axis is horizontal ; find the point where it will quit the curve. YIL Hydrostatics. 1. If TTbe the weight of a body in air, and w its weight in W water, show that its specific gravity =-rp ; and find the specific gravity of a piece of metal which weighs lOa-T^* grains in air, and 97 '29 grains in water. 2. The whole length of a cylindrical pontoon with hemi- spherical ends is 22 feet ; its diameter is 2 feet 8 inches ; and its weight with the portion of the bridge it supports is 1550 lbs. : what additional weight does it bear when the axis of the cylinder is on the level of the surface of the water 1 3. "What weight of fir-wood, specific gravity 0-57, must be attached to a piece of copper, specific gravity 8*79> a^^l weighing 10 oz., that the mass may just float ] 4. A prismatic diving-bell, of which the height inside is 8 feet, is sunk in the sea to the depth of 70 feet j find the height to which the water will rise inside the bell, and the density of the included air compared to that at the surface of the sea where the pressure of the atmosphere is 33 feet of sea water. 5. Describe the principle of the barometer, and how this instrument is applied to the determination of the difierence in the level of points on the earth's surface. 6. Describe the action of the Fire Engine. 7. Describe the action of the Bramah Press. 299 VIII. Differential Calculus. 1. Find the differentials of the following functions of x : — ■ ^ {d'-x') x~a u — log. , w = sin £C . cos x, ^'x + a' 2. If w is a function of x and u^ represents u when in it x becomes x + h, what is the value of u^ (Taylor's Theorem) ? 3. From the expression for the range, r = — • sA • ^ q cos * find the value of e which gives r a maximum on a given plane whose inclination is i, the shot being fired with a given velocity v. 4. Find the value of the subtangent in the ellipse. Integral Calculus. 5. Find the following integrals : x'^dx f x^dx . . I dx , . f 01? i 6. Find the area of a curve whose equation is ax = yj (a" — x^)y from x=0 to x = a. 7. Find the area of an elliptic segment in terms of a circular segment radius = a (the semi-axis major), and having the same abscissa x, 8. Find the volume of a paraboloid; and show that it is equal to half the circumscribing cylinder. 9. Find the volume of a ring in the form of a double cycloid, and having an elliptical section. ANSWERS. ANSWERS. 1. 67. 1, 317 generations and 2fif years. 2. 35. 3. ^gOO. 4. llhrs. 59 m. 26'S4s. 6. 10000. 6. .£98959.85. 5'17d. 7. 18. a 27*5. 9. 60. 10. 2000. 11. 15. 12. £85. 17s. 2ld. 13. £119. Il5. 5-lM 14. 75cwt. nitre, 12*5 cwt. charcoal, 12*5 cwfc. sulphur. 15. Sfin. 16. 98. 17. Shrs. 36 m. 18. 6ifg?-days. 19. 1016. 20. 9hrs. 49 m. 5i\s. 21. (1) 3hrs. 16 m. 21-82 s. 22. One twenty-fourth. (2) 3hrs. 49 m. 5*45 s. (3) 3hrs. 32 m. 43-64 s. 23. 24854-85. 24. Excess of French gun 1 cwt. 47*4 lbs, 25. Sdays8hrs. 26. 4100 days. 27. HI- 28. 3*84 fr. 29. 6hrs. 40m. 30. 12,22; Tgalls. 31. '880339 quart. 32- ^- 33- ^- 34. -018374. 35. -15. 36. i. 37. 22 shillings. .38.^. 39."^. 40. |. 15 103 41. Tr- 42. •858333*crown, --^ Napoleon, lo 384 43. «£54. Us. 9M- 44. £100. 45. To ^ 4 cwt. 2 qrs. 18 lbs. lOfoz. To^lScwt. Iqr. 16 lbs. 12^02. 46. £16000. 47. ^. 48. is. U. 49. .£ioo. 50. .£100. 51. 1418464-5125. 52. -163057. 53. 276-824165. 54. -049261 ; 00027004. 55. 2736-577; 63-216. 56. 9655-651. 57. 848-3968. 58. Sixth place of decimals. 59. 27*69957. 60. -000000009.976. 61. jy. 62. j^. 203 Q 63. ip.j 45. 8d.', lAinch. 64. -06568 = -^ X ^. 65. •00138251. 66.3^(jX^. 67. 23-11422; 8-114334. 304 ANSWERS. 68. 119. 68. -0021814. 69. 1-4142135; 1-259921. 70. 1-79256. 71. 2-571281599. 72. 30365889721. 73. 1-912931182. 74. 10921227. 75. 10000. 76. 12000. 77. 7^. 78. £702.93. 79. 51 square feet, 11 rectangles of 1 foot x 1 in. and 4 square in. 80. 5 sq. ft. 7 (ft. X in.) 1 sq. in. 6 (in. x line) 8 sq. lines. 81. 30-479. 82. 57 sq. feet, 2 (ft. x in.) 2 sq. in. 3 (in. x line) 1 1 sq. lines. 83. Iy8096021930. 84. 112344424214; 30332. 85. 46530163. 86. 131102101. 87. 155047. 88. 245. 89. -49825373. 90. 28. 91. (1)-^; (2)5; (3)f; (4) 0; (5) c». 92. 2-707. 93. Hh^l)(h^J){Sb^S)^ ^^ .^^333^^ ^^ ^.^^33^ o 96. a"; J«; 3".a«; dk 97. a.j^^x,/ ' 98. square both sides. 99. square both sides. 100. ii^^ 101.^??^. m.s-.j. 105.E.at„.ia,.esUon; <:;g;g-^:^' . Ans. X* + xy^, 4/(^2/") - 2a; V • H^^V)' 106. 2 "—. 107. (^-y)^(^'-/). 108. f^. m^T-f- 110.2^^:1^. 111.4^^ 112.^*. 113. 2x'. 114. i. 115.^. 4y^ iC ^T^r 11ft ^^ + ^" 117 ga^+y 11Q ^ + « ^^0' a-6* •^^'* 4a;» + 2a:y-2/"* ■^'^°' a;^+4aa; - 5a« ' 119. Question should be { -y(^'^) " V W}' ^ ^ ^ ^^^^ ^.^^ l-3a"»6*+3a"»6^-a"^6' ANSWERS. 305 120. 156. 120. ^' . 121. fill^. 122. ^ V{a - 6 V{- 1)}. 123. 1 - V(« + ^)' 124. 0. 125. ^ V(«^^ - f). x + y 126. 2-. 127. 1. 129. x'^-xSf + fx'^-y 128, y 130. -T^. 131. 2.^]^ 132. 2. 133. A * y--x^ 134. l+v/(l-^0 S8a;' 62a; 135. 21a;-'- 1010;'^+ 10a3'+136x*+12a;-27; 7«;'-^-^-42; remainder -490a; -279. 136. fk + el-cm- In. 137. v/(^''' - /")• 138. («' - ^^'f^^- 139. 4a'6' - 2a'6' - \^a^h' + 7«&' + 125« - 6a-^6^ 140. 2. 141, a'a;^-3a^6%y^ + Sa'5a;'V^-a'6^2/^. 142. \ y'z - 6y^^ - 1 ^^ 143. c«^- 1 xK i. 144. « - f «<^+ 1. c(a + 5 + c) + a6| /(c {G — a — h) + ah^ ~~ 2 J "^ V 1 2 J • 146. V3 . (V2 - 1) ; ^^' - ^ = 1. 147. ^3 . (^2 - 1). 148. \ . ^'20 . (V3 - ] ). 149. J {55) - 1. 150. 5 - 3 V2. 151. V5 + a/2. 152. 3 + 2^3. 153. 1 l + {a'-^h') -{a^-h') ^{a'+2a'h' + a'b') + {a' + b') + {a' + b') + (2a* + a'W - b') + {a' + 2a'b' + a%') .«. J{- l+{2a'-b'} +{a* + a'b') Quotient a;* + (2a' - b') x' + a' + a%\ 154. 6 (a; + 2/) -20. ^KK 6a ,^ 21057/ 120a;V 135a;V ^^^x\f , 155. —-10 + —-^ -^+ 5-^^^ ^+&c. xy 156. "7 r ^~* + -^ ^"^ + a;~* - 1 5a:~' - 2aj~" + &c» 4 4 C. 20 306 157. ANSWEr.S. 169 157. 1 + 7 -3 + 2 1-3-31+25+ 3-15- 8 + 19 + 3 + 10 + 7 + 28-21+ 7± + 14-(21)=fc(0)-(2l) + (14) + (56) rfc ± ± ± ± ± ± (0) ± (0) ± (D) ± (0) - 3-12+ 9- 3± 0-6+ (9)± (0)+ (9) + 2+8-6+ 2 db + 4 - (6)± (0) 1+4- 3+ l± 0+ 2- 3± 0-3+ 2 + 8 + 65 The coefficients in brackets are not used in determining the final remainder. ^ , ^ x" 4^x* 3 jc* 2 Quotient J + -g-- ^ -^ -3 + g' " Final remainder " - 3x' + 2 1 a:* - 3x + 1 4. Continuation of Quotient 3+7r-i+Q~5+ -^ + ^^' X X ' + ^ ; 2a' - 3a«6-' - 4a'6-'. 163. «'«' - 5xV - 1 la;y«' - 10a;y«' - 40xy*z' - 55y V ; 46la:yV-3502/V. 164. a'6~' + Sa'b-' - 5a*b-* + 7a'6~' - ^ct'b-' + 4a6-' + 6 ; 7a"'6'-4a-'6«+5a-'*6^ 165. a; V + s^V"" - sA"^ + 6j;V' + ^^^"'i 22x- V' + 2 la;-y - 6a;-y. 166. sc-'z'+ 2x-Y'^^+ 3?/- V- 4>xy-^z'+ 5xY*z'+ 6a;y V-7^y "^" i -10a;VV-2a;"2/-V. 167. 6a;-y + 4a;-y-2a;-y+7?/' + 9a;/-8a;V; - 6?/-^ + 4a!y-*. 168. 8a-.«Z;«- 7a"''6'+ 6a-T- 5a-'6'+4a-*6*-3a-'6'+2a-V-a-'6. 169. aa^y +3a;y-2a-'a;y +5a-V2/'+ 6a-Vy-4a-V+a"V V ; 5a- V - Sa-'xhj-' + 6a- V^"' + a" Vy-^ ANSWERS. 170. 307 193. 170. 1 - 2x-* + 9a;-' - 23x~^ + x^+^x-3- X-' -3 1 + 2 , ^ ^ ^o 71-60-23 ,, ,^ „ = 1-2 + 9-23 + ^ — z — :7-^=-15 + 12 = -3. 1+2-3-1 1+2-3-1 Arrange the work thus : -5 5-4±0=fc0-l *5 ± + 25 - 30 1_1±0±0-1+1 1 1 5 25 • -4-25 + 30-1 -4± 0-20 + 24 I 4 --^.0=.0--+l 14 ^1 --+ — =fcO±0 + — : 5 25 25 1 -25 + 50-25 or - 1 + 2 - 1 t-1+1 4 ^ 4 24 ±0 + — 25 5 25 or*-l±0-5 + 6 -1+2-1 1 + 2 -1 + 1-1 + 1-1 -2-4+6 -2+4-2 -8 + 8 or - 1 + It 172. ^ax~3y. 1286 . •. a; — 1 is the G. c. m. 173. x^ + ax''-5a'x + 3a\ 175. 889 ^w/v 653 179.0. = — ; y. 182. 3 (1 ± J3). 176. 35. 1102 59 180, 177. 19. 411 662 183. 4; 1. 185. 188. 5±J417. 5±V273 186. 184. _4a^ (1 + a) 174.. = ^. 181. f;i. 187. 4; 3. ,.y^.±^^ 189. WM::MI. 190. -3; 2. 191. * -^ {1 - Ji' 3)}. 192. ^5;^3J5. 193. 0; . v/3 20-2 308 ANSWERS. 194. 220. 194. 9; 16; 'J±^, 195. _ |(i . ^5). 196. "^i- • 197. 3; -f ; 8i^|>/27. igg. -3; -5. 199. 2; ^-6. 200. y^^. 201. a; i. 202. 8; ^^,. 203. 5; 2. 204. a! = l-V5;y=V5-l. 206. a; = 4or^; 2' = ||'"''- „«„ 653 1102 ^^^ 206. 'K=-5g ; 2' = ~59~" 207. a;=ll or 5; y = 5orll. 208 ..-.i^,,,^.^f. 209. a; = Yor4; 2/ = ^ or 8. 210. 35 = 4 or- 1 ±75; 7j = l or 1^J5. 212. a;=3i 2 or ^ ; y = 2; 3 or ^ 214. aj=-; 2^ = -. 2(^ + 5) + ! 3faV(4c)^ + l) . 2(a-h) + l^J{^a+l) 01 c — 7> ah + ac — hc _ , a6 + 6c — ac /w XS /«• 218. a3=8; 2/ = 5. ^ 35' ^ V{35 (4a« - 6°)} . - 36- ^^ ^{35 (4^^ - 6°)} 220. In question read x^ + y^; x = 6; y = 3 or - 2. ANSWERS. 309 221. 248. 221. .= 5, 2 or l^Zm, ^ = 2, 5 or Whi2i). 223. a; = ±2; y = ±2. 224. a; = =t5, ±3; y = ±3, ±5. 225. »J=J(V8«+ 756) ov\n{~a)+'J{-b)}; y=\ C^Sa- y55) or 1 {y{- a)- ^(-i)}. 226. a;=4or|; y = 2or-i. 227.«. = 9orlii±^^ti9), y = 3or -^^-/(-^9> . 228. aJ=4; 2/=l. 229. x=6', y=4. 230. aJ-3, 2, -2, -3; y = 2, 3, -3, -2. 231.0^ = 3,2, _^, -?, ^^=^^3, -3=tV3; y«2, 3, -^, -^, |=p^3, -3:^^^. 232. ^ = -^^-3' ^^^J3' ^^^- "^"T^ ^"T- 234. a? = 49 or 25 ; 3/ = 25 or 49. 235 x.^ ""^ ^ V{(a + 5-a&)V4a5} ^ y=.^LZ^ , a+b a+b a+b' 236. aj = 3V2; y = j2. 237. a; = l; ^ = ^2- 238. x^^2, ±i; y = =bl, =p^_. 239. a; = ±l, ±2; 2/ = =^2, ±1. 240. x = 64>; i/=8. 241. aJ = 2, -^; 2/ = 3, -1. 242. «J = 5, -3; y = 3, -5. 243. aJ = ±9; 2^ = ±4. 244. x = ^3, ±1; y = ±l, ±3. 245. x = 2; 2/ = l; « = 4. 246. x = l; 2/ = 2; ;2; = 4. 247. a; = 10j y = 5; z = 3. 310 ANSWERS. 249. 284, (c^a^ - c^a^) {\a^ - b^a^) - {c^a^ - cji^) {b^a^ - b^a^) ' 250, x = -] y^-^; z = -. 251. x = -5; y = 7; z=2. _ {ce - hf) (b^ + a c)-{bd + ae)(c^+ab) 252. « = (? + a6)(aV6c)-(6' + ac)« * 253. a5 = 6j y=12; ^ = 60. (6^ - ctQ (ac ^b')- (cl - 6m) {be - a*) 254. y- ^f,^_ ac) (ac - b') - {be - c') {c' - ab) * 255. x = \', y=2; z=3. 256. a;-lj 2/ = 2; 2;=4. 257. x = \', 2/ = 2; »=5. 258. x = 2; y = l; z = S. 259. a; = 30or^; y = =t 29 or 1 ^(^626) ; « = 36 or-^. ««^ a + c?-56 + 7c a + b- 5c + 7d 260. ^ = :j -; x = ; y 4 ' 4 b + c- 5d-^la c + d-5a + 7b z = 4 ' 4 261. 32; 40. 262. 6cwt.; 9cwt.; 8 cwt. 263. 4lirs. 30min. a.m.; 5lirs. a.m. 264. ^1- 25. 6c?. 265. 7lirs. 30min.; 150 miles. 266. 12. 267. 400. 268. 530gr. 490gr. 269. 24 days; 48 days. ^70. 131-25. 271. 1800. 272. 1296. 273. £2. los.; 135.40?. 274. 18/jl ; a + d = mh + d Q.nd.h + c = b+md (mh + d)-(b — md) = (m— 1) (6 — d) which must be +, since m>l and b>dj .\ a + d>b + c. 354. Express in front of ordinary train ^ — 15. 355. 25 and 100. 356. x — y-yX and x + y are in Arith. Progression; if they be also in Harm. Progression, we must have x—y x+y X .*. a -y; X and ar+ 2/ are identical. Also = ; .-. x — y,x and x + y would be in Greom. X x + y Progression. 357 ^-^=--- a-b _b — c 1 , 1 c~b + a , 1 1 a + c-b a c-o a(c-b) c a-b c(a-b)' .*.- + - + — j+ j. = fa + c-b) [—. iT + -7 n} = 0; a c c-b a-b ^ ^\a(c-b) c(a-b)) ' since c{a-b) = a{b -c). ANSWERS. 313 358. 403. 358. See Appendix. 359. Divide by (v + -5) tlms : 111- 5 ~3 ±0 -5 +2 -•5 - -5 + 2-75 + 0-125 -00625 + 2-53125 1 - 5-5 - 0-25 + 0-125 - 5-0625 + 4-53125 Hence the value of the given expression is 4-53125. 360. 158j 169; 56-92897104; 28-57983059. 361. -4161. 362. 2; 00. 363. 11*60768. 364. -l -1567768576. 365. 8*82489. 366. 10-5712248832. 367. - 12. 368. See Appendix. 369. See Appendix. 370. x' - 6x' + 6ic- + 34a; - 1 95 = 0. 371. - 1 3x\ 372. - ; c and - ad. 373. + 444a;. 374. 70. . 375. a;* -5x^-1 la;-' + 149a; - 230 = 0. 376. + 134a;. 377. 3 and - 4. 378. 4 and - 7. 379. 6 and - 2. 381. 3 and 2. 382. -14aj\ 383. See Appendix. 384. 1 ; 2 ; 3; - 1 ; - 3. 385. 3 is a root. 386. 2±V-5 and2±;y5. 387. n = 4<. 388. See Appendix. 389. ^ + 1 each equal to ^ ; 6 + 1 each equal to q; and so on. 390. Three, each equal to 2 ; and two, each equal to — . 1 =fc /5 391. Two pairs — ^- and -2. 392. Two pairs 4ino 3^-3 1, 1.3 , 1 , 1.3 3 498. ^^'' + -^^^'^ + j-^,'a-\x-'b'--~~^^ . a-^a;-^-. 6=^+ ... 1.3.3.5. 7...(2p~7) , „ •^'' ^^_^;^;-/ ^^ . »=-' . a; ^ . 6^- . (- ly-'. 499. .»-?«a;' + ^ + -|;- g-l.l-g--^-(2j>-^) ^ 2 8a I6a^ li^ - 1 . 2^"^ * a'^^"^ * 500. ^"^ + 4»"^^+32''"'-'' ■^128''"''^ "^ 3.7.11...(4j^- 5) ^-1^^ 501. a-n^a-V'.j^ + i^.^-V.jv. o 2.5 3.8.13.18...(5^-7) J-^ a «aj 1 . 3 cra;^ 1 . 3 . 5 ax^ 502. -,-^^ + -^--^ + --g-.-^ + 1.3.5.7...(2jo-3) CTo;^"^ 503. «-1--y+|f|--y-^^--^"-2/^^ 2. 5. 8. ..(3^-4) ,3^2, „(^,, \p-l. 3'-' ' ''^ ' 318 ANSWERS. 504. 512 13 4 _2( 504. a-^^-a~^.x 3 2.3 .^^.a-V.,., 4. 7. 10. 13. ..(g/^- 2) -^-^^ -_, 505. a^.aJ + 7a-".a:''+^—^,. a" '*.«;' + 4 2.4 1.5.9. 13.. .(4;^ -7) -^ ^^, |;>-1.4^'-' • ^^^ 2 , 2.5 , ,2.5.8 _8 8 506. ^ + 3^-'-y+^:y»«-'-2/^+7y:3^-«y L3 2.5.8...(3;>-4) _, a1««1.S r- 507. «^ + 2«-^.«''+^72«-«"'"^- I P- 1 .2^^ 5 ^2.5 1.3.5.7...(2/)-3) /^\^3^,, 2 1 _9 1 . f) _« - 508. x' + -^x ^^y-^-^rr-^-x ^-y 1 .6. 11. l6...(5;?-9) -''i^ -1.5- -^ -2^' 509. aj^+^a; ^•.y + — — aj -.2/' + 3 " 2.3=^ 1.4.7...(3/>-5) --? 510. .4-lff.--''-^ ...^.g;-., ..„.^, 511 .»i.-.J'.l^,rt-...!;ii!t:Azi>..H«.j^.i io5 la 'jR Qo 612. a-^ +3«-'' • 6e' + 2-^3%-' .5V [p - 1 ■ 3"-' ANSWERS. 319 513. 521. 3 2.3 6;—T 3p-2 3. 5. 7. 9.. .(2^0 -3) '-f *^-» 514. a^"^ 2/^ + o «^"''- 2/' + ^ • ^"' • 2/^ .a? , v 515. No. 509. 516. y-^ + z 2/~^ • ^' + 1^. • 2/-'"' • 2.4' 1.5.9. 13...(4jP-7) -'-?^' 3 [p - 1 . 4*^* • y • « (p-i) „.„ £C 1 aj' 1 a;' 1 . 3 »* 1 . 3 aj' , ^1^- ^-«^2-a^-2-^"'272-^-2T2-^^^^'- ^-* 2 9 3 6.2" 4 518. a^ +-5-a^.2/ +2^..»^.2/'+ 6.11. l6...(5/?-9).2^' '-^ [}o-l .5^' .^-^ il 1 _i 1.4 - _it „ 519. ic^ + ^a-'.aJ ^«+„— ^2.a~*.aj ^2;' + -a ,y 3(P-1) 4.7.10...(3;.-5 j 3,-.,^^^-^^^, | p-l . 3^-' 520. a^6Uia-^.6^c+...... 1.4.7.10...(3p-5) -?2f2 fc-' [p-1.3^- •" •* •" • 621. a*-la-^.i4-l:|.a-t.6- 1.2.5.8...(3y-7) „-'-^' ,'-?i i . ct . w , 320 522, 522. ANSWERS. 567. x^ 1.3 cc* but if a = 3 and £c = l; /^ — - = J2. -rr ,^,111111111 5 1 5 1 J 2.8 3° 2.8 3' 1 •333333 -333333 •111111 -x- -055556 •037037 •012346 •001372 •000457 3' 2 5 1 — X — ; ..•. ^023148 8 3"* 1 1 ... -001543 8''3* 10 1 .... ^000430 4.8'3'' 10 1 .... -000014 4.8 3^ ' 1-414024 524. 1956. 528. 420. 533. 645. 525. 109600. 529. 244. 530. 24. 534. 45. 535. 15. 537. 6720. 538. 725760. 539. 33, times. 541. 325. 13 39 523. --^-^'^ -y- 526. 24. 527. 2520. 531. 5040. 532. 170. 536. 4989600. ,.>./^ [90 , , 1 89 542. 28 days. 543, 84. 544. 63. 547. H- 548. 626. 549. 11879. 550. 56iO; 5i80. 551. 14492, 552. 6. 553. 120. 554. 60. 555. 4. 556, 16 557. 4095. 558. 1840. 559. ii940. 560. n-i 561. 190. 562. 1405. 563. 22. 564. 672. 565. 399; 728; 1218. 566. 25. 567. 5525. ANSWERS. 321 568. 623. 568. "^""^f^\ ' 569. 2639; 5019. 570. 4960 j 9455. 571. 140. 572. 5525 ; 2925. 673. 2689; 5019. 574. 50. 575. 5763. 577. Com. diff. ^ ^li!!lzl) . 580. 4970. o 581. m = ^^; 2024; 3795. 582. 20540. 583. 330. 586. 4250. 589. CL' = n. 592. « = € = 2-718281828459 963. 609. 0-39794. 610. 369897; 0.8115752. 611. -69897; 1-2552725; 2-1303338; 8750613. 613. n = 8; z = 2. z 1 615. If ^i= 1 and z=\. = - : and we get log. 2 and then z 1 logs 8 = 3 log, 2. Again, ii n = S and z=2, = - and /iiTii -{• z y -e get log, 10. 616.,-^^ = i. 617. 3-54413. 618. aJ= -8796686; y= -26285. 619. a; = _lO-73644; y = -l9'0ll. 620. x = \:, y^l. At A, 621. i» = - 2-91481; 2/ = 5-65678. 622. a:-6; «^=2(iog6 + 21oga)- 623. . 7iog6.]og^ (5 log a + 3 log h) log c/ - 8 log 6 log c ' 28^ ^^g^_ (5 log a + 3 log h) log c? - 8 log 6 . log c 21 322 ANSWERS. 625. 683. 625. The 11^ 626. 2-2872824. 627. '002431267. 628. -00543928. 629. -108603. 630. 6-19620102. 631. -0005218504. 632. -0180448. 633. 33035. log b log c 634. 1-317372. 635. 12-58341; 0:=^; y= ^^^,,^^_^,^ -^,' 636. 3-080336. 637. -5095667. 638. -000000311481. 639. 1-469493. 640. -04950141. 641. 43336970. 642. 1-739405. 643. -4403654. 644. ^519. 25. Id. 645. f. 646.^,. 647. S- 672. ^^__ -S.Jir,'-r^^Jl'^r,'-,r' _ /( ^ \ <^(^ + ^) h{a-\-h) ab(a + b) 673. ab . ^ [^^^) i J(a' + b') ' J{a' + 6^) ' a' + 6' * 675.^:{a^V(^3Va-)}; ^^^y^^.^ . 2« 676. Distance from angle = — . 677. The segments being e and /, and the ratio n : 1 ; the sides are ( — »— =^ ) and 7i ( —. I. \n-l/ V^i-1^ ^'°- 4>J{8{8-a)(s-b){8-c)}' 679. Length of line = < — ^ — j-— > where a is the longest line which can be drawn to the circle of wliich the radius is r. 680. «=[fw=^V(^-6^)}]^. ^ 2 681. Straight line meets base at distance ,^-- • 6 or from foot of pei-pendicular on base. «rtrt m • i . 1 .. « tan a — 5 683. Trigonometrical question : p.^-— — . ' ° ^ ^ i ^ + 5 tan a ANSWERS. 323 684. 810. 684. =t 20 or =fc 15 j ± 15 or ± 20. 709. 33" and 1 5". 711. 38" 1 V 49"-7. 713. ^ ; ^ ; 10 ^2 ; - 10 ^/3. 715. 2l"29'9"-6; -375. 716. f ; J3; l; ^l^^^f ; ^^JS; 2.j3. 717. 114" sr 33\ 720. 5; 10" 42' 18''. 732. 9^. 734. For COS 23" 30' read cos 22" 30'. 773 ± J^^"^^^) ^ 2 774. aJ=sm->d=l^(l7d=^869); 2^ = cos-^±|v(l 7=^=^869). 775. cos -<#>) = tan"' ^- i^ ; ^ + >^; ^-<^<0. 778. taiia;= ' • 779. 6=15°. 780. 6> = tan-^-— -^ tan (a + (3)- tan (a - -^)' 782. tan^_^~^cota. 7^ + 1 783. r!^,. 784. tan^ = ^or|. 785. TrfTM- 786. sin^A = U^^j21; sin^ ^ = 1 =. ^ ^6- 787. A = 4<5'; B = 30". 789. 71° 28' 6"; 50" 56' 10". 790. 17-321; 90" and 30". 791. C= 546-8625; ^ = 76" 44' 45"; 5 -64" 33' O". 793. 15" 57' 55"-2; 134" 54' 14"-3; 29" 7' 50"- 5. 794. 70079. 795. 381-14. 796. 288-7. 797. i6-7 799. 4009-5. 800. 466-94. 802. Pr=2132; P^= 1107-3. 803. 962-605. 804. 4188-597. 806. AG =1626-636; BE = 1106-8562. 808. 608-584. 809. 36 J3, 810. 438-36; 177-14. 21—2 S24 ANSWERS. 811. 873. 811. r^-l^^,- -T^l- 812. Equal. (sm(^-a) sm(y-a)j 813. 55ry765. 814. 5-8426; 3-8477; 8-3034. 815. 51-08; 280-96. 816. 390-27; 345-15. 817. 826-32; 9" 50' 58''. 818. 1967-4. 819. P^ = 926-35; FB = 783-5; PC -51079. 820. 287*2. 821. (^ circle can be described ^bout ABCD, .-. CBB is a right angle), CD = 353 5. 822. 1897-2. 823. 2208-6; 178-73. 824. 5-176; 10; 14-141. 825. 219-31. 826. 177-582. 827. 86-0904; 43-88534. 828. (Take the liorizontal plane, passing through the lower wire), 2^" 14'. 829. (By Simpson*s rule), 1 r. 13-1312 po. 830. 2421024. 832. 10 r. Op. 19*5536 sq. yds. 833. 44-85 acres. 834. 3 ^21. 835. 625-17. 836. 13-926 ch.; 23.21 ch.; 32-494 ch. 837. 5*78926. 838. 3; 33. 839. 110-45. 840. At base 30"; at vertex 120". 841. 20. 842. ^25. 6s. 2fc/. 843. '^ - 845. -2146025. * 4 846. '^.coti^; 25^3; 100; 172 05. 849. 625. 850. 3-36 feet. 851. (Distance lost, is difference of quadrantal arc and radius). 852. 280. 854. 119-37 sq.ft. 855. 78-604; 26-376. 856. 1253-88. 857. 61419. 858. 3698. 859. 200 J{3) - 150 J(3) = 50 J (3) = ^ {150 J{3)]. 862. 11-454. 863. 464-1. 864. 172-966. 865. 20731. 867. 761-811. 868. 184-2. 871. 2827-4 sq. yds. 873. 60769. ANSWERS. 325 874. 931. 874. (The number of circular sections in section of rope, is the same as the number of circumscribing hexagons) 2 875. 60 j so'-, 30"; 60". 876. 271-69. 878. Height : diameter :: 1 : 6. 879. 140-625 lbs. 882. 34-7812 cub. in.; 11-8338 cub. in. 883u 282454 in. 884. r = 4-64 in.; 869 cub. in. 885. 61-552. 886. 2664-2 gall. 887. 148-44 ft. 888. 531-128 cub.ft. 889. 3-09. 890. 25000. 891. 5 1 77-3. 892. ^.v/(4 893. i7-3i4lbs. 894. 930597; dWS^^ 895. 1-8522 in. 896. 7-32 tons. 897. 37811-3. 898. ~^ 899. 1-488 in. 900. 2-0124 oz. 902. 134-3029; 3-9274cwt. 903. 942-211 cub. in. 904. 207-88 lbs. 905. Iton 14cwt. 2qrs. 10 lbs. 906. 121-86 lbs. LP-d\ D~d D' + Dd^d'_ D' + Del + d' _ frustum ^^'- A^ - 8^ " A - 6 "" A^ + AS + E' " A' + A8 + 8^ ~fr^iIsUJ^i ' Sr 908. Height of cone cut off by tangent plane of sphere = — ; 3 5 Volume of cone = - irr^' volume of segment = — -n-r^ ; 8 ' ^ 24 ' .'. volume of charge = 7, Trr' = - x - tt/ = - volume of shot. o 8 3 8 909. 23-562 cub. in. 910. 11145-1013. 911. 294-8 lbs. 912. 390-88. 913. 86-837 in. 914. 2-427 in. 915. The segments of axis are -x^Q; « v9-{V2-l) and t(3-Vl8). 922. 1551-4 sq. miles. 931. A = 139" 58' 55"; B= 141" 28' 57". 326 ANSWERS. 936. 998. 936. 48" 11' 58"; 8V 8' 51"; 67' 21' 29". 937. 45^ 20' 38"; 6V 6' 54"; 86' 38' 56". 939. ^ = 31° 58' 49"; (7= ]44'» 57' 42"; c = 113" 50' 5&\ 940. 11' 30'. 941. 33' 0' 55". 943. 64" 20'; 73" 54'; 33' 40'. 944. 118" 48'. 945. 35' 16'. 946. 64" 20'. 948. 92"42'10"; 29" 59' 44"; 310-28 sq. miles. 951. 70" 32'; 258-8 sq. mi. 952. 8-38 sq. in. 953. 89*9 sq. ft. 955. 85" 28' 26". 956. Lat. 63' 38' 49" S., Dec. - 45" 51' 35", 957. ^ 62" 48' 29". 958. 73" 5' 54". 959. 55' 51' 35" 5', 47" 13' 30"-5. 960. 28" 10' 7''1 ; - 4" 9' 10"'l. 961. 305" 47' 28"-65; 22" 25' 17". 962. 42" 10' 12"-5; - 3" 46' 55", 963. 336' 12' 39"-7; - 4" 45' 10". 964. 23** 15'^ 39'. 965. IQ** 49" 41'. 966. lo*" lO"^ 15*. 967. 12'* 10"^ i8'-6. 968. + 4" 37". 969. + 8" 5S'-6. 970. 8** 42™ l*-2. 971. n** 57"" 26'-52. 972. 13'* 14"* 44''-02. 973. 27'' 15"* 32'-l4. 974. 20** 9" 39'-63. 975. 18" 57' 15" E. 976. 7'' 31™ 49'-87. 977. 58* 57' 17"; 8** 6™ 40'-8. 978. 41" 21' S6". 979. - 58" 56' 18". 980. 20" 48' 15"; 20** 40°* 40'-2. 981. - 60" S6' 40"; 43" 50' 50". 982. 21" 32' 14". 983. 3" 58' 16"; 56'' 38™ 50»-4 W. 985. 31" 27' 59"-4; 10** 55"^ 10'-4 W. 986. 64" 13' l6''* 987. 108" 28' 15". 988. 134" 37™ 13"-5 E. 989. 185" 49' 43"-5 W. 990. 155" 41' S6" E. 991. 102" 40' 15" E. 992. 27" 11' 36"-33 W. 993. 115" 9' 38" E. 994. 6o" i' 19"; 14'* 58™ l^^'S. 995. Lat. 26" 21' 11", Alt. 29" 38' 44". 996. Az. 51" 52' 10" R of K; Lat. 40" 20' 25". 997. 59" 6' 1"; 9" 3' 20" W. of S. 998. (1) 7^ 29" 37*; 60" 11' 34"; (2) 4'* 22" 35'; II9" 48' 26". ANSWERS. 327 999. 1053. 999. 51" 54' 42". 1000. 52" 23' 15" or 7" 47' 47" 1001. 4^^ 16"" 43^ 1002. 17*^ 38^ 53\ 1003. 9" 22' 24". 1004. N. 5" 2p' 36''-5 W. 1005. 60" 53' 14". 1006. 48" 4' 56". 1007. 27" 43' 50"; 117" / 10"-5 E. 1008. S. 1" 3' 26'' W. 1009. 49" 21' 52"; 81" 49' 3" W. Bearing K 105" 53' 25" E. Distance 516*88 miles. 1010. True dist. 20" 57' 34", .Lon. 10^ 50"^ 48 7 E. 1011. 35' 59' 14". 1012. 72" 33' 4". 1013. 80" 9' 34". 1014. 19" 49' 10". 1015. 28" 8' 24". 1016. 97" 45' 4". 10 1 ^ IS 1017. 120"1'45". 1018. 9y + 10a: + 13 = 0; -~; --; --. 1019. H V17 i tan-^ • 25. 1020. 5^ + 9x = —^, 1021. 35. 1022. 62/ + 1 la; - 35 - 0. 1023. (^ 0) ; (- 2, 0) ; (- 1 , ^) . 1024. 7-6. 1025. x-yj3 = 5 + j3. 1026. ^ . 1027. (i ; - 1). 1028. 22/ - a; + 17 = 0. 1029. tan-^ - ~ ; tan"^ " ^ J 5 i y^ • 1031. 2y - 7a5 + 25 = 0. 1032. -^ . 1033. tan"^ '778. 1034. tan-.-^. 1037. g. 3- 4 1038. 4'y + 3x-5a = 0. 1039. 42/-a; + 5 = 0. 1040. yy\/17. 1041. 15; 3^13. 1042. ({|; ^); at right angles. 1043. 2 V2. 1045. tan- ^ . 1046. gj ; - ^) tan- - i| . 62 1047. Equilateral A; area 9^3. 1048. 8-375 in. 1049. 5-^ . 1050.5. 1051. f. 1053. 2/ = -3a;. 328 ANSWERS. 1054. 1090. 1054. 3x" - 58x + sf = 0. 1055. I; i ^29 ; (o, ^)(i, - J) . 1058. If base =5; perpendicular =jo, and segment of base ^ q. The locus is 3af + 3i/'-2(b +q)x- 9.py - ^'^ = 0, a circle. 1059. (-f;)V(y-g)'= 26-21. 1060.(.-f)V(.-;J=^^ 1062.. 1063. (3,-1); Vi3; (-i, i); fi. 1065. (^^) $ = |a:; {BC) 51/ = - 3a; + 18. 1066. 8y - Q^x = 2.5. 1067. 2 ; 252/ 4- 2 (37 =F 3 ^41) a; = 0. 1068. tan"^ f . 1069. y + a;'' + 6y-2a; + ^ = 0. 1070. (1,7); (--5, --5); 1^234. 1071.-(i±^;-l±|^^). 1072.^^,. 1073. If AB^a and BC=b; Iocust/'-Ox" -b7/ + ax = 0. 1075. x'-2ax + 4>my = 0. 1078. J5 ; J3; I JlO. o 1079. Circle. 1080. 2/' - 1 2a; + 36 = 0. 1081. «'+?/'- ^'. 1083. 4a;^ + V = a«; radius^, 2 1084. (1) The ellipse becomes a circle; (2) it becomes a sti*aight line. 1086. -±^- ^^. 1087. ■ '"' 1089. 8a V2. 1090. - ^^^^^^±^ ■,2{aJ(a'+ h') - «'}. ANSWERS. 329 1091. 1116. 1091. Altitude of parabola being h] bisecting line cuts it at dist- ance Y7T frona highest point. 1093, ^ \^^ ~ ^' ^ si'* ' b 1094. 2/'=y . 1096. Circle. 1098. If {h; k) (//; k') be two points in a parabola (of which the latus-rectum is m) from which normals, perpendicular to one another, can be drawn, kk' = — 4m. "^ m ^ Squaring the first of these, and substituting, we get, finally, y' = m{x — 3m)y a parabola having the same axis as the original, and a latus-rectum one-fourth of that of the given curve ; its vertex being at a distance 3m from that of the original. The result may also be obtained by means of the equation to the normal in terms of the tangent of its inclination to the axis. qq 1104. P:Q:W::1 :J3:2. 1105. - 475. 1106. tan-^^; tan"'?. 1107. 67-2 lbs. 1108. 4^Z) acting at E; by constructing the several parallelo- grams the proposition is obvious. 1109 CA. 1110. If be the angle between P and B, sin ^= p. sin a; also 7? = - ^ cos a :^ J{F' - Q' sin^a). nil. 52-44; tan^ = -6-4. 1112. 32-247 lbs. inclination of pressure to horizon = 71" 10' 30''. 1113. AC = IaB; smC=?^-^; t= ^J{n0^6Jl37). 1114. i?' = P'+^'+2P^cosa; sin ^ = ^^ . sin a. ms. J3;e= 90". ui6. /a = ^ ; /. = ^^ •/.• 330 ANSWEI^. 1117. 1145. 14 1117. jgcwt. 1118. tan 5 = 3-27. 1119. 1682 lbs. vertically downwards. 1120. At A, F= WJ2. AtB, F= IF. ^{2 (4 + J3)} ; ^ = tan-^i^!-^. AtC,F=^2W;e^ = 60\ 1125. 12-5 lbs. 1126. 9-58 lbs.; 1-69 lbs.; 9-444lbs.; -SSeibs. 1127. 10-805 cwt.; 21-938cwt. 1128. k.%A,R = PJ{^ + J3); e= tan-^ ?i^ . 1129. Incl. to vertical = cos"' - ^ J2. 1130. 1 cwt. 1131. t = l6; «, = 12; «? = 12-8; w^ = 7'2. 1132. 9*68 cwt. W 1133. Tension AE = — . tan a . cosec 6 ; W pressure = — . sec a . sin (a + 6) cosec 0. 1134. 3273 yds. from centre of stream. 1135. The strings must be vertical, and they will then sustain an additional weight of 4 lbs. if suspended at a distance -r from the stronger string. W 1136. Tension of ^C= tension oi BD=-^', tension of CE = ten- TT W sion of ED = -,- ; compression of CD - . . If the string were continuous, the bar would be forced upwards by a W pressure on each side equal to — (2 - JS). 1137. 5 : 8. 1138. 6-276 cwt.; 37*4192. 1140. S = y;y=^. 1145. -589 in. ANSWERS. 331 1146. 1168. .^.^ _ 5a _ 5b 1146. a; = — ; 2/=^. 1148. From 1, ^J13; from 2, i^^O; from 3, ^J5. 1149. i^; i^ from base of cylinder; 1 : Js. 16 5 ^^ ^ ^ ^. ^^^-. w 6 X 16 X 17m- 7716 67rr^ 1150. 6l76in. 1151. j^ + 6.2^.18 "" IT ■ 1153. 0° and 60°. 1154. tan 6 = j^ tan o. iga 1155. 73-21 lbs. 1157. I>ist. from opposite angle ^^' ,^ . 1158. If the angular points and the points of bisection be joined, those lines -will bisect the sides of the interior triangle. Whence the centre of gravity of the latter may be proved to be identical with that of the original triangle. 7 1159. Distance from side a=—rh. lo 1160. Common side on the edge. 1161. Distance from side of square •7384a. 1162. 15-4 in. 1163. 4-8 ft.; 50 lbs. 1164. Must balance on circumf of common base, .'. r : h ::1 : ^3. 1165. 40. 1166. Draw through the centres of gi*avity lines parallel to the bases of the triangles. These will form a regular figure: whence the property in question may be deduced. 1167. Taking the sides as axes, _ - W -\- 3 IB or C) , = « or - ; when A=B = C. 1168. At point of bisection of the line drawn parallel to axes, at a distance -^ ^ from axis of exterior. 332 ANSWERS. 11G9. 1218. Q2r 1169. tan ^ = ^ . 1170. Distance from vertex 7*609 in. yon 1171, 7 from common base. 14 1172. 6-6 lbs.; 4-6 lbs.; 8-8 lbs. 1173. h=rj3. 3 48 1174. = tan~* -. Dist. of point from circumf. of base is tan"' •— . 8 09 1175. — r-TTz ^, . h from surface of water. lb {63 +m) 1177. 22 lbs. If ^C and AB be taken as axes x= — ;y=^. 1178. C-21 and 2379; 120°. 1179- 3 cwt. 1 qr. 1180. 14 in. from weight 2. 1181. 12 cwt. 1182. 20iiii. 1183. tan-' l\ J3 and 60" - tan"' II ^3 ; ^ in. from angle. 1184. Distance from point of suspension of P is tan"' ^^ ' —- . F + Q con ^ 1185. 70« 12' and 19" 48'. 1186. { V(^^^) - l}«' 1187. 45 lbs. 1188. 4-02 lbs. 1189. 56 lbs. 1190. W + nw + P or 2"F + (r-l) to. 1191. 33 lbs. exclusive of pulleys. 1192. -^ . iio>i ^tifn+ ft co%a- Cg ^^^- wl^-¥wl wll Ai^y^. • 1195. -^ ; — ; — '—.. 1196. 2 cwt. and 5-016 cwt. 1199. ^^9 lbs. ' ' if»AA r\ F'sina — P. cosf ^^^^ 1200. Q= — . 1201. 3-8203 cwt.; 96732 cwt. 1202. 113-2. 1203. A. cot 15". 1204. l -55 cwt. 1205. 13-467 lbs. 1206. S41-5. 1207. Q = 3P', P = 4F. 1208. W=5P; Tr=10P; Tr=12P. 1209. 8415 tons. 1210. 4-4563 lbs. 1211. tan-' 1-333. 1212. 47*124 tons. 1213. 38-27 cwt. 1214. -052 cwt. 1215. 215-42 cwt. 1216. 724. 1217. ^^^. 1218. i69'68lbs. ANSWERS. 333 1219. 1260. 1219. cot-i -33. 1220. Height = 2»- cot o. 1221. E = r . ^^^-,^ . 1222. ^ sec a. 1223. Distance of one of the weights from highest point .1 2 + cos a r . cot" sina 1224.^. 1225.cos^ = <"'"-'>r^'-^ ^. 1226. Greatest tension (when beam leaves wall) 1227. tan-^ ^^ . 1228. r-\ p + (2-^ - 1) «,. 1229. 942-477 lbs. 1230. 6xV 1231.20 42'. 1232. — . 1233. 5760 lbs. n 1234. TT-H.^-^.^-^:. m5.7-3feet. w 1236. 132-44 tons. 1237. ^ = ^^^' 1238. 1 : 6-88. 1239. 2847 tons. 1240- (| + ^) . ^- . 1241. ^^/7. 1242. sm - = -1-^ . 1244. ^ cot a. W 9.h 1247. Strain at each hinge = jj . JO^V + 9a^) r tan ^ - - . ♦ 46 ^ ^ ' ' 3a 12 1248. y V^. 1249. 4jcwt. 1250. Inclination to face of plane = - - 2a. 1251. cos A = ^^ . 1253. 4-814. 1254. COS-' -838; 52-22 lbs. 1255. Tf. sin )8 . cosec (a + /8). 1257. 9-265 cwt. 1258. Vertical angle = 2 tan-' fi. 1259. 2(l-/>t')tan'd-4/*tan»^-sec'^ = 0. 1260. W^.sin2t. 334 ANSWERS. 1261. 1291. 1261. The compound solid will first begin to topple over; and when the inclination of the base of the cylinder is tan~*-4 the cone will topple from the cylinder. 1263. tan~* -63. 1264. 5-905. 1265. 62cwt.; 72-739 cwt. 1266. 8-164 cwt. 1267. It will fall over. 1268. 353-5 lbs. 1269. -086. 1270. The centre of gi*avity is not the centre of figure. If P be the point of contact with the plane, C the centre of figure and (r the centre of gravity; the sphere will slide without rolling if GFC > i and < e ; and it will roll without sliding when GPC < i and i^. 1280. ll'0344feet. 1282. tan~^ ~^^ ; this inclination will be the same for either side if the position of the centre of gravity be symmetrical for all sides, i. e. if the board be uniform, but not otherwise. When the inclination is 45°, u = -, . ' 2 + /A 1283. Segments by centre of gravity are as 43 : 57 nearly. 1284. /oiz. '^^^y"^ . 1285. Not higher than 18-75 feet. 1287. cos- V • cot a). 1288. tan-'. ,,^^,^1 2w + W 1289. 93-3 lbs. 1290. 7-212 lbs. 1291. tan"^ -—— . /*. 1319. ^= \—ydy=-^. 1320. Circumference = 4-1 5 in. ANSWERS. 335 1294. 1358. 1294. 3-7 lbs. 1295. l-hVo. 1296. 10-615 and 13-385. '^r 3 1298. — from centre. 1299. j V' c . T 1 Qtyi 1300. Distance from centre = — ^ . 1301. ^a^ a 5 1302. 1 7rab\ 1303. x = y=z = ^r. 1304. lb; \a, 1305. 31 cwt. j tension would be converted into compression. 1307. Beam gives way at 8*38 tons; wire at 12-6 tons. 1308. 3-7138 tons. 1309. 3 -59 feet. 1310. 23712-216. 1311. 59''-14.: 60". 1312. 3-76. 1313. -008 in. 1314. 65-29. 1315. 16756 lbs. 1316. 57 tons 1 1 -65 cwt. 1317. -004012. 1318. 22-4 in. circumference; 3 tons 56 lbs. 1321. 2-98 miles. 1322. '85 in. inside. 1323. Diameter increased 1 : ^^2. 1324. 896OO. 1325. 47 minutes. 1326. 61 -3. 1327. 61-8. 1328. ^^3 cub. feet. 1329. 34-16 h. p.; 222-8 H. p. 1330. 427-8. 1331. -008 1^ . (2 Tf + a) - ir| • 1332. 61-09 cub. feet. 1333. 41 h. p. 1334. 1?222^. 'pLn 1335. 5-1 H.P. 1336. 85-3 H.P. 1337. 3-67 feet. I rit 1338. 2 ^ /-,- . 1340. 1500 tons. 1341. 43-2 bushels. 1342. 160-7 bushels. 1343. 361-4 bushels. iQdA /i^.+Zg^.+Za^B + ^g- . 90oo(/,^^, +/^^^+y;^z^^+&c.) ^'^**- 528 ' B • 1345. 230400. 1346. 16128O. 1347. 2-79 tons. 1348. 1-1 ton. 1349. 2083 millions. 1350. 3bl33. 1354. 38640. 1355. 57*14. 1357. 69 miles an hour. 1358. tan~^ -5 ; 768 feet ; No, in a parabola. 336 ANSWERS. 1359. 1390. 1359. .in^«-±|^. (7^ — 1360. 83-33 and 4l6'66 from starting points. 1361. 46*875 miles an hour; 18*75 miles an hour. 1362. 115920. 1363. 16-1. 1364. 567 lbs.; 850-5 lbs. 1365. 38400 tons at 1 foot per second. 1366. 4008-85 feet. 1367. 13600 lbs. 1368, 200 oz.; 41216 oz. 1370. A/{^(«^-^rt/;g^} ^ 1371^ 50' with line ; 10 miles an hour. 6 (2m> + p) 1372. 44 feet from line of original relative motion ; oblique velo- city 44. ;J373^ 1 1 -67 feet per second. 1374. 24-64 feet. 1375. 5-656 miles an hour; S.R 1376. 29° 13'; 8-84 feet 1377. 8-48 miles an hour. 1378. Both bodies being subjected to the same velocity of motion by stream, their relative motion is unaffected by it, and it may be disregarded. Angle with bank = 60". 1379. 101-62 feet from funnel. 1380. 33-9 feet * aft.' 1381. The motions are reversed, and each body moves with e times its original velocity. 1382. 6-9 feet. 1383. 14-44 oz. 1384. Velocities ^6l and ^91 at angles cot"' 9 J 3 and cot"'?^' with tangent at impact. 14 1385. ^ -. -B :: 4 : 3. VeL of -5 = — in direction of ^'s motion. o 1386. 64-506. 1387. Vel. = 10-45 ; tan"^ ^^^5^3^ ' 1388. Spheres will recede on lines symmetrical with original lines of motion. 1389. 2e- 1:3. 1390. Angle of incidence = tan"* — — 5—^ , 1 - e tan a ANSWERS. '337 1392. 1442. 1392. Velocity of C = ^^ . {a' + V + 2ab cos C)^ ; angle with AC^sui -77-72 — ^ — -7 7-. . 1393. 11 -02 feet from plane. 1394. 4 seconds; impossible. 1395. ^^Al±^}^ 1396^ 21.33 feet. 1397. 30 feet. 1398. 21-77; 3977. 1399. -.{a+gt^^{a'+2agt)]. 1400. 179-2. 1401. 150 feet. 1402. 1031. 1403. 100-3. 1404. 2-16 sec; 1-467. 1405. 100-625 feet. 1406. 205 feet below starting point. 1 + e' 1407. -878 foot; s. -^, 1408. 20-23 miles per hour. 1409. 4-443 below top of cliff; A will be falling, and £ rising. 1411. 375 feet. 1412. 7*3 sec. and 2-7 sec. 1413. Height of tower 193-2; greatest height 257-6; 6 sec. 1414. 115; 250; 4-15 sec. 1415. 135-2; 76-2 feet. 1416. 4-98 sec; 80-252. 1417. 229*056 feet from base. 1418. 49-8 miles an hour; 6"^. 5V-5, P — Q 2PQ 1419. Acceleration = -73 — -^ . g ; tension = -5 — -^, 1420. %. 1421. %. 1422. 32-2; 3-75 oz. o 7 1423.8-12. 1424. Y«^- 1425. 2 ^25r. 1426. 4-91 oz. 1427. 4-69 sec; 21-44. 1429. 6o-6. 1430. 1-77 sec; 9*022 feet. 1431. S5-S5, 1432. 2 tons; 89076 yards. 1433. I6-I. 1434. 47-1 miles per hour. 1435. 39*75. 1442. 3434-7 feet; 28-7 sec» c. 22 338 ANSWERS. 1446. 1446. tana: -h) ; F' = gh'(h'. .-? Kkih- -70- h'{k'- • 1477. 1456. 1447. The bodies will all be at equal distances ^gt' vei-tically be- low the extremities of equal lines drawn from the point of projection in the directions of projection. 1448. 3084 yards. 1450. 7764 feet; 3-66 inches. 1451. Resolve parallel to plane. 1452. 16307 feet. 1454. 14-314 feet. 1455. - 4" 54'; -503 inch. I, If side = 2a and base = 2c ; velocity = — — • . / ( T-i a ) • 1457. t = ' • { ^' sin (a - i) =fc Fr'. sin' (a - i) - 2ag cos i]. g cos I ^ 1458. The greatest distance is the same in each trajectory, viz. 2*8/ 'X sin \o. — i) ^ rjij^g times are also equal, therefore the veloci- 9.g cos I ties are in A. p. ; w = - cot (a - 1) cot i, 1460. J = cot a. 1461. x' = ^ . y. 1462. '79 inch. 1463. 4-0678 lbs.; 1574-9 yards. 1464. 618 feet. ' 1466. It returns to ^. 1467. -5 and 1 i 2. 1468. 67*1 below the obstacle. . 1469. s/{3gh). 1470. 9 feet below, and gr.feet horizontally from starting point, 1471. 3183-3 yards. . - . 1473. Below; error = — . sin' a. tan i. sect. 1474. BA^BG] .'. BAG < BGA = GlH, 1475. 13416-6 feet. 1476. ^^r^^^ 1477. -2^23'. ANSWERS. 339 1478. 1520. 1478. ffg sec 1479.12-27. 1480. If„w', y be the actual numbers of vibrations, n and jo being the true numbers, then n—n : p' —p :: n : p. But if a and B be the corrections of the pendulums — ^^^ = — and n 21 p' — pB - — - = ^, nearly; whence a : p :: I : T. 1481. Time of revolution 1 -^S hours. 1483. S : 5. 1484. 8-34 feet from ceiling. 1485. 3980 feet; 144-85 feet. 1486. 4000 miles. 1488. 8-64 sec. 1489. 9787 inches. 1490. 18°^ l^ 1491. S9-l 5 inches; 4-32 sec. 1492. 1292-5 feet. 1493. 24" 25'. 1494. 17 times. 1495. 3-036 inches. 1496. If ^ he the measure of the force of gravity, when the mile and hour are taken as units ; and 6 the required angle : tan^ = -^ ; tension =^A/(?^+l)i number of oscilla- tions pel* mmute = — /I — -r^ — I , where A = . ^ TT V \ -^ / , 5280 1497.1:^2. 1498. 344 yards. 1499. -112. 1500. 12096 feet. 1501. V^ : 1. 1502. cos-^-25. 1503. y . ^^ + ^^ + ^^ ^ . 1504. j^^ — . 1506. j^^r^^y 1507.-^ = 4^, 1508. 231 : 89. 1509. 1 : 17-28. 1510. 32-5 oz. 1511 ^ 1512. 27 : 10. 1513. Copper 443*4 lbs.; tin 56-6 lbs. 1514. y| «' + (^ + ^) y^. 1515^ 678-8 grs. 1517. 35 : 3. 1518. 5 : 17. 1519. 1 : 7. 1520. 527 : 623. 22-2 340 ANSWERS. 1521. 1563. 1521. 4 V2 - 3 : 2 V6. 1522. -3-. 1523. I • r, . _ Sirh — ^r 4r 1524. When the depth of the centre is h] ^ 1 , ±j. ^ "" • I 1525. l + j3:2 + ^3, 1526. 1:2. 1527. r - g . 1538. |; |«. , 1529. ^-p- 4 1 4c'— 56' 1530. If side -2a, and base = 26; ratio = — . ^^^ _y t '> tension -^"^ 14c'-56•• ^;^^. 1532. 6-1728 lbs. 1531. 3,-«. 1533. 2^ =|^5|g • »=• 1534. 8-67 feet. 1535. f . p. q Oft f^—fi 1536. -1 5. 1537. 1 20 lbs. in direction of axis -^ . '^ Y, . 1538. 416-7 lbs. 1539. -4653 inch. 1540. 7168 sq. feet. 1541. Hemisphere in fluid; neutral. 1542. 6*7; 5*924. 1543. 10-665 inches. 1544. 13 tons l-p cwt. 1545. 1 ton 15-6 cwt. 1546. 4 tons 77 cwt. 1547. l{^ + 3at), 1548. 4-65 feet. 1549. 82-83 feet o 37 V 1550. Volume increased by -^^ ; 4685*1 feet. 1551. 399-21 feet. 1552. 2*5 sec. 1553. 36 8 feet. 1554. 107". 1555. ^-op; 933''4>5. ' 1556. 39 lbs. 1557.^-. 1558.^. 1559. e'. x\ {{x + S) tan a; + a; sec'ic}. 1562. ^ 1 • {\-x){\~xy 1563. -^-T. (1 + x'f ANSWERS. 341 156i. 1692. 1565. <^-{i-xy 1566. ^.^^^-^.ia'-^x')^. (a'-xj 1567. ^ ^. 1568.. 7?^^. ^^^^•'{^TxT- 1570. K + a^r*. 1571. »'(» + te')^. 1572. i^^l^. 1573. :f^3. 1574. ^1:^^, 1575.^-1^,. 1576. ^-sino:. 1577. aj"*-' e«'°"' .{m + x cos «). 1578. i J • 1579. n smT-'x, sin (ti + 1) a;. 1580. - -2„— ^ . X T J. 1581. e' (cot a; + log sin a;). - 1582. - ^r:^. . 1583. ,._,'_^ ' 1584 1585 sm cc a - 6 cos £c * "*-^°*' 1 _ a;2 ^ ic' sin2ic ^^{2 (cos 2a - cos 2x)} * 1586. e^- (log sin a; + 2 cot x - cosec^cc). ^^n^ since . , ^ e^* ooqpoS^'*^^ 1587. rr + sm-^a; . cos a;. 1588. -cosec^e ^ (1 - x'> s/x 1589. ^F^TT)- 1590. 2(«'-.')i--^,-^^ 1591. ../... . 1592. ' X- sc' + iB + l *"•'*'• a (1 + a + x" + a") • 342 ANSWERS. 1594. 1613. 1594. -^-^- = 2a. PM ANSWEKS. 1631. . 1651. 1631* An asymptote parallel to axis of a;, at a distance a; two infinite branches towards positive y's j points of inflexion (a a\ . / a a\ 1632. Two infinite branches; one towards positive x and negative y ; the other towards negative x and positive y. The other branches run into one another at the origin which is a 1 1 point of inflexion. Greatest ordinate at cc = -y- ; p = ^ JS 1633. ^ r. 1634. (2rx-x')\ 3(a + 3xy ; 1 , 2cx + h-(b'-4ac)^ .^,. ^ or -jTj^ — - — -.log ^^ '-' iH'>4ac. 1636.log ^ ^. X637.log-- X ' ^'^" """x + l x + l 9 1638. log (^-3)J + ^)' . 1639. I (a' - ^rK 1640. ^^^i^. 1641. Iog(^)* 1642. <-'-"')^f-^"-> . 1643. - J. (^^;:^l<«a-^=^ 16^- '°g »+Vri+»') - 1647. - 1 (x'+ 2a')(a'-«')*. 1648. - 1 (3 - 2*0*- 1649. a sin^' ^ + (»' - '«')*• 1650. (a + 5a;')". 165L log (a - 2) ^ a!-2 ANSWERS. 345 1652. 1674. 1652. |log(a! + 3) + |log(a;-2). 1654. ^^ (2ax-xf + |(4a-3)vers-^. 1655. J^^J^. 1656. log-^^ii^. 1657. r'Yers-'^-(2ri/-tf)K x^.{x-2y r ^ '' 1658. -r . log [2cx + 6 + 2 V{c {cx^ + hx + «)}]. 1659. - or vers — . (2ax - x"")'. 1661.;^.logi^^ 1662.>nog.-^*.«. 1663. ^(log.-^). 1664. (,_^);^^^)„,. > 1665. e^ (x' -2x + 2). 1666. log (C^)*". 1667. e'^''. (2aj* - 6(« + 12a;^ - 12). 1668. -tan^a;. 1669. seca; + cos£c. 1670. Q tan® X-- tan'* a; + - tan^a; + log . cos x, 1671. ?^sin-^ + |(l-.»)^. 1672. - T - ^ (1 - «'')' «i^"' ^^ + i («i^"' '^)'- "° 1673. 7 sin' x-^l sin^ a;. 1674. - + t sin 2x. 5 7 2 4 346 ANSWERS, 1675. 1G89. 1675. tan x(l+- tan'' xj . 1676. - tan a; sec a; + - log tan (7 + ^) • 1677. X ( log tan - - cot jc . cosec x j . 1678. „ tan' X - tan x + x. 1679. ^ • 1630. f^.dx = 2{2ax)K 1681. ^; | ^ («» + a6 + ft'). 1682.f.log^±(^-|'(a'-.'). 1683. jydx = a vera-' ^ + (2aa; - xy+ C ; 1684. Jyc/^ = |'log^ + a; AjVc^^^.logS. 1685. /2/c?^=f (a;-a)^.(^^ + |a) + (7; •• //^=^«^ 1686. [yJaj=rvera-'- + (2ra;-a;'f4-C; .\ rydx = r\tr + 2) = area of generating circle + - square on diameter. 1687. /2^& = J (1 _g + (7; .-. fjdx = i . 1688. fy(fe = oMog(4<(''+a!") + a'.tan-'^ + C; 1689. j2/c/a; = |ai(3a-ic) + C; .-. j\dx = ^J, ANSWERS. 347 1690. 1696. 1690. j'n-x''di/ = -7rf^ + ^x^+G; .-. volume =-,^ Tral .*. / 4 ^ - ll . (b'- h'f=2^ve2.DFE, Vol. generated by DFIJ about FE=\ tt (2/ - 7if dx, •^ lir^y —Tifdx = I Try^dx — tt / 2hy dx + irh^x = 'jr j^,(a'-x')dx-27rh. j-{a'-x')'^dx+'jrh'x 73 7 = TT. 6''a; - TT —-i cc^ — SttA. - . (circ. area, rad. a) + irh^x 3a a ^ ' = TT. (6'' + ^*) . a; — TT — . •— - Stt^ . - (circ. area, rad. a) Oi CL -'7r{lf + h')x- TT— ^ x^ - irhah sin"^ ~ +irh-x{a^- x^)^. 1694.-a'g-ilog2}. 1695. 2,rTO (J' -ay, 1 7m* {(6 + m)* - (a + m)^}. 1696. |«:y<& = - |(r'-:.')i + C; Jy& = { sin- ^ + 1 (/ - aO^i .*. X = — , and from symmetry F= •-- . 348 ANSWERS. 1697.^ 1702. 1697. jj0os6.de=^.sme; jld6 = l6; X=\t.^^; , TT , 47* if a = - , for semicircle, X = — . 'i Sir 1698. J2x7jdx = ^^{7^- a^)^+ C; j^ydx = r» bLt' ^ + a; (r* - xi^ Taking these between the limits x = a and x = r and re- ducing, we get X= ^^^^^ '^^ P=o base; or, if h be the height of the segment, a = r-h', ©■ 3 p r —h 3 ' a — sin a . cos a a — - . • r r 4 i I T^ 3a 1699. \xydx = - m^ x^ ; I yc/; 1700. [a; ^ . ^o; = - r (r' - x')^ + C ; \'^'dx = r sin"' ^ . Hence, if the limits of integration be and r, we have for semicir- . And, if the limits I {r^-1f)h rad. x chd. 9>r cular arc, X= — . And, if the limits be h and r, TT COS r 1701. /-V»sinW.../^(l-^). 1713. M2r\ (i -^) . 1714. M |-' . 1715. il/ ^' . 350 ANSWERS. 1716. 1724. 1716. ^--^' 1717. ^i\- 1718. i^^. 1719. By 1710 moment of lamina about axis = w.-^ = — . ( -7- ) > , ,, Trr^.dx ,^ 3a;' - and m = M. i. ^M.,-rr^dx\ irrh hr 10 1720. I^et moment of sphere radius r = MK'j r=^K^ shell (radii r and i2) = i/;^,/, then MK; + MKJ = MK% but from 1718^/ = ^, and K'=~-y .-. J/,^^^; = Jf.^r'-if.^r;, butJ/,Jf andir, are as r^ r/ and /-r/; .-. (r' - r/) Z"^/ = | (r' - O ; 1721. ^^. . 2 1722. Expand 1720 and find the limit when r = r/, MK* = - Mr\ o 1723. ifr». 1724. If the elementary mass m be referred to the intersection of the diagonals by lines parallel to the sides, and if r be its radius of gyration ; mr* = wa;' + my' + 9.mxy cos a, and therefore MK^ = :§ (ma;') + S {my^) + 22 (ma;?/) cos a, but ,^ dx.dy , 3f ^ y m = M. , . sina= -^ . c/a;.g?/: 4a6 bin a 4a6 'ANSWERS. 351 1725. 1726. f+a r+b jif C+<^ f^^ M n -6 4a6 4ao cJ 3 r+<^ r+b Ji j2 f^"" f-^^ M 1725. Moment about axis of parabola = if - , and moment about 5 tangent at vertex = M — - ; therefore required moment 6 „ 1 ^(?«'4^0 1726. If intersections of diagonals of opposite faces be joined, these will be the three principal axes, and therefore, as re- gards these axes, 2 (mxy), "^ {mxz) and 5 imyz) are seve- rally zero. Also, moment of side about edge = moment of edge about edge = M —■] therefore moment of cube about o edge (= moment of side about perpendicular edge) = ^ ; therefore moment of cube about one of the principal axes = M. —— — M,— = M. — - . And diagonal makes with the o 2 o principal axes, the angles cos"^ -y-; hence the general form I- A cos^ a + B , cos* P + G . cos' y= 3 A cos' a gives, 2-1 2 moment about diagonal = 3 M -^ x --=M-7r. 352 ANSWERS. 1727. 1731. 1727. Since the line drawn through centre of cube parallel to diagonal of face makes angles of 45" ; 45° and 90" with the principal axes, moment about that line = Jf ^ (cos* 45 + cos' 45 + cos" 90) = M%, o Hence moment about UU = —^- + — — = if , — , 6 4 12 1728. See 1726. 1729. Fmd the moments, M — — ; M — ; M , ~- about the three principal axes, and ai)ply the general form as in 1726: M. — - x — - — rj * a being the altitude and b ' 20 a* + 6' ® the radius of the base of the cone. 3x' 1730. Mass of element = M. -jrdij; Qi •. moment =^,j_^-<^y; substitute for x* irom equation to ellipse, and we get 2 moment about 26 = i/" . - a'. 5 1731. Taking elliptical section parallel to axis 2a and calling the minor axis of this section z, its mass will be represent- 3xz ed by M. — j- dy. Also moment of ellipse about an axis through its centre perpendicular to its plane will be found to be Mass x — , Hence moment of ellipsoid about 4 a6 = ^ / (a;' + «') xzdy. Substituting the values of x and z derived from the equations cc' = yg (^' ~ V^) ^^^ «' = 71 (&' - y^ we get ultimately : — moment about 26 = — —— . ANSWERS. 353 1732. 1744. 1^32. fV37- i'33-VS' "34 ..yg- 1735. -. ^( ^^.^(^.^ig,.) }- 1736. By 1713 and 1700. 1739. If resistance for unit of velocity =^: —- = — ~ ^ q\ at w . , = i /'^ i^„ Jyj + ^Jk 9.g\l k' "="' Jw-vjk' 1740. t7 = ;^. .^; .*. S = ;r7- . log, j-^. 1742. ,^..04. -I r»). 1743.,^..o4^^i). -iff A A . 2r. sina , ^ ■.rr-.« *» . /%Fcosa.^ \ 1744, t = ; and by 1743 r = ^, \og,i^-^—- + 1 j; 23 TABLES. MEASURES OF WEIGHTS. English. Thoy Weight. grs. 24 = 1 dwt. 480 = 20 = 1 oz. 5760 = 240 = 12 = 1 Tr. lb. Avoirdupois Weight. drs. 16 = 256 = 7168 = 28672 = 1 oz. 16= lib. 448 = 28 = 1 qr. 1792= 112= 4= 1 cwt. 573440 = 35840 = 2240 = 80 = 20=1 ton. French. 1 Milligramme = '001 gramme = •0154326^ 1 Centigramme = '01 gramme = •154326 1 Decigramme = •! gramme = 1-54326 5' 1 Gramme = !• gramme = 15-4326 ■ w 02- 1 Decagramme = 10* grammes = ] 54-326 1 Hectogramme = 100* grammes = 1543-26 1 Kilogramme = 1000* grammes = 15432-6 zy" 1 Myriagramme = 10000- grammes = 151-32-6 , TABLES. MEASURES OF LENGTH. 35 English. inches. 12 = 1 foot. 36 = 3 = 1 yard. 198 = 16J = 51 = 1 rod. 792 = 66 = 22 = 4 = 1 chain. 7920 = 660 = 220 = 40 =10=1 furlong. 63360 = 5280 =1760 =320 = 80 = 8 = 1 mile. French. 1 Millimetre = •001 m^tre 1 Centimetre = •01 metre 1 Decimetre = •1 metre 1 Metre 1- m^tre = 39*37009 English inches. 1 Decametre = 10- metres 1 Hectometre = 100- metres 1 Kilometre = 1000- metres 1 Myriametre = 1 10000- metres MEASURES OF SURFACE. English. sq. inches. 144 = 1 square foot. 1296 = 9 = 1 square yard. 39204 = = 2721 : = S0| = 1 pole. 1568160= = 10890 = 1210 = 40 = 1 rood. 6272640 = = 43560 = 4840 =160 = 4=1 acre. Iso; 1 acre = 4840 sq. yds 5. = 10 X (22)^ sq. yds. = 10 sq. chains. French. 1 Centiare = •0: I are 1 Declare = •1 are 1 Are 1- are = -0247105 English acre. 1 Decare = 10- ares 1 Hectare = 100- ares 1 Kiliare = 1000- ares 1 Myriare = 10000- ares 23- -2 356 TABLES. MEASURES OF CAPACITY. English. gills. 4= 1 pint. 8=2= 1 quart. 32 =» 8 = 4 = 1 gallon. 64= 16= 8= 2= 1 peck. 256- 64= 32= 8= 4= 1 bushel. 2048= 512= 256- 64= 32= 8 = 1 quarter. 10240 = 2560 = 1280 = 320 - 160 = 40 = 5 = 1 load. The Imperial gallon = 277*274 cubic inches = 10 lbs. water. French. 1 Centilitre - -01 litre 1 Decilitre = -1 litre 1 Litre » i- litre = 1-76068 English pint 1 Decalitre « IC litres 1 Hectolitre - 100- litres 1 Kilolitre = 1000- litres 1 Myrialitre- 10000' litres o *^ 'a -g s length is the Mdtre = 39-37009 Eng. inches, weight is the Gramme = weight of Centimetre cubed of distilled water, surface is the Are =10 Metres squared, capacity is the Litre == 1 Decimetre cubed, volume is the Stere = 1 Cubic Metre. FORMULA. 357 FOEMUL^. Arithmetical Progression. 8=^{2a + {n-l)h}-; l = a + {n-l)h; s = n——-; b = -^. Geometric Progression. r"-l , „_, rl-a a 8 = = „.__; Z = „^-..,= __j,^ = _ Piles of Spherical Shot. Triangular "<"^^^^"^^) ; Square ^i-^^f«^^) _ . , n(n + l)(2n + l+ 3m) Rectangular — ^^ ^-^ ' , Permutations and Combinations. n things, r together, all different ; No. of Permutations = n{n—l)...{n — r + \), n{n— \)...{n — r+ 1) No. of Combinations \r [n n things, all together j a alike, h alike, c alike j , — r; , . Binomial Theorem. «-i n(n-l) „_2 2 n{n-~\){n-2) „_- 3 (a->rxY = ar^nar \x+^-^ — -'a"-'a;' + -^ r^ -^ a" ^aJ^. . . . H j .a .X \r—l 358 FORMULA. Exponential Theorem. a=l^Ax + -^ + ^^ + _, where A = a-l-l(a-iy + -{a-lf-&c. = \og,a, x' x" g= 1 + 1 + 1 + i + i =2718281828459, 2 [3 [n log,^^ = 2(«+3 + 5+ -;• Plane Trigonometry. sin (il ± -B) = sin ^ . cos 5 ± cos ^ . sin B, cos (il ± ^) = cos ^ . cos i5 T sin ^ . sin B, sin 2il = 2 sin -4 . cos A, cos 2 A = cos' A - sin** ^ = 1 - 2 sin* A = 2 cos' ^ - 1 , sin^ + sin 5 = 2 sin ^(A + B) cos ^ (A-B), sin^- sin^ = 2 cos ^(A + B) sin i{A- B), sin (ti + 1) -4 = 2 sin riil . cos A - sin {n - 1) A, cos (n + 1) ^ = 2 cos 71^ . cos ^ - cos {n - 1) A, 2sin^-4 = l — cos 2^, 4 sin" -4 = 3 sin -4 — sin SA, 8 sin* -4 = 3 — 4 cos 2^ + cos 4^, 16 sin* -4 = 10 sin ^ — 5 sin SA + sin 5A, &c. = &c. 2cosM = l+cos2-4, 4 cos' -4 = 3 cos ^ + cos SA, 8 cos* ^ = 3 + 4 cos 2-4 + cos 4i4, 16 cos® -4 = 10 cos -4 + 5 cos SA + cos 5^, &c. = &c. sm FORMULAE. 359 .21 I - COS 2A , , . _,. tan A ± tan B tan A = —-. ; tan (A=i=B)= , , 1 + cos 2^ ' ^ '' i =F tan ^ . tan ^ ' O fn-n d tan 2 A = Y4:4nM ' ^'"^ ^^ ^ "^^ "''' (^ - ^) = ^^^' ^ ' ^^^' ^ = cos^ B — cos^ A, In any plane triangle, a : b : G :: sin A : sin B : sin C ; 2 bo ' 2 be ' 2 s,{s-a) Spherical Trigonometry. Napier's Rules. (1) Sine of middle part = product of tangents of adjacent parts, (2) Sine of middle part = product of cosines of opposite parts. In any spherical triangle, cos a = cos 6 . cos c - sin 6 . sin c . cos -i ; . , , , sin (« — 6) sin (s- c) „ , . sin s . sin (s - a) sm' A^ = \ / ■ 'I cos'4^= ^-r-^ '\ /^ sin 6 . sm c sm 6 sm a ^^ sm^£-6)_sin(^c) * Sin s . sm {s — a) sin a : sin 6 : sin c :: sin A : sin B : sin C ; * ^ ' COS i (as + 6) * ' ^ ^ ' sm i (a + 6) ^ tan 1 (« + 6) =^^iii|z4) . tan 1 c, 1 / r\ cos i (^ - ^) , , 360 FORMULA. Table op Astronomical Mean REFRAcmoNS. Zenith Distance. Mean Kefraction- Zenith Distance. Mean Eefraction. Zenith Distance. Mean Refraction. O 1 1 / // 1 73 30 3 15 85 10 10 11 10 10 74 3 21 85 20 10 28 20 21 74 ^Q 3 28 85 30 10 46 SO 34 75 3 S5 85 40 11 6 ^5 41 15 SO 3 42 85 50 11 26 40 49 76 3 50 86 11 47 45 58 76 SO 3 59 86 10 12 10 50 1 10 11 4 8 86 20 12 34 52 1 15 77 30 4 18 86 30 13 54 1 20 78 4 28 86 40 13 27 bQ 1 26 78 SO 4 40 86 50 IS 56 58 1 ^S 79 4 52 87 14 27 ()0 1 41 79 20 5 1 87 10 15 61 1 45 79 40 5 10 87 20 15 S5 ^^ 1 49 80 5 20 87 30 \6 12 63 1 54 80 20 5 31 87 40 16 52 64 1 b^ 80 40 5 42 87 50 17 S5 ^5 2 5 81 5 54 88 18 21 m 2 10 81 20 6 7 88 10 \9 11 67 2 17 81 40 6 20 88 20 20 5 68 2 24 82 6 35 88 30 21 3 69 2 31 82 20 6 50 88 40 22 5 69 30 2 ^5 82 40 7 7 88 50 23 13 70 2 39 83 7 25 89 24 27 70 30 2 44 83 20 7 45 89 10 25 47 71 2 48 SS 40 8 7 89 20 27 14 71 30 2 bs 84 8 30 89 30 28 50 72 2 58 84 20 8 55 89 40 30 SS 72 30 3 3 84 40 9 23 89 50 32 15 73 3 9 85 9 54 90 34 32 FOEMUL^. 361 Mensuration. Triangle = - 6;? = - a6 sin C = J{8 .{s-a){s- h) {s - c)}. Trapezoid = ^ (a + ^)P' At Area of regular polygon of n sides n a ^ 180*' 1 ^2 . 360 J. 180° - a' cot = - nK sin = nr tan , 4 9^ 2 ** w i? and r being the radii of tlie circumscribed and inscribed circles. 180° Perimeter of regular polygon = 2wr tan . Perimeter of circle = Sttt : area of circle = ttt ~ —— = -— . ' 4 47r Area of sector of circle = -^ — '^aFs> ^^ (^ being the length of arc). Area of segment of circle = - (a - r . sin ^°) ; ring = tt (72^ - r'). Vol. of prism = £'.h {B^ being area of base). h Vol. of cylinder = irr^h j vol. of cone = tit* - . Vol. of pyramid = B\\', vol. of frustum = \{A^ + B^-¥A. B). o o Surface of prism =^9^. + 2-5^ (p being perimeter and B^ the area of base). Surface of cone = irrh + irr^. Surface of sphere = ^irR^; surface of segment = 9.irEh. Vol. of sphere = - "rri^j vol. of segment = — - {SR — h). o o Vol. of circular ring (having circular section rad. a) 362 FORMULA. Co-ordinate Geometry. Equations to Lines, (fee. (Rectangular Co-ordinates.) Straight line ; y = ax-^h. Straight line through point (A, ^) ; y-k = a{x— h). Jc-k Straight line through points (k, k), {h\ k'); y-k = j' _j (^ ~ ^0* Straight line perpendicular to y = oa; + 6 is y = — x-\-h\ oil parallel io y - ax -k-h is y = ax -\-l) . Perpendicular ovLy = ax + h from (/t, k) — —jr. j. • Circle; origin at centre; y*-f-a;' = r'. circumference, centre on axis of a;; y* + x*- 2rx = 0. circumference, centre not on axis of x ; y^ + x'- 9.ax - 2l3y = 0. anywhere, {y- PY + {x — a)' = r'. tangent, origin at centre ; ky + hx - r", (A, ^) being a point on the circumference. Circle; chord of contact to tangents drawn from exterior point (7i', 1c\ k'y 4- h'x = r'. Conic sections, y''=px + qx^ origin at vertex ; axis on axis of X, ^ = latus-rectum ; in ellipse q is negative, in hyperbola q is positive; in parabola y is 0; in circle g' is — 1. Ellipse, origin at centre; t/' = -2 («' - ^^)i a vertex; y* = — {2ax-x^. a FORMULA. 363 Hyperbola, origin at centre ; ^" = — (^^ - <^^)y Oj vertex ; 3/^ = — (^^ + Saa?). Parabola, origin at vertex; y^=4imx. Cissoid ; y^ - ; concboid j x^'if = (b^ - x^) {a + xf. AtOj — X Witcb; xy^ = 4ia^ {2a-x). X {origin at base j x = a vers~^ — \/(2a2/ - y^), origin at vertex ; y = a vers"^ - + J(2ax - x^). Tangent to any curve; y — h = -y (x — h). Subtangent = y-=-. dx Normal to any curve 'iy—h = — -^{x-l}). {y — x-j- ^ wben aj = 00 , Intercepts of asymptotes; J a; - v -T- » wben cc = 00 . ^ ^ dy' If equation to curve be reducible to tbe form T c d . y = ax + -^ 1- — + (fee, X X tben equation to rectilinear asymptote is y = ax + b. At a singular point, being a point of contrary flexure, 3"! = 0> ^^ 5 > ^^^ changes its sign ; a multiple point, ^= 2, and ^ or ^ will have multiple values. 364 FORMULA. A conjugate point; -7^ = -, and is impossible £orx = a^ h. A cusp; ~ will have but one value; -7-^ two, or more. The radius of curvature is p = .g . To find the evolute; IS-y = pl. (£^)"*, and a - a; = ^ (2^ - yS). Statics. Forces in one Plane. Conditions of Equilibrium. ■CI .1 1 . X r/, cos a, +/„ cos a„+/, cos a, + (fee. = 0, Forces through one point i-^-'i . ' / . ' \f . ^ . (/j sin ttj +/g sin o, +/3 sin ag + &c. = 0. Forces not all meeting in one point; /, cos a, 4-/2 cos ttg +f^ cos a^ + &c. = 0, /j sin a^ +/j sin a^ 4-/3 sin a^ + &c. = 0, Ay, cos a, 4-/^2/8 cos a^ +/32/3 cos «3+ <^c.) ^ - /jOJ, sin ttj -/^£c^ sin a^ -f^x.^ sin 03 - &c. j Centre of Gravity. ._ wx, + wjc^ + to..a;„ + and ^ - -j^^ FORMULiE. 365 - 2 - h In triangle ; x = -h; parallelogram J a; = ^ . Trapezoid : x~-, j- : semicircle; ^ = — . — (from centre). Sector of circle \x = , Segment of circle; x = -t . -. — ~ (measured from centre). . . . - _ r. chord 2a . . . - _ 2r Arc of circle : x — — : arc oi semicircle a; = — . 2a TT Arc of semicyeloid ',x = -r', ^=(t — -jy*. Area of cycloid; y = ^ r ; area of parabola; ^ = --h, o Area of semi-ellipse; ^ = ^ from centre. „ - 8h ., _ SA Cone; a; = — ; pyramid; x= —-» oil Om Conical surface; x = -—', volume of hemispliere ; » = — , 3 8 Volume of spherical segment; X'=~. j- (from centre). Surface of spherical segment; ^ = „ (from base). Spherical sector; £C = - . (2r - A) from centre, 8 Come frustum; aJ =i / , ^ f — i^ (from Tra'), — 2 Paraboloid; a; =» ^ A* Sm FORMULA. Parabolic frustum; x = -. ^,^^a (froi» 'ra ). _ 2 arc j a; = - r, _ 2r 157r-8 Cycloid j I surface of soUd of revolution ; «; = ^5 * 3^_4 » I vol. of solid of revol. x = -. x^Tzr^ i^^^^ ^^®)- Machines. Pulley; IF, = nP; F, = 2-P; Tr3 = (2--l)P; and, if weight of each pulley be w, W^=^{r - I) . F + (2'' -1 -n)w. sin t Inclined plane, without friction; ^-^- 3^» . • _ .^ 8in(i±e) ^ithfriction;P=Tf.— ^-^^^. Screw, without friction; F=W. ^^^j _ —. h^fxr r with friction; F=W. ^r:^' ^ Dynamics. Impact, direct; { „(i_,') . ( r„ - T.) EORMULiE. 3G7 Impact, oblique; F. . cos a = F„ cos a ^ ^—^ r ^' , [F/sina = F«. sintt. 1 v^ 1 Uniform acceleration : s = - ft^ = ^^= - tv^ _V _ 2s _ /2s '^~t~ 2s~¥' If the body be projected^ with velocity F, in line of action of/, s=F«±l/i^; k;^f^±2/,. Variable acceleration : /= -^ ; ^ ~ ;r ^ '^~ T^ * 2 Projectiles : 3/ = a; tan a - gpra^^^gg^ ; 2 . F. sin (a — z) 2 F^. sin (a - i) cos a < = ^-i ^ j r = ^ — —. ; g . cos I g . cos ^ sin (2a — 1) = ^ cos^ z + sin i. Simple pendulum : ^ = tt . /- ; Conical pendulum : t = 2Tr /-, Pressure on a curve = ( — ± ^ cos j . m. 368 FOBMUL^ Moment of Inertia. For an arc of a plane curve; -^ ^Y"^ \dxJ ] * r 1 ^^ For an area of a plane curve; -ir =V' dM ( fdy\')i For a surface of revolution; -^ = 27ry • U + I ^ ) f • For a solid of revolution ; -^ = iry*. For any solid; -jZ = jj^i/' ^^* Centre op Oscillation. When moment of inertia about axis of suspension =:MK% and distance of centre of gravity from the same axis = h, length of simple pendulum = -j- . TABLES. 369 Specific Gravities. Cork -240 Larch -522 Elm -588 Teak -657 Fir -753 Beech -852 Oak '934^ Lignum Yi tee 1*334 Coal 1-270 Sand 1-886 Clay 1-919 Brick 2-000 Portland Stone 2-145 Chalk... 2-315 Granite 2*651 Zinc 7*190 Tin 7-291 Iron (cast) 7-207 Iron (wrought) 7-778 Steel 7-816 Brass 7-824 Copper (cast) 8-78S Copper (wire) 8-878 Silver 10*474 Lead 11-352 Mercury 13-568 Gold (standard) 17-486 Gold (pure) 19-258 Platinum (pure) 19*500 Platinum (laminated) .. 21*041 One cubic foot of brickwork weighs 1 cwt. One linear foot of rope weighs in lbs. *045 x (circumf )^ in inches. Proof strength of rope in cwt. = 2 x (circumf ) ^ in inches. Elasticity. Tenacity. Resistance. Expansion. W 9.^: V ^^^ ^^^ those of the re- mainder (which are unaffected by the change in the divisor) By this example it is seen that the coefficients of the terms of the modified quotient are identical with those of the first terms of the successive remainders (converted into sums by the change of the signs of the divisor), and that they can readily be obtained without the intervention of the subordinate summations : thus the 31 . . 9-1 coefficient — can be obtained from its components 7 + ^ + 4- with- 23 out the intervention of the -- , and so of all others, to whatever extent we choose to caiTy the division. To put the matter in a general form let the coefficients of the divisor be represented by 1 + 6 + c, and those of the dividend by A+B + C + B + U + F+G; wo may then symbolize the operations thus, Form 1. l+h + O A + B +C + D + F + F +G(^AA-B+C^^-i-D^+F^^ A +Ab + Ac B^+ C+D B+Bb + Bc C^+B^ + F C^^ + C^b+C^c B^ + F^ + F D+Db + Be F^^+F^Jb + F^c F +G APPENDIX I. 377 using j5^, C'^, (fee. to express the sums of tlie terms immediately above them ; so that B=B+Ah D^^=^D + Bc + Ch E =E + C c + D h And the operation might be arranged, more concisely, thus : Form, 2. l + h + c) A+B + C + J) + B + F+G A +Ab + AG B +Bb+Bc E +E b+E c where each line, except the last, is obtained by multiplying its first term into the several terms of the divisor, those successive first terms being identical with the coefficients of the quotient. It thus becomes unnecessary to take the trouble to again write these coefficients of the quotient in their ordinary position on the right of the dividend. If now the successive lines be written diagonally instead of horizontally, and the divisor be written vertically downwards, the process will be precisely the same as before; but the arrangement will be very much more convenient, and the quotient will be exhibited in a more intelligible form, thus : Form 3. 1 + b + c A + B + C + J) + E + F + G + Ab + Bp + Cb+I)b +E^p + Ac ■\-B^c + C^c +D^c +E^e A + B^+C^^ + D^^ + E^^ + F^^ + duct-s of the roots, taken two and two^ with their signs changed. The coefficient of the^bwr^A term is the sum of the products of the roots, taken three and Hiree, with signs changed, and so on. The coefficient of aj", or the last term, being the product of all the roots with their signs changed. Cor. 5. Every equation has the same number of roots as there are units in the highest exj^onent of the unknown quantity in it. APPENDIX II. 3So Prop. III. If the signs of the alternate terms in any equation be changed, the signs of all the roots will be changed. For if, in any equation, we substitute - x for x, all the terms involving odd powers of x will be changed; while all the terms involving even powers of x will retain their original signs. Con- sequently the effect of such substitution will be to change the alternate signs of the equation ; but the substitution of —x for x is equivalent to changing the signs of all the roots. Hence, if an equation have its alternate signs changed, the signs of all the roots will be changed. Prop. TV. Every equation, all whose roots are real, that is, which are not of the form a^fi J —\, has the same number of positive roots as there are variations of sign, and the same number of negative roots as there are permanences of sign in passing successively from term to term of the equation. It is evident that the number of variations together with the number of permanences make up one less than the number of terms, and are therefore equal to the number of roots. Now, supposing the series of signs in any equation to be + + _ + + 4._4. if we introduce a new positive root, by multiplying by x — r, wo shall have + +-+++-+ - + + - + -■ — + -* db + -4-±=t- + - in which result the doubtful signs will depend upon the relative magnitude of the contiguous coefficients of the original equation. But we can see that the number of these doubtful signs will be. c. 25 o8G APPENDIX ir. tlie same as the number of permanences in the orisjinal, and that they will moreover correspond to them ; consequently these signs can never tend to diminish the number of variations. It is also evident that the additional sign introduced by tlie multiplication must be different from the last of the original series and must therefore give an additional variation at the end of the result. Each positive root therefore introduces at least one varia- tion which did not previously exist in the equation. Similarly, taking the same series of signs and performing tlie multiplication by x-^-r, which is equivalent to introducing a ncNv negative root, we have + +-+++-+ + +-+++-+ + =k — ^=bife4- + ^±4- Here the doubtful signs are the same in number and corresjMDnd to the variations of the original, and they cannot therefore tend to diminish the number of permanences; and the additional sign at the end of the series must have the effect of introducing a new permanence. - Consequently this result must contain at Iciist one more permanence than the original. "VVe therefore find that each positive root introduces at least one variation, and each negative root at least one permanence ; but the number of variations and the number of permanences together, being equal to the number of roots, it follows that each positive root will introduce one, and only one, vai-iation, and each negative root one, and only one, permanence ; that is, there must be the same number of positive roots as there are variations of sign, and the same number of negative roots as there are perma^ nences of sign in the successive terms of the equation. Degua's Criterion for Imaginaiit Eoots. Cor. It follows from the above proposition, that if any coefficient of an equation 5e =t and the eion of the j^'^'^cediny APPENDIX II. 38? term he the same as the sign of that which follows it, the equation must have two roots which {under the condition of all the roots being real) would he hoth positive and negative, and which must therefore he imaginary. Pkop. Y. Imaginary roots enter an equation in pairs ; that is, if a root of an equation be a + p J — 1, then a — ^ ^ — 1 must also be a root of that equation. Let ax'^ + hx^ + ca; + c? = 0, be an equation one root of which is a + P ^J -1, then, dividing by x-{a + p J -1) we have a+h +c +d (a+/3^-l but, since a + )8^— 1 is a root of the equation, we must have a"'+j8"V-l = 0, which can only be the case when a''= and /3"'= 0. If now we proceed to divide by x — (a — l3^—l) we shall have a+h +c +d (a-)8.^-l «+(«-^V-l)+K-^V-l) +{a-rj-l) in which it will be observed that a, a ', a'', p, f^'\ P'" are precisely the same as in the preceding divisor ; but it has there been shown that a'= and (^"=0', hence a'- P^" J - 1 = 0, and therefore the equation is divisible by x-(a-/3J-l); and consequently, if a + /3^-l be a root, a-jS ^ -1 will also be a root. Examples. 1. Determine whether 8, 6, 4 or 2 are roots of the equation X* - 19iC^ + 128:z;' - 356x + 336 = 0. Ans. 2, 4 and 6 are roots. 25—2 388 APPENDIX II. 2. Two roots of the equation x^ - Ox* - 67 x^ + 200a;' + 588a; - 1440 = 0, are 2 and 6; determine the equation containing the remaining roots. Ans. a;' + Gx** - 3 la; - 1 20 = 0. 3. One root of the equation X*- 1 9a;' + 132a;"- 302a; + 56 = 0, is 4 ; what is the equation containing the remaining roots 1 Ans. a;'-15a;' + 72a;-14 = 0. 4. Two roots of the equation X* - 16a;* + 86a;' - 176a; + 105 = 0, are 1 and 5 ; what are the other roots? Ans. 3 and 7. 5. Has the equation a;' — 2a;'— 15a; + 36 = any equal roots ] Ans. 2, each equal to 3. 6. Determine the equal roots of the equation a;« + 3a;' - 6a;* - 6a;^ + 9a;' + 3a; - 4 = 0, and thence the other roots. Ans. 1, 1, 1, —1, —1, -4. 7. Determine the equal roots of the equation a;*- 6a;' + 12a;'- 10a; + 3 = 0. Ans. 1, 1, 1. 8. By the application of the law of the coefficients, form the four equations whose roots are respectively (1, 2 and 3); (1, 2 -3 and 4); (2, 3,-4 and 5) and (3 and 2 ± 3 ^^l). (1) a;'-6a;'+lla;-6 = 0; (3) a;*- 6a;' + 9a;'- 94a;- 120 = 0; (2) a;*-4a;'-7a;'+34a;-24 = 0j (4) a;' - 7a;' + 25a; - 39 = 0. 9. One root of the equation x' - 12a;* 4- 67a;' - 212a;' + 366x - 260 = 0, is 2 + 3^-1; determine the equation containing the remaining roots ; and show, by division, that one of them is 2 - 3 ^- 1. APPENDIX II. 389 Pkop. VI. If the coefficients of an equation be taken in inverted order, the roots of the new equation will be the reciprocals of those of the original. Let the given equation be aa:" 4- Jaj"-' + ca3"-^ + +hx' + lx-\-m = ....(1), then the new equation will be mif^l'if-^^'ky''-^ +c?/V62/ + a = (2), or, dividing throughout by y"? a h c hi -. + ^ + ^-^...... + -. + - + ^ = 0... (3), and, comparing (3) with (1) we see that «; = -, hence the values of if y in (2) are the reciprocals of the values of a; in (1). Cor. 1. If the coefficients of the proposed equation be the same when taken in inverted order as when taken in direct order, it is clear that the new equation will be identical with the original and the roots of the equation and their reciprocals must furnish the same series of numbers j that is, to each root there must be a corresponding root which is its reciprocal and the roots must therefore be of the form : 1 1 1 1 . ^' a' ^1' TT' ^2> -» «3» -<^C. a a^ a^ a^ Equations of the above form are called " reciprocal equations," or " recurring equations." Cor 2. In a recurring equation of an odd degree one root will be + 1 or - 1 according as the sign of the last term is — or + ; for the equation must have one root which is the same as its recipro- cal and it must therefore be =»= 1 ; since also the other roots consist of pairs which have the same sign, the last term of the equation. 390 •appe:ndix ir. which is the product of all the roots with their signs changed, must have a contrary sign to the root unity. Cor. 3. Let rw)" + 5a3"~* + cx"''^ .,... + ex' + hx + a = (1) be a recurring equation ; then dividing by x* we have .(Ai.)._.(j-.i)..(j-.4-> k + fx* + —^ + lfx + -j -1-171 = (2) if n be even, which can always be made the case, since if n be odd one of the roots is known. But if a; + = y, we have X and so on. Hence the values of a;^ + -3 : a' + -^ &c. can be determined in ar X terms of y"; /; 2/'"'; /, and those being substituted in the equation (2) the resulting equa- tion will be of the form 0^2/'+ Py" +yy* ...... +X2/' + /Liy + v=0y and from the values of y in this we shall obtain the values o^ x in the original (1) by means of the equation a; 4- - = y. X APPENDIX II. 391 Hence, {/* n he even, tlie solution of tlie original equation can be made to depend upon that of an equation of tlie degree . If n be odd, one root of the equation is +1 or — 1, and the solution of the original may thei'efore be made to depend upon n— 1 . tJiat of an equation of the degreee — - — , since w — 1 is even. Prop. VII. To transform an equation into another whose roots shall be greater or less by a given quantity, than those of the original. Let ax^ + bx^ + ex* + dx' + ex^ +/x + ^ = 0, he the given equation, it is required to determine the coefficients -—' ^ ^^^ a being the coefficients of the second and first terms of the original equation. Thus, to eliminate the second term from the equation 2x'-12ii;'-7a; + 2 = (1), 12 we have r = = 2, and, dividing continuously by a; - 2, we get 2_12- 7+ 2 { + 4-16-46 .2 - 8-23-44 + 4-8 - 4-31 + 4 2=fc 0-31-44 So that the new equation 2a;j*- 31a?j — 44 = 0, will have its roots less by 2 than those of the original (1). Method op Solution. If we reduce the roots of an equation by a number which is greater than one, only, of its positive roots, that root will appear as a negative root in the reduced equation, and consequently, as- suming all the roots to be real, the number of variations in the reduced equation will be one less than the number in the original. Again, if we reduce the roots of this equation by a number which, added to the former number, is greater than two, only, of the roots of the original, another root will become negative, and we shall therefore, in this reduced equation, find one variation less than in the last, that is, two less than in the original. And so on. APPENDIX II. 393 "We can therefore determine the positions of the positive roots in the scale of numbers by continually diminishing them by unity; for when the roots have been diminished by a number which is greater than any one or more of the positive roots, we shall immediately be made aware of the fact by the loss of a cor- responding number of variations in the reduced equation. If we change the alternate signs of the equation, the negative roots will become positive and the positive negative, we can therefore determine the positions of the negative roots in precisely the same manner as we can those of the positive roots. If, in any such reduction of the equation, we should lose two or more variations, the roots thus indicated may be either equal, nearly equal, or, an even number of them may be imaginary. If they be nearly equal, we shall obtain their positions by reducing the roots of the equation by a number less than that last em- ployed. If they be equal, they will be determined by the aid of the derived equation. And if they be imaginary, we can have recourse to the equation whose roots are the reciprocals of those of the original : and the following considerations will show how it may be thus employed. Sudan's Criterion for Imaginary Roots. If the roots of an equation be reduced by a number p which is greater than m of its roots, the reciprocals of those roots will be greater than the reciprocal of that number, or - , and there- fore, if those roots be real, the number of variations left when the reciprocal equation is reduced by - will correspond to the number of variations lost when the original is reduced by p : hence if ni be the number of variations lost in the latter case and q the number left in the former, there must he q — m roots which are imaginary, since, if real, they must be less than p and their reci- procals also less than - , which is absurd. 394 APPENDIX It. Extension of Budan's Criterion; and Method of Separation OF Nearly Equal Koots. It does not follow that, when q — m = Oy (that is, when the number of variations left in the reduced reciprocal equation is the same as the number lost from the original) there are no imaginary roots indicated by the lost variations : but these will seldom fail to be made evident by continuing the reduction of the reduced reciprocal equation until all the variations left disappear, and then applying the same test as before; thus Let the given equation be a;* - 2ic* - 13ic' + S^aj* - 20a; +4 = 0. Reciprocal equation. 1- 2 -13 + 39- 20 4- 4(1 4-20 + 39-13- 2+ 1 (1 - 1-14 + 25+ 5+ 9 -16 + 23 + 10+ 8+ 9 ± 0-14 + 11 + 16 -12 + 11+21 + 29 + 1-13- 2 - 8+ 3 + 24 + 2-11 Two var. lost m = :2. - 4- 1 Two var. left g = 2 1+ 3-11- 2 + 16+ 9(1 4± 0- 1+24 + 29+ 9(1 + 4- 7- 9+ 7+16 + 4 + 3 + + + Two lost, + 5- 2-11- 4 Reciprocal of reduced reciprocal + 6+4-7 equation. + 7 + 11 9 + 29 + 24- 1± 0+ 4(1 1+ 8+11- 7- 4+16(1 + 38 + 62 + 61+61 + 65 + 9 + 20 + 13+ 9 + 25 None left, .-. roots imaginary. Two lost 16- 4- 7 + 11+ 8+ 1 (1 + 12+5+ + + None left. Should it prove necessary to apply this extended criterion, it is quite certain that the roots, if real, must be very nearly equal ; and this process, if carried far enough, cannot fail to separate them ; for suppose two roots r and r, , each less than 1 , to be very nearly equal, then the difference of their reciprocals , or r r^ APPENDIX II. 395 is evidently greater than their difference r^-r; and so it will be also in every successive reciprocal transformation. It will appear moreover that the positions of these corresponding roots in the successive reduced equations will enable us to determine their actual values; for, calling the original equation -4, we will suppose that two variations are lost between 3 and 4; then, calling the reciprocal equation 5 or — , (the subscript 3 indicating that the roots of A have been reduced by 3) we will suppose that the corresponding roots of this reciprocal equation lie between 2 and 3, and taking the equation which has the roots of equation B reduced by 2 and indicating it as before by B^^^ its reciprocal equation may be indicated by C or -— . Supposing the roots of 2 G to be between 5 and 6, we should have the roots of the ori- ginal, A^ between the values expressed by the two continued fractions 3 + — — and 3 + 2+i 2+i 5 . 6 38 45 that is, between the values -— and -- , or between 3*45 and 3*46. By continuing this process we must at length arrive at a stage where one variation is lost between two particular numbers, and the other between two other numbers; and the position of each root can therefore be obtained. Thus, supposing the above opera- tion had been carried a step further, and we found that one of the variations in C disappeared between 2 and 3, and the other between 4 and 5, we should then have the first root limited by the fractions 3 + and 3 + 1 1 + 2- ^ 5 + - 396 APPENDIX II. and the other by the fractions 1 3 + 2 + Li, and 3 + 2 + -!• that is, one of the roots would lie between 3*458 and 3*457 ; and the other between 3*4565 and 3 -4560. The roots would thus be separated, and we could obtain their actual values, either by con- tinuing the process, or by using the initial figures above obtained, in Horner's process upon the original equation. Horner's process for obtaining the roots of a numerical equa- tion consists in finding figure by figure a value which, when used in diminishing the roots of the given equation, will tend, more nearly than any other value, to make the last term of the reduced equation equal to zero, without necessitating the loss of the parti- cular variation which is dependent upon the root of which we are in search. It is thus, in fact, a mere extension of the process al- ready pointed out for the determination of the position of the roots. Thus, supposing we had to find one of the roots of the equation a;' - 7a; + 7 = 0. For the determination of the positions of the roots we have : Positive roots. (A) l±0-7 + 7 (1 +1-6+1 + 2-4 1+3-4+1 {1 + 4=t0+l + 5 + 5 1+6+5+1 Two variations lost. (A.1 Negative 1:^0- 7- 3+ 9- root. -7(3 f6 + 3+ 2- + 3+18 -1 + 6 + 20 + 3 (Bor -i-'\i-4 + 3+l (1 ^ ^^^ -3^0 + 1 -2-2 1-1-2+1 Two variations left. l+P + 20-1 The root is evidently between - 3 and - 4. .*. these roots are not imaginary. APPENDIX ir. 397 In order to find further figures of the negative root, to which we will first turn our attention, we must employ a figure in the first place of decimals which will diminish the terminal coefficient - 1 as much as possible without changing its sign. A little consider- ation will show that -1 is too large a value for this purpose, for, proceeding as before ; we have 1 + 9+20 - 1 (^-1 + -1+ -91 +2-091 9-1 +20-91 + and the variation upon which everything depends is gone. We must therefore in this case use a figure in the second place of decimals, and a few trials made mentally will show that -04 is the value to be used; thus, 1+9 +20 -1 (-04 + -04+ -3616 + -814464 + 9*04 + 20-3616 --185536 •04 + -3632 9-08 + 20-7248 •04 9-12 So that the reduced equation having its roots less by 3*04 than those of the original, is x^+9-\2x^^ + 20-7248a;j - '185536, and, if this last term had been reduced actually to zero, the value of the root would have been exactly 3-04. "We may continue the same process to any extent, and the next stei3 would be the employment of the digit 8 in the third place of decimals, since the -185 of the last coefficient divided by the 20-7 of the preceding sum gives -89 as an approximate multi- plier. Using this figure we now get : 398 APPENDIX II. 1+9-12. +20-7248.. --185536. .. ('008 •008 -073024 + -166382592 + 9-128 + 20-797824 - -019153408 + -008+ -073088 + 9-136 + 20-870912 -008 9-144 + 20-870912 - -019153408 It is obvious that a continuation of this process will encumber us with a large number of decimals which have no influence on the result, unless it be required in a very extended form ; we may therefore contract the work by cutting off the same number of figures from each coefficient, allowance being made for the increase in each which would otherwise take place, that is, 3 in the last, 2 in the last but one, and 1 in the last but two, or second coeffici- ent. So that in order to prevent any further increase in the figures of the last term we have only to cut off one figure from the last term but one; 2 from the last but two; 3 from the last but three, and so on, according to the degree of the equation ; thus. 1 + 91 44 + 20-87091|2- -019153408 {^'0009 823J + 18791226 20-87914| - 362182 823 20-88737 Since the two remaining figures of the second term will bo cut off at the next step, the process now becomes one of simple contracted division, and we have 20-8;8|7|3|7 - -000362182 (-00001734 208874 153308 146211 7097 6266 831 835 and the complete root is -3-04891734. APPENDIX II. 399 This is the process for the determination of any real root of an equation, and it may be carried out to any required extent : but it is obviously unnecessary to interrupt the continuity of the operation at each successive figure of the root by bringing down into a horizontal line the coefficients of the reduced equation ; omitting this the work would stand thus : -7 (,3-04891734. + 6 1±0 ~ 7 + 3 + 9 + 3 + 2 + 3 + 18 + 6 + 20 + 3 + 3616 + 9.. + 20-3616 + 04 + -3632 + 904. + 20-7248 . . + 04 + 73024 + 9*08 + 20-797824 + 04 + 73088 + 9-12 + 20-870912 8 823 + 9-128 + 20-87914 8 823 + 9-136 8 + 20-818i7l3|7 -1 814464 185536. .. 166382592 19153408 18791226 362182" 208874 153308 146211 7097 6266 831 835 + |9-1|44 The successive double lines in the several columns indicate the terminations in those columns, of the successive stages of the ope- ration as the root is evolved figure by figure. Iteverting now to the two positive roots between 1 and 2, we 400 APPENDIX ir. may attempt to separate them either by the use of decimal reduc- tions or by employing the extension of Budan's Criterion, which has ah-eady shown them to be probably real and nearly, if not quite, ec[ual. By the first method we have only to look for a change of sign in the last term, since, if the roots be separable, one of the variations must disappear without the other, and this too will be shown by the change of sign above mentioned. Using therefore decimal reductions, we have 1+3 -4 +1 (-1 •1+ -31- -369 + 3-1 -369+ too small, 1 + 3-4 +1 (^-2 + -2+ -64- -672 + 3 -2 - 3 -36 + too small, 1 + 3-4 +1 (,-3 + -3+ -99- -903 + 3-3 -3-01+ too small, 1 + 3-4 +1 (^'4 + -4 + 1-36- 1-056 + 3-4 - 2-64 - too great, .*. root between 1 3 and 1-4, 1+3-4 +1 (^-5 + -5 + 175 -1-125 + 3-5 -2-25- too small. 1+3-4 +1 (-6 + -6 + 2-16 -1-104 + 3 -6 - 1 -84 - too small. 1 + 3 -4 +1 (-7 + -7 + 2-59- -987 + 3*7 - 1-41 + too great, .*. root between 1*6 and 1-7- APPENDIX II. 401 By the method of reciprocals, we have, in continuation of that portion of the process already worked at p. 396 : (Bj) 1 — 1-2 + 1 (1 One root of this equation less than 1 ; ±0-2-1 /. one root of reciprocal equation (B) at p. SQG, + 1-1 less than 2. (Bg) 1 + 2-1-1 {J. One root of this equation less than 1; + 3 + 2+1 .'. one root of reciprocal equation (B) at p. S96. less than 3. The roots of the original are therefore separated, and are con- sequently neither equal nor imaginary; and, in order to find their approximate distinguishing values, we have only to revert to the original equation and, by noting the successive steps of our trans- formations, find the continued fractions which express the approxi- mate values of the two roots : we thus find one of the roots to be nearly a; = 1 + - , and the other nearly, x=l+-; or x=l'5 and x=\-3. These values cannot of course be viewed as more than a some- what rough approximation to the true values; but if we choose to carry the process a step further we can readily obtain more com- plete results ; thus, (^Oor^-^ 1-2-1 + 1 (1 (Doy^ 1 + 1-2-1(1 ^ ^^^ -1-2-1 ^ ^'^ +2d.0-l ±0-2 One root of (C) less than 1, and one (Cj) 1 + 1-2-1 (^1 root between 2 and 3 ; the first of + 2 ± - 1 these refers to the positive root of (B J and the latter to that of (B^). c. 26 402 APPENDIX II. And the approximate values will now be x= 1 + — . and x=l + — , OT x= l'6l and x= 1"J3. The equation (D) is the reciprocal of (B^), but it happens to be identical with (Cj) ; hence it appears that from this [)oiut the process for both roots gives identical results. It will be found in fact that one of the roots is represented by the continued fraction 111J_1111|1 ^~ "^1 + 2 + 4 + 20 +2+3+H-6+1+2' which gives the value x = 1*692021473 ; And the other by the continued fi-action a;=i+l iiJ_lll^ll 2 + 1+4 + 20 +2 + 3 + 1+ i5 + 1 + 2' which gives the value x= 1*35689587. The sum of these values is 3 0489 1734, the value of the nega- tive root, as it ought to be, since the coefficient of the second term of the original equation is =*= 0. To obtain these roots by Homer's operation is now quite sim- ple; sin.ce we liave previously determined the fii-st decimal of each root, we have only to make use of it, and act, afterwards, according to circumstance, as in the determination of the negative root. The operation will stand thus : APPENDIX II. 403 1±0 + 1 + 1 + 1 +¥ + 1 4-3-3 + -3 + 36 + -3 ■f 3-95 + 5 + 4-00 + 5 + 4-056 + 6 + 4-062 + 6 + 4-0)68 -7 + 1 + 2 -4 .. + w -301 + 1-08 '36 -l'9S + 197^ -1-732^ + 200C -1-532^ + 24S -1-508164 + 24372 -1-48379 2 + 325 -1-48054 + 325 -1-4772 + 3 9 -1-4769 + S + 7 -6 + 1... 903 (^1-356895867 97.. _86625 10375.. . 9048984 1326016 1184429 141587 132922 8665 7382 1283 1181 102 89^ 13 10 3 -1 -47616 -404 APPENDIX II. And, for tlie root between 1*6 and 1-7, tlius 1±0 -7 + 1 + 1 + 1 -6 + 1 + 2 + 2 -4.. + 1 + 2-16 + 3-6 -1-84 + 6 + 252 + 4-2 + 68.. + 6 + 4401 44 + 4-89 + 9 + 11201 + 4482 + 4-98 + 9 + 15683 + 101 + 5 072 4- 2 + 1578444 + 10148 5-074 2 + 15885 + 1 92 i5-o;76 15886 1 + 7 0*69202147 -6 + 1 ... 1104 •lOi.. . 100809 3191... 3156888 34112 31772 2340 1589 "75! 635 T16 111 158871 For further examples, tlie student is referred to the body of the work. cambeidqb: printed at the university press. October, 1873. A Catalogue of Educational Books, with a Short Account of their Character and Aim^ Published by MACMILLAN AND CO.. Bedford Street, Strand, London, CLASSICAL. iEschyluS.— ^SCHYLI EUMENIDES. The Greek Text, with English Notes and English Verse, Translation, and an Introduction. By Bernard Drake, M.A,, late Fellow of King's College, Cambridge. 8vo. 3J-. 6d. The Greek text adopted in this Edition is based upon that of Wellauer. 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This method he has found to be eminently successful as a means of furnishing the students with a copious vocabulary, training them to use it promptly, confidently, and with correct articttlatiott, and instilling into them an accurate and intelligent knowledge of Greek Grammar, which he hopes may aid otJier teachers in realizing the same ends. The presetii little volume furnishes a series of twenty- five gradu- ated dialogues in parallel columns of Greek and English on a great variety of interesting subjects. The author has had the advantage of submitting his work to the judgment of several scholars of repute, both English and Scotch. The Globe says: *^ Professor Blackie s system is sensible; his book is likely to be useful to teachers of Greek; and his suggestions valuable to the learners of any language." CLASSICAL. 3 Cicero.— THE SECOND PHILIPPIC ORATION. With an Introduction and Notes, translated from the German of Karl Halm. Edited, with Corrections and Additions, by John E. B. Mayor, M.A., Fellow and Classical Lecturer of St. John's College, Cambridge. Fourth Edition, revised. Fcap. 8vo. 5^. This volume opens with a List of Books useful to the Student of Cicero ^ and some accaunt of various editions^ mostly German, of the works of Cicero. The Introduction is based on Halm. The English editor has further illustrated the work by additions drawn, for the ?nost fart, (l) from the ancient authorities ; (2) from his own private marginal references, and from collections; {'^ from the notes of previous com- mentators. A copious ^argument'' is also given. ** On the whole we have rarely met %vith an edition of a classical author which so thoroughly fulfils the requirements of a good school-book.'" — EDUCATIONAL TiMES. *• ^ valuable edition,'' says the Athen^UM. THE ORATIONS OF CICERO AGAINST CATILINA. With Notes and an Introduction. Translated from the German of Karl Halm, with many additions by A. S. Wilkins, M.A. Professor of Latin in Owens College, Manchester. 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This book may be used in connection with Mayor's '■'■Greek for Beginners.''^ ^^ After a careful examination we are inclined to consider this volume unrivalled in the hold which its pithy sentences are likely to take on the memory, and for the amount of true scholarship embodied in the annotatiojts." — Educational Times. Mayor (Joseph B.)— greek for BEGINNERS. By the Rev. J. B. Mayor, M.A., Professor of Classical Literature in King's College, London. Part L, with Vocabulary, is. 6d. Parts II. and HI., with Vocabulary and Index, -t^s. 6d., complete in one vol. Fourth Edition. Fcap. 8vo. cloth, 4J. 6d. The distinctive method of this book consists in building up a boy's knowledge of Greek upon the foimdation of his knowledge of English and EDUCATIONAL BOOKS, Latin^ instead of trusting everything to the unassisted memory, Greek •words have been used in the earlier part of the book except such as have connections either in English or Latin. Each step leads naturally on to its successor; grammatical forms and rules are at once applied in a series of graduated exercises ^ accompanied by ample vocabularies. Thus the book serves as Grammar, Exercise book^ and Vocabulary, 7he ordinary ten declensions are reduced to three, which correspond to the first three in Latin ; and the system of stems is adopted, A general Vocabulary, and Index of Greek words, completes the work. " We know pf no book of the same scope so complete in itself, or so well calculated to make the study of Greek interesting at the very commencement.^'' — Standard. Peile (John, M.A.)— AN INTRODUCTION TO GREEK AND LATIN ETYMOLOGY. By John Peile, M. A., Fellow and Assistant Tutor of Christ's College, Cambridge, formerly Teacher of Sanskrit in the University of Cambridge. New and Revised Edition. Crown 8vo. \os. 6d. These Philological Lectures are the result of Notes made during the author's reading fjr sroeral years. These Notes were put into the shape of Lectures, delivered at Christ'' s College, as one set in the ^'Intercollegiate'^ list. They are now printed with some additions and modifications. * * The book may be accepted as a very valuable contribution to the science of language.''' — Saturday Review. Plato.— THE REPUBLIC OF PLATO. Translated into English, with an Analysis and Notes, by J. Ll. Davies, M.A., and D. J. Vaughan, M.A. Third Edition, with Vignette Portraits of Plato and Socrates, engraved by Jeens from an Antique Gem. i8mo. 4J. (id. An introductory notice siipplies some account of the life of Plato, and the translation is preceded by an elaborate analysis. ** The translators have,'' in the judgment of the Saturday Review, '^ produced a book which any reader, whether acquainted with the original or not, can peruse with pleasure as well as profit. " Plautus (Ramsay) — the MOSTELLARIA OF PLAU- TUS. With Notes Critical and Explanatory, Prolegomena, and Excursus. By William Ramsay, M.A., formerly Professor of Humanity in the University of Glasgow. Edited by Professor George G. Ramsay, M.A., of the University of Glasgow. 8vo. i\s. CLASSICAL, 7 " The fruits of thai exhaustive research and that ripe and well-digested scholarship which its author brous;ht to bear upon everything that he undertook are visible throughout. It is furnished with a complete apparatus of prolegomena^ noteSy and excursus; and for the use of veteran scholars it probably leaves nothing to be desired" — Pall Mall Gazette. Potts (Alex. W., M. A.)— HINTS towards latin PROSE composition. By Alex. W. Potts, M.A., late Fellow of St. John's College, Cambridge ; Assistant Master in Rugby School j and Head Master of the Fettes College, Edinburgh. Third Edition, enlarged. Extra fcap. 8vo. cloth. 3^. An attempt is here made to give students^ after they have mastered ordinary syntactical rules, some idea of the characteristics of Latin Prose and the means to be employed to reproduce them. Some notion of the treatment of the subject may be gathered from the * Contents.^ Chap. I. — Characteristics of Classical Latin, Hints on turning English into Latin ; Chap. II. — Arrangement of Words in a Sentence; Chap. III. — Unity in Latin Prose, Subject and Object ; Chap. IV. — On the Period in Latin Prose ; Chap. V. — On the position of the Relative and Relative Clauses. The Globe characterises it as ^' an admirable little book which teachers of Latin will find ofveiy great service.''^ Roby.— A GRAMMAR OF THE LATIN LANGUAGE, from Plautus to Suetonius. By H. J. Roby, M.A., late Fellow of St. John's College, Cambridge. Part I. containing : — Book I. Sounds. Book II. Inflexions. Book III. Word-formation. Appendices. Second Edition. Crown 8vo. %s, 6d. This work is the result of an independent and careful study of the writers of the strictly classical period, the period embraced between the t if ?ie of Plautus and that of Suetonius. The author's aim has been to give the facts of th; language in as few words as possible. This is a Grammar strictly 0ftheL.21.tm language ; not a Universal Grammar illustrated from Latin, nor the Latin section of a Comparative Grammar of the Indo' Etcropean languages, nor a Grammar of the group of Italian dialects, 0/ which Latin is one. It will be found that the arrangement of the book and the treatfnent of the various divisions differ in many respects from those of previous grammars. Mr. Roby has given special prominence to the treatment of Sounds and Word formation ; and in the First Book he has done much towards settling a discussion which is at present largely engaging the attention of scholars, viz., the pronunciation of the classical languages. EDUCATIONAL BOOKS: ** The book is marked by the clear and practised insight of a master tn his art. It is a book that would do honour to any country y — ATHENAEUM. Rust.— FIRST STEPS TO LATIN PROSE COMPOSITION. By the Rev. George Rust, M,A. of Pembroke College, Oxford, Master of the Lower School, King's College, London. New Edition, i8mo. \s. 6d. This little work consists of carefully graduated vocabularies and exercises, so arranged as gradually to familiarise the pupil with the elements of Latin Prose Composition, and fit him to commence a more advanced work. Sallust.— CAII SALLUSTII CRISPI CATILINA ET JUGUR- THA. For Use in Schools. With copious Notes. By C. Merivale, B.D. (In the present Edition the Notes have been carefully revised, and a few remarks and explanations added.) New Edition. Fcap. 8vo. 4J. dd. This edition of Sallust, prepared by the distinguished historian of Rome, contains an Introduction, concerning the life and works of Sallust, lists of the Consuls, and elaborate Notes. ** A very good edition, to which the Editor has not only brought scholarship but independent judgment and historical criticism:^ — S pectator. The JUGURTHA and the CATILINA may be had separately, price 2.S. 6d. each. Tacitus. — THE HISTORY OF TACITUS TRANSLATED INTO ENGLISH. By A. J. Church, M.A., and W. J. Brodribb, M.A. With Notes and a Map. New and Cheaper Edition. Crown 8vo. 6s. The translators have endeavoured to adhere as closely to the original as was thought consistent with a proper observance of English idiom. At the same time, it has been their aim to reproduce the precise expressions of the author. The campaign of Civilis is elucidated in a note ofsotne lengthy which is illustrated by a map, containin^^ the names of places and of tribes occurring in the work. There is also a complete account of the Roman army as it was constituted in the time of Tacitus. This work is characterised by the Spectator as "a scholarly and faithful translation.'''' THE AGRICOLA AND GERMANIA OF TACITUS. A Revised Text, English Notes, and Maps. By A. J. CHURCH, M.A., and W. J. Brodribb, M.A. New Edition. Fcap. 8vo. 3^. dd. CLASSICAL. Tacitus — continued. " We have endeavozired, with the aid of recent editions^ thoroughly lo elucidate the text, explaining the various difficulties, critical a?id gramma- tical, which occur to the student. We have constdted throughotd, besides the older commentators, the editions of Ritter ajtd Orelli, but we are under special obligations to the labours of the recent German editors, Wex and Kritz." Two Indexes are appended, (i) of Proper Names, (2) of Words and Phrases explained. "A model of careful editing,''^ says the Athenaeum, " being at once compact, cojnplde, and correct, as well as neatly printed and elegant in style.^^ THE AGRICOLA and GERMANIA may be had separately, price 2.S. each. THE AGRICOLA AND GERMANIA. Translated into English by A. J. Church, M.A., and W. J. Brodribb, M.A. With Maps and Notes. Extra fcap. 8vo. 2s. 6d> The translators have sought to produce such a version as may satisfy scholars who demand a faithfid rendering of the original, and English readers who are offerided by the baldness and frigidity which commonly disfigure translations. The treatises are accompanied by Introductions, Notes, Maps, and a chronological Sunimary. The Athen^UM says of this work that it is " a version at once readable atid exact, which may be perused with pleasure by alh^ ajtd considted with advantage by the classical student." Theophrastus. — the characters of theo- PHRASTUS. An English Translation from a Revised Text. With Introduction and Notes. By R. C. Jebb, M.A., Public Orator in the University of Cambridge. Extra fcap. 8vo. ds. 6d. The first object of this book is to make these lively pictures of old Greek manners better known to English readers. Bid as the Editor and Trans- lator has been at considerable pains to procure a trustworthy text, and has recorded the results of his critical labours in an Introduction, Notes, and Appendices, it is hoped that the work will p'ove of value even to the scholar. *' We must not omit to give due honour to Mr. Jebb's trans- lation, which is as good as translation can be ... . Not less commendable are the execution of the Notes and the critical handling of the Text. " — Spectator. The Saturday Review speaks of it as '^ a very hatidy lo EDUCATIONAL BOOKS. and scholarly edition of a loork which till now has been beset with hindrances and difficultieSy but zukich Mr. Jebb's critical skill and judgment have at length plcuai within the grasp and compreJiaision of ordinary readers. " Thring.—Works by the Rev. E. THRING, M.A., Head Master of Uppingham School. A LATIN GRADUAL. A First Latin Construing Book for Beginners. New Edition, enlarged, with Coloured Sentence Maps. Fcap. 8vo. 2J. dd. The Head Master of Uppingham has here sought to supply b^> easy steps a knowledge of grammar ^ combitud with a good Vocabulary. Passages have been selected from the best Latiti authors in prose and verse. These passages are gradually built up in their grammatical structure^ and finally printed in full. A short practical manual of common mood con- structionsy with their English equivalents, forms a second part. To the New Edition a circle of grammatical Constructions with a Glossary has been added ; as also some coloured Sentence Maps , by niectns of which the different parts of a sentence can easily be distinguished, and the practice of dissecting phrases carried out with the greatest benefit to the student. A MANUAL OF MOOD CONSTRUCTIONS. Fcap.Svo. \s. dd. Treats of the ordinary mood constructions, cts found in the Latin, Greeks and English languages. The EDUCATIONAL TIMES thinks it ^'very well suited to young students." A CONSTRUING BOOK. Fcap. 8vo. 2s. 6d. Thucydides. — the SICILIAN EXPEDITION. Being Books VI. and VII. of Thucydides, with Notes. A New Edition, revised and enlarged, with a Map. By the Rev. Percival Frost, M. A., late Fellow of St. John's College, Cambridge. Fcap. 8vo. $s. This edition is mainly a grammatical one. Attention is called to the force of compound verbs, attd the exact meaning of the various tenses employed. " The notes are excellent of their kind. Mr. Frost seldom passes over a difficulty, and what he says is always to the point. " — Educational Times. Virgil.— THE WORKS OF VIRGIL RENDERED INTO ENGLISH PROSE, with Notes, Introductions, Running Analysis, and an Index, by James Lonsdale, M.A. and Samuel Lee, M.A. Second Edition. Globe 8vo. 3^. da. \ gilt edges, 45. dU. CLASSICAL. II The original has been faithfully rendered, and paraphrase altogetJier avoided. At the same time, the translators have endeavoured to adapt the book to the use of the English reader. Some amount of rhythm in the structure of the sentence has been generally maintained ; and, when in the Latin the sound of the words is an echo to the sense i^as so frequently happens in Virgil), an attetnpt has been made to produce the same result in English. The general introduction contains whatever is known of the poet's life, an estimate of his genius, an account of the principal editions and translations of his works, and a brief view of the influence he has had on modern poets ; special introductory essays are prefixed to the * * Eclogues, " * * Georgics, ' ' and * '^Eneid. " The text is divided into sections, each of which is headed by a concise analysis of the subject ; the Index contains references to all the characters and events of any importance. " A more complete edition of Virgil in English it is scarcely possible to conceive than the scholarly work before us."*' — Globe. Wright. — Works by J. WRIGHT, M.A., late Head Master of Sutton Coldfield School. HELLENICA ; OR, A HISTORY OF GREECE IN GREEK, as related by Diodorus and Thucydides ; being a First Greek Reading Book, with explanatory Notes, Critical and Historical. Third Edition, with a Vocabulary. i2mo. 3^. dd. In the last twenty chapters of this volume, Thucydides sketches the rise and progress of the Athenian Empire in so clear a style and in such simple language, that the editor has doubts whether any easier ar more instruc- tive passages can be selected for the use of the pupil who is commencing Greek. This book includes a chronological table of the events recorded. The Guardian speaks of the work as ^^ a good plan well executed.'''' A HELP TO LATIN GRAMMAR ; or, The Form and Use of Words in Latin, with Progressive Exercises. Crown 8vo. 4^. 6d. This book is not intended as a rival to any of the excellent Grammars now in use ; but as a help to enable the beginner to understand them. THE SEVEN KINGS OF ROME. An Easy Narrative, abridged from the First Book of Livy by the omission of Difficult Passages ; being a First Latin Reading Book, with Grammatical Notes. Fifth Edition. Fcap. 8vo. 3J. With Vocabulary, 3J. 6^. 12 EDUCATIONAL BOOKS. Wright — continued. This work is intended to suffply the pupil with an easy construing book^ which may at the same time be made the vehicle for instructing him in the rules of grammar and principles of composition. The notes profess to teach what is commonly taught in grammars. It is conceived that the pupil will learn the rules of construction of the language much more easily from separate examples, which are pointed out to him in the course of his reading, and which he may himself set down in his note-book after some scheme of his own, than from a heap of quotations amassed for him by others. ** The Azotes are abundant, explicit, andfull of such grammatical and other information as boys 7'equire." — Athenaeum. ** This is really," the Morning Post says, *' what its title imports, and we believe that its general introduction into Gramnuir Schools would not only facilUate the progress of the boys beginning to learn Latin, but also relieve the Masters from a very considerable amount of irksome labour . . . . a really valuable addition to our school libraries.''* FIRST LATIN STEPS; OR, AN INTRODUCTION BY A SERIES OF EXAMPLES TO THE STUDY OF THE LATIN LANGUAGE. CrowTi 8vo. 5j. The following points in the plan of the work may be noted: — I. The pupil has to deal with only one construction at a time. 2. This construction is made clear to him by an accumulation of instances. 3. As all the constructions are classified as they occur, the construction in each sentence can be easily referred to its class. 4. As the author thinks the pupil ought to be thoroughly familiarized, by a repetition of instances, with a construction in a foreign language, before he at- tempts himself to render it in that language, the present volume contains only Latin sentences. 5. The author has added to the Rules on Prosody in the last chapter, a few familiar lines from Ovid^s Fasti by way of illustration. In a h'ief Introduction the author states the rationale of the principal points of Latin Grarnmar. Copious Azotes are appended, to which reference is made ifi the text. From the clear and rational method adopted in the arrangement of this elementary work, from the simple way in which the various rules are conveyed, and from the abun- dance of examples given, both teachers and pupils will find it a valuable help to the learning of Latin, CLASSICAL. 13 CLASSIC VERSIONS OF ENGLISH BOOKS AND LATIN HYMNS. The following works are, as the heading indicates, classic renderings of English Books. For scholars, and parti- cularly for writers of Latin Verse, the series has a special value. The Hymni Ecclesiae are here inserted, as partly falling under the same class. Church (A. J., A.M.)— HOR^ TENNYSONIAN^, sive Eclogae e Tennysono. Latine redditse. Cura A. J. Church, A.M. Extra fcap. 8vo. 6s, Latin versions of Selections from Tennyson. Among the authors are the Editor, the late Professor Conington, Professor Seeley, Dr. Hessey, Mr, Kebbel, and other gentlemen, Latham. — SERTUM SHAKSPERIANUM, Subnexis aliquot aliunde excerptis floribus. Latine reddidit Rev. H. Latham, M. A. Extra fcap. 8vo. 5^-. Besides versions of Shakespeare this volume contains, among other pieces, Gray's '' Elegy" Campbell's '' Hohenlinden," Wolfe's'' Burial of Sir John Moore" and selections from Cowper and George Herbert. Lyttelton. — the COMUS of MILTON, rendered into Greek Verse. By Lord Lyttelton. Extra fcap. 8vo. 5^. THE SAMSON AGONISTES OF MILTON, rendered into Greek Verse. By Lord Lyttelton. Extra fcap. Svo. 6j. (>d. 14 EDUCATIONAL BOOKS. Merivale. — KEATS' HYPERION, rendered into Latin Verse. By C. Merivale, B. D. Second Edit. Extra fcap. Svo. 3^-. dd. Newman. — HYMNI ECCLESI^E. Edited by the Rev. Dr. Newman. Extra fcap. Svo. is. 6d. Hymns of the Meduroal Church. The first Part contains selections from the Parisian Breviary ; the second from those, of Rome ^ Salisbury ^ and York. Trench (Archbishop). — SACRED LATIN POETRY, chiefly Lyrical, selected and arranged for Use ; with Notes and Introduction. Fcap. Svo. 7J. In this work the editor has selected hymns of a catholic religious sentiment that are common to Christendom^ while rejecting those of a distinctively Romish character. MATHEMATICS. 15 MATHEMATICS. Airy. — Works by Sir G. B. AIRY, K.C.B., Astronomer Royal :— ELEMENTARY TREATISE ON PARTIAL DIFFERENTIAL EQUATIONS. Designed for the Use of Students in the Univer- sities. "With Diagrams. Crown 8vo. cloth, t^s. dd. It is hoped that the methods of solution here explained^ and the instances exhibited^ xmll be found siifficiettt for application to nearly all the important problems of Physical Science, zvhich require for their co7npleie investigation the aid of Partial Differential Equations. ON THE ALGEBRAICAL AND NUMERICAL THEORY OF ERRORS OF OBSERVATIONS AND THE COMBINA- TION OF OBSERVATIONS. Crown 8vo. cloth. (>s. 6d. In order to spare astronomers and observers in natural philosophy the confusion and loss of time zvhich are produced by referring to the ordinary treatises embracing both branches of probabilities (the first relating to chances which can be altered only by the changes of entire units or in- tegral midty>les of units in the fundamental conditions of the problem ; the other concerning those chances which have respect to insensible grada- tions in the value of the element measured), the present tract has been drawn up. It relates only to errors of observation, and to the rules, derivable from the consideration of these errors, for the combination of the results of observaiiotis. 1 6 EDUCATIONAL BOOKS. Airy (G. B.) — continued. UNDULATORY THEORY OF OPTICS. Designed for the Use of Students in the University. New Edition. Crown 8vo. cloth. The nndidatory theory of optics is presented to the reader as having the same claims to his attention as the theory of gravitation : namely^ that it is certainly true, and that, by mathematical operations of general elegance, it leads to results of great interest. This theory explains with accuracy a vast variety of phenomena of the most complicated kind. The plan of this tract has been to include those phenomefia only which admit of calculation, and the investigations are applied only to phenomena which actually have been observed. ON SOUND AND ATMOSPHERIC VIBRATIONS. With the Mathematical Elements of Music. Designed for the Use of Students of the University. Second Edition, Revised and Enlarged. CroviTi 8vo. 9J. This volume consists of sections, 7vhich again are divided into numbered articles, on the follo2uing topics: — General recognition of the air as the medium which conveys sound ; Properties oj the air on which the forma- tion and transmission of sound depend ; Theory of undulations as applied to sound, &^c. ; Investigation of the motion of a wave of air through the atmosphere ; Transmission of waves of soniferous vibrations through dif- ferent gases, solids, and fluids ; Experiments on the velocity of sound, dr'c. ; On musical sounds, and the manner of producing them ; On the elements of musical harmony and melody, and of simple ?nusical composi- tion ; On instrumental music ; On the human organs of speech and hearing. A TREATISE OF MAGNETISM. Designed for the use of Students in the University. Crown 8vo. qj. 6d. As the laws of Magnetic Force have been experimentally examined with philosophical accuracy, only in its connection with iron and steel, and in the influences excited by the earth as a whole, the accurate portions of this work are confined to the investigations connected with these matals and the earth. The latter part of the work, however, treats in a more general way of the laws of the connection between Magnetism on the other hand and Gal- vanism and Thermo-electricity on the other. The work is divided into Twelve Sections, and each section into numbered articles, each of which states concisely the subject of the following paragraphs. AIA THEMA TICS. 1 7 Airy (Osmund).— A treatise ON GEOMETRICAL OPTICS. Adapted for the use of the Higher Classes in Schools. By Osmund Airy, B.A., one of the Mathematical Masters in Wellington College. Extra fcap. 8vo. 3J. 6^. " This is, I imagine, the first time that any attempt has been made to adapt the subject of Geometrical Optics to the reading of the higher classes in our good schools. That this should be so is the more a matter for remark, since the subject 7vould appear to be peculiarly fitted for such an adaptation / have endeavoured, as much as possible, to avoid the example of those popular lecturers 7uho explain difficulties by ignoring them. But as the nattire of my design necessitated brevity, I have omitted entirely one or two portions of the subject which I considered unnecessary to a clear understanding of the rest, and which appear to me better learnt at a more advanced stage.''' — Author's Preface. " This book,'' the Athen^UM says, **w carefully and lucidly written, and rendered as simple as possible by the use in all cases of the most elementary form of investigation. " Bayma. — THE ELEMENTS OF MOLECULAR MECHA- NICS. By Joseph Bayma, S.J., Professor of Philosophy. Stonyhurst College. Demy 8vo. cloth. los. 6d. Of the Twelve Books into which the present treatise is divided, the fir it and second give the demonstration of the principles which bear directly on the constitution and the properties of matter. The next three books contain a series of theorems and of problems on the laws of motion of elementary substances. In the sixth and seventh, the mechanical constitution of mole- cules is investigated and determined : and by it the general properties of bodies are explained. The eighth book treats of luminiferous cethei'. The ninth explains some special properties of bodies. The tenth and eleventh contain a radical and lengthy investigation of chemical principles and relations, which may lead to practical results of high importance. The tivelfth and last book treats of molecular masses, distances, and powers. Beasley.— AN ELEMENTARY TREATISE ON PLANE TRIGONOMETRY. With Examples. By R. D. Beasley,. M.A., Head Master of Grantham Grammar School. Fourth Edition, revised and enlarged. Crown 8vo. cloth. 3^. dd. This treatise is specially intended for use in schools. The choice of matter has been chiefly guided by the requirements of the three days' examination B 1 8 EDUCATIONAL BOOKS. at Cambridge. About four hundred examples were added to the second edition, mainly collected from the Examination Papers of the last ten years. In this edition several nnv articles have been added, the exatnplcs have been largely increased, and a series of Examination Papers appended. Blackburn (Hugh). — ELEMENTS OF PLANE TRIGONOMETRY, for the use of the Junior Class of Mathematics in the University of Glasgow. By Hugh Blackburn, M.A., Professor of Mathematics in the University of Glasgow. Globe 8vo. IS. 6d. The author having felt the want of a short treatise to be used as a Text-Book after the Sixth Book oj Euclid had been learned and some knowledge of Algebra acquired, which should contain satisfactory de- monstrations of the propositions to be used in teaching Junior Students the solution of Tria?tgles, and should at the same time lay a solid foundation for the study of Analytical Trigonometry, thinking that others may have felt the same want^ has attempted to supply it by the publication of this little work. Boole. — Works by G. BOOLE, D.C.L., F.R.S., Professor of Mathematics in the Queen's University, Ireland. A TREATISE ON DIFFERENTIAL EQUATIONS. New and Revised Edition. Edited by I. Todhunter. Crown Svo. cloth. 14J. Professor Boole has endeavoured in this treatise to convey as complete an account of the present state of knowledge on the subject of Differential Equa- tions as was consistent with the idea of a work intended, primarily, for elementary instruction. The earlier sections oj each chapter contain that kind of nuitier which has usually been thought suitable for the beginner, -uhile the latter ones are devoted either to an account of recent discovery, or the discussion of such deeper questions of principle as are likely to present themselves to the reflective student in connection with the methods and processes of his previous course. " A treatise incomparably superior to any other elcmenLary book on the same subject with which we are acquainted. " — riiiLusoriiiCAL Magazine. A TKEATLSE on DIFFERENTIAL EQUATIONS. Supple- mcr)ta:y Volume. Edited by 1. ToOHUXTEK. Crown Svo. cloth. 8j. Gd. MATHEMATICS. 19 Boole — conti7i iced. This volufne contains all that Professor Boole wrote for the purpose of enlarging his treatise on Differential Equations. THE CALCULUS OF FINITE DIFFERENCES. Crown 8vo. cloth. \os. 6d. New Edition, revised by J. F. Moulton. In this exposition of the Calculus of Finite Differences^ particular attettiion has been paid to the connection of its methods with those of the Differential Calculus — a connection which in some instances involves far more than a merely formal analogy. The work is in some measure designed as a sequel to Professor Boole's Treatise on Differential Equa- tions. ^^ As an original book by one op the first matheinaticians of the age, it is out of all comparison with the mere second-hand compilations which have hitherto been alone accessible to the student. ^^ — PHlLOSOrniCAL Magazine. Brook -Smith (J.)— ARITHMETIC IN THEORY AND PRACTICE. By J. Brook-Smith, M.A., LL.B., St. John's College, Cambridge ; Barrister-at-Law ; one of the Masters of Cheltenham College. Complete, Crown 8vo. 4^. dd. Part I. 3J. dd. Writers on Arithmetic at the present day foel the necessity of explaining the principles on which the rides of the subject are based, but few as yet feel the necessity of making these explanations strict and complete. If the science of Arithmetic is to be viade an effective instrument in developing and strengthening the mental powei's , it ought to be worked out rationally and conclusively ; and in this work the author has eitdeavoured to reason out in a clear atid accurate manner the leading propositions of the scie7tce, and to illustrate and apply those propositions in practice. In the practical part of the subject he has advanced somewhat beyond the majoiity of preceding writers; particularly in Division, in Greatest Common Measure, in Cube Root, in the Chapters on Decimal Money and the Metric System, and more especially in the applicatiom of Decimals to Percentages and cognate subjects. Copious examples, origitial and selected^ are given. *' This strikes us as a valuable Manual of Arithmetic of the Scientific kind. Indeed, this really appears to us the best we have seen.^'' — Literary Churchman. " litis is an essentially practical book, providing veiy definite help to candidates for almost eveiy kind of com - petUive examination.''— ^KYYi%Yi. Quarterly. B 2 20 EDUCATIONAL BOOKS. Cambridge Senate-House Problems and Riders, WITH SOLUTIONS :— 1848-1851.— PROBLEMS. By Ferrers and Jackson. 8vo. cloth. 15^. (>d. 1848-185 1. —RIDERS. By Jameson. 8vo. cloth, ^js.dd. 1854. — PROBLEMS AND RIDERS. By Walton and Mackenzie. 8vo. cloth. \os. (>d. 1857. — PROBLEMS AND RIDERS. By Campion and Walton. 8vo. cloth. 8j. 6d. i860.— PROBLEMS AND RIDERS. By Watson and RouTH. Crown 8vo. cloth. 7J. 6d. 1864.— PROBLEMS AND RIDERS. By Walton and Wil- kinson. 8vo. cloth, los. 6d. These volumes will be found of great value to Teachers and Students, as indicatbig the style and range of matheviatical study in the University of Cambridge. CAMBRIDGE COURSE OF ELEMENTARY NATURAL PHILOSOPHY, for the Degree of B.A. Originally compiled by J. C. Snowball, M,A., late Fellow of St. John's College. Fifth Edition, revised and enlarged, and adapted for the Middle- Class Examinations by Thomas Lund, B.D., Late Fellow and Lecturer of St. John's College, Editor of Wood's Algebra, &c. Crown 8vo. cloth. 5j. This work will be found adapted to the wants, not only of University Students, but also ofmatty others who require a short course of Mechanics and Hydrostatics, and especially of the candidates at our Middle Class Exatninaiions. At the end of each chapter a series of easy qttestions is added for the exercise of the student. CAMBRIDGE AND DUBLIN MATHEMATICAL JOURNAL. The Complete Work, in Nine Vols. 8vo. cloth. 7/. 45-. Only a few copies rcTnain on hand. Among Contributors to thts work will be found Sir W. Thomson, Stokes, Adams, Boole, Sir W. R. Hamilton, De Morgan, Cayley, Sylvester, Jellett, and other distinguished mathematicians. Candler.— HELP TO ARITHMETIC. Designed for the use of Schools. By H. Candler, M.A., Mathematical Master of Uppingham School. Extra fcap. 8vo. 2s. 6d. MA I HE MA TICS. 2 1 This zvork is intended as a companion to any text- book that may be tn use. *' The main difficulties which boys experience in the different rules are skilfully dealt with and removed.^^ — Museum. Cheyne. — Works by C. H. H. CHEYNE, M.A., F.R.A.S. AN ELEMENTARY TREATISE ON THE PLANETARY THEORY. With a Collection of Problems. Sfccond Edition. Crown 8vo. cloth, ds. 6d. In this volume an attempt has been made to produce a treatise on the Planetary theory, which, being elementary in character, should be so far cotnplete as to contain all that is usually required by students in the University of Cambridge. In the New Edition the work has been carejully revised. The stability of the Planetary System has been more fully treated, and an elegant geo?netrical explanation of the formulce for the secular variation of the node and inclination has been introduced. THE EARTH'S MOTION OF ROTATION. Crown Svo. 3^. dd. The first part of this work consists of an application of the method of the variation of elemetits to the getteral problem of rotation. In the second part the general rotation formulce are applied to the particular case of the earth. Childe.— THE SINGULAR PROPERTIES OF THE ELLIP- SOID AND ASSOCIATED SURFACES OF THE Nth DEGREE. By the Rev. G. F. Childe, M.A., Author of " Ray Surfaces," " Related Caustics," &c. Svo. \os. 6d. The object of this volume is to develop peculiarities in the Ellipsoid ; and, further, to establish analogous properties in the unlimited congeneric series of which this remarkable surface is a constituent. Christie. — a COLLECTION OF ELEMENTARY TEST- QUESTIONS IN PURE AND MIXED MATHEMATICS ; with Answers and Appendices on Synthetic Division, and on the Solution of Numerical Equations by Horner's Method. By James R. Christie, F.R.S., late First Mathematical Maste. at the Royal Military Academy, Woolwich. Crown Svo. cloth. Sj. 6d. This manual is nozv extensively used in Schools and Colleges y both in England and in the Colonies. It has also been found of great service for students preparing for the Middle Class and Civil and Military Service Examinations y from the care that has been taken to elucidate the principles of all the rules. The present edition has been carefully revised. ' ' To all those whose minds are sufficiejttly developed to comprehend the simplest mathematical reasoning, and 7vho have not yet thoroughly mastered the principles of Arithmetic and Algebra, it is calculated to be of great advantage. " — Athen^um. Of this 7Uork, also, one of the highest possible authorities, the late Dean Peacock, writes : ^^ Mr. Smith's work is a most useful publication. The rules are stated with great clearness. The examples are well selected, and worked out with just sufficient detail, ivithout being encu??ibered by too minute explanations ; and there prevails throughout it that just proportion of theory and practice which is the crowning excellence of an elementary work.'*' ARITHMETIC FOR SCHOOLS. New Edition. Cro%\Ti Svo. cloth. 4^. dd. Adapted from the atithor's work on ^^ Ainthmetic and Algebra,^' by the omission of the algebraic portion, and by tlu introduction ot ne^u exercises. The reason of each arithmetical process is fully exhibited. The system cj Decimal Coinage is explained ; and ans7uers to the exercises are appended at the end. The Arithmetic is characterised as ** admirably adapted for instruction, combining jtid sufUciejtt theory 7uith a large and well-selected collection of excercises for practice. ""^ — ^Journal of Education. MATHEMATICS. zz Barnard Smith — continued. A KEY TO THE ARITHMETIC FOR SCHOOLS. Tenth Edition. Crown 8vo. cloth. 8j. dd. EXERCISES IN ARITHMETIC. With Answers. Crown 8vo. imp cloth. 2J-. 6^. Or sold separately, Part I. \s. ; Part II. is.; Answers, 6d. These Exeixises have been published in order to give the pupil examples in every rule of Arithmetic. The i^r eater number have been carefully compiled Jrom the latest U?tiversity and School Examination Papers. SCHOOL CLASS-BOOK OF ARITHMETIC. i8mo. cloth. 3^-. Or sold separately, Parts I, and 11. \od. each; Part HI. is. This jnanual, published at the request of many schoolmasters^ and chiefly intended for National and Elementary Schools, has been pre;^. on the same plan as that adopted in the author's School Ai'ithmetic, which is in extensive circulation in England and abroad. The Metrical Tables have been introduced, from the conviction on the part of the author that the knowledge of such tables, and the mode of applying them, zuill be of great use to the rising generation. KEYS TO SCHOOL CLAS^-BOOK OF ARITHMETIC. Com- plete in one volume, i8mo. cloth, 6^. dd.-, or Parts L, II., and III., is. 6d. each. SHILLING BOOK OF ARITHMETIC FOR NATIONAL AND ELEMENTARY SCHOOLS. i8mo. cloth. Or separately, Part I. 2d.; Part II. 3^.; Part III. ^d. Answers, 6d. THE SAME, with Answers complete. i8mo. cloth, is. 6d. This Shilling Book of Arithmetic has been pi'epared for the use oj National and other schools at the urgent request of numerous Masters oJ schools both at home and abroad. The Explanations of the Rules and the Examples zvill, it is hoped, be found suited to the most elementary classes. KEY TO SHILLING BOOK OF ARITHMETIC. i8mo. cloth. 4J. (id. C 34 EDUCATIONAL BOOKS. Barnard Smith — continued. EXAMINATION PAPERS IN ARITHMETIC. i8mo. cloth. I J, dd. The same, with Answers, i8mo. \s. ()d. The object of these Examination Papers is to test students both in the theory and practice of Arithmetic. It is hoped that the method adopted will lead students to deduce results from general principles rather than to apply stated rules. The autJiQr believes that the practice of giiing examples under particular rules makes the working of Arithvietic quite mechanical, and tends to throw all but very clever boys off their balance when a general paper on the subject is put before them. KEY TO EXAMINATION PAPERS IN ARITHMETIC. l8mo. doth. 4^. dd. THE METRIC SYSTEM OF ARITHMETIC, ITS PRINCIPLES AND APPLICATION, with numerous Examples, written expressly for Standard V. in National Schools. Fourth Edition. iSmo. cloth, sewed. 3^/. In the New Code of Regulations issued by the Council of Education it is stated ^^ that in all schools children in Standards V. and VI. should know the principles of the Metric System, and be able to explain the advantages to be gained from uniformity in the tnethod of forming multiples and sub-multiples of the unit.'''' In this little book, Mr. Smith clearly fiLnd simply explains the principle of the Metric System, and in con- siderable detail expounds the French system, and its relation to the ordinary English method, taking the pupil on as far as Compound pivision. The boc>k contains numerous Examples, and two wood-cuts ■flji^tratit^g the Al^ric Table^ of Surface and Solidity. Answers to the A CHART OF THE METRIC SYSTEM, on a Sheet, size 42 m. by 24 l«|j on Roller, mounted and varnished, price 3^. 6d. Fourth ^ditioi?, ^.'.'^r^ '- J^y the Mro) JEducationat Code ii ii drdaitted that a Chart of the Metric System be conspicuously hung up on the ImUs of every school under Government inspection. The publishers believe that the present Chart unll be found to a?iswer all the requirements of the Code, and afford a full and perfectly intelligible view of the principles of the Metric System. The principle of the system is clearly stated and illustrated by exam p^-^ : *^'-' MATHEMATICS. 35 Barnard Smith — co7itinued. Method of Forming the Tables is set forth ; Tables follo7U, clearly showing the English equivalent of the French measures of— I. Length ; 2. Surface; ■^.Solidity; 4. Weight; ^.Capacity. At the bottom of the Chart is drawn a full-length Metric Measure^ subdivided distinctly and intelligibly into DeoimetreSy Centimetres^ and Millimetres. " We do not remejuber that ever we have seen teaching by a chart more happily carried out. " — School Board Chronicle. Also a Small Chart on a Card, price \d. EASY LESSONS IN ARITHMETIC, combining Exercises in Reading, Writing, Spelling, and Dictation. Part I. for Standard I. in National Schools. Crown 8vo. 9^. Diagrams for School-room walls in preparation. From the novel method and the illustrations used this little book cannot but tend to make the teaching of Arithmetic even to very young children interesting and successful. If the book be used according to the directions of the author, the method of instruction cannot but prove sound and easy, and acceptable to teacher and child. The Standard of Examination fixed by the Education Department for 1872 has been adhered to. The West- minster Review says : — " We should strongly advise everyone to study carefully Mr. Barnard Smithes Lesso7ts in Arithmetic, Writing, and Spelling. A more excellent little work for a first introduction to know- ledge cannot well be written. Mr. Smithes larger Text-books on Arithmetic and Algebra are already most favourably known, and he has proved no7c> that the difficulty of writing a text-book which begins ab ovo is really sur- mountable ; but we shall be much mistaken if this little book has not cost its author more thought and mental labour than any of his 7nore elaborate text-books. The plan to combine arithmetical lessons with those in reading and spelling is perfectly novel, and it is worked out in accordance with the aims of our National Schools ; and we are convinced that its general in- troduction in all elementary schools throughout the country will produce great educational advantages'^ THE METRIC ARITHMETIC. This book will go thoroughly into the principles of the System, intro- ducing the money tables of the various countries which have adopted it, and containing a very large number of Examples and Examination Paters. [Nearly ready. C 2 35 EDUCATIONAL BOOKS, Snowball. — THE elements of plane and spheri- cal TRIGONOMETRY; with the Construction and Use of Tables of Logarithms. By J. C. Snowball, M. A. Tenth Edition. Crown 8vo. cloth, "js. 6d. In preparing- the present edition for the press^ the text has been subjected to a careful revision ; the proofs of some of the more impor- tant propositions have been rendered more strict and general ; and more than two hundred examples, taken principally from the questions set of late years in the public Examinations of the University and of individual Colleges, have been added to the collection of Examples and Problems for practice. Tait and Steele.— a treatise on dynamics of a PARTICLE. With numerous Examples. By Professor Tait and Mr. Steele. New Edition, enlarged. Crown 8vo. cloth. loj. dd. in this treatise will be found all the ordinary propositions, connected fioith the Dynamics of Particles, which can be conveniently deduced without . the use of D^Alembert^s Principle. Throughout the book will be found a . number of illustrative exarnples introduced in the text, and for the most part completely worked out ; others with occasional solutions or hints to . assist the student are appended to each chapter. For by far the greater portion of these, the Cambridge Senate-House and College Examination Papers have been applied to. In the new edition numerous trivial errors, , and a few of a more serious character, have been corrected, while many .. nezo examples have been added. Taylor. — geometrical CONICS ; including Anharmonic Ratio and Projection, with numerous Examples. By C. Taylor, B. A., Scholar of St. John's Coll. Camb. Crown 8vo. cloth. Is. 6d. This work contains elementary proof s of the principal properties oj Conic Sections, together with chapters on Projection and Anharmonic Ratio. Tebay.— ELEMENTARY MENSURATION FOR SCHOOLS. With numerous Examples. By Septimus Tebay, B.A., Head Master of Queen Elizabeth's Grammar School, Rivington. Extra fcap. Svo. y. 6d. The object of the present work is to enable boys to acquire a moderate knowledge of Mensuration in a reasonable time. All difficult and useless .'natter has been avoided. The examples for the most part are easy, and the rules are concise. " A very compact useful manual." — Spectator. MATHEMATICS. 37 WORKS By I. TODHUNTER, M.A., F.R.S., Of St. John's College, Cambridge. \ " They are all good, and each volume adds to the value of the rest" — Freeman. " Perspicuous language, vigorous investigations, scrutiny of difficulties, and methodical treatment, characterise Mr. Todhunter' s works. ^^ — Civil Engineer. THE ELEMENTS OF EUCLID. For the Use of Colleges and Schools. New Edition. i8mo. cloth, '^s. 6d. No method of overco?ning the difficulties experienced by young students of Euclid appears to be so useful as that of breaking up the demonstrations into their constituent parts ; a plan strongly recommended by Professor De Morgan. In the present Edition each distinct assertion in the argu- ment begins a new line ; and at the ends of the lines are placed the necessary references to the preceding principles on which the assertions depend. The longer propositions are distributed into subordinate parts, which are distinguished by breaks at the beginning of the lines. Notes, Appendix, and a collection of Exercises are added, MENSURATION FOR BEGINNERS. With numerous Examples. New Edition. i8mo. cloth, zs. 6d. The subjects included in the present work are those which have usually found a place in Elementary Treatises on Mensuration. The mode of treatment has been determined by the fact that the work is intended for the use of beginners. Accordingly it is divided into short independent chapters, which are followed by appropriate examples. A knowledge of the elements of Arithmetic is all that is assumed; and in connection with most of the Rules of Mensuration it has been found practicable to give such explana- tions and illustrations as will supply the place of formal mathematical demonstrations, which would have been unsuitable to the character of the work. ^^ Eor simplicity and clearness of arrangement it is unsurpassed by any text-book on the subject which has come under our notice.''^ — Educational Times. 38 EDUCATIONAL BOOKS, Todhunter (I.) — continued. ALGEBRA FOR BEGINNERS. With numerous Examples. New Edition. i8mo. cloth. 2s. dd. Great pains have been taken to render this work intelligible to young stttdents, by the use of simple language and by copious explanations. In determining the subjects to be included and the space to be assigned to each, the author has been guided by the Papers given at the various examinations in elementary Algebra which are now carried on in this country. The book may be said to consist of three parts. The first part contains the elementary operations in integral and fractional expressions ; the second the solution of equations and problems ; the third treats of various subjects which are ititrodticed but rarely into Examination Papers, and are more briefly discussed. Provision has at the same time been made J or the introduction of ectsy equations and problems at an early stage— for those who prefer such a course. KEY TO ALGEBRA FOR BEGINNERS. Crown 8vo. cloth. 6j. dd. TRIGONOMETRY FOR BEGINNERS. With numerous Examples. New Edition. i8mo. cloth, is. (>d. Intended to serve as an introduction to the larger treatise on Plane Trigonometry, published by the author. The same plan has been adopted as in the Algebra for Beginners : the subject is discussed in short chapters, and a collection of examples is attached to each chapter. The first fourteen chapters present the geometrical part of Plane Trigonofnetry ; and contain all that is necessary for practical purposes. The range of matter included is such as seems required by the various examinations in elementary Tri' gonometry which are now carried on in this country. Answers are appended. MECHANICS FOR BEGINNERS. With numerous Examples. New Edition. i8mo. cloth. 4-f. dd. Intended as a companion to the tnvo preceding books. The work forms an elementary treatise on demonstrative mechanics. A knowledge of the elements at least of the theory 0/ the subject is extremely valuable even for those who are mainly concerned with practical results. The author has accordingly endeavoured to provide a suitable introduction to the study of applied as well as of theoretical mechanics. The work consists of two parts, namely. Statics and Dynamics. It will be found to contain all that is usually comprised in elementary treatises on Mechanics, togethei' with some additions. MATHEMATICS. 39 Todhunter (I.) — continued. ALGEBRA. For the Use of Colleges and Schools. Fifth Edition. Crown 8vo. cloth. 7^. dd. This work contains all the propositions which are usually iticluded in elefnentary treatises on Algebra, and a large number of Examples for Exercise. The author has sought to rejider the work easily intelligible to students, without impairing the accuracy of the demonstrations, or con- tracting the limits of the subject. The Examples, about Sixteen hundred and fifty in number, have been selected with a view to illustrate every part of the subject. Each chapter is complete in itself; and the work will be found peculiarly adapted to the wants of students who are without the aid of a teacher. The Answers to the Examples, with hints for the solution of some in which assistance m-ay be needed, are given at the end of the book. In the present edition two New Chapters and Three \i\mAxedi miscellaneous Examples have been added.. The latter are arranged in sets, each set containing ten Examples. ^^ It has merits which unquestionably place it first in the class to which it belongs.''^ — EDUCATOR. KEY TO ALGEBRA FOR THE USE OF COLLEGES AND SCHOOLS. Crown 8vo. \os. 6d. AN ELEMENTARY TREATISE ON THE THEORY OF EQUATIONS. Second Edition, revised. Crown 8vo. cloth. 7j. 6d. This treatise contains all the propositions which are usually included in elefnentary treatises on the theory of Equations, together with Examples for exercise. These have been selected from the College and University Examination Papers, and the results have been given when it appeared necessary. In order to exhibit a comprehensive view of the subject, the treatise includes investigations which are not found in all the preceding elementary treatises, and also some investigations which are not to be found in any of them. For the Second Edition the work has been revised and some additions have been made, the most important being an account of the researches of Professor Sylvester respecting Newton! s Rule. *' A thoroughly trustworthy, complete, and yet not too elaborate treatise^* Philosophical Magazine. PLANE TRIGONOMETRY. For Schools and Colleges. Fourth Edition. Crown 8vo. cloth, ^s. The design of this work has been to render the subject intelligible to beginners, and at the same time to afford the student the opportunity of 40 EDUCATIONAL BOOKS, Todhunter (I.) — continued. obtaining all the information which he will require on this branch oj Mathematics. Each chapter is followed by a set of Examples : those which are entitled Miscellaneous Examples, together with a few in some of the other sets, may be advantageously reserved by the student for exercise after he has made some progress in the subject. In the Second Edition the hints for the solution oJ the Examples have been cotisideradly increased, A TREATISE ON SPHERICAL TRIGONOMETRY. New Edition, enlarged. Crown 8vo. cloth. 4^. 6d. The present work is constructed on the same plan as the treatise on Plane Trigonometry, to which it is intended as a sequel. In the cucouni of Napier's Rules of Circular Parts, an explanation has been given of a method of proof devised by Napier, which seems to have been overlooked by most modern writers on the subject. Considerable labour has been bestowed on the text in order to render it comprehensive and accurate, and the Examples {selected chiefly from College Examination Papers) have all been carefully verified. *^ For educational purposes this work seems to be superior to any others on the subject.^'' — Critic. PLANE CO-ORDINATE GEOMETRY, as applied to the Straight Line and the Conic Sections. With numerous Examples. Fourth Edition, revised and enlarged. Crown 8vo. cloth, is. 6d. The author has here endeavoured to exhibit the subject in a simple manner for the benefit of begintiers, and at the same time to iiulude in one volume all that students usually require. In addition, therefore, to the propositions which have always appeared in such treatises, he has intro' duced the methods of abridged notation, which are of 7nore rece7it origin ; these methods, which are of a less elementary character than the rest of the work, arepUued in separate chapters, and may be omitted by the student at first. A TREATISE ON THE DIFFERENTIAL CALCULUS. With numerous Examples. Sixth Edition. Crown 8vo. cloth. loj-. dd. The author has endeavoured in the present work to exhibit a compre- hensive view of the Differential Calculus on the method of limits. In the more elementary portions he has entered into considerable detail in the explanations, zvith the hope that a reader who is without the assistance of a tutor may be enabled to acquire a competent acquaintance with the subject. The method adopted is that of Differential Coefficients, To the different MATHEMATICS. 41 Todhunter (I.) — continued, chapters are appended examples sufficiently numerous to render another book unnecessary ; these examples being mostly selected from College Ex- amination Papers. ^^ It has already taken its place as the text-book on that subject'^ — Philosophical Magazine. A TREATISE ON THE INTEGRAL CALCULUS AND ITS APPLICATIONS. With numerous Examples. Third Edition, revised and enlarged. Crown 8vo. cloth. \os. 6d. This is designed as a work at once elementary and complete, adapted for the use of beginners, and sufficient for the wants of advanced students. In the selection of the propositions, and in the mode of establishing them ^ it has been sought to exhibit the principles clearly, and to illustrate all their most important results. The process of summation has been repeatedly brought forward, with the view of securing the attention of the student to the notions which form the true foundation of the Calculus itself as well as of its most valuable applications. Every attempt has been Tnade to explain those difficulties which usually perplex beginners, especially with refere7ice to the limits of integrations. A new method has been adopted in regard to the transformation of multiple integrals. The last chapter deals with the Calculus of Variations. A large collection of exercises, selected from College Examination Papers, has been appended to the several chapters. EXAMPLES OF ANALYTICAL GEOMETRY OF THREE DIMENSIONS. Third Edition, revised. Crown 8vo. cloth. A TREATISE ON ANALYTICAL STATICS. With numerous Examples. Third Edition, revised and enlarged. Crown 8vo. cloth. \os. 6d. In this work on statics {treating of the laws of the equilibrium of bodies') will be found all the propositions which usually appear in treatises on Theoretical Statics. To the different chapters examples are appended, which have been principally selected from University Examination Papers. In the Third Edition many additions have been made, in order to illus- trate the application of the principles of the subject to the solution of problems. A HISTORY OF THE MATHEMATICAL THEORY OF PROBABILITY, from the time of Pascal to that of Laplace. 8vo. 8 J. 42 EDUCATIONAL BOOKS. Todhunter (I.) — coiitifiued. The subject of this volume has high claims to consideration on account of the subtle problems which it involves ^ the valuable contributions to analysis 2ohich it has produced, its important practical applications, and the emi- nence of those who have cultivated it. The subject claims all the interest which illustrious names can confer : nearly every great mathematician xvithin the range of a century and a half comes up in the course of the history. The present work, though principally a history, may claim the title of a comprehensive treatise on the Theory of Probability, for it assumes iti the reader only so much knowledge as can be gained from an elementary book on Algebra, and introduces him to almost ez>ery process and every species of problem which the literature of the subject can furnish. The author has been careful to reproduce the essential elements of the ofiginal works which he has analysed, and to corroborate his statements by exact quotations from the originals, in the languages in which they were published. RESEARCHES IN THE CALCULUS OF VARIATIONS, principally on the Theory of Discontinuous Solutions : an Essay to which the Adams Prize was awarded in the University of Cam- bridge in 187 1. 8vo. 6j. The subject of this Essay was prescribed in the following terms by the Examiners : — "^ determination of the circumstances under which dis- continuity of any kind presents itself in the solution of a problem of maximum or minimum in the Calcultts of Variations, and applications to particular instances. It is expected that the discussion of the instances should be exemplified as far as possible geometrically, and that attention be especially airected to cases of real or supposed failure of the Calculus.''^ The Essay, then, is mainly drooled to the consideration of discontinuous solutions; but incidentally various other questions in the Calculus of Variations are examined and elucidated. The author hopes that he has definitely contri- buted to the extension and improvement of our knowledge of this refined department of analysis. Wilson (J. M.) — ELEMENTARY GEOMETRY. Books I. II. III. containing the subjects of Euclid's First P'our Books following the Syllabus of Geometry prepared by the Geometrical Association. Third Edition. Extra fcap. 8vo. 3^. bd. By J. M. Wilson, M.A., late Fellow of St. John's College, Cambridge, and Mathematical Master of Rugby School. MATHEMATICS. 43 Wilson (J. M.) — continued. SOLID GEOMETRY AND CONIC SECTIONS. With Appen- dices on Transversals and Harmonic Division. For the use of Schools. By J. M. Wilson, M.A. Second Edition. Extra fcap. 8vo. 3J. dd. This work is an endeavour to introduce into schools some portions of Solid Geometry which are now very little read in England. The first twenty-one Propositions of Euclid'' s Eleventh Book are usually all the Solid Geometry that a boy reads till he meets with the sttbject again in the course of his analytical studies. And this is a matter of regret, because this part of Geometry is specially valuable and attractive. In it the atten- tion of the student is strongly called to the subject matter of the reasoning ; the geometrical imagination is exercised ; the methods e?nployed in it are more ingenious than those in Plane Geometry, and have greater diji- culties to meet ; and the applications of it in practice are more varied. Wilson (W. P.) — A TREATISE ON DYNAMICS. By W. P. Wilson, M.A., Fellow of St. John's College, Cambridge, and Professor of Mathematics in Queen's College, Belfast. 8vo. 9J. dd. *^This treatise supplies a great educational need.'^ — Educational Times. Wolstenholme. — A BOOK OF MATHEMATICAL PROBLEMS, on Subjects included in the Cambridge Course. By Joseph Wolstenholme, Fellow of Christ's College, some- time Fellow of St. John's College, and lately Lecturer in Mathe- matics at Christ's College. Crown 8vo. cloth. 8j. 6d. Contents: — Geometry (Euclid) — Algebra — Plane Trigonometry — Geometrical Come Sections — Analytical Conic Sections — Theory of Equa- tions — Differential Calculus — Integral Calculus — Solid Geometry — Statics — Elementary Dynamics — Newton — Dynamics of a Point — Dynamics of a Rigid Body — Hydrostatics — Geometrical Optics — Spherical Trigonometry and Plane Astronomy. ** jhidicious, symmetrical, and well arranged." — Guardian. 44 ££K/CATIONAL BOOKS. SCIENCE. ELEMENTARY CLASS-BOOKS. The importance of Science as an element of sound educa- tion is now generally acknowledged ; and accordingly it is obtaining a prominent place in the ordinary course of school instruction. It is the intention of the Publishers to produce a complete series of Scientific Manuals, affording full and accurate elementary information, conveyed in clear and lucid English. The authors are well known as among the foremost men of their several departments ; and their names form a ready guarantee for the high character of the books. Subjoined is a list of those Manuals that have already appeared, with a short account of each. Others are in active preparation ; and ^e whole will constitute a standard series specially adapted to the requirements of be- ginners, whether for private study or for school instruction. ASTRONOMY, by the Astronomer Royal. POPULAR ASTRONOMY. With Illustrations. By Sir G. B. Airy, K.C.B., Astronomer Royal. New Edition. iSmo. cloth. 4J. ()d. This work consists of six lectures, which are intended " to explain to intelligent persons the principles on which the instruments of an Observa- tory are constructed (omitting all details, so far as they are merely sub' sidiaty), and the principles on which the observations made with these instruments are treated for deduction oj the distances and weights of the bodies of the Solar System, and of a few stars, omitting all minittivE of SCIENCE. 45 Elementary Class- Books — contiiiued. formulcB, and all troublesome details of calculation.^^ The speciality of this volume is the direct reference of every step to the Observatory, and the full description of the methods and instruments of observation. ASTRONOMY. MR. LOCKYER'S ELEMENTARY LESSONS IN ASTRO- NOMY. With Coloured Diagram of the Spectra of the Sun, Stars, and Nebulas, and numerous Illustrations. By J. NoRMAN LocKYER, F.R.S. New Edition. i8mo. ^s. dd. The author has here aimed to give a connected view of the whole subject, and to supply facts., and ideas founded on the facts, to serve as a basis for subsequent study and discussiojt. The chapters treat of the Sia?s and Nebuhe ; the Sun; the Solar Sy stern ; Apparent Movements of the Heavenly Bodies; the Measurement of Time; Light ; the Telescope and Spectroscope ; Apparent Plcues of the Heavenly Bodies ; the Real Distances and Dimen' sions; Universal Gravitation. The most recent astronomical discoveries are i?icorporated. Mr. Lockyer' s work supplements that of the Astronomer Royal mentioned in the previous article. " The book is full, clear, sound, and worthy of attention, not only as a popular exposition, but as a scientific ' Index.' " — Athen^UM. " The most fascinating of elementary books on the Sciences.'" — NONCONFORMIST. QUESTIONS ON LOCKYER'S ELEMENTARY LESSONS IN ASTRONOMY. For the Use of Schools. By John Forbes- Robertson. i8mo. cloth limp. \s. 6d. PHYSIOLOGY. PROFESSOR HUXLEY'S LESSONS IN ELEMENTARY PHYSIOLOGY. With numerous Illustrations. By T. H. Huxi-EY, F.R.S., Professor of Natural History in the Royal School of Mines. New Edition. i8mo. cloth. ^. 6d. This book describes and explains, in a series of graduated lessons, the principles of Pluman Physiology ; or the Structure and Functions of the Human Body. The first lessoti supplies a general view of the subjtci. This is followed by sectiotis on the Vascular or Venous System-, and- the Circulation ; the Blood and the Lymph; Respiration ; Sources of J oss and of Gain to the Blood ; the Ftmction of Alimentation ; Motion and Locomotion ; Sensatons and Sensory Organs; the Organ of Sight ; the 48 EDUCATIONAL BOOKS. Elementary Class-Books — continued. from the distinct objects and ideas treated in the natural and experimental sciences have been generally substituted. At the efid o/ almost every Lesson will be found references to the works in which the studettt will most frofitablv continue his reading of the subject treated, so that this little volume may serve as a guide to a more extended course of study. The Guardian thinks '■'• nothing caji be better for a school-book,^^ and the Athen^UM calls it "a manual alike simple, interesting, and scientific.''^ PHYSICS. LESSONS IN ELEMENTARY PHYSICS. By Balfour Stewart, F.R.S., Professor of Natural Philosophy in Owens College, Manchester. With numerous Illustrations and Chromo- liths of the Spectra of the Sun, Stars, and Nebulae. New Edition. iSmo. 4f. 6(/. A description, in an elementary manner, of the viost important of those laws which regulate the phenomena of nature. The active agents, heat, light, electricity, etc., are regarded as varieties of energy, and the woik is so arranged that their relation to one another, looked at in this light, and the paramount importance o/ the laws of energy, are clearly brought out. The volume contains all the necessary illustrations, and a plate reptresent- ing the Spectra of Sun, Stars, and Nebula, forms a frontispiece. The Educational Times calls this " the beau ideal of a scientific text-book, clear, accurate, and thorough.'''* PRACTICAL CHEMISTRY. THE OWENS COLLEGE JUNIOR COURSE OF PRAC- TICAL CHEMISTRY. By Francis Jones, Chemical Master in the Grammar School, Manchester. With Preface by Professor RoscoE. With Illustrations. New Edition. i8mo. is. dd. This little book contains a short description of a course of Practical Chemistry, which an experience of mayiy years has proved suitable for those commencing the study of the science. It is intended to supplement, not to supplant, instruction given by the teacher. 77ie siibject-matter has been very carefully compiled, and many useful cuts are introduced. ANATOMY. LESSONS IN ELEMENTARY ANATOMY. By St. George MiVART, F.R.S., Lecturer in Comparative Anatomy at St. Mary's Hospital. With upwards of 400 Illustration.s. j8mo. 6j. dd. SCIENCE, 49 These Lessons are intended for teachers and students of both sexes not already acquainted with Anatomy. The author has endeavoured, by certain additions and by the mode of treatment, also to fit them for students in medicine, and generally for those acquainted with human anatomy, but desirous of learning its more significant relations to the structure of other animals. The Lancet says, ^^ It may be questioned whether any other work on Anatomy contains in like compass so proportionately great a mass of information." The MEDICAL TiMES remarks, " The work is excellent, and should be in the hands of every student of human anatomy.''^ MANUALS FOR STUDENTS. Flower (W. H.)— an introduction to the oste- ology OF THE MAMMALIA^ Being the substance of the Course of Lectures delivered at the Royal College of Surgeons of England in 1870. By W. H. Flower, F.R.S., F.R.C.S., Hunterian Professor of Comparative Anatomy and Physiology. With numerous Illustrations, Globe 8vo. 'js. 6d. Although the present work contains the substance of a Course of Lectures^ the form, has been changed, so as the better to adapt it as a handbook for students. Theoretical views have been almost entirely excluded : and xvhile it ts impossible in a scientific treatise to avoid the employment of technical terms, it has been the author's endeavour to use no more than absolutely necessary, and to exercise due care in selecting only those that seem most appropriate, or which have received the sanction of general adoption. With a very few exceptions the illustrations have been drawn expressly for this work from specimens in the Museum of the Royal College of Surgeons. Hooker (Dr.)— the STUDENT'S FLORA OF THE BRITISH ISLANDS. By J. D. Hooker, C.B., F.R.S., M.D., D.C.L., Director of the Royal Gardens, Kew. Globe 8vo. 10s. 6d. The object of this work is to supply students and field-botanists with a. fuller account of the Plants of the British Islands than the manuals hitherto in use aim at giving. The Ordiiial, Generic, and Specific characters have been re-written, and are to a great extent original, and drawn from living or dried specimens, or both. * ' Cannot fail to perfectly fulfil the purpose for which it is intended.'' — Land and Water.. * • Containing the fullest and most accurate manual of the kind that has yd appeared.''— Vaia. Mall Gazette. D 50 EDUCATIONAL BOOKS, Oliver (Professor). — FIRST BOOK OF INDIAN BOTANY. By Daniel Oliver, F.R.S., F.L.S., Keeper of the Herbarium and Library of the Royal Gardens, Kew, and Professor of Botany in University College, London. With numerous Illustrations. Extra fcap. 8vo. ds. 6d. This manual is, in substance, the author^ s ^^ Lessons in Elementary Botany" adapted for use in India. In preparing it he has had in vie^o the want, often felt, of some handy resume of Indian Botany, which might be serviceable not only to residents of India, but also to. any one about to proceed thither, desirous of getting some preliminary idea of the Botany of that country. ** // contains 'a well-digested summary of all essential know- ledge pertaining to Indian botany, wrought out in accordance vnth the best principles of scientific arrangement." — Allen's Indian Mail. Other volumes of these Manuals will follow. Ball (R. S., A.M.)— EXPERIMENTAL MECHANICS. A Course of Lectures delivered at the Royal College of Science for Ireland. By Robert Stawell Ball, A.M., Professor of Applied Mathematics and Mechanics in the Royal College of Science for Ireland (Science and Art Department). Royal 8vo. 1 6 J. The author's aim has been to create in the mind of the student physical ideas corresponding to theoretical laws, and thus to produce a work which may be regarded either as a suppletnent or an introduction to manuals 0/ theoretic mechanics. To realize this design, t/ie copious use of experimental illustrations was necessary. The apparatus used in the Lectures, and figured in the volume, has been principally built up from Professor IVi/lis's most admirable system. In the selection of the subjects, the question 0/ practical utility has in many cases been regarded as the one of paramount importance. The elementary truths of Mechanics are too well kno^vn to admit of novelty, but it is believed that the mode of treatment which is adopted is more or less original. This is especially the case in the Lectures relating to friction, to the mechanical powers, to the strength of timber and structures, to the laws of nwtion, and to the pendulum. The illustrations, drawn from the apparatus, are nearly all original, and are beautifully executed. SCIENCE, 51 Clodd — THE CHILDHOOD OF THE WORLD: a Simple Account of Man in Early Times. By Edward Clodd, F.R.A.S. Second Edition. Globe 8vo. 3J-. Professor Max Muller, in a letter to the Author^ says: ^^ I read your book with great pleasure. I have no doubt it will do good, and I hope you will coittinue your work. Nothing spoils our temper so much as having to unlearn in youth, manhood, and even old age, so tnany things which we were taught as children. A book like yours will prepare a far better soil in the child'' s mittd, and I was delighted to have it to read to my children.^'' Cooke (Josiah P., Jun.)— FIRST PRINCIPLES OF CHEMICAL PHILOSOPHY. By Josiah P. Cooke, Jun., Ervine Professor of Chemistry and Mineralogy in Harvard College. Crown 8vo. \2s. The object of the author in this book is to present the philosophy of Chemistry in such a form that it can be made with profit the subject of College recitations, and furnish the teacher with the means of testing the studenfs faithfulness and ability. With this view the subject has been developed in a logical order, and the principles of the science are taught independently of the experimental evidence on which they rest. Guillemin.— THE FORCES OF NATURE: a Popular Intro- duction to the study of Physical Phenomena. By Amedee Guille- min. Translated from the French by Mrs. Norman Lockyer, and Edited, with Additions and Notes, by J. Norman Lockyer, F.R. S. With II Coloured Plates and 455 Woodcuts. Second Edition. Royal 8vo. cloth, gilt. 31^. dd. '• Translator and Editor have done justice to their trust. The text has all the force and flow of original writing, combining faithfulness to the author's meaning with purity and independence in regard to idiom ; tuhile the histoncal precision and accuracy pervading the work throughout, speak of the tvatchfd editorial supervision which has been given to every scientific detail, . . . Altogether, the work may be said to have no parallel, either in point of fulness or attraction, as a popular manual of physical science.^^ — Saturday Review. D 2 52 EDUCATIONAL BOOKS. Lockyer. — THE SPECTROSCOPE AND ITS APPLICA- TIONS. By J. Norman Lockyer, F.R.S. With Coloured Plate and numerous illustrations. Second Edition. Crown 8vo. This forms volume one of "Nature SerieSy" a Series of Popular Scientific Works now in course of publication, consisting of popular ami instructive works, on particular scientific subjects — Scientific Discovery, Applications, History, Biography — by some of the most eminent scientific men of the day. They 7vill be so written as to be interesting and intelli- gible even to non-scientific readers. Mr. Lockyer' s 7Vork in Spectrum Analysis is widely known. In the present short treatise will be found an exposition of the principles on which Spectrum Analysis rests, a description of the various kinds of Spectroscopes, and an account of what has already been done with the instrument, as well as of what may yet be done both in science and in the industrial arts. Roscoe (H. E.)— SPECTRUM analysis. Six Lectures, with Appendices, Engravings, Maps, and Chromolithographs. By H. E. Roscoe, F.R.S., Professor of Chemistry in Owens College, Manchester. Third Edition, revised throughout. Royal 8vo. 2 1 J. " In six lectures he has given the history of the discovery and set forth the facts relating to the analysis of light in such a way that any reader of ordinary intelligence and information will be able to understand what ' Spectrum Analysis^ is, and what are its claims to rank among the most signal triumphs of science of which even this century can boast." — NON- CONFORMIST. *' The illustrations — no unimportant part of a book on such a subject — are marvels of wood-printing, and reflect the clearness which is the distinguishing merit of Mr. Roscois explanations.^' — Saturday Review. ** The lectures themselves furnish a most ad- mirable elementary treatise on the subject, tvhilst by the insertion in appendices to each lecture of extracts from the most important published memoirs, the autJior has rendered it equally valuable as a text-book for advanced students." — Westminster Review. Thorpe (T. E.)_a series of chemical problems, for use in Colleges and Schools. Adapted for the preparation of Students for the Government, Science, and Society of Arts Ex- aminations. With a Preface by Professor RoscoE. i8mo. cloth, is. :scjencjs. 53 In the Preface Dr. Roscoe says — ** My experience has led me to feel more and mo7'e strongly that by no method can accuracy in a knowledge of chemistry be more surely secured than by attention to the working of well- selected problems^ and Dr. Thorpe's thorough acquaintance with the wants of the student is a sufficient guarantee that this selection has been carefully made. I intend largely to use these questions in my own classes, and I can confidently recommend them to all teachers and students of the science.''^ Wurtz.— A HISTORY OF CHEMICAL THEORY, from the Age of Lavoisier down to the present time. By Ad. Wurtz. Translated by Henry Wa'ETS, F.R.S. Crown 8vo. 6j. ** The treatment of the subject is admirable, and the translator has evidently done his duty most efficiently'^ — Westminster Review. '* The discourse, as a resume of chemical theory and research, unites singular luminousness and grasp. A few judicious notes are added by the translator,'^ — Pall Mall Gazette. S6 EDUCATIONAL BOOKS. MISCELLANEOUS. Abbott.— A SHAKESPEARIAN GRAxMMAR. An Attempt to illustrate some of the Differences between Elizabethan and Modern English. By the Rev. E. A. Abbott, M.A., Head Master of the City of London School. For the Use of Schools. New and Enlarged Edition. Extra fcap. 8vo. 6^. The object of this work is to furnish students oj Shakespeare and Bacon laith a short systematic cu:count of some points of diffa'ence between Eliza- bethan syntax and our own. A section on Prosody is added, and Notes and Questions. The success which has attended the First atid Second Editions of the ** SHAKESPEARIAN GRAMMAR," fl«^ the detnand for a Third Edition within a year of the piibli cation of the First, have encouraged the author to endeavour to make the work somewhat more usefid, and to render it, as far as possible^ a complete book of reference for all difficulties of Shakespearian syntax or prosody. For this purpose the whole of Shake- speare has been re-read, and an attempt has been made to include within this Edition the explanation of every idiomatic difficulty that comes within the province of a grammar as distinct from a glossaty. The great object being to make a useful book of reference for studetits, and especially for classes in schools, several Plays have been indexed so ftdly that with the aid of a glossary and historical notes the refer etues will serve for a complete com- inentary. "A critical inquiry, conducted with gj'eat skill and knowledge, and with all the appliances of modern philology .... We venture to believe that those who consider themselves most proficient as Shakespearians wUl find something to learn from its pages. '^ — Pall Mall Gazette. " Valuable not only as an aid to the critical study of Shakespeare, but as tending to familiarize the reader with Elizabethan English in general.^' — Athen^UM. Berners.— FIRST LESSONS ON HEALTH. By J. Ber- ners. i8mo. IS. Third Edition. This little book consists of the notes of a number of simple lessons on sanitary subjects given to a class in a National School, and listened to MISCELLANEOUS, 57 with great intei'est and intelligence. They have been made as easy and familiar as possible, and as far as they go may be deemed perfectly trust- worthy. One of the author's main attempts has been, to translate the concise and accurate language of science into the colloquial nursery dialect comprehensible to children. The book will be found of the highest value to all who have the training of children, who, for want of knowing what this little book teaches, too often grow up to be unhealthy, defective men and women. The Contents are — /. Introductory. II. Fresh Air. III. Food and Drink. IV. Warmth. V. Cleanliness. VI. light, VIL Exercise. VIII. Rest. Besant.— STUDIES IN EARLY FRENCH POETRY. By Walter Besant, M.A. Crown 8vo. ?>s. 6d. A sort of impression rests on most minds that French literature begins with the ^^ sihle de Louis Quatorze f^ any previous literature being for the most part unknown or ignored. Few know anything of the enormous literary activity that began in the thirteenth century, was carried on by Rulebeuf Marie de France, Gaston de Foix, Thibault de Champagne, and Lorris ; was fostered by Charles 0/ Orleans, by Margaret of Valois, by Francis the First ; that gave a crowd of versifiers to France, enriched, strengthened, developed, and fixed the French language, and prepared the way for Corneille and for Racine. The present work aims to afford information and direction touching these early efforts of France in poetical litei'ature. ' ' In one moderately sized volume he has contrived to introduce us to the very best, if not to all of the early French poets .''^ — Athen^UM. ^^ Industry, the insight of a scholar, and a genuine enthusiasm for his subject, combine to make it of very considerable value." — Spectator. Calderwood.— HANDBOOK OF MORAL PHILOSOPHY. By the Rev. Henry Calderwood, LL.D., Professor of Moral Philosophy, University of Edinburgh. Second Edition. Crown 8vo. 6 J. While in this work the interests of University Students have been con- stantly considered, the author has endeavoured to pivduce a book suitable to those 7vho wish to prosecute privately the study of Ethical questions. The author has aimed to present the chief pf-oblems of Ethical Science, to give an outline of discussion under each, and to afford a guide for private study by references to the Literature of the Science. The uniform- object has been to give a careful representation of the conflicting theories, supplying the reader with matenals for independe7it judgment. 58 EDUCATIONAL BOOKS. Cameos from English History. — See Yonge (C. M.) Delamotte. — a BEGINNER'S DRAWING BOOK. By P. H. Delamotte, F.S.A. Progressively arranged, with upwaids of Fifty Plates. Crown 8vo. Stiff covers, zs, dd. This work is intended to give such instruction to Beginners in Drawing, and to place before them copies so easy^ that they may notjind any obstacle in making the first step. Thenceforward the lessons are gradually progressive. Mechanical improvements, too, have lent their aid. The whole of the Plates have been engraved by a new process, by means of which a varying depth of tone — up to the present time the distinguishing character- istic of pencil drawing — has been imparted to woodcuts. * * We have seen and examined a great many drawing-books, but the one now be/ore us strikes us as being the best of thefn all." — ILLUSTRATED Times. 'M concise, simple, and thoroughly practical work. The letter -press is throughout intelligible and to the point." — Guardian. D'Oursy and Feillet. — a FRENCH GRAMMAR AT SIGHT, on an entirely new method. By A. D'Oursy and A. Feillet. Especially adapted for Pupils preparing for Ex- amination. Fcap. 8vo. cloth extra. 2s. (>d. TJie method folloived in this volume consists in presenting the grammar as much as possible by synoptical tables, which, striking the eye at once, and following throughout the same order — ** used — not used ; " "changes — does not change " — are easily remembered. The parsing tables will enable the pupil to parse easily from the beginning. The exercises consist of translations fro?n French into English, and from English ijito French ; and of a number of grammatical questions. Green.— A HISTORY OF THE ENGLISH PEOPLE. By the Rev. J. R. Green, M.A. For the use of Colleges and Schools. Crown 8vo. 8j. 6d. Hales. — LONGER ENGLISH POEMS, with Notes, Philological and Explanatory, and ati Introduction on the Teaching of English. Chiefly for use in Schools. Edited by J. W. Hales, M.A., late Fellow and Assistant Tutor of Christ's College, Cambridge, Lecturer in English Literature and Classical Composition at King's College School, London, &c. &c. Extra fcap. 8vo. 4J. 6d. MISCELLANEOUS. 59 This zvork has been in preparation for some year's, and part of it has been used as a class-book by the Editor. It is intended as aft aid to the Critical study of English Literature, and contains one or more of the larger poems, each complete, of prominent English authors, from Spenser to Shelley, including Burns' *■'■ Cotter's Saturday Night '^ and '■' Twa Dogs,''' In all cases the original spelling and the text oj the best editions have been given : only in one or two poems has it been deemed necessary to make slight omissions and changes, ^^ that the reverence due to boys might be well observed." The Introduction consists of Suggestions on Teaching of English. The latter half of the volume is occupied with copious notes, critical, etymological, and explanatory, calculated to give the learner much insight into the structure and connection of the English tongue. An Index to the Notes is appetided. Helfenstein (James), — a comparative grammar OF THE TEUTONIC LANGUAGES. Being at the same time a Historical Grammar of the English Language, and comprising Gothic, Anglo-Saxon, Early English, Modern English, Icelandic (Old Norse), Danish, Swedish, Old High German, Middle High German, Modem German, Old Saxon, Old Frisian, and Dutch. By James Helfenstein, Ph. D. 8vo. i8j. This work traces the different stages of development through which the various Teutonic languages have passed, and tfie laws which have regulated their growth. The reader is thus enabled to study the relation which these languages bear to one another, and to the English language in particular^ to which special attention is devoted throughout. In the chapters on Ancient and Middle Teutonic Languages no grammatical form is omitted the knowledge of which is required for the study of ancient literature, tuhether Gothic, or Anglo-Saxon, or Early English. To each chapter is prefixed a sketch showing the relation of the Teutonic to the cognate languages, Greek, Latin, and Sanskrit. Those who have mastered the book will be in a position to proceed with intelligence to the more elaborate works of Gritnm^ Bopp, Pott, Schleicher, and others. Hole.— A GENEALOGICAL STEMMA OF THE KINGS OF ENGLAND AND FRANCE. By the Rev. C. Hole. On Sheet. \s. The different families are printed in distinguishing colours, thus acilitating reference. 6o EDUCATIONAL BOOKS. Jephson.— SHAKESPEARE'S "TEMPEST." With Glossarial and Explanatory Notes. By the Rev. J. M. Jephson. Second Edition. i8mo. is. It is important to find some substitute for classical study^ and it is believed that such a substitute may be found in the Plays oj Shakespeare. For this purpose the presetit edition of the " Tempest" has been prepared. The introduction treats briefly of the value of the study of language^ the fable of the play, and other points. The notes are intended to teach the student to analyse every obscure sentence and trace out tke logical sequence of tlie poefs thoughts ; to point out the rules of Shakespear^s versification ; to explain obsolete words and meanings ; and to guide the students taste by directing his attention to such passages as seem especially worthy of note for their poetical beauty or truth to tiature. The text is in the main founded upon that of the first collected edition of Shakespeari s Plays. Kington-Oliphant.— THE SOURCES OF STANDARD ENGLISH. By J. Kington-Oliphant. Globe 8vo. ds. Martin.— THE POET'S HOUR : Poetry Selected and Arranged for Children. By Frances Martin. Second Edition. i8mo. 2.S. dd. This volume consists of nearly 200 Poems selected from the best Poets, ancient and modern, and is intended mainly for children betiueen the ages of eight and hoelve. SPRING-TIME WITH THE POETS. Poetry selected by Frances Martin. Second Edition. iSmo. 3J. dd. This is a selection of poetry intended mainly for girls and boys between the ages oft%velve and seventeen. Masson (Gustave). — a FRENCH-ENGLISH AND ENG- LISH-FRENCH DICTIONARY. By Gustave Masson, B.A., Assistant Master in Harrow School. Small 4to. 6j. M'Cosh (Rev. Principal).— For other Works by the same Author, see Philosophical Catalogue. THE LAWS OF DISCURSIVE THOUGHT. Being a Text-Book of Formal Logic. By James M'Cosh, D.D., LL.D. Svo. 5j. MISCELLANEOUS, 61 In this treatise the Notion (jvith the Term and the Relation of Thought to Language^) will be found to occupy a larger relative place than in any logical work written since the time of the famous ^^ Art of Thinking^ * ' PVe heartily zvelcome his book as one which is likely to be of great value in Colleges and Schools.'" — Athen^um. Morris.— HISTORICAL OUTLINES OF ENGLISH ACCI- DENCE, comprising Chapters on the History and Development of the Language, and on Word-formation. By the Rev. Richard Morris, LL.D., Member of the Council of the Philol. Soc, Lecturer on English Language and Literature in King's College School, Editor of ** Specimens of Early English," &c. &c. Third Edition. Extra fcap. 8vo. 6^'. Dr. Morris has endeavoured to write a work which can be profitably used by students and by the tipper forms in our public schools. English Grammar, he believes, without a reference to the older forms, must appear altogether anomalous, inconsistent, and unintelligible. His almost un- equalled knowledge of early English Literature renders him peculiarly qualified to write a work of this kind. In the writing of this volume, moreover, he has taken advantage of the researches into our language made by all the most eminent scholars in England, America, and on the Continent. The author shows the place of English among the languages of the world, expounds clearly and with great minuteness ^^ Grimni's Law,^"* gives a brief history of the English language and an account of the various dialects, investigates the history afid principles of Phottology^ OHhography , Accent, and Etymology, and devotes several chapters to the consideration op the various Parts of Speech, and the final one to Deri- vation and Word-formation. " It makes an era in the study of the English tongue.''^ — Saturday Review. ^^ He has done his work with a fulness and completeness that leave nothing to be desired.'^ — NON- CONFORMIST. "A genuine and sound book. "— Athene UM. Oppen.— FRENCH READER. For the Use of Colleges and Schools. Containing a graduated Selection from modern Authors in Prose and Verse ; and copious Notes, chiefly Etymological. By Edward A. Oppen. Fcap. 8vo. cloth. OgS. 6d. This is a Selection from the best modern authors of France. Its dis- tinctive feature consists in its etymological notes, connecting French with the classical and modern languages, including the Celtic. This subject has hitherto been little discussed even by the best-educated teachers. 62 EDUCATIONAL BOOKS. Pylodet. — NEW GUIDE TO GERMAN CONVERSATION; containing an Alphabetical List of nearly 800 Familiar "Words similar in Orthography or Sound and the same Meaning in both Languages, followed by Exercises, Vocabulary of Words in frequent use. Familiar Phrases and Dialogues; a Sketch of German Literature, Idiomatic Expressions, &c. ; and a Synopsis of German Grammar. By L. Pylodet. i8mo. cloth limp. 2s. 6d. Sonnenschein and Meiklejohn. — the ENGLISH METHOD OF TEACHING TO READ. By A. Sonnenschein and J. M. D. Meiklejohn, M.A. Fcap. 8vo. Comprising : The Nursery Book, containing all the Two-Letter Words in the Language. id. (Also in Large Type on Sheets for School Walls. 5 J. ) The First Course, consisting of Short Vowels with Single Consonants. 3i)hich it tvas intended. Although full of instruction and calculated highly to interest and even fascinate children^ it is a work which may be and has been used with profit and pleasure by all. **I have, I hope,^* the author says, ^' shown that it is perfectly easy to teach childrm, from the very firs^y to distinguish trice history alike from legend and from wilful inven- tion, and also to understand the nature of historical authorities and to weigh one stateme?it against another^ I have throughout striven to connect the history of Kn^land with the general history of civilized Europe, and I have especially tried to make the book serve as an incentive to a more accurate study of historical geography.'^ In the present edition the whole has been carefully revistd, and such improroements as suggested themselves have been introduced. * ' The book indeed is full of instruction and interest to students of all ages, and he 7nust be a well-informed man indeed who will not rise fro??i its perusal zuith clearer and more accurate ideas of a too much neglected portion of English History." — Spectator. Historical Course for Schools.— Edited by Edward A. Freeman, D.C.L., late Fellow of Trinity College, Oxford. The object of the present series is to put forth clear and correct views of history in simple language, and in the smallest space and cheapest form in which it could be done. It is meant in the fust place for Schools ; but it is often found that a book for schools proves useful HISTORY. 67 for other readers -as well, and it is hoped that this may be the case with the little books the first instalment of which is now given to the world. The General Sketch will be followed by a series of special histories of particular countries, which will take for granted the main principles laid down in the General Sketch. In every case the results of the latest historical research will be given in as simple a form as may be, and the several numbers of the series will all be so far under the supervision of the Editor as to secure general ac- curacy of statement and a general harmony of plan and sentiment ; but each book will be the original work of its author, who will be responsible for his own treatment of smaller details. The Editor himself undertakes the histories of Rome and Switzerland, while the others have been put into the hands of various competent and skilful writers. The first vdnme is meant to be introductory to the whole course. It is intended to give, as its name implies, a general sketch of the history of the civilized world, that is, of Europe, and of the lands which have drawn their civilization from Europe. Its object is to trace out the general rela- tions of different periods and different countries to one another, without going minutely into the affairs of any particular country. This is an object of the first iinportance, for without clear notions of general history, the history of particular countj'ies can never be rightly understood. The narrative extends from the earliest movemejtts of the Aryan peoples, doxvn to the latest events both on the Eastern and Western Continents. The book consists of seventeen moderately sized chapters, each chapter bein^ divided into a number of short numbered paragraphs, each zuith a title i>refixed clearly indicative of the subject of the paragraph. ^^ It supplies the great want of a good foundation for historical teaching. The scheme is an excellent one, and this instalment has been executed in a way that promises much for the volumes that are yet to appear."*^ — Educational Times. /. GENERAL SKETCH OF EUROPEAN HISTORY. By Edward A. Freeman, D.C.L. Third Edition. i8mo. cloth. 3J. dd. IL HISTORY OF ENGLAND. By Edith Thompson. i8mo. 2.S. 6d. " Freedom jroin prejudice, simplicity of style, and accuracy 0/ statement, are the characteristics of this Utile volume. It is a trustworthy text-book E 2 68 EDUCATIONAL BOOKS. and likely to be generally serviccible in schools." — Pall Mall Gazette. * ' Upon the luhole, this manual is the best sketch oj English history for the use »f young people we have yet met with.'" — AxHENiiiUM. ///. SCOTLAND. By Margaret Macarthur. 2j. JV. ITALY. By the Rev. William Hunt, M.A. y. The following will shortly be issued : — FRANCE. By the Rev. J. R. Green, M.A. GERMANY. By J. Sime, M.A. Yonge (Charlotte M.)— -a parallel history of FRANCE AND ENGLAND : consisting of Outlines and Dates. By Charlotte M. Yonge, Author of "The Heir of Redclyffe," *' Cameos of English History," &c. &c. Oblong 4to. 3^. 6a. This tabular history has beat draxvn up to supply a want felt by many teachers of some means of making thei^ pupils realize what events in the two countries were contemporary. A skeleton luirrative has been con- structed of the chief transactions in either country, placing a column between for what aff'ccted both alike, by which means it is hoped that young people may be assisted in grasping the mutual relation of events. ** We can imagine few more really advantageous courses of historical study Jor a young mind than going carefully and steadily through Miss Yonge* s excellent little book.''' — Educational Times. CAMEOS FROM ENGLISH HISTORY. From RoUo to Edward IL By the Author of "The Heir of Redclyffe." Extra fcap. Svo. Second Edition, enlarged. 3^. dd. The etuleavour has not been to chronicle facts, but to put together a series of pictures oJ persons and events, so as to arrest the attention, and give s)me ind'iv'iduality and distinctness to the recollection, by gathering together details at the most memorable momerits. 71ie " Cameos''^ are intended as a book for young people jtist beyond the elernentary histories of England, and able to enter in some degree into the real spirit of events, and to be struck with characters and scenes presented in some relief. " Instead oj dry details,'' says the NONCONFORMIST, ^' we have living pictures^ faith- ful, vivid, and striking."" A Second Series of CAMEOS FROM ENGLISH HISTORY. The Wars in France. Extra fcap. Svo. pp. xi. 415. ^j. HISTORY, 69 This nnu volume, closing zuiih the Treaty of Arras, is the history of the struggles of Plantagenet and Valois. It refers, accordingly, to one of the most stirring epochs in the mediccval era, including the battle of Poictiers, the great Schism of the West, the Lollards, Agincourt and Joan of Arc. The atitJioress reminds her reada's that she aims 7?ierelyat " collecting from the best authorities such details as may present scenes and personages to the eye in some fulness f her Cameos are a ^^ collection of histoiical scenes and portraits such as the young might find it difficult to form for themselves •without access to a very complete library T " Though mainly intended,''^ says the John Bull, ^^ for young readei's, they will, iftve mistake not, be found very acceptable to those of more mature years, and the life and reality imparted to the dty bones of history canjtot fail to be attractive to readers of every age. " EUROPEAN HISTORY. Narrated in a Series of Historical Selec- tions from the Best Authorities. Edited and arranged by E. M. Sewell and C. M. Yonge. First Series, 1003 — 11 54. Third Edition. Crown Svo. 6^. Second Series, 1088 — 1228. Crown 8vo. 6s. When young children have acquired the outlines of IJistory from abridg- ments and catechisms, and it becomes desirable to give a more enlarged view of the subject, ijt order to render it really 7(sefil and interesting, a difficulty often arises as to the choice of books. Two courses are open, either to take a general a?id consequently dry history of facts, such as EusseVs Modern Europe, or to choose some work treating of a particular period or subject, such as the zuorks of Macaulay and Froude. The formei' course usually renders history uninteresting ; the latter is 7insaiis'nctory because it is not S7ifjicienily comprehensive. To remedy this difficiiUy, Selections, continuous and chronological, have, in the present volume, been taken from the larger works oj Freeman, Milniafi, Palgrave, and others, zvhich 7nay serve as distinct landmarks of historical reading. " We know of scarcely anything,^'' says the Guardian of this volume, "which is so likely to rais to a higher leierage standard of English education,^'* 70 EDUCATIONAL BOOKS. DIVINITY. * J^ For other Works by these Authors, see Theological Catalogue. Abbott (Rev. E. A.)— Works by the Rev. E. A. Abbott, M.A., Head Master of the City of London School : — BIBLE LESSONS. Second Edition. Crown 8vo. 4J. (id. This book is xvritten in the form of dialogites carried en behveen a teacher and pupil, and its main object is to make the scholar think for himself. The great bulk of the dialogues represents in the spirit, and often in the words, the religious institiction 7uhich the author has been in the habit of giving to the Fifth and Sixth Fonns of the City of London School. The author has endeavoured to make the dialogues thoroughly unsectarian. ** Wise, suggestive, and really profound initiation info religious thought."— Guardian. ♦* / think nobody could read them unthout being both the better for them himself and being also able to see hoio this difficult duty of imparting a sound religious education may be effected." — F7-om Bishop of St. David's Speech at the Education Conference AT Abergwilly. THE GOOD VOICES ; A Child's Guide to the Bible. Crown 8vo. cloth extra, gilt edges, ^s. Mr. Abbott is already knoxvn as a most successful teacher of religious truth ; it is believed that this little book iinll sho7V that he can make Biblt lessons attractive and edifying ez>en to the youngest child. The book is quite devoid 0/ all conventionality and catechetical teaching, and only en- deavours in simple language and easy style, by means of short stories and illiistratiotts from every quarter likely to interest a child, to ifu print the rudiments of religious knozulcdge, and inspire young ones 7uith a dcsi^'e to love and trust God, and to do what is right. The attthor wishes to imbue them tuith the feeling that at all times and in all circiunstances, whether in totv7i or cotmtry, at work or at play, they are living in the presence of a heavenly Father, 7vho is continually speaking to thevi with the Good Voices of Nature and Revelation. The volume contains upwards of 50 woodcuts. PARABLES FOR CHILDREN. With Three Illustrations. Crown 8vo. , gilt edges. 3J. 6d. DIVINITY. 71 ^^ Contains a number of really delightfully written and yet simple Parables, to be read out to little children as an introduction to Bible reading. They are certainly admirably adapted for the furpcse. The style is colloquial and will be understood and appreciated by the youngest child, and the parables themselves are very interesting afid well chosen.'*^ — Standard. Arnold. — a BIBLE-READING FOR SCHOOLS. The Great Prophecy of Israel's Restoration (Isaiah, Chapters 40 — 66). Arranged and Edited for Young Learners. By Mat- thew Arnold, D.C.L., formerly Professor of Poetry in the University of Oxford, and Fellow of Oriel. Third Edition. i8mo. cloth. IS. ^' Schools fr the people,''' the po7ver of letters — 7!'hich ejubraces nothing less than the whole history of the human spirit — has hardly been brought to bear at all. Mr. Arnold, in this little volume, attemp^ts to remedy this defect, by doing for the Bible what has been so abundantly done for Greek and Roman, as well as English atithors, viz. — taking " so7ne whole, of admirable literary beauty in style and treatment, of manageable length, within defined limits ; and presenting this to the learner in an intelligible shape, adding such explanations and helps as may enable him to grasp it as a connected and complete work'^ Mr. Arnold thinks it clear that nothing could more exactly suit the purpose than what the Old Testament gives us in the last twenty-seven chapters of the Book of Isaiah, beginning " Comfort ye,-' ^'c. He has endeavoured to present a perfectly correct text, maintai?iing at the same time the unparalleled balance and rhythm of the Authorised Version. In an Introductory note, Mr. Arnold briefly sums up the events of yrcvish history to the starting-point of the chapters chosen ; and, in the copiotis notes appended, e^-ery assistance is given to the complete understanding of the text. There is nothing i)t the book to hinder the adherent of any school of interpretation or of religious belief from using it, and fi'om putting it into the hands of children. The Preface contains nuich that is interesting and valuable on the relation of " letters " to education, of the pri^iciples that ought to guide the makers of a nnv version of the Bible, and other important matters. Altogether, it is believed the voluine will be found to form a text-book of the greatest value to schools of all classes. ' ' Mr. Arnold has done the greatest possible sei-vice to the public. We never read any translation of Isaiah which interfered so little ivith the musical rhythm and associations of our English Bible translation, while doing so much to display the missing links in the con- nection of the partsP — Spectator. 72 EDUCATIONAL BOOKS. Cheyne (T. K.)— the BOOK OF ISAIAII CHRONO- LOGICALLY ARRANGED. An Amended Version, with Historical and Critical Introductions and Explanatory Notes. By T. K. Cheyne, M.A., Fellow of Balliol College, Oxford. Crown 8vo. *]$. dd. The object of this edition is simply to restore the probable meaniug of Isaiah^ so far as this can be expressed in modern English. The basis of the version is the revised translation of i6i i, but no scruple has bten fill in introducing alterations^ wherever tJie true sense of tJie prophecies appeared to require it. *' A piece of scholarly luork, vety carefully and considerately done'' — Westminster Review. Golden Treasury Psalter.— students' Edition. Being an Edition of "The Psalms Chronologically Arranged, by Four Friends," with briefer Notes. i8mo. 3^. 6d. In making tfiis abridgment of^^ TJie Psalms Chronologically Arranged, * tfie editors fiave endeavoured to meet the requirements of readers of a different class from those for whom the lirger edition was intended. Some wf 10 found the large book useful for private reading, Jiave asked for an edition of a smaller size and at a louver price, for family use, while at the same time some Teacho's in Public Schools have suggested tfiat it would be convenient for tJiem to fiave a simpler book, which they could put into tfie hands of younger pupils. *^ It is a gem,'" says tfie Nonconformist. Hardwick. — a HISTORY OF THE CHRISTIAN CHURCH. Middle Age. From Gregory the Great to the Excommunication of Luther. Edited by William Stubbs, M.A., Regius Professor of Modern History in the University of Oxford. With Four Maps constructed for this work by A. Keith Johnston. Third Edition. Crown 8vo. los. 6d. Although the ground-plan of this treatise coincides in many points with th.it of the colossal zvork of Schrockh, yet in arranging the materials a very different course has frequently been pursued. IVitfi regard to his opinions the late autfior avowed distinctly tfiat fie construed history with the specific prepossessions of an Englishman and a member of tfie English CfiurcJi. The readei' is constantly referred to the autfiorities , both original and critical, on which the statements are founded. For this edition Professor Stubbs fias carefully rnised both text and notes, making such corrections of facts, dates, and the like as tfie results of recent research zvar'rant. The doctrinal, historical, and generally speculative views 0f the late autfior have been preserved intact. "As a m anual for DIVINITY, 73 H ar d wick — co7iti?med. the shtdent of ecclesiastical history in the Middle Ages, we know no English work which can be compared to Mr. Hardwiclzs book." — Guardian. A HISTORY OF THE CHRISTIAN CHURCH DURING THE REFORMATION. By Archdeacon Hardwick. Third Edition. Edited by Professor Stubbs. Crown 8vo. los. 6d. This volwne is intended as a sequel and companion to the ^'^ History of the Christian Church dicring the Middle Age.^'' The author's earnest wish has been to give the reader a trnstivorthy version of those stirring incidents tvhich mark the Reformation period^ withotit relinquishing his former claim to characterise peculiar systems, persons, and events according to the shades and colours they asstcme, xvhen contemplated from an English point of viei.v and by a member of the Church of England. Maclear.— Works by the Rev. G. F. MACLEAR, D.D., Head Master of King's College School. A CLASS-BOOK OF OLD TESTAMENT HISTORY. Seventh Edition, with Four Maps. i8mo. cloth. 4^'. (yd. This volume forms a Class-hook of Old Testament History from the earliest times to those of Ezra and Nehe^niah. In its preparation the most recent authorities have been cons7clted, and wherever it has appeared usefiil. Notes have been subjoined illustrative of the Text, and, for the sake of more advanced students, references added to larger works. The Index has been so arranged as to form a concise dictionary of the persons and places mentioned in the course of the narrative ; zvhile the Maps, which have been prepared with co7isiderable care at Stanford'' s Geographical Establish- niifit, will, it is hoped, materially add to the value and tisefulness of the Book. ^' A carefid and elaborate though brief compendiuvi of all that modern research has done for the illustration of the Old Testament. We knoxv of no work which contains so much important information in so sinall a compass' — British Quarterly Review. A CLASS-BOOK OF NEW TESTAMENT HISTORY, Including the Connexion of the Old and New Testament. With Four Maps. Fourth Edition. l8mo. cloth. 5^-. (^d. A sequel to the author's Class-book of Old Testament History, contimiino the narrative from the point at which it there ends, and carrying it on to the close of St. Pauls second imprisonment at Rome. In its preparation , 74 EDUCATIONAL BOOKS. Maclear — continued. as in that of the former volume, the most recent and trustrvor thy authorities have been consulted, notes subjoined, and references to larger works added. It is thus hoped that it may prove at once an useful class-book and a convenient companion to the study of the Greek Testament. *'A singularly clear and orderly arrangement of the Sacred Story. His work is solidly and completely done. " — Athen^^um. A SHILLING BOOK OF OLD TESTAMENT HISTORY, for National and Elementary Schools. With Map. i8mo. cloth. New Edition. A SHILLING BOOK OF NEW TESTAMENT HISTORY, for National and Elementary Schools. With* Map. i8mo. cloth. New Edition. TTtese works have been carefully abridged from the author's larger manuals. CLASS-BOOK OF THE CATECHISM OF THE CHURCH OF ENGLAND. Second Edition. i8mo. cloth. 2s. 6d. • This may be regarded as a sequel to the Class-books of Old and Nero Testament History. Like them, it is flemished with notes and references to larger works, and it is hoped that it may be found, especially in the higher forms of our Public Schools, to supply a suitable manual of instruction in the chief doctrines of the English Church, and a useful help in the preparation of candidates for Confirmation. ''It is indeed the work of a scholar and dimne, and as such, though extremely simple, it is also extrejnely instructive. There are fnv clergymen who 7vould not find it useful in preparing candidates for Confirfnation ; and there are not a feiv who would find it useful to themselves as well.'' — Literary Churchman. A FIRST CLASS-BOOK OF THE CATECHISM OF THE CHURCH OF ENGLAND, with Scripture Proofs, for Junior Classes and Schools. i8mo. 6d. New Edition. THE ORDER OF CONFIRMATION. A Sequel to the Class Book of the Catechism. For the use of Candidates for Confirma- tion. With Prayers and Collects. iSmo. 3^/. New Edition. Maurice.— THE LORD'S PRAYER, THE CREED, AND THE COMMANDMENTS. A Manual for Parents and School- masters. To which is added the Order of the Scriptures. By the Rev. F. Denison Maurice, M.A. Professor of Moral Philosophy in the University of Cambridge. i8mo. cloth limp. is. DIVINITY. 75 Procter. — a history of the book of common PRAYER, with a Rationale of its Offices. By Francis Procter, M.A. Tenth Edition, revised and enlarged. Crown 8vo. IOJ-. 6d. In the cotirse of the last tiventy years the whole qnestiott of Liturgical knoztiledge has been reopened with great learning and accurate research : and it is mainly with the vinu of epitomizing extensive publications^ and correcting the errors and misconceptions which had obtaiiied curre?tcy, that the present volume has been put together. " We admire the author's diligence, and bear zvilling testimony to the extent and accuracy of his reading. The origin of every part of the Prayer Book has been diligently investigated, and there are fexv questions of facts connected with it xvhich are not either siiffi.ciently explained, or so referred to that persons interested may 7Uork out the truth for themselves. ''' — Athen^^UM. Procter and Maclear. — an elementary INTRO- DUCTION TO the book of common prayer. Re-arranged and supplemented by an Explanation of the Morning and Evening Prayer and the Litany. By the Rev. F. Procter and the Rev. G. F. Maclear. Fourth Edition. i8mo. 2s. 6d. As in the other Class-books of the series, No*es have also been subjoined^ and references given to larger works, and it is hoped that the volume ivill be fotcnd adapted for use in the higher forms of our Public Schools, and a suitable manual for those preparing for the Oxford and Cambridge local examinations. This Ne%v Edition has been considerably altered, and several important additions have been made. Besides a re-arrangement of the tvork generally, the Historical Portion has been supplevtented by an * Explanation of the Morning and Evening Prayer and of the Litany. Psalms of David Chronologically Arranged. By Four Friends. An Amended Version, with Historical Introduction and Explanatory Notes. Second and Cheaper Edition, with Additions and Corrections. Crown 8vo. Sj. dd. 7o restore the Psalter as jar as possible to the order in 7vhich the Psalms were written, — to give the division oj each Psalm into strophes, of each strophe into the lines which composed it, — to amend the errors of translation, is the object of the present Edition. Professor Ezvald's works, especially that on the Psalms, have been extensively consulted. This book has been used with satisfaction by masters for private work in higher classes in 76 EDUCATIONAL BOOKS. schools. The SPECTATOR calls this ** one of the most instructive and valuable books that has been published for many years.*' Ramsay. — THE CATECHISER'S MANUAL; or, the Church Catechism Illustrated and Explained, for the use of Clergymen, Schoolmasters, and Teachers. By the Rev. Arthur Ramsay, M.A. Second Edition. i8mo. is. 6d. A clear explanation of the Catechism^ by 7vay of Question and Answer. " This is by far the best Manual on the Catechism^ we have met with." — English Journal of Education. Simpson.— AN EPITOME OF THE HISTORY OF THE CHRISTIAN CHURCH. By William Simpson, M.A. Fifth Edition. Fcap. 8vo. 3^. 6d. A compendious summary of Church History. Swainson.— A HANDBOOK to BUTLER'S ANALOGY. By C. A. Swainson, D.D., Canon of Chichester. Crown 8vo. is. 6d. This manual is designed to sen'e as a handbook or road-book to the Student in reading the Analog)', to give the Student a sketch or 07itlinemaf> of the country on which he is entering, and to toint out to him matters oj interest as he passes along. Trench.— SYNONYMS OF THE NEW TESTAMENT. By R. Chevenix Trench, D.D., Archbishop of Dublin. New Edition, enlarged. 8vo. cloth. 12s. The study of synonyms in any language is valuable as a discipline for iraitiing the mind to close and accurate habits of thought : more , especially is this the case in Greek — **« language spoken by a people of the finest and subtlest intellect ; who sa^v distinctions where others saw none, 7uho divided out to diffei-ent 7vords what others often were content to huddle confusedly under a common term. This work is recognised as a valuable companion to aiery student of the New Testament in the ofiginal. This, the Seventh Edition, has been carefully revised, and a considerable number of new synonyjHS added. Appended is an Index to the Synonyms, and an Index to many other words alluded to or explained throughout the work. '^Ileis,*^ the ATHENi^UM says, *' a guidi in this department of knmtf- ledge to whom his readers may intrust themselves 7tnth confidence. Bis sober judgment and sound sense are barrieis against the misleading influence op arbitrary hypotheses.** DIVINITY. 77 WestCOtt.— Works by BROOKE FOSS WESTCOTT, B.D., Canon of Peterborough. A GENERAL SURVEY OF THE HISTORY OF THE CANON OF THE NEW TESTAMENT DURING THE FIRST FOUR CENTURIES. Third Edition, revised. Crown 8vo. loj. dd. The author has endeavoured to connect the history of the New Testament Canon with the growth and consolidation of the Chzirch, and to J)oint out the relation existing between the amoicnt of evidence for the authenticity of tts component parts, and the whole mass of Christian literature. Sicch a method of inquiry will convey both the truest Jiotion of the connection of the written Word with the living Body of Christ, and the surest conviction oj tts divine authority. Of this work the Saturday Review writes : " Theo- logical students, and not they only, but the general public, owe a deep debt of gratitude to Mr. Westcott for bringing this subject fairly before them , in this candid and comprehensive essay As a theological tvork it is at once perfectly fair and impartial, and imbued with a thoroughly religious spirit; and as a manual it exhibits, in a lucid form and in a narrow compass, the results of extensive research and accurate thought. We cordially recommend it." INTRODUCTION TO THE STUDY OF THE FOUR GOSPELS. Fourth Edition. Crown 8vo. los. 6d. The author's chief object in this work is to show that there is a true mean between the idea of a formal harmonization of the Gospels and the abandonment of their absolute truth. The treatise consists of eight chapters: — /. The Ft-eparation for the Gospel. II. The Jezvish Doctrine of the Messiah. III. The Origin of the Gospels. IV. The Charac- teristics of the Gospels. V. The Gospel of St. John. VI. &^ VII. The Differences in detail and of arrangement in the Synoptic Evangelists. VIII. The Difficulties of the Gospels. ''^ To a learning and acctiracy which commands respect and confidence, he tmites what are not always to be found iti union with these qualities, the no less valuable factdties of lucid arrangement and graceful and facile expression.''^ — London Quarterly Review. A GENERAL VIEW OF THE HISTORY OF THE ENGLISH BIBLE. Crown 8vo. loj. (>d. Second Edition. 78 EDUCATIONAL BOOKS, We St c olt — continued. " The first trustixjorthy accouni wt have had of that unique and mar- vellous monument of the piety of our ancestors** — Daily News. " A briej, scholarly, and, to a great extent, an original contribution to theological literature. He is the first to offer any considerable contJtbu- tions to uihat he calls their internal history, ivhich deals with their relation to other tt-xts,7oith their filiation one on another, and with the principles by which they have been successively viodifiedf^ — Pall Mall Gazette. THE BIBLE IN THE CHURCH. A Popular Account of the Collection and Reception of the Holy Scriptures in the Christian Churches. Xew Edition. iSnio. cloth. 4J-. dd. The present book is an attempt to ans^uer a request, which has been made from time to time, to place in a simple form, /or the use of general readers, the substance of the author's '^History of the Canon of thcNcw Testament." An elaborate and comprehensive Introduction is fallowed by chapters on the Bible of the Apostolic Age ; on the Growth of the New Tatament ; the Apostolic Fathers ; the Age of the Apologists ; the First Christian Bible ; the Bible J Proscribed and Restored ; the Age of Jerome and Augustine ; the Bible of the Middle Ages in the West and in the East, and in the Sixteenth Century. 7 wo Appendices on the History of the Old Testament Canon before the Christian Era, and on the Contaits of the most ancient A/SS. of the Christian Bible, complete the volume. *'' IVe would recommend ei'cty one who loves and studies t/ie Bible to read and ponder this exquisite little book. Mr. IFestcott's account of the * Canon * is true history in its highest sense." — LITERARY CHURCHMAN. THE GOSPEL OF THE RESURRECTION. Thoughts on its Relation to Reason and History. New Edition. Fcap. 8vo. 4J. 6d. This Essay is an endeavour to consider some of the elementary truths 0/ Christianity as a miraculous Reuelaiion, from the side of History and Reason. If the arguments which are here adduced are valid, they will go far to prove that the Resurrection, with all that it includes, is the key to the history of man, and the cotnplement of reason. Wilson.— THE BIBLE STUDENT'S GUIDE to the more Correct Understanding of the English translation of the Old Testament, by reference to the Original Hebrew. By William Wilson» D.D., Canon of Winchester, late Fellow of Queen's College, Oxford. Second Edition, carefully Revised. 410. cloth. 25J. DIVINITY. 79 This work is the result of almost incredible labour bestoived ondt during many years. Its object is to enable the readers of the Old Testament Scriptures to penetrate into the real meaning of the sacred writers. All the English zvords used in the Authorized Version are alphabetically arranged , and beneath them are given the Hebrew equivalents, with a careful expla- nation of the peculiar signification and construction of each term. The knowledge the Hebrezu language is not absolutely necessary to the profit- able use of the work. Devout and accurate students of the Bible, entirely unacquainted with Hebrew^ iiiay derive great advantage from frequent reference to it. It is especially adapted for the use of the clergy. * * For all earnest students of the Old Testament Scriptures it is a most valuable Manual. Its arrangevient is so simple that those who possess only their mother-tongue, if they will take a little pains, may employ it with great profit." — Nonconformist. Yonge (Charlotte M.)— SCRIPTURE READINGS FOR SCHOOLS AND FAMILIES. By Charlotte M. Yonge, Autlior of **The Heir of Redclyffe." Globe 8vo. is. 6d. With Comments, Second Edition. 3^-. 6d. A Second Series. From Joshua to Solomon. Extra fcap. IS. 6d. With Coinments, 3J. 6d. Actual need has led the author to endeavour to prepare a reading book con - venient for study with children, containing the very words of the Bible, with only a few expedient omissions, and arranged in Lessons of such letigth as by experietice she has found to suit with children's ordinary power of accurate attentive interest. The verse fortn has beeji retained, because of its con- venience for children reading in class, and as more resembling their Bibles ; but the poetical portions have been given in their lines. When Fsabns or portions from the Prophets illustrate or fall in with the narrative they are given in their chronological sequence. The Scripture portion, with a veiy fezu notes explanatory of mere words, is bound up apart, to be used by children, while the same is also supplied with a brief comment, the purpose of which is either to assist the teacher in explaining the lesson, • or to be used by more advanced young people to whom it may not be possible to give access to the authorities whence it has been taken. Professor Huxley, at a meeting of the London School Board, partictilarly mentioned the selection made by Miss Yonge as an example of hozo selections might be made from, the Bible for School Reading. See Times, March 30, 1 87 1 . Catalogue of Works on Education^ Physical and Mental, General and Special. Arnold.— A FRENCH ETON : OR, MIDDLE - CLASS EDUCATION AND THE STATE. Fcap. 8vo. cloth, zs. U, This intei-esting little volume is the result of a visit to Fiance in 1859 by Mr. Arnold, authorized by the Royal Commissioners^ ivho were then inquiring into the state of popular education in England, to seek^ in their name, information respecting the French Primary Schools. ** A very interesting dissertation on the system of secondary instruction in France, and on the advisability of copying the system in England.^^ — Saturday Review. HIGHER SCHOOLS AND UNIVERSITIES OF GERMANY. Crown 8vo. ds. Jex-Blake.— A visit to some American schools AND COLLEGES. By Sophia Jex-Blake. Crown 8vo. cloth. 6>r. " /« the following pages I have endeavoured to give a simple and accurate account of what I saxu during a series of visits to some of the Schools and Colleges in the United States, . . . I wish simply to give other teachers an opportunity of seeing through my eyes what they cannot perhaps see for the?nselves, and to this end I have recorded just such parti- culars as I should myself care to knoxv." — Author's Preface. ^'Miss Blake gives a living picture of the Schools and Colleges themselves in which that education is carried on."— 'Paia. Mall Gazette. Maclaren.— TRAINING, IN THEORY AND PRACTICE. By Archibald Maclaren, the Gymnasium, Oxford. 8vo. Handsomely bound in cloth, 7^. 6d. The ordinary agcfits of health are Exercise, Diet, Sleep, Air, Bathing, and Clothing. In this work the attthor examines each of these agents PHYSICAL AND MENTAL, ETC. in detail, and from two different points of vieiv. First, as to the manner in which it is, or should be, administe^'ed under ordinary circumstances : and secondly, in zvhat manner and to what extent this mode of adminis- tration is, or should be, altered for purposes of training ; the object of ^^ training," according to the author, being ^^ to put the body, with extreme and exceptional care, under the influence of all the agents which promote its health and strength, in order to enable it to meet extreme and excep- tional donands upon its energies.'''^ Appended are various diagrams and tables relating to boat-racing, and tables connected with diet and training. " The philosophy of human health has seldom received so apt an exposi- tion.''^ — Globe, '■^ After all the nonsense that has been written about training, it is a comfort to get hold of a thoroughly sensible book at last.^^ —John Bull. Quain (Richard, F.R.S.) — ON SOME DEFECTS IN GENERAL EDUCATION. By Richard Quain, F.R.S. Crown 8vo. y. 6d. Having been charged by the College of Surgeons with the delivery of the Hunterian Oration for 1869, the attthor has availed himself of the occa- sion to bring under notice some defects in the general education of the country, 7vhich, in his opinion, affect injuriously all classes of the people, and not least the members of his o%vn profession. The earlier pages of the addrets contain a short notice of the genius and labours of John Hunte?; but the subject of Education will be found to occupy the largei' part. ''An i'lfercsting aadition to educational literature.^^ — Guardian. Selkirk.— GUIDE TO THE CRICKET-GROUND. By G. II. Selkirk. With Woodcuts. Extra fcap. Svo. 3^. 6^/. The introductory chapter of this little work contains a history of the iVational Game, and is follo7ved bv a chapter giving Definitions of Terms. Then follo7v ample directions to young cricketers as to the proper style in which to play, information being given on every detail connected with the ^anie. The book contains a number of useful illustrations, including a specimen scoring- sheet. " We can heartily recommend to all cricketers, old and young, this excellent Guide to the Cricket-ground. ''' — Sporting Life. FOURTEEN DAY USE RETURN TO DESK FROM WHICH BORROWED This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. 5Jan'56PLY Wli$ REC'D LD ^€9 LIBRARY USF JUN 3 iab9 78 J AN '54VC Ml 2 - L - Iw w J REC'D LD ^H«*^^^^^\N14'64-9PM LIBRARY USE SE?1 0^956 REC'D LD SEP 10 1956 «I»«PTX IRCCT* Cg - «rtn§ir -tt495a K LD 2l-100m-2,'55 (B139s22)476 General Library University of California Berkeley f YB i /o3i ,■ ^*