MEMORANDA MATHEMATICA A SYNOPSIS OF FACTS, FORMULAE, AND METHODS IN ELEMENTARY MATHEMATICS BY W. p. WORKMAN, M.A., B.Sc. HEAD MASTER OF KTVGSWOOD SCHOOL, HATH LATE FET.LOW OF TRINITY COLLEGE, CA^fBRIDOE OXFORD AT THE CLARENDON PRESS 1912 HENRY FROWDE, M.A. PUBLTRHER TO THE UNIVERSITY OF OXFORD LONDON, EDINBURGH, NEW YORK TORONTO AND MELBOURNE . .*'!'t.''tt' PREFACE Most students and teachers of Mathematics have at times wished for a book to which they might speedily refer for a particular formula or method. Such a want the author attempts to supply in the present volume, which collects in convenient compass the facts, formulae, and results of Elementary Mathematics and is intended, not to replace existing text-books, but to be a com- panion to them all, useful for revision and handy for reference. The book embraces the subjects usually read in a school course with the exception of Arithmetic, which is only fragmentarily treated, and the more elementary parts of Geometry, which have less need of a summar}' than other subjects, since they are always presented in proposition form. The other branches of Elementary- Mathematics are treated exhaustively, so that, as far as those subjects are concerned, the summary given should include almost all the requirements of a University course. It is hoped, therefore, that the book will be of special value to candidates for mathematical scholar- ships and to students in their first year who are reading for Mathematical Honours at the Universities. Formulae with their deductions and applications are presented synoptically in accordance with a scheme enabling a reader to perceive analogies which he would possibly miss if dependent solely upon text-books. The 261065 iv PREFACE methods by which the results are obtained are often indicated liy a reference to other theorems upon which they depend, or by a summary of the proof, especially where the customary discussion calls for comment or modification. Notes are frequently added on matters of interest. Elaborate cross-references and a full index are provided ; and, where subjects overlap, some repetition has been considered advisable. The text has been set in two types. Matter is printed in larger type when, in the author's judgement, it should be committed to memory (but not, of course, verbatim). Matter in small type is confined to less important results, corollaries, and the solution of illustrative examples. Abbreviated proofs are also given in this form, together with data which are only occasionally required. By arrangement with the author and publishers, the small but excellent collection of Mathematical Tables compiled by Mr. W. E. Paterson (Five-figure Logarithmic and Trigonometric Tables, Clarendon Press) has been appended to the book, and it is believed that this will increase its usefulness. Every care has been taken in the correction of the proofs, but faults can scarcely be absent from a com- plicated work compiled in the intervals of a busy life, and the author will at all times be thankful to hear of any errata which may be detected, or to receive any suggestions for the improvement of the book. Bath, July, 1912. CONTENTS Arithmetic .... page 1 Mensuration .... 6 Algebra 11 Theory of Equations 38 Determinants .... 64 Plane Trigonometry 68 Spherical Trigonometry 89 Geometry .... 99 Geometrical Conics 130 Analytical Geometry 157 Differential Calculus . 197 Elementary Statics 232 Elementary Dynamics . 243 Elementary Hydrostatics 258 Index 266 Mathematical Tables 273 _ AEITHMETIC * 1. The Nine Remainder. Add up the digits of a number, rejecting all nines and all digits whose sum makes 9 and subtracting 9 whenever the sum is greater than 9. The single digit remainder obtained is the remainder after division of the number by 9. Ex. 3782615. The work runs 5, 6, 12, [3], 5, 13, [4J, 11, [2], 5. Ans. 5. The numbers in [ ] occur after subtracting a 9. It is best for the beginner to form the habit of beginning with the units digit. The Eleven Remainder. Begin with units digit, add alternate digits, when the left of the number is reached return, subtracting from the sum obtained the digits which still remain. If when the right of the number is again reached the result is < 11 it is the remainder after dividing by 11. If it is greater than 11 subtract ll's until it becomes less. Elevens may be rejected or added at any stage. i:a:. 7 8 6 5 3 8 1. The work is 1, 4, 10, 17, (begin sub- tracting) 9, 4, (add 11) 15, 7. Ans. 7. Proving by Nines and Elevens. To test a multiplication result, MN = P. Find the remainders m, n, p after dividing If, iV, P by 9 or 11. Then the remainder of mn must be^. Notes. (1) The student is urged to test every completed multiplica- tion result this way. The time required soon becomes infinitesimal. (2) Instead of the Eleven Remainder the Nine Remainder may be used, but is not so satisfactory. (3) A product which satisfies the test may still be wrong. * The summary of this subject does not profess to be complete, and embraces only some points which are commonly forgotten. Enuncia- tions are sometimes taken from the author's Tutorial Arithmetic. 1312 B AEITHMETIC * 1. The Nine Kemainder. Add up the digits of a number, rejecting all nines and all digits whose sum makes 9 and subtracting 9 whenever the sum is greater than 9. The single digit remainder obtained is the remainder after division of the number by 9. Ex. 3782615. The work runs 5, 6, 12, [3], 5, 13, [4J, 11, [2], 5. Ans. 5. The numbers in [ ] occur after subtracting a 9. It is best for the beginner to form the habit of beginning with the units digit. The Eleven Kemainder. Begin with units digit, add alternate digits, when the left of the number is reached return, subtracting from the sum obtained the digits which still remain. If when the right of the number is again reached the result is < 11 it is the remainder after dividing by 11. If it is greater than 11 subtract ll's until it becomes less. Elevens may be rejected or added at any stage. i:a:. 7 8 6 5 3 8 1. The work is 1, 4, 10, 17, (begin sub- tracting) 9, 4, (add 11) 15, 7. Ans. 7. Proving by Nines and Elevens. To test a multiplication result, MN = P. Find the remainders m, n, p after dividing M, N, Phy 9 or 11. Then the remainder of mn must be^. Notes. (1) The student is urged to test every completed multiplica- tion result this way. The time required soon becomes infinitesimal. (2) Instead of the Eleven Remainder the Nine Remainder may be used, but is not so satisfactory. (3) A product which satisfies the test may still be wrong. * The summary of this subject does not profess to be complete, and embraces only some points which are commonly forgotten. Enuncia- tions are sometimes taken from the author's Tutorial Arithmetic. 2 ' '* ' ' . AKITHMETIC 2.' TfiE 'Metic -System. Multiples (Greek prefixes) : deca-, hecto-, kilo-. Submultiples (Latin prefixes) : deci-, centi-, milli-. (1) Length. Unit : 1 metre = 39-37 inches. Approximations: 1 centimetre = f inch. 1 kilometre = f mile. (2) Area. Unit : 1 Are = 100 sq. metres. Approximation : 1 Are = J^ acre. Note. But scientific measurements are usually made in sq. cm. or sq. m. (3) Volume. Unit : 1 litre = 1 cub. decimetre. Approximation : 1 litre = If pints. Note. But scientific measurements are usually made in c. c. (4) Mass. Unit', 1 gram = wt. of 1 c.c. of pure water at4C. = 15-432 grains. Note. The accurate result is easier to remember than the ap- proximation 15f. Approximation: 1 kilogram = 2J^lb. (5) The C. G. S. system of units is the system in which the centimetre, gram, second are the units respectively of length, mass, and time. 3. Terminating Decimals. The following should be known by heart : 2 = A .5 = i .25 = i .75 = I 125 = i .375 = I .625 = f -875 = |. j^ote. ^These are of great use in simplifications, e. g. -6875 = .61 = ^1. 4. Kecurring Decimals are subject to the following laws: Latv i. If N/D is a proper fraction in its lowest terms and D is a prime number, other than 2 or 5, then N/D ARITHMETIC 3 gives rise to a pure circulator and the number of places in the recurring period is either B-l or one of the factors of 2)-l. Note. If D = 7, 11, 17, 19 (mod. 40), N/D runs to the full period of D 1 figures or to an odd submultiple of that period. If -^ = 1) ^5 ^> 13, N/D runs to an even submultiple of (D 1) figures. The last group of figures is easily remembered if desired : 1, 3, 9 and 1 + 3+9. Latv ii. If D and E are different prime numbers other than 2 and 5 and 1/D has a period of p figures, while 1/E has a period of q figures, then N/DE [N prime to BE) has a period of r figures where r is the L. C. M. of ^ and 2 Note. Similarly for three or more factors. Law iii. If Z) is a prime number other than 2 and 5 and 1/D has a period of i? figures, l/D^ will usually have a period of^D figures, 1/D^ of pl)^, and so on. Note. The only exceptions at present known to this are D = 3 and i) = 487 ; 1/487 and 1/4872 both have periods of 486 figures. Latv iv. If D is a number prime to 2 and 5 such that 1/D has a period of p figures, the fraction 1/2'** 5** D is a mixed circulator with either m or n non-recurring figures (according as m or n is greater) and with p recurring figures. Law V {The Cyclic Law). If 2) is a prime number other than 2 and 5 and 1/D has a period of ^ figures, then the fractions 1/D, 2/Z), 3/Z) ... (2)- 1)/Z) can be divided into {D - l)/p groups, each group containing p fractions giving rise to a circulating period made up of the same figures, for all fractions of the group, taken in the same order round a circle, but starting at different points of the circle. Ex. 1/13 circulates in 6 figures, .*. there are (13 1)/6 = 2 groups. 076923, 769230, ^692307, -923076, -230769, -307692 represent re- spectively ^, i, ^, if, 1%, and j%, while the circulators formed from 153846 = i5 give the remaining 6 fractions with denominator 13. Laiv vi (Tlie Complementary Law). If N/D is a fraction in its lowest terms which gives rise to a circulator having an B 2 4 ARITHMETIC even number of places in its recurring period (say 2n), then the last n figures of the recurring period may be found from the first n figures by subtracting these successively from 9, provided D is either (1) a prime, or (2) the product of primes, none of which is capable separately of giving a period of n figures, or (3) the product of such primes with a power of 2 or 5 or both. Ex. Thus in the Ex. of Law v, 076 + 923 = 999, &c. 5. Recurring Decimals. The following results should be learnt : 16; = 142857 with cyclical interchanges giving f, &c. ; = -076923 with cyclical interchanges, &c. [ 4, Law v, Ex.], and with easy deduction, mentally, of j^. Notes. (1) As iu 3, -8076923 = -83^ = { = ||, &c. (2) Many otlier results are not so much learnt as worked mentally when required, e. g. -3, -83, &c. (3) To remember the periods for ^ and ^^3 note that 7 = 1+4+2, and 18 = + 7 + 6. The Complementary Law gives the rest of the period. 6. Involution and Evolution. (1) The following squares should be known by heart : 1 _ JL 13 1 2 4 3 9 4 16 5 25 6 7 8 9 1 36 49 64 81 1 121 144 169 196 225 256 289 324 361 2 441 484 529 576 625 729 (2) The following cubes should be known by heart 1 1 2 3 4 5 6 7 8 512 9 729 1 1331 8 1728 27 64 125 216 343 ARITHMETIC 5 (3) The following square roots should be known : 72 =14142 ^S = 1.7320[5]. 7. Logarithms. The following logarithms should be known : log 2 = .3010[300] log 5 = -6990 log 3 = 4771[213] log 7 = -8451 log 77= 4971[5]. 8. Numerical Data. The following numerical data are worth retaining. They are for the most part approximations only: (1) A cubic foot of water weighs 1000 oz. A pint of water weighs a pound and a quarter. (Exact.) A gallon of water weighs 101b. (Exact.) A gallon contains 277J cub. inches. (2) A year is a leap-year when the last two figures of its number form a number divisible by 4, except for centuries, in which case the number of the century must be divisible by 4. (3) The diameter of a halfpenny is 1 inch. Ten farthings, five halfpennies, or three pennies weigh 1 oz. Av. (Exact.) (4) The radius of the Earth = 4000 miles = E, and its mean density is 5^. (5) The mean distance of the Moon = 60 E, its radius is Yi E, and its angular diameter = ^. (6) The mean distance of the Sun = 93,000,000 miles, and its angular diameter = J. (7) The velocity of a shot from a big gun is from 1500 to 3000 ft. per second, the velocity of sound in air is 1100 ft. (332 m.) per second, and the velocity of light is 186,000 miles (300,000 Km.) per second. MENSURATION LENGTHS OF CUKVED LINES 1. Circumference of Circle (radius r) = 2TTr. Values of tt: 3-14159... Approximations-. 22/7, 355/113. Note. The approximation 355/113 easily remembered as 113)355( gives 6 decimal places, 3-141592... correct, and is almost as easy to work with as 22/7, which only gives 2 places. 2. Arc of Circle, subtending an Z^ radians (= A^) at the centre, Br = A-nr/lSO = i {8 chord ^ arc - chord arc} approximately. jVofe. If A < 60, the percentage error of the approximate formula is -1 or less. 3. Ellipse, Quadrant 2 L W 2!2! V2y 3!3!V2/ *" J* Note. - ^TT (a + &) gives the first two terms of this series and is a better approximation than \v */ab. If e < | the error is < -1%. 4. Parabola (vertex A). If P be a point whose abscissa is x \^x + \/a + X Arc AP = ^/ax + x'^-\'a\og -y/a Areas of Plane Figures (Rectilinear Boundaries). 5. Triangle. A = -J base x i' height, = ^ product of any two sides x sine of contained angle, = Vs(s- a) (s - &) (s - c), where s = -| (a 4- & + c). MENSURATION 6. Rectangle. Area = length x breadth. Square. Area = (side)^. Parallelogram. Area = any side x j.^ dist. to opp. side, = ah sin 0. [a, h adjacent sides con- taining Z^.] 7. Tkapeztum. Area = (mean of !l sides) (J.^ distance be- tween them). 8. Quadrilateral. Divide into triangles. [See also Ti^ZG^. 14.] 9. Regular Polygon. Divide into triangles from the centre. ITRIG, 15.] For reference. Tlie area of a regular polygon, side a, is kct^, where k has the following values: Triangle (-4330), Square (1), Pentagon (1-7205), Hexagon (2-5981), Heptagon (3-6339), Octagon (4-8284), Nonagon (6-1818), Decagon (7.6942). 10. Irregular Polygon. If non-reentrant draw -L^'^ from all angular points upon the longest side, thus dividing into 2 triangles and w - 3 trapezia. If reentrant by triangles. Areas of Plane Figures (Curvilinear Boundaries). 11. Circle. Area = nr^. Circular Ring. Area = tt (i?^ - r^), where R is outer radius, r inner. = T:{R-r){R + r). Sector of Circle subtending an /.A{= 6 radians) at centre. A ^^^^ = 360"^"^ iOr^ Segment of Circle. Subtract A formed by radii and chord from sector. ^pji 12. Ellipse. Area = irah. Segment of Ellipse. Area APN Area AP^N Sector of Ellipse. Area ACP = AresisANP+CNP. 8 MENSURATION 13. Parabola. The area cut off from a parabola by any chord = I of that of a parallelogram with the chord as base and with the side parallel to the chord touching the parabola. Note. In particular if a double ordinate be drawn perpendicular to the major axis at a distance x from the vertex it cuts off from the curve an area ^xy = a^ x^, where 4a is the Latus Rectum. 14. Simpson's Approximate Rule for any curve. Area i?PZ^ = ^{yi + ?/2n + l + 2(?/3 + /5+ ... +2/2n-l) s= J [sum of first and last ordinates, twice the other odd ordinates, and four times all even ordinates] x distance between the ordinates. y >^ y^r^yzM Fig. 2. Notes. (1) By far the best of the approximate formulae is the remarkable Weddle's Rule : Divide the area by parallel and equidistant ordinates into groups of 6 spaces. For each group of 6 add together the odd ordinates (the first and the last are odd ordinates), six times the middle ordinate, and five times the two other even ordinates. Multiply the sum by three-tenths of the distance between the ordinates and the result is the area. (2) If the ordinates are measured, and not calculated, the errors of Weddle's Rule are far inferior to the errors of measurement. (3) Both Weddle's and Simpson's Rule give their best results when none of the ordinates are zero, and when they are numerous. 15. Similar Plane Figures. Areas proportional to the squares of corresponding sides. MENSURATION 9 Areas of Curved Surfaces. 16. Right Circular Cylinder. Curved surface = perimeter of base x height = 2T:rh. 17. Right Circular Cone. Curved surface = ^ perimeter of base x slant height = Tirl. = -nrJi cos 6. (/^ = ^ height, = semi- vertical angle.) = irr^ sin 0. 18. Sphere. Curved surface = 4^7: r"^. Segment of Sphere. Curved surface = 27rrh. [h = ^ height of segt.] Belt of Sphere (cut off by II planes). Curved surface = 2TTrh. [h = ^ dist. of planes.] Note. These results are particular cases of an important general theorem. If a sphere just fits inside a cijlinder tivo planes perpendicular to axis of the cylinder will cut off curved areas from the sphere and from the cylinder which are equal (and have the same centroid) . Volumes of Solids. 19. Rectangular Parallelepiped. Vol. = length x breadth x height = dbc. Cube, edge a. Vol. = a^. 20. Pyramid on any base : Tetrahedron. Vol. = J base x height. 21. Regular Solid, edge a. Volume = ka^, where k has the follow- ing values : Tetrahedron (J^ V^)> Cube (1), Octahedron (^ \^2), Dodecahedron f (15 + 7 \/5), Icosahedron -^^ (^^ + ^ V^). 22. Prism. Vol. = cross-section x length. 23. Frustum of Prism. Vol. = cross-section x average length of edges. 24. Cylinder on any base (Right or Oblique). Vol. = area of base x i.^' height. 10 MENSURATION 25. Cone (Right or Oblique). Vol. = i base x _Li' height. = J Tir'^li (for rt. circular cone). 26. Frustum of Cone or Pyramid [i. e. figure cut off by two planes || to base]. If A^ A.^ be areas of ends of frustum and /* the J_'' dist. between them, Vol. = ih [^1 + A^+ VA^ A^\ 27. Sphere. Vol. = f tt/-^. Zone of Sphere. Vol. = f ti/j {/i^ + 3 (rj^ + /-g^)}, where r^ r^ are the radii of the li ends and 7i is the J-^* dist. between them. Note. If Xj X.J be distances of the parallel ends from the centre and a be theradius of the sphere this result may also be written ^n (x^ rcj) { 3a' (Xj2 + x^x^ + a;^^^ j. Segment of Sphere. Put r,^ = in 'zone'. 28. Ellipsoid. Yol. = ^-nahc. Spheroid. Put h = c. 20. Paraboloid of Revolution. The volume of a para- boloid of revolution cut off by a plane -L^' to the axis = h^f that of the cylinder on the same plane base and of the same height. 30. Similar Solids. Volumes proportional to the cubes of corresponding edges. 31. Pappus's (Guldinus's) Theorems. (1) If an arc of a plane curve be made to revolve round any line in its plane which does not intersect it, the area of the surface thus formed = length of arc x length of path described by ' centroid ' of arc. (2) If a closed plane cui-ve be made to revolve round any line in its plane which does not intersect it, the volume of the solid thus formed = area of curve x length of path described by the ' centroid ' of area. Note. Hence surface and volume of Anchor Ring. ALGEBEA 1. Multiplication and Division. (1) Multiplication and Division should usually be per formed using coefficients only. Ex. 3x2-x+l is written 3-1 + 1, -a^ + 2ab + 3b^ as -l->r 2-{S. (2) Homers Synthetic Division. Ex.1. Dividea^ + l by a;'-2a:;+l 2 -1 1+0+0+0 2 + 4+6 -1-2 1+2+3+4 + 0+1 + 8 -3-4 + 5-3 The divisor is written to L. of L.H. vertical line without its first terra (if the coefficient of this is 1) and with the sign of the rest changed. Subtraction replaced by- addition. Ans. x^ + 2x2 + 3x + 4, and remainder 5x 3. Ex. 2. Divide 2x^ + x^y ix^ y^ + 5x7/ + y* hy 2 x^ xy + 1/^ as far as the term involving l/x^. First term of divisor placed as divisor in last line but one and used before any term of that line is used as a multiplier. Other terms of divisor have signs changed as usual. In practice the last line but one can be omitted. 2+1-4+5+1 1+1-2+1 -1-1+2 2+2-4+2+4 3+1-2+1+2 2y^ Ans. x'^ + xy~2y''--\ f- [See also TH. EQNS. 1.] (3) -^^-^ x-a n integral. + ax* + a^x' + . exactly, always, = x^' x-a remainder 2 a**. + ax'^^~^ + a'^x^ ^+ ... but always with 12 ALGEBRA x + a remainder -2a^, n odd. af^ + a'' = x*^-^ - ax'^-^ + a'^x''-^ - exactly, n even, = a;'-i-aic^i-2 + a2^w-3 x-\-a remainder 2 a**, n even. Proof. By Induction. 2. Identities. (1) (x-\-a)(x + h) = x^-(a + h)x-\-db, (X'\-a)(x + l)) ...(n factors) = x'^ + x^-'^%a + x^^~-%db-\- exactly, n odd. \-dbc.. (2) (a + 6) (a - &) = a^ - h\ i. e. sum x diff . = diff . of squares, (3) W (a 6)2 (2a)2 (a +6)3 (a -If (a + 6 + c)3 (a + 6)^ = a^2db + h\ = 2o2 + 2 2a6 = sum of squares + twice product of each term into sum of all that follow it. = a3 + 3a26 + 3a62 + ^3^ = a3-3a25 + 3a62-63^ = a^ + W^c^ + ^{Jb + c)(c + a)(a-\- h) = :Sa3 + 32a26 + 6a6c, a" + na = a** - nci' n-l6+*!i|_l)^n-2?,2+.,. [See 19. The coefficients are determined by Newton*s Rule, which is shown in the diagram. Each number in the diagi-am is the sum of the number immediately above it and of the number imme- diately to the left of that. The diagram gives the cofficients up to (a + 6)5. 1 5 10 10 1 3 6 i 10 ALGEBRA 13 (5) a^ + h^ = {a + h){a^-aJ) + h^) = {a+J)){a + aih)(a + coH), a^^h^ = {a- h) (a2 + a6 + &2) = (a - ft) (a - o) b) {a - co2 h), a^ + h^ + c^-Sal)c = ia + h + c) (^a^ - :S&c) = (a + h + c){a + oih + oi'^c)(a + oi^b + coc), i^T^ofe. For oj, , =, or < 4 ac. (3) oc + /3^-^^, ^ a a The equation is a{x- oc) {x- (3) = 0. Ex. To find the equation whose roots are l+$/(X, l + Oi/0. 003 = 2/ = l+i3/a = (a + i3)/a = -h/aot, Butaa2 + &a + c= O,-. &c. a = b/ay. Caution. When involution has occurred in solving any equation the roots obtained may not satisfy the equation and must be tested. (4) The expression ax^ + 6^ + c has always the same sign as a unless a and /3 are real and different and x lies between them. (5) Maxima and Minima of ax^ + hx + c. Write it <^-'2a) 6x2 iac-h^ 4a Or put ax'^ + bx + c = y, solve for x and express condition that roots are real. Maxima and Minima of {ax"^ + hx + c)/{px'^ -\-qx + r). Put exp. = y, solve for x and express condition that x is real. Then use (4) above. (6) Eliminani of two Quadratics. To eliminate x between ax^ + l>x+c = and px'^ + qx + r = 0, use 7 above to obtain x'^ _ X _ 1 hr -cq cp- ar aq - hp ALGEBRA 17 9. Equations, Numerical Solution. [See also TH. EQNS. 8.] Graphical methods are available if approxi- mate answers only are required, and can be applied to : (a) Simple Simultaneous Equations in two variables. The point of intersection of the straight lines ax -^ ty -\- c = 0, px+qi/ + r= 0, will give the solution of these two equations taken simultaneously. (b) Quadratics. For ax^ + 'bx + c = find the intersections of ^ = ^^ and ap + bx + c = 0. (c) Cubics and all higher equations. Thus x^ + 'bx + c = can be solved by / = a;^, y + hx + c =0. Note. The curves ij = x^, y =x^ can be obtained accurately cut in celluloid. (Messrs. Macmillan, or J. J. Griffin.) 10. Equations, Artifices in Solution. ,^. x + a x + h x + c x+d -r^. ., . , n (1) h = h . Divide in each trac- er; -a x-l) x-c x-cl x + a . 2a tion : = 1 + x-a x-a (2) ax^ + V ax^ -hx + c = hx + d. Put ax'^ -hx + c = y^. (S) + - = a, -^ + ^ = h. First solve for - and - ^ ' X y x y X y A ', -' x,y are roots of P - a + 6 = 0. xy = h) ^ ^ ^ (5) x^y'^ = a^ '^11 = n.^ 0^2 4. ^^2 ^ X y'^ = a) xy = a) x'^ + y^ = a) y = hy xy = h) xy=h) e x + y and x-y. Zl\- Use (6). Determine x + y and x-y. x^ -y-^ ^ a xy (6) If * homogeneous ' in x and y put y = mx and find m. (7) x- + y^ = a) Vse{x + yf = x^ + y'' + Bxy(x + tj) to x + y =0) find xy, then (4). Similarly for ^ ^ ^ _ ^ J * 1372 C 18 ALGEBKA (8) yz = a^, zjc = ft^, xy = c^ -, .'. xyz = + ahc. (9) x(x-\-y-\-z) = a and two similar equations ; .*. x->fy + z = \^a + 'b-irc. (10) x^-yz = a' y'^-zx 1) z^-xy = c ) X y J hc-a^ = x(S xyz -x^-y^ - z^) ' ' hc~a^ ca-h^ ah (11) x^- + 2yz= a y^ + 2zx = b z^ + 2xy = c -c' -V yz-x^ a{Sahc-a''-W-c'] = &c. x + y + z = Va + h + Cj x + (idy + co^z = + Va + oi^h + oiC, x + ay^y + ct)Z = Va + a)& + uy^c. Add these equations and remember 1 + co + co^ = 0. (12) x + y + z = a x'^ + y^- + z^=^ h x^ + y^-\- z^ _ Q Determine yz + zx + xy = q, xyz = r. Then rr, y, ^ are roots of 11. Equations of Higher Orders. [See also 9, and TH. EQNS.] (1) If the roots of p^^x'"'+PiX'^~'^ + ... +p^^ = are Oil, 0^2, ... a,i, the equation is jPo(^-o^i)(^- 0^2) (^-^^n) = ^> and ! + a2 + + ^n = "i'l/i'o* ia2-..an = (-Ti'n/i'o- (2) If Va + \/& is a root, and coefficients are rational, + Va a/6 are also roots. (3) Beciprocal Equations. [See also TH. EQNS. 10.] (a) If of odd degree test for roots 1 or - 1. {b) If of even degree, say x'^ + ax^ -{ hx'^ + ax + 1 = 0, divide by x^, and put x + - = ^, /r^ + -^ = P - 2. X X (c) Similar method for x^ + arc^ + hx^ ^aW'x + k^ = 0. (4) If no other method is available endeavour to guess ALGEBRA 19 a root X = oc (guided by the consideration that it is a factor of the last term), and divide by x-a (5) If elimination is required between two higher equa- tions, Bezout's Method is recommended. [TH. EQJSfS. 16.] 12. Irrational and Complex Expressions. (1) If a+ Vb = c+ Vd, a = c, h = d, provided these quantities are known to be rational and h and d are not squares. (2) If a + &i = c + (^i, a = c, 6 = d, provided these quan- tities are known to be real. [^ = \/ - 1. ] (3) An identity containing i remains true if - i be written for i, provided that the coefficients involved are known to be real. (4) va+ Vh = \^x-\- Vy is satisfied if x and y are so chosen that x + y = a, i:xy = 6. (5) ^a ^ Vh. If this can be expressed as ic + \^y, then x^ + Zxy = a. But also Va^ ~h = {x+ Vy) (x - Vy). Elimi- nate y to determine x. (6) V i' + ^^ + ^** + ^^ ^^^ ^ expressed as Vx + Vy + ^8 if it is possible to find x, y, z to satisfy the four equations x + y + s = p, 2 Vyz = i>, 2 Vzx = q, 2 s/ xy = r. (7) Rationalizing Factor. To rationalize a^ + 6 9' : let 1 j_ aP = X, h'i = y, and let n be L. C. M. of p and q. Divide x^ y^hy xy and substitute. (8) Cube Roots of Unity. The complex cube roots of 1 are ~ ~^ They are denoted by w, co^ and have the following properties : (a) Either is the square of the other. (b) 1 + 0) + 0)2 =0. (c)o)3 = 1. c 2 20 ALGEBRA 13. Ratio and Proportion. (1) If a/h = c/d^ a + h:b = c + d:d, a-h:h = c-did, a+h:a-h = c + d:c-d, pa + qh ipa -qb ^pc + qd :pfc - qd, (2)Ifa/!, = cMeach = (^-^J. (3) If Ui/b^, ajb^, ^J^z ... ^J^n ^^^ ^^ ascending order of magnitude, (4) A fraction is brought nearer to unity by adding the same (positive) quantity to both numerator and denominator. 14. Variation. (1) If a oc &, a = mb. (2) If a a inversely as &, a oc !/&, a = m/b. (3) If a oc 6 when c is constant, and oc c when b is con- stant, then when both vary a oc be. 15. Scales of Notation. (1) A number in scale r is divisible by r- 1 if the sum of the digits ^Q +i)i +^2 + i^ so divisible. (2) A number is divisible by r + 1 if jPo "i'l +1^2 ~ is so divisible. 16. Progressions. (1) Arithmetical. I = a + {n~l)d, s = -^ (a + 1) = ^(2a + n - 1 . d). Mean {A) between a and & is J (a + &). (2) Geometrical I = ar"~^ r 1 1 - r*^ Ir s = a T- ^ So. = r-1 1 a \-r Mean (G) between a and b is Vah. ALGEBRA 21 Note, The proof given for s^ is usually slipshod. It is not obvious that the limit of a?" = ( = oo ) if ?" < 1. Cf. the second limit in 22 (4) when a= 1. A geometrical illustration is easy and valuable. (3) Harmonical, If a, h, c are in H. P., 1/a, 1/6, 1/c are in A. P. a:c = a-h:h-c. 2 ah Mean [H) between a and & is a + h Note. To construct Jf geometrically take AM = a, MB -= 6 in a straight line, and at A and B erect X" AK = a, BL = b. Join KL. At ilf erect MN ^ AB meeting KL in N, Then MiV = H. (4) A> a> H and G'^ = AH. 17. Interest. If P be the principal, r = rate per cent., R=l + (r/100). Simple Interest for n years = Pnr/ iOO. Compound,, ,, =FB>^-F. 18. Permutations or Arrangements. (1) P/ = ^(n-l)... (n-r+l) = ^-^^j; P/= w! (2) Permutations of n things, p alike, q alike, &c. _ n ! (3) Permutations of n things, r together, repetitions allowed = n''\ Combinations or Selections. n\ on ^ n{n-\)...{n-r^\) _ n\ _ \^) ^r - ^j ~(w-r)!r! ^''-'' (2) Total number of selections from n things = 2^-1. (3) Selections of n things, r together, repetitions allowed, (4) Vandermonde's Theorem. [Proof 19 (4) (a).] 22 ALGEBRA (5) No. of selections, r together, from objects p alike of one kind, q of another, &c., is the coefficient of x"^ in . 1 X 1-x 19. Binomial Theorem. (1) (a + hf =a + wa-i& + ^^?^i-^a^-2&2+..., (l-x)-^=l- x+ a^+... tooo, [C. ifa; *' ^ "~9~ ^^ "o ~ * (3) Homogeneous Products of rth degree of n quantities, ^' 1.2... r Proo/. Consider H. P. of degree 7 of 4 letters a, b, c, d. Denote a^hd^ by ...|.||... The vertical strokes divide the 7 dots into groups, first group for a's &c., none in the c-group. H^ is the no. of ways of arranging the 10 symbols, 7 dots, 3 strokes. The reasoning is perfectly general. (Whipple. ) ALGEBKA 23 The sum of these products is the coeflficient of x"^ in 1/(1 - ax) (1 - Ix) {\-cx)., [See 4 (2) (d) (4) Derived Binomial Series are obtained by equating coefficients in such identities as (a) (1 + Xp X (1 + X)^ = (1 + X)'+^, [giving Vander- monde's Theorem. (l + x)'^x(l-x)'^ =(i-x^)'^, {i + xf^x {i-xy^ = )- ^ = - -i&c. ^ ^ ^ ^ (1-a;)^ {1-x)^ (b) {l-(6x-6x^)}-'^= 3(l-3a;)-i-2(l-2a;)-A. 20. Multinomial Theoeem. The general term of {aQ + aiX+a2x'^+ ...)** qlrlsl... 12 3 ' where /x = q + r + s+ ... . Note. Wherever possible avoid the use of this theorem. Thus , {l + 2x + 3x'^ + ...) = (l-a;)-2 21. Logarithms. x (1) log xy = \ogx + log y, log - = log : - log i/, if log x'^ = r log X, log v^a; = - log x. r (2) log^ b X logj, a = 1, log^ X = log^ 6 X log^ X. Mnemonic x/a = b/a x x/b. (3) If the base is 10 the characteristic of a number with n figures in its integral part is n-l; of a number with n ciphers before a significant figure is reached after the decimal point the characteristic is -n-1. 22. Exponential and Logarithmic Series. (1) Exponential Theorem. e = 1 + 1 + 1/2! + 1/3 !+...= 2.71..., e^= l + xA-x'^/2\ + x^/3l+ ..., a^= 1 + a? logg a + (ic logg a)V2 ! + .... 24 ALGEBRA Proof. Let/ (x) = l+x + xy2! +xV3! +.... Prove the series always convergent. Using Binomial Theorem deduce /(w) x/(n) =f{m + n). Deduce /(m) = [/(I)]"* = e*", m integral. Prove next for/(p/g). Last prove for/( m). (2) Logarithmic Series. log,(l-x)= -:,- J-'J-.... [C.if^Pn - OAs-n - 2 + ^ M - 1) when r has all integral values from to a are called the Intermediate Convergents between Pn-2/ln-i and Pn/o.ni which form the first (r = 0) and last (r = a) members of the series. If P,./Q,., P^-i/Qr-i be any consecutive pair, P,. Q,._i -P,._iQ,. = 1. The series of fractions Pi/q^ . . . Pa/Qs . . . Pj% ..., the gaps being filled 26 ALGEBRA by the intermediate convergents, is a series continually increasing towards C. F., while the series i^g/gg --Vi/^ii Pe/Qe is a series con- tinually decreasing towards the C. F. It is impossible between any consecutive pair of either series to insert a fraction of smaller denominator. Note. Hence the solution of such problems as : Find the fraction of denominator not more than 2 figures which is the best approximation to TT. [Ans. 311/99.] ii. Becurring Continited Fractions. Caution. These results apply only to V'ivyjf and not to {P+^/n)/M unless P = 0. /.^ m . . 1111 (1) To evaluate x = :j t , write ^ ' a+ h+ a+ b+ ...^ _ 1 1_ ' ~ a+ h + x and solve the quadratic. (2) If VN/M is converted into a C. F. it necessarily takes the form pi 1 1^_ JL_ ?^ 1_ I 1 L_ ^ L^i+ ?>2+ &3+...+63+ ^2+ ^1+ 2ai + j6i+...' i. e. after a while the partial quotients reappear, but in the reverse order ; a partial quotient double the integral part will then follow; after the appearance of 2a, recurrence takes place and the recurring part is shown by brackets. If a^ = the last partial quotient of recurring period is 2 fe^. The middle quotient may or may not be repeated. (3) If VN is converted into a C. F. and if Pn/q.yi is the penultimate convergent of any recurring period, i.e. the convergent formed from the partial quotient immediately preceding 2ai, (4) Pn/^n ^ing the penultimate convergent of a recurring period, a2n 2 Iq,, p^ J ALGEBRA 27 Note. An exceptionally good method for finding square roots rapidly. Thus for >/l7, p^/q^ = 4, p^/q^ = 3^, pjq^ = ^^-, the error in which does not aflfect the first 6 places. [See 23, i. (5).] iii. Evaluation ofp^Jq^^, (1) 0. R - - . i?n = o^n-i + ^n-2. Hence jp^ is nth term of a recurring series whose scale is 1-ax- Ix^, [See 30(8) and 31.] (2) C. F. . Obtain relation connecting a+ c^- + c'+ ... ^ (3) C. F. . A general form ior p^/q^ can be found in certain cases. 12 22 32 Pn = Pn-i + n^Pn-Pn-(n+l)pn-i = -n[Pn-i-npn-2'\- r2_ r(r+l) r(r+2 ) , ^ ^ 1+ 2+^ 3+ ... Pn-{r + n)pn-i = -r[^,i_i-r + ?t-li?_2]. iVbte. From any C. F. capable of evaluation thus another may be iPn-2' Putp = 22.13^2 42.3 12 3 derived. Thus - ^r 5 gives Pn = npn-i + npn-2- Put pn = Pn/w ! 1+ ^"^ >+ ... We get P = n2 P_i + w^ (n - 1 ) P_2 ; . '. the fraction ^ ^ -^ can be evaluated. So also we may put Pn = Pn/'>''* where r is any integer. iv. Conversion of Series into G. F. (1) - + -++ - u 1 w/2 '^n 1 ^.,2 %^ %-l^ %- UI + U2- ^2 + W3 - 2*n-l + W^ Proo/. To recall this solve the equation l/My4-l/w,.+i = !/(, + ^r) 28 ALGEBRA (2) The following are added for reference (deduced from (1)) : Mi+ U^ Uy+ Wg H2 + 1 , a; X2 1 Mo2 11^2 ~ -r + 1- .. Uq u^ Ua Uq iiqX-^u^ u^x + u^ ... (3) The quotient of two convergent series S^, Sg may be expressed as follows : Si = 1 \-PyX+p.,x'^-ir ... S2 = l+giX + g2a;H ... Sj-Si = (gi-Pi)a;S3, where .S'3 =^ '[+riX + r^x'^+ ...; S3 S2 = ('"1 31)35 Si, where S^ = l + 2xy. (2) See 13 (3). ajh >^a/'2,l)>aj\. (3) A.M. of any number of + quantities > G.M. Proof. (x + y)' = {x yf + ixy. Hence if any two of the quantities are unequal we increase their product without altering sum by making them equal. (4) (SftW)/^ > {(^a)/n]^, where ^a extends to the n quantities a, &, c, .... Proo/. First prove for 2 letters. Expand (a" + 6")/2 = {(a4-6)/2 + (a-&)/2}*+ .... Three cases, (i) m + integer or , (ii) m between and 1, (iii) m not integral but +, which is deduced from case ii. Then extend to > 2 letters. If any two are unequal replace by (a + &)/2, then 2 a un- altered, 2a"* diminished ; .'. min. val. 2a"* is when all letters are equal. (5)IfiP>2, (a) (xP-l)/p >(x3-\yq. w(i%-f>(^+if- 27. Maxima and Minima. (1) See 8 (5). (2) {a + xf' (h - xf. Apply 26 (3) to {(a^-x)/m}^ {(h-x)/n\\ (3) a^l^cP..., subject to the condition a + h + c+ ... = constant. Apply 26 (3) to (a/m)'(&/w)(c/i?)^... (4) Most ' Inequality ' Theorems give maxima or minima results. Note. e. g. the minimum value of (& + c) (c + a) {a-J[-'b)/abc is 8. 28. Theory op Numbers. [See also 24 (5).] (1) Ii M = N (mod. m), P=Q (mod. m), M + P=N+Q, M-P=N-Q, MP = NQ, M^ = N^ {k integral). 32 ALGEBRA (2) The product of r consecutive integers is divisible by r ! "Proof. Induction, or use (6) to show that any prime is contained in (n-f r) ! at least as often as in ?i ! r !. (3) Fermafs Theorem. (a) NP~N=0 (mod. p) if j? be prime. (h) NP~'^ -1 = (mod. p)p prime, N prime toi?. Proof. N, 2N,...{p^l)N= (notin order) 1, 2, ... (i?-!) (mod.p),&c. (c) 2r^'^ 1 = (mod. p). iV prime tojp, 0(p) = no. int.

semi-convergent (l-i + J~J:+"-)> oscillating (1 - 1 + 1 - 1 + ...). Semi-convergent = conditionally convergent.] ALGEBEA 83 (1) S ^ 1/1P+1/2P + 1/SP-]- ...isCiip > 1, Difi>> 1. ^ ^ ^ ^ ^ ** \ (all terms + ). {a) If ujv^ is always finite and does not approach in limit, S and 2 are C and D together. (&) S is G when 2 is if u^/u^+i > v^/v^+i always ; (3) /S = Wi - ^2 + W3 - ... is (7 if % >i^2 >^3 ) provided Lt^w^ = 0. If Lt w,j 4= ^j '^ i^ oscillating. This series may be semi-convergent (see Note). (4) S = Mi + t*2+ ... +Mn+... SisC. SisD. S is doubtful. 1. When Ltrt,=x of w is 2. nUn is 3. Un/Un^x is 4. w(M/M+i-l)is >1 >1 not not <1 <1 = = = 1 = 1 (5) (1 + v^ (1 + v^ (\-^v^ ... is finite if v^-^v.2^ + v.+ ... is C. Note. It is important to note that the terms of a series must not be regrouped unless the series be absolutely convergent. 30. Summation of Series. [/S^ = % + %+ -..] (1) Method of Differences. [% - % = A % , A Wg - ^ ^1 = A . A % = A^ Wj, &c.] 2! (&) ((l + Af-l| w('/^-l)^ (c) Montmorfs Theorem. U S = Uq + UiX + ^2^^ + [^ < 1 J ^x = ^o- j^T^ +Awo A 9 + .... /S^ may be deduced from S^^ . 34 ALGEBRA Cor. Ui-u^ + u.^-... =22^1-22^*^1+ p ^ % " Notes. (1) A is an operation. Beware of supposing that AMi = (Ami)2 (2) The A's are best found by a tabular arrangement as shown. In the example given m^ = 1, Am = 7, A^Mj = 12, AS Ml = 6, and all higher A's vanish. 1 8 27 64 125 (3) The method leads to finite results only 7 19 37 61 when is of the form An'^ + BnP-'^+ ..., and 12 18 24 leads to no useful result when Un contains 6 6 terms involving n in the index (e. g. G. P.). (4) If however m is known, it is best to ex- press it as a + bn + cn(n+l)+ ..., and use (5) below. This is better than to use (2) or (3) below. (2) Powers of Natural Nunibers. Sy = 1*" + 2^ + ... + w*'. n{n+l) ^ n{n+l)(2n+l) ^ n{n+l) ]' ^1 = 2 ' ^^ ^ 6 ' ^ ^ I 2 ) ^^^ ' The following results are added for reference : , ^4 = 5^jn(n + l)(2w + l)(3n2 + 3n-l); Se = iVn\n+l)2(2n2 + 2n-l) ; ^6 = 42w(n + l)(2n+l)(3n* + 6n'-3n + l); -^7= 2*4:w2(n + l)2(3n< + 6n3-n2-4n + 2); and the following are interesting : (S,-S,)/(S3-S0 = |(2n + l) ; S5 + S7 = 2 Sx*. [See also DIFF. CALG. 7 (6) (d)]. (3) Undetermined Coefficients. Series to which (1) (h) is applicable may also be summed, if u^^ is known, by assuming 8^ = a-\-hn + cn^+ ... ; . . u^^ = /S+i -Sy^= b + c{2n+l)+ .... Equate coefficients. (4) If w,j can be split into the difference of two terms, one of which (%i_i) is the same function of w- 1 as the other (v^) is of n, the sum is v^^_Vq. Note. One of the most general series coming under this head is S + r + iq + r){q + 2r) + {q + rj {q + 2r) {q + 'Sr) ^ "" ALGEBRA 35 The series referred to under (5) and (6) also come under this group. But in fact any series which can be summed is in this group, since (5) w,^ = {an + b)(a.n+l + h) ... {a .n + m-l + b). Rule : To find the sum place one additional factor at the end of u^, divide by the number of terms so increased, and by the common difference {a), and add an undetermined constant to the result, determining this constant by ^ = 1 or 2, &c. Caution. The first factoi's in successive terms must be part of the same A. P, as are the various factors in any term. Thus the rule can- not be applied to u,i = n(n + 2) (w + 4). To evaluate this write n{n + l){n + 2) + Sn{n + l) + Bn, and use rule, or as n^ + Qri^ + Sn and use (2). (6) u^ = l/{an + b){a.n+l + b) ... (a .n + m-l + b). Rule: To find the sum strike off the first factor, divide by the number of factors so diminished and by the common difference (a), change the sign, and add an undetermined constant, determining this as before. Caution. As in (5) Un = l/w(w + 2)(n + 4) = (M+l)(w + 3)/n(w + l) (n + 2) (w + 3) (n + 4), then write (w+l)(n + 3) as n{n + 'i)+Sn + S. m is replaced by the sum of three terms to which the rule may be applied. (7) S = a + {a + b) X + (a + 2b) x^ + ..., one factor proceed- ing by A. P., the other by G.P. Multiply by x and subtract. Note. A particular case of (1) (c) above, also of (8). Many series, all of which are included in (1) (c), may be summed by a repetition of the operation. (8) Recurring Series. (a) If 2r terms are given, assume a G. F. of the type {a + bx + cx^+ ...)/(l +px + qx'^+ ...) with r terms in the numerator and r + 1 in the denominator. Cross-multiply and equate coefficients. (5) The G.F. is S^ when series is C. D 2 36 ALGEBRA (c) To find S^ split G. F. into partial fractions. These fractions allow u^ to be found. (a) If there are no repeated factors in the denominator of the G.F. the partial fractions allow S^ to be written, for the part of S^ corresponding to the G.F. A/{l (Xx) is (/3) If there are repeated factors, find u^ and write : Sn = U0 + U1X+ ...+UnX*', PX Sn = PUqX + ... +pu^.i X^^ + pU^X^^"^^ qo(P-S^^= ... and add to obtain S^. (9) Series derived hy Differentiation, cf;c. If rhx + co(?+ .,. can be summed to n terms or to 00 , then such series as . a + &. V .x-^c.2'',x''+ ... can also be summed to n terms or to 00, by repeated differentiations, &c. Ex. a + &.12a; + c.22.x2+ ... = a-\-xD{xD(p]. [D =- cl/dx]. Note. While ' calculus-dodging ' is not recommended it is of real help sometimes to notice that the results of differentiation may always be obtained by substituting x + h for x on both sides of an equation and equating coefficients of x. To a limited extent the integral calculus is available to sum series of the types a + bx/1^ + cx^/2''+ ..., &c. (10) Binomial, Exponential, and Logarithmic Series. [See 19(4), 22(5).] If a, h, c,... represent the coefficients of either the Binomial, the Exponential, or a Logarithmic Series, then the series can be feummed if p and q are unequal positive integers (.q^-p), and (n) be a positive integral algebraical function of n, and the method is applicable when the denominators contain any, the same, number of unequal factors of the type. To start the summation : (a) Binomial and Exponential Series. Complete the factorials which ALGEBRA 37 already occur in the denominators of a, 6, c... bringing them up in the successive terms to g!, (3 + I) !, (9 + 2) ! ... by suitable multiplica- tions for both numerator and denominator. (b) Logarithmic Series. Split the coefficient (coiu) + * (a, /3 unequal) (2) If i?^ + ciPn~i + ^1^^1-2 + ^i'n-s = ^> ^J^(x). (h) Equation with any specified term removed. Use (c), and choose 1i to make specified term vanish. (0 Homographic Tramformation. x = , p^y + q Note. If a, /3, 7, 8 be any 4 roots of either equation, is unchanged by the transfonnation. a_)3 7-5 (3-y ' S-a THEORY OF EQUATIONS 41 4. Depression of Equations. If ^ = (a) the degree of the equation can be depressed 2. For the equations /(a) = 0, /{()} = have a root in common. Find the H. C. F., hence a, hence /?. j^ote. Special artifices often available as in Ex. Ex. a;^ 2a;3 8a;2 + 23a; 14 = has 2 roots whose sum is 3; solve. We must have a;4-2a;3-8x2 + 23x-14 = (x^-Sa^+m) (cc^ + a+w). Equate coefficients and solve, m = 2, w = 7. 5. Delimitation of Roots. (1) Superior Limits Positive Roots. Method i. Grouping. Ex. 1. a;3(x2-5x-13) + 2a;Ha;-70 = 0. Determine by mental solution of a;^ 5a: 13 = the limit for the first bracket. This isx = 7. It makes the second bracket + also, and is the required limit. Ex. 2. a^ + 2x* + 3x^ + 8x^-62x-80 = 0. Group as a;5 - 80 + 2 a; (x3 - 26) + x2 (3 a; - 8) = 0. Limit 3. Ex. 3. x3-9a:-2 = 0. a;(x2-10) + a;-2 = 0. Limit x = >/lO. Method ii. If x^~^ be the highest power of x which has a negative coefficient, and p be the greatest negative coeffi- cient, the limit is 1 + Vp. Method iii. If each negative coefficient taken positively be divided by sum of all preceding + coefficients, the greatest of all the fractions thus formed, increased by 1, is the limit. Proof. Use the substitution a;" = (x-l) (x-i + a;"-2+ ... +1) + 1 in all + terms. Notes. (1) Methods ii and iii are rarely as serviceable as Method i. (2) Better results will be obtained by both methods on putting x = 2y, or X = 3y and obtaining the limits for y. Method iv (Newton). Let the derived functions be /^ {x\ fi (^) fn-ii^)' Determine the least value of x which makes f^_i + . If this makes /^_2 + test it on^_3. If not, increase it by 1 at a time until a value is reached which makes f^-2 + ' Test this on /^-s, increasing it if necessary. The 42 THEORY OF EQUATIONS final value which makes all the /*s including f(x) + is the limit. Method V (RoUe's Theorem). A real root of fi{x) = lies between every adjacent two of the real roots of f{x) = 0. If the real roots of/i(x) = in descending order be (Xj, 0t2,...0ln-i, then if the series of values co , (Xi, (X^,.. ttn-i> ~ oo be substituted for x in /(as), the changes of sign in the results will determine the number and position of all the real roots of /(a;) = 0. No more than one real root of f(x) = can lie between two adjacent roots of /j (x) = 0. If /,. (x) = has k imaginary roots/(x) = has at least k. Ex. Hence x' + Sx+p = has only one real root. Method vi (Waring). Form the equation whose roots are the squares of the differences of the roots of f(x) = 0. [See 11 (1), (4).] Let h be the inferior limit of its + roots. Take Tc = Vb, and let s = superior limit for f{x). If the series of values 5, s-A;, 5-2A;, ... be substituted for x in f{x), the changes of sign in the result will detect the number and position of all the real roots off(x) = 0. Theoretically perfect, but practically useless except for cubics. k as thus determined usually much too small. (2) Inferior Limit, Positive Roots. Put x = - and deter- mine superior limit for /. (3) Negative Roots. Put x = ~y and determine + limits for y. (4) Fourier's Theorem. The number of real roots of f{x) = which lie between a and /3 {jS > oi) is not greater than the excess of the number of changes of sign in the series /(a), f {(x), f^ {oc), ... /^(a) over the number of changes in the series /(A /i (y8), /<, {(3), ... f^ {/S) and differs from this excess always by an even number. i^ofe. Descartes' Eule [ 2 (11)] is a particular case. Budan's Statement. Write x + Oiior x in f(x) [see 1 (iii)], and count the number of changes of sign in the result. THEORY OF EQUATIONS 48 Similarly write x + (i and count. The number of real roots between ex and P is either the difference of these two counts or differs from it by an even number. (6) Sturm's Theorem. Let f(x)= be an equation cleared of equal roots. [See 6.] Perform the operation of finding the H. C. F. of f(x) and /i (a?), but change the sign of every final remainder hefore it is used as a divisor. Call these modified remainders F^ (x), F,(x) Let the last remainder, also with its sign changed, be Then the number of real roots between OL and jS (fi > Oi) is equal to the excess of the number of changes of sign in the series /(a), /i (a), -JP^2(^)> ^m(^)' ^^^^ ^^ number of changes of sign in the series f{i3),fAP],F,m,...F^m- Notes. (1) F,a is independent of x and cannot be zero unless equal roots are present. (2) If equal roots are present the theorem gives the no. of distinct roots. (3) If in the process of finding the J"s we reach one, Ff., which cannot change sign [e.g. 3 0:^ + 5] we can stop the process at this point and apply the theorem to/,/i, F.^,,..Fy. (4) In applying the H. C. F. process we may multiply or divide either divisor or dividend by any positive quantity, and this mul- tiplication may take place at any stage during the process of division provided that it is applied to the whole of what is left of the dividend. (5) We rarely need calculate i^,, for since a value which causes Fm-i to vanish will give opposite signs to F^-i and F, we can determine its sign by inspection. Proof. One change is lost between/ and /^ when a; passes through a root. No change is lost in the series when x passes through a root of Fy, because F,._i, F,,+i have opposite signs if F,. = 0. Ex. f{x) = a;-5a;-l. /i (rejecting 5) = x*-l. F^ = 4a;-f-l. F3= + [Note (6)]. Hence the changes are : 44 THEORY OF EQUATIONS / /i -^2 ^3 00 + + + + change 2 + + + + ell langes 1 - + + 1 - - + + n 1 -1 + - + V 2 -2 + - + n 3 00 + - + 3 ly real roots lie between 2, 1 ; 0, - 1 ; ; -1, -2. 6. Equal Roots. The equal roots of the equation f{x) = are given by the various factors of the H. C. F. of f{x) and /^ (x) each repeated once more frequently than in this H. 0. F. 7. Commensurable Roots (Newton). Calculate /(I), /(-I). If either of these vanish, divide by ic - 1 or ic + 1, noting roots 1 or - 1. Make coefficient of highest term unity. [Substitute X = Jcy.] Determine possible limits for roots of resulting equation. Write down in a horizontal line all divisors (except 1,-1) of absolute term which lie between these limits. Note. If a is a root, a 1 divides/(l), a + 1 divides /( I). Heject any divisors which do not satisfy this condition. Divide the absolute term by each of these divisors, adding to each quotient the last coefficient but one, and writing the result for each divisor immediately below it. If any number results which is not divisible by the number at the head of the column, reject that column. Divide those numbers which are divisible and add to quotient the next coefficient on L.H. Again reject or divide, and so on. If finally, after n-1 divisions, the quotient - 1 is obtained, the number at the head of the column is a root. Notes. (1) If the coefficient of the highest term be 1 there are no fractional commensurable roots. (2) If the equation cleared of commensurable roots thus obtaiue4 THEORY OF EQUATIONS 45 have an absolute term wliich is divisible by any one of them, it must be tested to see whetlier such root be repeated. Ex. x*~x^-x'^ + 19x-i2 ^ 0. Limits for roots 3, -5 (Method of Grouping): /(I) = -24,/(-l) = -60. Divisors of 42 : 2, 2, 3, 4. Of these 4 is rejected by /(I) test. 2 -2 -3\ -2 40 33 _2 21 12 I ^oots 2, 3. _2 * 3 I Equation (a;-2)(a; + 3)(a;2-2a; + 7) = 0. -1 -ij 8. Numerical Evaluation of Roots. [Of these methods i and ii are most generally useful. i can be relied on to 2 significant figures, or, with very care- ful drawing, to 3, and may be supplemented by iii when greater accuracy is required. Method iii is to be recom- mended as more speedy than v if 7-fig. accuracy will sufiice, and is rather less subject to error. Method iv is novel, as far as English textbooks are concerned, and will often give 3 and even 4 figures with much less mental work than any other. Method vi is little known. Method ix deserves more attention than it has received from English textbooks. Method X is almost useless.] Method i (Graphical). Ex. 1. x3-9x-2 = 0. Trace the curves j/ = x^, t/ = 9a; + 2. Or, Trace the curves y = x^, (y 9)05 = 2. Their intersections give all the real roots. Note. The curve y = x^ can be obtained in celluloid [Messrs. Griffin], as also other curves. Ex. 2. Sin X a; cos a; = 0. Trace the curves y =^ x, y = tanx. Note. The general quartic can be graphically solved by the inter- sections of the parabola y = x^ + 2ax+p and the circle x2 + 2/H2gx + r = 0. Method ii {Interpolation). The rule is similar to that em- ployed in the use of logarithms and trigonometrical tables. If a, y3 ( > a) be two values of x between which a root oif(x) 46 THEORY OF EQUATIONS lies, then, if /(a) = A,f{^) = -B, an approximation to the value of X will be A , , AB + Boc Notes. (1) This method is sometimes called the Regida Falsi. (2) It is particularly adapted to exponential and similar equations, though it may be applied to any. (3) For a further example of it see Method xii. Ex. Solve a;* = 3. It is obvious that a root lies between 1 and 2, nearer 2. Write the equation loga; + logloga5 = log log 3 = 1-6786 and tabulate L.H. for the values 1'5, 1-6, 1-7, 1*8, 1'9. Applying interpolation to this table we obtain as an approximate value of X 1-825. Again using 4-fig. logs tabulate for values 1-824, 1-825, 1-826. We get 1-8254. Now using 7-fig. logs tabulate for 1-8253, 1-8254, 1-8255 and we obtain 1-825455. Method iii (Newton). If c be an approximate root of f{x) = 0, then c -f(c)/fi (c) is usually a better approximation. Ex. x' 9a; 2 = 0. The successive approximations to one root are : 3, 3-1, 3-105, 3-10548.... Notes. (1) Will usually succeed unless there are two roots near c. (2) (Fourier) Will not, however, succeed with certainty unless for the values over which the approximation is proceeding /(x)//'i(x) is , and also/(x)//2(x) is +. (3) Particularly valuable for trigonometrical equations, &c. Ex. Sinx xcosx = 0. The approximations to one root are : 4-5 n-, 4-477 ir^ 4:' mil n, Method iv can also be applied to x == tan-i x. (4) Applicable to two or more simultaneous equations : f(x, y) = 0,

(^,V) + Hi+fiy9a; + 2 4 88 3-86 3-36 32-24 3-18 [Use tables for 8180] 3-18 30-62 8-128 3-128 30-152 8-112 3-112 30-008 3-108 3-108 29-972 3-106 3-106 29-954 8-1056 [Mental interpolation] 3-1056 29-9504 3-1055 (6) For the next root take x a little lower than the first root and substitute in a; = (x^ 2)/9. Thus starting with x = S the successive approximations are 2-8, 2-2, -95, --13, --222, --223, --2235. (c) For the next root take a number a little less than the root just found and use x = 'v/9x+2 again. The successive approximations arex = --23, --41,-1-19, -2-06, -2-55, -2-75, -2-83, -2-86, -2-874, -2-881, -2-8816, -2-8818,.... Notes. {1) The first four or five steps should be done with rough accuracy mentally until the numbers begin to converge. (2) The method has the curious property that numerical mistakes will not usually prevent eventual success, though they may transfer the evaluation from one root to another. (3) Seven-figure accuracy may be obtained by the use of logarithms, but if this is desired it is better to apply Newton's Method (iii) when 4 figures are obtained. (4) The method may be applied in the form x = \/9+ -. Thus successive approximations are 3, 3-11, 3-1053, .... 5 1 (5) An equation such as x^ = 5x 1 can be treated as x^ = j. The successive values are 2, 1.3, 1-48, 1-430, 1*443, 1-440,... the correct value being 1-44050.... (6) It is applicable to exponential equations, &c. Thus e^ = 2 x^ is treated as x = loge 2 + 2 logg x, and starting with the value x = 3, a rather slow convergence produces the value 2-63, which Newton's Method changes into 2-61787 ... . (7) No preliminary separation of the roots is required. Even the delimitation of them is not necessary. 48 THEORY OF EQUATIONS (8) The method appears to have been first given in Ei)glish, at all events with any complete explanation, by Ross {Nature, vol. 78, p. 663), but it is substantially identical with the method which German text- books call Iferation (Runge, Praxis der Gleichungen, pp. 56-9, 81 If.). (9) It may be applied to simultaneous equations, and where applicable is often easier than Method iii. x8_8_25 = ,/ Ex. ' , /,. Write x = 4^t/ + 25, y = i-x. /=l is ap- x + y 4: = J 7 ^ proximate solution. This gives x = <^26 = 2-96. This (second equa- tion) y = 1*04. This (first equation) x = 2-967. Hence successively y = 1-033, x = 2-9664, y = 1-0336, and, using 7.fig. tables, x - 2-96646, y = 1-03355, x = 2-966444, y = 1-033556, x - 2-966446, y = 1-033655, which figures are probably correct. Metlwd V (Horner). Determine by trial, or by 5 (5), two integers between which the root lies, or, if a decimal, its first figure. Write the equation with coefficients only, and with missing terms shown by zeros. * Operate ' [ 1] across with the first figure (or integral part) of root. Operate again, stopping at the last column but one. Operate again, stopping at the last column but two, and so on until the operation is performed on one column only. Affix to second column on L. H., 00 to next, 000 to next, and so on. An estimate of the next figure of the root may now be obtained by dividing the figure which now stands in the last column by the figure in the last but one, making a rough calculation as to the effect upon this figure (in the last but one) of the operations performed by the quotient, and choosing, usually, the highest integer which after the operations will not result in a cJumge of sign in the last term. Eepeat the series of operations with this integer, again affix O's, and so on. After 3 or 4 figures have been obtained we may obtain several others correctly by the following approximation: THEORY OF EQUATIONS 49 Instead of adding O's cut off one fig. from R.H. of last column but one, two from last but two, &c., and repeat the series of operations. Then again abbreviate until all the columns have been cut away. Ex. x3-9x-2 = 0. 10-9 -2 I 3-105482616;} ' 3 -2000 6 1800 - 109000000 90 1891 - 9617375 91 19830000 - 1646659 92 19876525 9300 19923075 The process is now com- 9305 1992307 pleted by the contracted 9310 1992679 division of 199813 into 9315 1993051 1646659. The last figure ,0 93 199305 thus obtained is in- 1... 199313 accurate. The roots of this equation are : 3.1054826165..., -.2234620717..., -2-8820205448.... Notes. (1) The case where 2 roots are nearly equal requires care until they are separated. Until this point is passed a better estimate of the next figure is given by twice the last column divided by the last but one. Or use Method iv or vii (see Note). (2) If the root which is being evaluated is less than 1, use 3 (&) to multiply the roots by such a figure as to bring it between 1 and 10. Thus to evaluate the second root of the above equation, put y = lOx, i. e, find the root of 2/3-900?/ + 2000 = which lies between 2 and 3. Method vi (Weddle). The method will be explained in its application io x^ -^x-2 = 0. It is, of course, perfectly- general. Determine two integers between which a root lies, 3 and 4. Put x = '^y. The equation in y, f(y) = 0, has a root between 1 and 2. An approximation to this root may be obtained by putting y = 1 + c; .'. /(I + c) = ; .-. /(l) + c/'(l) = if c is small; .'. c = -f(l)/f{l). In our case c = .03. Put y = 1-03 ^f, and determine ^ in the same way, 1.005. Put s = 1-00^ w and determine w, &c. Finally the root is 3 x 1-03 x 1-005 x 1-00001 x 1-0000005. 1372 E 50 THEORY OF EQUATIONS The whole work for the determination of 7 figures is exhibited below: a;3 27' ^81 : -9x -27 27 x3 = -27 X 1 = -2 81 -27 54 ' Kadix ' 3 27-81 -8343 -81 -27-81 -2 28-6443 859329 1-03 [^1 + ,^, 29-m362Q 147518": 29-503629 x 3 = -27-81 X 1 = - 2 89 -28 61 29-651147 j 148256 13905 29-799403 148997 -27-94905 - -306371 1-005 [=1 + -^^^/ 29-948400 29^948400 x 3 = -27-94905 xl = : - 2 90 -28 62 99 29-948699 299 279 29-948998 299 -27-949S29 I - -000650 1-00001 29-949297x3 = -27.949329x1 = - 2 90 28 62 29-949297 - -000032 1-0000005 100 Explanation. The dotted lines are inserted to divide the work into stages and render it easier 1-0000105 to follow. After putting x = By, thus using 50001 ' radix ' 3, the equation becomes 27 x^ 27 a; 2 = 0, and the work in the top space shows the evalua- 1-0050106 tion of /(I) and of/ (1), namely 27 x 3 - 27 x 1, and 301503 the deduction of the second radix 1-03. The work in the next space shows the formation of the new 1-0351609 equation 27 (l.OBfa^-27 (1-03) a; -2 = 0, the first 3-1054827 column giving the successive multiplications of 27 by 1-03 three times, the second giving the second term in the THEORY OF EQUATIONS 51 equation, the third giving /(I), and the fourth f (1), leading to the next radix 1.005. Finally at the right of this explanation the five radices are multiplied together to give the root 3.1054827. The successive forms assumed by the equation are shown by the figures in italics. Notes. (1) This method, which has been strangely neglected, is due to T. Weddle (R. S. Trans,, Ap. 29, 1841). (2) If Compound Interest Tables are available, the work in the first column may be dispensed with and the method is exceedingly brief. In any case it is less subject to error than most. Method vii (Lagrange). Determine a the integer next below a root of f{x) = 0. Substitute a + \/y for x. If there is only one root between a and a + 1, the resulting equation in y will have only one positive root ( > 1). Let 6 be the integer next below this root. Substitute y = h + Xfz and so on. The value of ic is a + :; &+ c+ .... Ex. x^- 9a; 2 = 0. One root is between 3 and 4. 10-9-2 3 0-2 6 18 _> 9 Operator 3 The equation in y is therefore 2?/3-f l82/2+9i/+l = 0, and the Horner operation is applied to this after finding a root between 9 and 10. Thus the operation is worked alternately from L. to R., and from R. to L. The safest arrangement for the work is as follows : 1 9 18 -2 82 9 ,e -153 -18 Operator 9 Operator 2 82 -153 11 175 339 -36 -2 -14 -30 336 82 339 336 -30 and so on. The next o perator is 1 12 and the root is "^9+2+ 12+ 6+ 1+ 2+ 1+ 1+ 1+ 11+ ... On a rough average each convergent involves accuracy to a fresh decimal place. The value of the fraction written is 3-10548261657 ... . E 2 52 THEORY OF EQUATIONS Note. The method is more risky numerically than Horner's, but has the great advantage that separation of the roots is not required. If one of the subsidiary equations has more than one positive root, the work may be continued with either. If a subsidiary equation repeats itself, the root may be completely determined [see ALG 23 ii (1)]. The process may at any stage be continued by Horner's Method or may replace Horner's Method with advantage until separation of the roots has been effected. Method viii (Lagrange). Ex. ir^-9ic-2 = 0. Let ic = a be a root, then x^ -^x-2 = (x-a){x'^ +px + q) ; Since \og{q-\-px-\-x^) contains only + indices of x, --co.iinlog[l-i(^ij:-2j]; 2 2^ 2^ 2^ ... a =----- 3 - - 12 ^ - ... , a very rapidly converging series giving a = '223462 .... Notes. (1) This method is applicable only to equations of the type x"4-ca; 6 = 0, and then only produces a rapidly convergent series if c > 6 and c > 1. The solution in this case is _ & _ J>_ 2n b2-i _ 8M(3n-l) 63n-2 ^~ c c"+i "^ "2" 6-2 "+i 2-3 c3+i (2) It gives the numerically least root of the equation and this only. (3) A similar method applied to a;2"+i +ca;2 b = gives a + j8 and a2 4-/32 jind thus a and ^ where a and /3 are the two numerically least roots. (4) The theorem in the Differential Calculus known as Lagrange's can often be applied to the solution of equations by series. See BIFF. CALC. 6 (8). Method ix [Graflfe]. (a) Let / = be the equation. Form the equation, /^ = 0, whose roots are the squares of the roots of the given equation [ 3 (e)]. Kepeat the operation, f^, and so on. Unless the roots are nearly equal in value numerically (and even then eventually), one or more of the THEORY OF EQUATIONS 53 terms at R.H. of equation will soon become small com- pared with the terms immediately preceding. When this occurs, say with /^, the equation formed by excluding these smaller terms will give an approximation to the rth powers of the larger roots, that obtained by the smaller terms together with the last of the larger terms to the rth powers of the smaller roots. (b) Further, ii f = PoX'^-p^x'^-'^+p^x^--- ..., and if the roots are all real, approximations to the rth powers of all the roots in descending order are given hy Pi/pQ, P2/P1, Proof. (a) Consider (ccp) (x q) (x r) = 0. Ifp and 3 be large com- pared with r this is approximately x \x^ (p + q) x+ pq]pqr = 0, &c. (&) If a, /3, 7, .. be the roots of/'* in descending order, = 2a = a Po nearly, - = SOfiS = OiP nearly, . . ^ = /3 nearly. Po Pi Ex. / = -9x- -2 = 0. The coefficients are f 1 -9 -2 P 1 -18 81 -4 J' 1 -162 6417 -16 P 1 -13410 41172705 -256 At this stage it is easy to see that the fourth column will not affect the first five figures of the third column in f^^. The roots are there- fore separated and correct to 5 figures, the eighth power of the smallest root is given by 41172705a; 256 = 0, while to the same accuracy the eighth power of the other two roots is given by x2-13410 + 41172705 = 0. Notes. {V) For fuller details see Encycl Math., Bd. i. 1, pp. 440 flF., or Runge, Praxis der Gleichungen, pp. 157-82. (2) If all the roots of an equation are required, this method is pre- ferred by German writers as the quickest. It is difficult, however, to tell beforehand to what equations it is suitable. If the roots are numerically close together, or if the imaginary roots have a modulus near in value to the real roots, a high / will be required before the roots are separated, and the labour of finding this is great unless a calculating machine (for which the method is eminently suitable) is available. The method may be tried in a few minutes up to/* and 54 THEORY OF EQUATIONS will indicate with great exactness any root materially smaller or larger than all the rest. (3) If the work is taken beyond /% only 7 figures, at most, should be retained, and the notation 4-1172707 may be employed to indicate 4'117270x 10'. This notation has the advantage of showing easily when one term ceases to affect the others. (4) In the writer's experience the series Z', /^, Z^' .. . is to be pre- ferred to Graffe's series for equations of degree not higher than 5. It is almost as easy to calculate [ 3(/)], more rapidly effective, and has the great advantage of not obscuring the sign of the roots. (5) Wiiile great labour is often required to separate the roots for the (6) method, the (a) method will often suffice. Thus in the case given p can be found mentally, and all three roots from it, correct to 4 places, if Barlow's Tables are available, in less tluxn one minute. Method X [Bernoulli], If s^ denotes the sum of the ^th powers of the roots [see 11 (2)], the series of numbers s^^/s^y 83/^21 h/^i7"' will usually converge towards the value of the numerically greatest root. Note. This method is always tedious, and valueless in practice it a second root is numerically near in value to the greatest root (as e. g. in x' 9 X 2 = 0), or if there be a modulus of imaginary roots near in value to the greatest root. If such modulus be greater than the greatest root, the fractions have no limiting value. Method xi [For Cubic Equations], To solve x^-hpx + q = in the case when all the roots are real, put x = ky, . '. 1c^ y^ -^-pky \-q = 0. But cos^cx-f cosa- Jcos^a = 0. Hence the equation is satisfied hy y = cos a if 1^= - 7' h= ~ jcos^a. h^ i: 4 Eliminate h Determine cos^a, three values for cos oi, iovy, for X. Notc.Oi little practical value. A similar method involves cosh a. Method xii [Complex Boots]. Ex. x^-6x-l = 0. (a) Here the real roots are --200064 [best by Lagrange's series, or Graffe], -1-440501, +1-541652. The sum of these is 098913. But the sum of all the roots is 0, .. if r (cos ^ + 1 sin (p) THEORY OF EQUATIONS 55 bo the two complex roots 2rcos = '098913. But the product of all the roots is 1, .'. r^ = 2-250769. Hence r, and . (b) The following method does not require the real roots to be known and is (theor3tically) available when there are two or more pairs of complex roots. Put x= r (cos^ + t sin0), equate real and imaginary parts, and obtain r = r : ~ ^^^ (^sin5(/))6 sin^4(/) sin^ = = !(<{>), say. A little consideration will show that there is a value of near 90". Method ii is now employed. By the aid of 4:-fig. log. tables calculate F (89), F (88), F (87), F (86). The calculation is not tedious if the four F's are evaluated simultaneously. Plot the values on squared paper. A value slightly above 88 is indicated. Again with 4:-fig. tables calculate F for 87 54', 88, 88 6', 88 12'. A value near 88 6' is indicated. With 7-fig. tables plot F for 88 6', 88 6' 20'', 88 6' 40", 88 7', and we obtain ^ = 88 6' 39-2", whence the complex roots are 049455... + (1-499442...) i Note.GriiSe's Method can also be used to find the modulus (r) of the complex roots, provided that this is not close in value to one of the real roots. 9. Binomial Equations. (1) If a is a complex root of x"^-! = 0, a'^ will also be a root (m integral). (2) Further, if n is prime, the whole series of roots is (3) If m, n are prime to each other, o;*^ - 1 = 0, ^^ - 1 = have no common root except 1. (4) If m, n have H. C. F. 7c, the roots common to ^r*^ - 1 = 0, ic^ _ 1 = are all roots of a;^ - 1 = 0. Note. (3) is a particular case of (4). (5) li n = p^q^r^... {p, q, r... all primes), and a is any root of xP^-1 = 0, then (X, p,y... all satisfy ^^-1 = 0, and all the roots of ic**- 1 = are n products of the type aySy .... (6) Special Roots (Primitive Boots). (a) If w is a prime, all the complex roots of a;** - 1 = are ^ special '. 56 THEORY OF EQUATIONS (b) If n = p% the special roots are determined by {x^-l)/{x^JP-l) = 0. (c) If n = p^q^ r^..., the special roots are given by x^-1 n {x'^lPQ - 1) n (x'^ip - 1) * n (x/p^r - 1) " ~ ' and there are w^l - -)(l )(l ) of them. Ex. x0- 1 = 0, 60 = 2*. 3 . 5. There are 60 (1 -i) (1 -|) (1 -|) = 16 special roots, determined by (x60-l)(a:i"-l) (x6-l)(x*-l) (a;30_i)(a:20-l)(xi2-l)(x2_i) = 0. (7) The roots of binomial equations can all be com- pletely determined by De Moivre's Theorem, and are 2^7r . . 2A;7r X = cos + i sin [k = 0,1, ...n-1], n n ^ (8) Gausses Equation, x" 1 = 0. The roots of this equation can be completely determined by quadratic equations only. Let a be any complex root. All the complex roots are included in a, a^, ...a^^ j-g^e (2)]. Divide these into two groups These groups are easily formed if it is remembered that each term is the square of the preceding and that Oi^'' = 1. Prove ^i, 6^ roots of 62 + 0-4 =0. By taking alternate terms divide the 0's into two groups ^11 = a4-a* + ai6 + ai3; e^^ = as+a^ + a^ + a"; Prove 011 > ^n roots of e"^- 9^9-1 cups and e^-e,i.e~e,i = o. Finally (X, a^^ are roots ofe^-Oii^.O + l = 0. Further details on this problem, which Gauss considered his most remarkable discovery, and various methods of construction of the regular 17-gon geometrically in Enriques, Fragen der Elemcntargeomeirie, ii. 171-88. THEOKY OF EQUATIONS 57 10. Reciprocal Equations. These are of two classes : i. Coefficients of terms equidistant from the end equal and same sign. Root - 1 if degree is odd. ii. Coefficients of terms equidistant from the end equal and opposite sign. Root + 1 if degree is odd, 1 if even. On removing factors x+1, x-1, or x^-1, we have in every case a R. E. of class i, degree even (2m). The degree can now be halved by dividing throughout by x^ and using the substitution y = x+ -^, f-2 = x^- + -, &c. Notes. (1) Other equations can sometimes be reduced to the re- ciprocal form, e.g. 8x3 + 20x2 + 10 x + 1 _ q . p^t 2x = y. (2) The successive values for x^ + 3 = ^3 may be calculated from the relation tp = titp_-^ tp_2 ; thus t^ = y{y'^'^)y = y^'^y, &c. 11. Symmetric Functions. (1) Method hy Subsidiary Equations. Illustrated by cubic. (a) Functions of 7c > -^ - 7^ \ ^ ' /3 + y y + Oi OL + ji ^ /3 + y a + 13 + y-oc Pi+Po'^ Eliminate oc between this and (^Jqj Pi^ Pi-> PsX*^? ^Y == ^ (&) Functions of (fi-yf, {y-Oif, {oi-pf\ y = {/3-yY={/3 + yy-4.py y t^ ^ J Oi ^ Po ^ PoOi Note. Best, however, by first removingthe x^ term of cubic [ 3(/*)]. (2) Sums of Powers. [Po is taken as 1.] ()/iW//W = 2 1/(^-ot) = 2i(l-?f = ^ + ^4-^ + ...; X^ X^ X x^ x^ -1 1 / ^\~ = -2-(l ) = -s.-xs^o-x^s. 58 THEORY OF EQUATIONS To obtain the first divide /i by /as they stand. To obtain the second divide /^ by/, rewriting both from R. to L. (&)/iW=2/W/(a^-) = ^2X'*-i + (si + npj) x'^-^ + (2 +P1S1 + np^) x^-^ + ... ; also =^ nx^~^^r{n-\)p^x^~'^^ ..., Hence s^ +i?i = 0, 52+iJiSi + 2i?2= 0, mole, From these equations s,. may be written in terms of the p'a only, in determinant form, or vice versa. To continue the series, use 2a/(a) = 0, 2a2y(a) = o, &c. Sums of negative powers may be obtained from 2i/(a() = 0, si/() = 0,&c. Hence s's directly in terms of j)'s. W i+JPi2/+jP22/^+-.= (i-a^)(i-^^)- ; .-. log (l+i9i2/+i?2/^ +...) = -Si/-|s22/^- -.; * l+i'i^+i?22/^+ = e-'Ay-2^y' -. Hence i?'s directly in terms of s's. (8) 0^/?er Integral Functions can generally be obtained from 2/a p y = Syy^s^Sp Sf^Sy^+p s^Sp+^ SpS^+^ + s^+^+p* Hence 62(a^y)^ = V^- 3s^52m+ ^Ss^. THEORY OF EQUATIONS 59 (4) Special Devices. (a) For differences of roots of biquadratic determine cubic whose roots are \ = 0y + (XS, fx = yOl + fid, v = Oi0 + yS, i.e. (fJL-v) - (5-a)(/3-7), &c. (6) Equation of Difference-Squares. Let (i8-7)2, &c., be the roots of x^+qiX'"*-^+ ... = 0, m = in(M-l), and 2,. = sum of rth powers of these. 2 (x-a)2'* = nx2'--2rsi x2r-i^ , Fxit x = Oi, x = ^... and add. .*. 2 2,. = nsa,. 2rSiS2,._i+ .... Hence 2 's. Hence g's. (c) /(x) = (x-oL){x-0)... /'(a) = (a-i8) (a-7)...; .-. 3i=/'(a)/'(i3).... 12. Algebraic Solution of Cubic Equation. (aQaitt^a^ 5] ic l)*^ = 0. (1) Associated Functions. *h ^ applying to sum- mation with all possible changes of suffixes, after affixing proper signs. It is also often conveniently written as 1. Rules of Formation. i. The diagonal element (with suffixes in the proper order) is + . ii. Any interchange of 2 suffixes (whether consecutive or not) alters the sign of the element, the letters retaining their order. iii. Any interchange of 2 letters (whether consecutive or not) alters the sign of the element, the suffixes retaining their order. iv. Sarrus's Rule. For a three-row determinant the rule indicated by the diagram is convenient. i 'X ^2 3 t?3 Cg Arrows sloping downwards indicate terms with positive signs, upwards negative. i^ofe. ii and iii are equivalent statements, not distinct. DETERMINANTS 65 Ex. Sign of 03 hj Cg d^ lf^ g^ in j a^ b^ Cg d^ e^f^g^ \ . If two pairs of suffixes are interchanged, sign is unaltered. Hence sign of 3765142 Is (interchanging 3 and 1, and 7 and 2) same as 1265347, i.e. as 1234657. One change only needed now. Sign minus. 2. Law I. If any two ^parallel sets of terms (not necessarily consecutive) he interchanged^ the sign of A is changed. Proof. From 1, ii or 1, iii. Note. The 'sets' must be complete, i.e. whole rows or whole columns. 3. Law II. If any two ^parallel sets of terms are identical or proportional, A = 0. Proof. Use Law I. 4. Law III. A is unchanged if all the rows te written as columns, in the same order, or vice versa all the columns as rows. Proof. From the Kule of Formation. 5. Law IV. // every constituent in any row (or column) is multiplied (or divided) hy the same factor, A is multiplied (or divided) hy that factor. 6. Minors. The minor [J.^] of any term a^ is the de- terminant formed by striking out from A the row and column in which that term occurs and affixing the proper sign. Bule of Sign : Start at L. H. top corner and pass by rows or columns (never diagonally) to a^, counting + , - alternately at each term until a,, is reached. Note. Slight variations in text-books in regard to this definition. Some omit * Kule of Sxgn ' from definition of minor. 7. Laws of Minors. (1) A^ = bA^fbaf.. (2) A = I^a^A^, the 2 extending either to letters, suffixes unaltered, or to suffixes, letters unaltered. = ^hj.Bj. = ... ,, ,, (3) 0= 2a,^(r^s) = 2fc,J5, = .... Proof. Show that all terms of 2 occur in A and vice versa. 1878 F 66 DETERMINANTS c B A D C n 8. Higher Minors. Tlie second minors of A are formed by striking out the rows and columns in which any two selected terms (not in the same row or column) occur and affixing the proper sign. Similarly for third and higher minors. Rule of Sign. The sign is the product of the signs of the first minors of the terms selected. 9. Laws or Higher Minors. (1) Ap, g = c)2 A/'bttp ."daq. (2) A = ^'. apbq] Ap, q, the S extending either to letters, suffixes un- altered, or to suffixes, letters unaltered, and I fip bq\ = apbq aq bp, being known as the complementary to Ap,q. [Laplace's Theorem.] (3) A = 2 I ttj, bq c,. I Ap, q, ,.. (4) If in A which the figure represents diagrammatically A be any square block of terms, and all the terms in D (or in C, C) be zero, A = A\BB\. 10. Law V. If every constituent in any row or column cm he resolved into the sum of tivo otJiers, A can he resolved into t/u sum of ttvo others. Ex. If fti = /3i + )3/, &a = /S.^ + iS/... aihCi...ln\ = |ai/32C3.../| + \ a^^^' ^^^-.-hi [- 11. Law VI. A is unchanged when to the constituents of am row {or column) are added those of one or more other rows {of columns) multiplied respectively hy constant factors. Note. The beginner is cautioned against applying two or more o these operations simultaneously to the same rows or columns for thi simplification of a numerical determinant. Thus if he adds th( second row to the third row, and simultaneously the third row t( the second, he will obtain a A with two rows alike, which will van is] identically. It is wise also, in any case, to leave one row or on column unaltered in each operation. 12. Law VII. Multiplication of Determinants. ! a.2 a^ Pi P2 Pi = ^ap ^aq ^ar h h h 0.1 02 Oz ^hp ^hq ^hr Ci C2 C, n ^2 ^3 ^cp :^cq 2cr where Sop = a^^Pi + ^2^2 + ^i^a- DETERMINANTS 67 Proof. Consider the 27 determinants into which R. H. splits up. Notes. (1) Square of A = asymmetrical determinant of same order. (2) The process is perfectly general for A's of any order. (3) To multiply A's of different order, bring them to the same order by * bordering ' with zeros and unity. 13. Solution of Simultaneous Equations. / ajflj + &!?/ + Ci^+ ...= mj ^ J a2^ + ^22/+ = *^2 \ a X+ ... = m-> where the number of equations is also the number of variables (n\ 14. Eliminant of above system when m, = ^2 = "^h = ' = is \(^ihc^ '" IJ = ^^ 15. Determinants op Minors [Reciprocal Determinants]. (1) ^'^A,B,...L,\ = A-i. m A ^^^ _ ^A ^ _ ^ ^ 16. Skew Symmetrical Determinants, i.e. A's with lead- ing diagonal zeros, and the terms symmetrically placed thereto equal but with opposite signs. (1) Skew Symmetrical Determinants of an odd order vanish. (2) Skew Symmetrical Determinants of an even order are perfect squares. F 2 PLANE TKIGONOMETEY 1. Methods of Angular Measurement. (1) If the same angle measures D, G^, 6^, D/180 = a/200 = e/TT. (2) The unit of circular measure {radian) is an angle at the centre of a circle which subtends on the circumference an arc equal to the radius. Its measure is (180/7r) =57 nearly. More exactly 57-29578. (3) 6 = (arc)/(radius), measured in radians. 2. Projections. The sum of the projections on any straight line of the parts of any broken line joining any two points P and Q is independent of the manner in which P and Q are joined. Ex. Thus projection of PN + projection of NQ = projection of PQ wherever N is. 3. Ratios. perp. base , perp. (1) sine = ~r^ , cos. = ^ - , tan. = ^^-^ , ^ ' hyp. hyp. base hyp. hyp. ^ base cosec. = -^^^ , sec. = r^^^ , cot. = . perp. base perp. Note. But as soon as possible the student should adopt the projection definition : If the angles be described by a stn\ight line OP revolving from an initial position 01, cos = ratio of projection of OP on 01, to OP, sin= ,, Oi? (at right angles to 0/), to OP. PLANE TRIGONOMETRY R 69 (2) PN= OP sin ^ = OiV tan A, 0N= OP cos A = PN cot A. (3) cosine^ = complement- sine A = sin (90 - A), cot A = tan(90-^), cosec A = sec(90 -A), versin A = 1- cos A, covers A = 1 - sin J. . (4) Sin2^ + cos2^ == 1, sin^ tan -A = Fig. 3. cos J.' sec^ J. = 1 + tan^^, cosec^ A =- 1 + cot^ A. (5) If tan A = - , sin J. = , cos A = & Va^ + h^ Va^ + h^ Note. The ambiguity of the root must be remembered. (6) The student should thoroughly practise himself in the formulae, giving all the ratios in terms of one selected ratio. These are recalled mentally as follows : To write all the ratios in terms of the cosecant think of a right-angled A with hypotenuse c and perpendicular 1, then the base is vc^ 1, and we have the results . , 1 yc2_l 1 sin^ = -, cos^ = , tan^ = , &,c. Here c stands for cosecant. The ambiguity of the root must always be remembered. 4. Values op Special Ratios. (1) sm cos 30 45 60 90 1 2 1 V2 73 2 1 Vs 1 1 /\ 1 2 A 2 1 1 v/3 00 tan - = 70 PLANE TRIGONOMETRY Note. The first line may be remembered by the mnemonic v^l, a/|, VI VI VI The second line is the first written in reverse order. The third is the first divided by the second. But as soon as possible these results must be individually known. V'o + v/2 i^ofe. Since 3 = 18 - 16 the ratios of all multiples of 3 can be explicitly expressed in quadratic surds. 71 5. Angles of Form w ^ A. 77 (1) Any trigonometrical ratio of w ^ ^ will be the same l- r a j- even ^ ^ complementary '"^^^^ ^^ ^ accordmg as n is ^^^ . To affix the sign, suppose A acute, the sign will be that of the ori- ginal ratio in the quadrant in which the compound angle lies. Notes. (1) Too much importance cannot be attached to this formula. It must be practised until the results can be written without hesita- tion. The sign determined by supposing A acute holds when it is not. (2) In applying it to determine such expressions as sin 225, &c., refer to the nearest even multiple of 7r/2, e.g. treat this angle as 180 4 45 and not as 270 -45. (3) There are two distinct types of proof, according as n is even or odd. (2) Simplest Gases : sin (-A) =-smA, cos ( - J.) = cos A, sin (90-^) - cos^, cos (90''-^) - sin^, sin (180 - ^) - sin A, cos (180^ - ^ j = - cos A, PLANE TRIGONOMETRY 71 6. Angles with the same ratios as A. (1) If sin ^ = sin ^, 6 = nii + i- )^A. So for cosecants. (2) If cos C = cos A, =2n'nA. So for secants. (3) If tan^ = tsLiiA, 6 = nir + A. So for cotangents. Note. Always use these values in writing the solution of trigonome- trical equations, avoiding, where possible, the sine formula in favour of either the cosine or the tangent formula. 7. Addition Formulae. (1) sin {AB) = sin AcosB cos A sin B, cos (AB) = cos A cos J5 + sin A sin B, , fA,^)^ ta n.A+tanjg tan{A+B) = ^r^: t-t ^* ^ -^ 1 + tan ^ tan B /i^o J. 1 + tanJ. Notes. {1) The cotangent formula should not be learnt. It is cot A cot B + 1 cot (A + B) = ^- J" ^ ' cot 5+ cot -4 (2) The proof should be the Projection Proof. (3) The sine and cosine results follow easily also from Ptolemy's Theorem {GEOM. 24 (4)). ,^. . . . ^ . ^ + B A-B (2) sm ^ + sin jB = 2 sin ^ ^^^ sm J. - sin B = z sin ^ cos u4. + jB cos J. + COS 5 = 2 cos 5 cos Li AB cos A - cos 5 = 2 sin ^r sin These results wiW5^ be remembered in words : Sum of two sines = 25m [half sum) cos (half diff.), Biff, of two bines = 2 sin (half diff.) cos (half sum), Sum of two cosines = 2 cos (half sum) cos (half diff.), Diff. of two cosines = minus 2 sin (half diff .) sin (half sum). Notes (1) The student must be able to write mentally and without intermediate steps such results as tan A + tan B = sin (-4 + B)/cos A cos B, 2 A + B 2 A-B 2 A+B 72 PLANE TRIGONOMETEY tan ^ + cot 3.4 = cos2^/cos^ sin 3^, &c. They are not to be learnt as formulae. (2) The following transformation is often valuable : sin ^ + cos ^ = -/2 sin {Aio) = \/2cos (^ + 45). (3) sin AainB = i {cos (A-B)- cos {A + B)} , cos AcosB = ^ {cos {A-B) + cos {A + B)] , sin ^ cos ^ = "I {sin (A-B) + sin (A + B)}. These results must be remembered in words : Product of two sines = ^ {cos diff. - cos sum}, Product of two cosines = ^ {cos diff. + cos sum] , Product of sin and cos = J {sin sum + sin diff.}. jV'ote. In the phrase * difference ' the angles are to be taken in order. Thus if in the last formula the sine has the smaller angle, the differ- ence and its sine will be negative. (4) In a trigonometrical simplification involving pro- ducts of sines or cosines the proper procedure is almost always to use the formulae of (3). (5) sin (^ + 5 + 0+...) = Si-Sg-f^s-..., cos(^ + +C+...) = 2o-22-f24-..., where 2q = cos A . cos JB . cos C . and \ = the sum of all possible terms obtained by changing r of the cosines of 2 into sines. where S^. = the sum of the products of tan A, tan B, tan C, ... taken r together. 8. Multiple Angles. 2t (1) sin 2^ = 2 sin J. cos J. = r-^ J [^ = tan^] cos 2 A = cos^ J. - sin^Jl = - -^ , = 2cosM-l; l + cos2^ = 2cosM, = l-2sinM; l-cos2^ = 2sinM, tan2ul = . 7^; tanM = -r. 1-t^ 1 + cos 2 J. PLANE TRIGONOMETRY 73 (2) sin^A - sin^B = sin (A - B) sin {A + B). (3) sin 3^= 3s-4s3, [s = sin A J cos3^ = 4c^-3c, [c = cosJ.] tan 3^ : See 7 (5). Fnt A = B = C. For higher multiples see 24. (4) In a trigonometrical simplification involving squares or higher powers of sines and cosines the proper procedure is almost always to transform the powers into multiple angles. [23(1).] ,^^ . A A J- ; -. (5) sm -^ + cos = + V 1 + sin J., sin -cos = + yi-sin-^. To remove the ambiguity see 7 (2), 'Hote (2). . A sin J. 1 - cos A /I - cos A tan-r = =v.^ 2 1 + cos J. sin tI A' 1 + cos J. 9. Inverse Functions. (1) Sin~^;r + cos'^^ = ^. So for all complementary functions. Note, It is understood here that sin~ix means the least value of the angle. But the symbol is also used for any value, so that sin-i|= M7^^-(-)"g (2) tan~^ a + tan~^ 6 = tan~^ -^ 7 . l-dh Note. The student must practise the reading of other formulae as inverses, e.g. sin'^cc + sin~^t/ = sin~^ {a;\/l y'^ + j/V 1 x*]. 10. Properties of a Triangle. (1) A + B+C= IT. Hence sin J. = sin(5+ Q, &c. (2) 2 tan A = tan A tan B tan G, ^ ,A A B ,G ^ cot p = cot -^ cot ^ cot ^ J 2 cos^ J. = 1 - 2 cos ^ cos i? cos G. 2 COS ^ = 1 + 4 sin 77 sin -K sin ^, 74 PLANE TRIGONOMETKY Notes. (1) The following are for reference only : 2 sin A =4 cos cos - cos - , ^ a A A B . C 2 ^^"2 2 sin 2A = i sin A sin B sin C, 2 cos 2 A = 1 4 cos A cos B cos C. With a little practice they can easily be obtained mentally, as required. (2) Since (r-2^) + (7r-2B) + (7r-2C) = rr, the values n~2A, n-'2B, ir-2C can be substituted for A, B and C. So also {2iT-hA), &c. sm.4 sm^ sin (7 iVo.'es. (1) The substitutions a = d sin -4, 6 = dsin B, c d sin C are the most valuable of all in triangle-identities. (2) d is (see 12 (1)) the f?iameter of the circumcircle = 2R. (4) a = 6 cos G+ c cosB, &c. (5) a2 = h^ + (^-2hccosA, &c. h^ + c'^-a^ 26c ^a\ ' ^ /{s-ms-c) . A /sis- a) , ^ /(s-6)(s-c) . 2 , sin J. = Vs(s -a){s- h) (s - c). (7) tan^ = |:i-%ot4, &c. ^ -^ 2 6 + c 2 (8) J.rea o/ A = A = i&csin.4 = &c. = Vs(s-a)(s-'b)(s-c). 11. Solution of Teiangles. (1) Given a, h, c. Use 10 (6) or 10 (5). (2) Given A, b, c. Use 10 (7) to find i?-C; 5+ = 180-^; or 10 (5) for a. PLANE TRIGONOMETRY 75 (3) Given A, B, c. C = 180 -A-B, then use 10 (3). (4) Given A, B, a. As in (3). (5) Given a, h, A we have the ^ Ambiguous Case ', of which there are three treatments possible, all of w^hich should be familiar. {a) Geometrical. The circle, centre C, radius a, either does not meet the knoven direction of c~no solution ; or touches it one solution, B = 90 ; or cuts it in two points on same side of A two solu- ' tions, and two values of B, i. e. B and 180 - B ; or cuts it in two points on opposite sides of A one solution. (h) Use sin B = -sin A. Data make - sin ^ > 1 (no A), = 1 (one A), < 1 (two A's or one as 6 > or :)> a). (c) Use Or' = h^ + c'^-2hc cos A to give quadratic for c. 12. Related Circles. (1) Gircumcircle. II = ^^ = = JA* . B . G ^ asm-sin- (2) Incircle. r = = 2 = &c. S a. cos- ^^ . A . B . C , ., A = 4i? sm "^ sin -^ sin -^ = {s- a) tan 7? O (3) A-excircle. r^ = = s-a A cos- A-o ' ^ i? C .A = 4it sin cos cos -^ = s tan J J J CI Note. The formulae may be derived from the incircle by regarding the excircle as the incircle of a ' triangle ' whose angles are these being the angles bisected to obtain the centre. 76 PLANE TRIGONOMETRY (4) Distance of Cireumcentre and Incentre = ^W - 22?n Distance of Cireumcentre and Excentre = Vr^ + 2Tlr^. Note. Hence r rhc)/{ah + cd), {ab-^-cd) {ac-^hd)/{ad^-hc) and the circumradius is j^\/(a6 + crf) {ac-\-bd) {ad + bc). 15. Regular Polygons. The following results should be obtained mentally when required : a IT (1) J? = ^ cosec - ^ ^ 2 n (2) fcot^. ^ ^ 2 w 4 n 16. Maxima and Minima. (1) asinx + bcosoc^ Va^ + fe^sin (ic + a), where cot a = a/l, Hence, &c. 78 PLANE TRIGONOMETRY (2) Given 2^ = a, to investigate the max. or min. values of : A + B A ~ B 2 sin A. sin ^ + sin 5 = 2 sin cos -- ^r , .-. increased with- out altering A + B hy making A = B. Max. A = B = C ... n sin ^. 2sin(^ + B) - cos (^-5) -cos (-4 + J5). Max. A=B = C... 2tan^. tan^ + tanB= 2 sin {A + B) mn.A^B^-0. cos {A-B) + cos [A + B) n tan A. tsinA tan S _ cos (A^B)- cos (A + B) _ 2 cos (A + B) ^^ " cos{A-B) + cos{A + B)~ ~ cos {A-B) + coa {A + B)' 17. Limiting Values. (1) If ^ < J 77, sin 0, 6, tan are in ascending order. (3) ^ >sin^ > 0-^e'\ 1 > cos > 1 - i d-\ (4) As 6 increases from to Jtt, sin 6/6 continually diminishes and tan 6/6 continually increases. (5) Euler's Theorem : e e d sine cos 2 cos 22 cos p .. to CO = -j- 18. Series. [See also 21 (7).] ,. , sin sin , ^, , , (1) 0C+ {oc + B)+ ... to n terms, cos cos ^' c, sin (number x Adiff.)sm , i. n i. ^ t l\ Sum = ^^ - ,, ...^^ (mean of first and last). sin (^ dm.) cos (2) ^8iT^{oi + n^)coa^{(X + nP). Convert into multiple angles. [See 8 (4) and for method 23 (1).] e. g. 2 sin2 a. Use sin^ a = |(1 - cos 2 a). (3) sin a - sin (a + y8) + sin (a + 2y8) ..., is treated as 2 sin (oc + n/S + mr). PLANE TRIGONOMETRY 79 (4) ^"' a + c^'''{a + /3) + c^ ^'"^ (a + 2/?) + ... . The sum to ' cos COS COS '^^ infinity beginning at any term First Term - c^ x < Term before first ' l-2ccosdiff. + c^ Notes. (1) 'Term before first' in given series would be - sin (a j8), &c. (2) Deduce sum to n terms, namely 22 1 n+l Proof. It is a recurring series whose scale of relation is l-2ccos/3 + c2. (5) Uq cos OL + Ui cos (cx + yS) + ^2 COS (a + 2^) + . . . , where w,. is a rational integral function of r. To sum, multiply by cos (3 and subtract ; the result will be a series of one lower degree in r in the coefficients. Similarly for sine series. e.g. cos0 + 2cos2e + 3cos30 +.... (6) (a) cosec x + cosec 203+ .... Use cosec 2aj = cot x cot 2a;. (6) tana; + itan - + .... Use tan x = cot oj 2cot 2cc. 2 , 2 . ,2 + .... 2 (c) tan-i 2 +tan-i22 +tan-ig2+- Usetan-i(x+l) tan-^(x-l) = tan-i (d) tan X tan 2x + tan2a;tan 3x+ TT i. ^ i. T1 tan^ tanJB ^ . Use tan A tan B = 1, &c. tan(^^-jB) 19. De Moivre's Theorem. (1) (cos a + i sin ok) (cos ^ + i sin /3) = cos (oc + 13) + i sin (a + /?) and so for any number of factors. (cos a + i sin a)/(cos /3 + i sin ^) = cos {a-/3) + i sin (a - P). (2) cos 6 + i sin is one of the values of (cos w^ + i sin nOyi^ and the remaining n-1 values are obtained by writing 2rT: + n9 for nd and r= 1, 2, 3... (w- 1). 80 PLANE TRIGONOMETRY (3) cos nS and sin nO expanded homogeneously in cos 6, sin 6, Method : In the identity cos w^ + i sin nO = (cos 6 + i sin ^)" expand and equate real and imaginary parts on the two sides. Results : cos w0 = cos" $ ^-j - cos"-^ $ sin^ $ + -^ ^ ^ ^ -^cos"-'*esin4 0..., sinn^ = wcos"-i0sine- ^ q.^~ cos"~gsin''g + ... , ^ ^ n(n-l)(n-2), ,, tannO = l_"J|^tan'+,.. A^ote. These results hold for all integral values of n, and for all values of n provided lies between ^n. When n is integral they can be deduced from 7 (5). The series terminate in this case only. (4) Trigonometrical Identities. Any algebraical identity in a, &, c, &c., gives rise to two trigonometrical identities by substitution, a = cos 2a + i sin 2a, b = cos 2y8 + i sin 2y8, &c. Ex. 2a^ (^ c) = (b c) (c a) (a &) gives 2cos(/3 + 7 + 2a)sin(/3-7) = 4sin(i3-7) sin (7-a) sin (a-iS) cos (2a + 2^ + 27), 2 sin (i3 + 7 + 2 a) sin (iS - 7) = 4 sin (18-7) sin(7-a)sin (a-/3) sin(2a + 2i3 + 27). 20. Expansions of sin 6, cos 6, &c. Notes. (1) These results are true for all values of 6, provided $ be in radians. (2) In the proof it is necessary to prove that Lt (sin 6/6)'^ = 1, and also that Lt (cos 6)^ = 1 when n 00 and 6 = 0. Another point of diffi- culty is generally omitted in elementary treatment. [Hobson, 99.] (3) To expand sin^e, sin^^, sin^^cos^, &c., use Multiple Angles. [8(4)0 PLANE TRIGONOMETKY 81 03 205 1707 2) tan 6 = ^+ 3 + ]5 + 315 + cot0 - 1 e 03 2 05 3 45 945 cosec0 = 10 7 03 31^5 "^ 6 "^ 360 "^ 15120 02 56* 6106 sec0 = l+2 + 24+^72O+-- L^^^^^^^' ^'^^^' ^ (6) (6).] 21. Exponential Values. (1) cos^ + isin^ = e^\ (2) cos (9 = ^ (e' + e-^% sin ^ = ^ (e*' - g-^^- (3) Hence exponential values for all the ratios. (4) Complex Quantities. The following is a summary of method : If X is complex define e* as l+x + x^/^ ! + a:;3/3 ! + ... after proving the series convergent. Prove e^.eV = e*'+^. Define sin x, cos x by the series values. Deduce results of 21 (2) for complex values. From these deduce addition theorems for sin {x + y) and all formulae founded thereon, thence periodicity of sin x and cos x. Tan x is defined as sin rc/cos x. For logs define x as a log of a where a = e*. Hence since fannc ^ cos2n7r + isin 2nit, x-\-2nm = log a. Define a"' as e^^'os"'. Deduce ordinary seties for log (1 + y) when y is complex, the series holding only under certain convergency conditions. (5) Gregory's Series, tan"^ x = x-^x^ + \ x^ - .... [I>a;>-1.] (6) Evaluation of tt. Apply Gregory to the following : Euler, -Jtt = tan"^ -I + tan~^ ^, MacJiin. J tt = 4 tan ~ ^ i - tan" ^ 239- Clausen. | tt = 2 tan " * + tan " * |. Rutherford. ^ir = 4tan-i| tan'^^ + tan-i g'g. (7) Summation of Series. [See also 18.] If the sum of the series ao + aiX + a^x'^ + ... is known, f{x) say, that of the series a^ cos oc + a^^ cos(a + 6) + a^x^ cos (a + 2^) + . . . can be determined. Similarly for the sine series. Isole. The cosine series is the real part of e** / (xe^^). 1378 G 82 PLANE TRIGONOMETRY (8) Expansions. The exponential values are applied to such ques- tions as : (a) Expand e"* sin {hx+c) in ascending powers of x. (b) Given sin x = k sin (x + a) expand x in powers of A- (< 1). (c) If (1 /c)tanx = (1 + k) tan?/(A;< 1) expand x in terms of y or y in terms of x. 22. Hyperbolic Functions. (1) Befinitions, sinh ^ = i(c^ - c"^), cosh ^ = i (e^ + e'^), tanh ^ = sinh ^/cosh ^, &c. (2) cosh2 Q _ sinh2 ^ = 1. (3) cosh B = cos 9i, sinh ^ = - i sin ^/. Hence all results in ordinary trigonometry may be turned into hyperbolic results. Some of these are added for reference below. (4) The hyperbolic sines and cosines are periodic with period 2 rre. The hyperbolic tangent is periodic with period iri. (5) cosh-^a: = log (cc + \/a;2 1). sinh-i X = log (x+ \/x'^ + l). tanh-ix = I log [(1+ :)/(! -a;)]. (6) For Reference : cosh = 1; sinh 0=0. sinh {xy) = sinh x cosh y cosh x sinh y. cosh {xy) = cosh x cosh ?/ + sinh x sinh y. tanh(x + ,+ ...; = ^^3p^-^;-^-^. sinh 2x = 2 sinh a: cosh x. cosh 2 a; = cosh^ x + sinh^ x. sinh 3x = 3 sinh x-\-i: sinh^ x. cosh 3x = 4 cosh^ a; 3 cosh x. All the above and many more may be derived from the ordinary form by the rule that if sinh occurs explicitly or implicitly (as in tanh x) twice or thrice, it occurs with a sign opposite to that in the corresponding formula in Plane Trigonometry. Notes. (1) Some curious logarithmic series arise from the expan- sions of sinh-i x and (sinh-ia;)2 [see 25 (1) (2)] by the use of 22 (5). (2) Just as, if 0^ = a be the radius of a circle, centre 0, and PN, ON be ordinate and abscissa of a point P, Z. AOP = 6, ON = a cos 9, FN = a sin e, area AOP = ^a^O, so if OA be the semi-major axis, length a. of a R.H. centre 0, and P be a point on it with ordinate and abscissa PN, ON, ON = a cosh 0, PN = a sinh 0, area AOP = ^a^f, but is no the angle AOP. Hence the name hyperbolic. PLANE TEIGONOMETRY 83 23. Expansion of sin^^, cos**^ in multiple angles. (1) sin**^, cos^^t^, sin^^ cos**^ and such expressions are expanded by the use of the results that if x = cos 6 + i sin ^, ic + 1/ic = 2 cos 0, x^ + 1/x^ = 2 cosn9y x-l/x = 2 i sin e, x^-i/x^ = 2i sin n 0. Care is needed in the last ^erms, which are sometimes without the factor 2 which occurs in the other terms. 1\2 1_3+3_1 1-2+0+2-1 1-1-2+2+1-1 1 1 / 1\^ / 1\^ = -a4.[-i.-e-iO-K^-D] = -iV(sin50-sin 3^-2 sine). (2) Be&idts (for reference) : 2-icos"e = cosnd+ncos(n-2)0+ '^ '^~ cos (n-4) g+ .... 2"-isin"ecos -^ =cosne-ncos(n-2)0+ ^^^^ cos(w-4)e- ... [n even] mr n (h 1 ) 2tt-isin"0sin = Hinne-nsin(n-2)e + -^ - sin (w-4)e-... ^ 2 ! [w odd]. Notes. (1) The above results are true when n is integral, provided that the last term as given by the regular sequence is halved if it is not accompanied by sin-fl or cos 9. (2) They are also true if n is not an integer if the L.H. is dmbled, i. e. 2" written for 2"-i and if 6 lies between it/2 and 0. (3) The complete investigation when n is not integral is difficult. [Hobson, 222.] A rough investigation may start for first series from R.H. = realpartofx + wx-2+ ^^^~^^ x-H ... , i.e. of (a; + l/x) &c. 24. Expansion of sinw^, cos?^^ in powers of s and c. A. Descending Powers. Methods, (i) (1 - 2x) (1 - ^/x) = l-si{2coae-^) if X = cos d + i sin 9. Take logs and equate coefficient s^* for cos nB in powers of cos 0. Deduce sin nd by writing ^ir O for 6 or directly from (1zx) (l + z/x) = 1-s {z + 2isine). G 2 84 PLANE TRIGONOMETRY sill sin (ii) Sum the series x 0-{-x- 2^+ ... [18 (4)1, and ^ ' cos cos *- - equate coefficients of oi^ in the expanded sum. (iii) From 19 (3) by sin^^ = 1 - co^'^B, [Hohsmi, 78.J Results : (1) smn^/sine=(2c)-i-(n-2)(2c;-3+ ^^~^^ [''~^\ 2c)"-6~.... (2) 2cosn^ -(2c)..-.2c;-H*i^\2c)-.-'i(!iz|i!iri)(2;.-. + ..,. (3 ; Expansions in terms of sin 6 by putting ^ir6 for 6. They dififer as n is even or odd. Note. The expansions are to cease just before any index becomes negative. B. Ascending Powers, Methods. A. (i) and (ii) are available, or the results of A may be rearranged. (iii) Assume cos w^ = o + ^2^^ + ^4** + ... . Differentiate twice and equate coefficients of s-^ to obtain a relation between the coefficients. Similarly for other results. Note. This is on the whole the easiest way. For Diflferentiation see ALG. 30 (9) Note. Results : n2 ^ n2(n^-22) ^ nevm: (1) cosn^ = 1 ~ s^+ s*-...; (2) sinne=ncLs--g^s3+ ' ^^ -^-...j, and results for cosn^, sin n^/sin in terms of c by writing ^tt 5 fore. OM : (8) eos. . . [l - !fzl%, + ('-l')(n^-8^) ^,_ j ^ and results for sinnO/sinO, cosnd in terms of c by writing |ff for e. Notes. (1) The results given, B (1), (2), (3), (4), hold for any value of $ between +f t. Also for all values of w, but terminate only when n is even or odd integer respectively. () Note results when 6 =7r/6, ir/4, ir/3, 7r/2. PLANE TRIGONOMETRY 85 (3) Many interesting deductions. Multiply B. (4) by t and add to B. (1). Put t sin 6 ^ X. Hence expansion in x of [a;+ ^/(l +x2)]". This can be proved independently by double differentiation as in B. (iii). In this result put x = cot 2 ({>. Hence expansion of eot" 1 -X^)- = 1 + 1..T2+ ^^X<+.. . /^\ t / ' 1 \o ^ ^ X u , ^ X r \ -t (2) i(s,n^Jx)2 = _ + _._ + ^_^. _+.... r^:t.l Easily remembered if (1) be written ^= - = r + o ' o" + '^y 1 1 J o increasing every number by unity, turning 1 into 2, 2 into 3, &c. Proof. Expand sin n5 = n^ |n36'+ ... in 24 JB. (4) and equate coefficients of n. For (2) take 24 B. (1) and equate coefficients of n^. Notes. {1) sin-icc/\/l-x2 is derived from 24 B. (2) and (sin-ia;)V-v/l-x2from 24 B. (3). (2) Note series for it, tt^, &c., which arise from the formulae of 25, by substitutions x = 1, |, l/\/2, &c. 26. Symmetrical Functions. (1) S. F. of cos7r/*^> cosSir/n, cos 5 tt/w, . . . cos (2m- l)iT/n are roots in c, = cos ^, of cos w^ = - 1 [ 24 A. (2)]. These will pair off into repeated roots with a possible middle term c = - 1. Hence S. F. such as S cos Ti/n, 2 sec ir/n, 2 sec^ it/n, 2 tan^ tt/w. (2) If the equation giving cosw^ in powers of c is regarded as equation in c its roots are cos 6, cos (6 + 2T:/n), cos (0 + 47r/w)... . Hence S. F. of these. (3) So for sin 6, sin (^ + 2 ir/w), sin (6 + 4: ir/n) .... 86 PLANE TRIGONOMETRY (4) The roots in f of . ,1 f-i ** (^^ ~ 1) ^2 1 J n(n-l){n - 2) ., are tan 0, tan (^ + tt/**), tan(^ + 27r/).... Hence S. F. of these and of sec^^, sec2(^ + tt/w)..., and of cot ^, cot (^ + -n-Z^w) .... (5) For functions of a portion only of the complete set of angles, e.g. ^ = xr/lS, X = cos . Prove [ 7 (4), 8 (4)] 2a;2 = Z-2y-x, 2^/2 = ^-y-2x and deduce ^jV = l (1 V'lS). For another illustration see TH. EQNS. 9 (8). 27. Factors. (1) cosn^ = 2"-i (cos cos7r/2w) (cos0 cos37r/2n)... = 2-i r** ^^^1 (sin2 7r/2w-sin2 e) {sm^S7r/2n-sin^6)... [stopping short of sin^ it/2. Proof. cos nO is function of cos 6 which vanishes when cos ^ = cos7r/2n, cos37r/2n.., . Then see ALG. 3 (1) and to determine coefft. 2-i use 24 A. (2). For second line couple first and last factors of first line, &c. (2) sin n9 - 2"-i p ^^^i^l sin $ (sin2 Tr/n-sin^ 6) (sin2 2 vr/n - sin2 6).,. [stopping short of sin2 tt/2. nr = n-l ,.^0 {cos e - cos (<^ + 2 rff/w)j. (4) n = 2''-'^ sin^ TT/n . 8m^27T/n ...y 1 = 2^-isin27r/2w.sin23Tr/2w... [stopping short of 7r/2. (5) sino = 9n;'.:r{i-^v^'^}, cos = II':r {l-4(9V(2r-l)27r^ Proofs. (i) Prove first 27 (2) or (1). Put 6=0 and deduce (4), Divide (2) or (1) by corresponding results of (4). Put nd = OC and w = 00 . This proof does not involve complex numbers. For more exact proof, Hohson, 284. PLANE TRIGONOMETRY 87 B -i- It (ii) sin ^ = 2 sin ^ sin jr = 23 sin e/4 . sin fe + 7r)/4 . sin (0 + 27r)/4 . sin {e + Zn)/i = 2" - ^ sin 0/n . sin {6 + v)/n . . . sin (^ + n 1 n)/n [n a power of 2]. Couple second and last terms. Deduce 27 (2), &c. This proof does not involve complex numbers. Remember uncoupled term, cos 0/n. (iii) From 27 (6). Write 26 for 6. Put x = 1. Deduce +sinw9 = 2"-^ sin e sin (^ + 7r/n) Couple second and last terms. Deduce 27 (2) with ambiguity. Put 6 = and deduce 27 (4), thus settle ambiguity. Rest as before. (6) x^-''-2x''cosne + l = ItZT^ {x'^-2xcoa{e + 2r'n/n) + l]. Proofs. (i) Solve for a;" and use De Moivre. (ii) If M = L.H. prove w 2xm_i cos^ + cc'^Wn.a divisible by w^. Hence by induction prove m divisible by u^. Hence, &c. (7) x"-l = (a;2_l)|j[^. ^^ {x2~2a;cos2r7r/n+l} (woven) = (x-l)Il'.~j " {x2-2xcos2r7r/w + l} (n odd) x" + l = II',.~ o"~ {a;2-2a: cos (2r+ 1)77/^ + 1} (m even) =^ {x+l)'nlZl^'~^\x^-2xcos{2r+l)n/n + l} (n odd) (8) sin w 5 = 2"- ^ sin 6 sin [6 + ir/n) . . . , n factors. cosnO = 2^- 1 sin (a + 77/2 w) sin (^ + 3 77/2 w).... -^i factors. iv'o^e. If the first of these is thought of as 2ir/2n, 47r/2w,... the second is in a sense complementary to it, completing the series w/2n, 2Tr/2n, Sn/2n,.... 28. Deductions from Factor Results. (1) J-2 + 22 + - to^ =6'''- Similarly X V^ = ^'S X V(2'-i)" = 1"^ X VCar-D' = ^-^^ Proo/. Use 27 (5) and series for sin 0. Take logs and equate co- efficients. 88 PLANE TRIGONOMETRY (2) Series for log (sin 0/6), log cos 6, cot 6, cosec 6, tan 6, cosec2 e, sec2 6, &c. See BIFF. CALC. 7 (6). iVoo/: Use 27 ^5). Take logs and differentiate (or 6-\-h for d and equate coefficients of /j). The result for cosec^e = ^^._._^ l/{d-\-rn)^ gives interesting series for special values of d. (3) Summation of 2;.';," ;^2. Sr (^2)" Use 27 (5). Put 9 = vxi. Take logs. Differentiate. To proceed from 2 (r2 + x2)-i to 2 (r2 + a;2)-2, differentiate result of 2 (r2 + x2)-i and divide by x. (4) Factorization of cosh w 60, e. g. all 90, ' Quadrantal Triangle.' (3) The greater of two angles of a A is subtended by the greater side and conversely. (4) It is not true that the exterior angle of a A is greater than the interior and opposite angles, or that the sum of the three angles is 2-B. (5) LexelVs Theorem. If the base BG and the area of a A ABC be given, the locus of ^ is a small circle through the antipodes of B and (7.* (6) Speaking generally, no theorems of Plane Geometry will hold in Spherical Geometry which depend for their proof on the properties of parallels, on the sum of the angles of a A, or on the ratios of lines. * Cf. Euc. i. 37. 90 SPHERICAL TEIGONOMETRY Small Circles. (7) Given the base JBG of a A and E-A (where E is the excess), the locus of J. is a circle through J5 and C* (8) li IBAC be the Z in a semicircle, AB+IC ^^ lA.-f (9) If a chord (great circle) through a point P meet a circle in A and B, then tan ^ PA tan ^ PB is constant for all positions of the chord. I 2^ote. Two tangents to small circle from external pt. are equal. (10) A great circle Radical Axis (from any point on which tangents to two small circles are equal) exists, and a Radical Centre for three circles. All circles orthogonal to a Coaxal System pass through two Limiting Points on the Line of Centres, equidistant from the R.A. (11) The relations oi Poles and PoJars can also be developed, and a theory of Polar Reciprocation exists. Notes. (1) Be careful not to confuse this sense of the vfovdi pole with the one more usual in Spherical Trigonometry. (2) The following are examples of ' reciprocal theorems': If the vertical angle and the If the base and area of a A are perimeter of a A are given, the given, the locus of the vertex is a envelope of the base is a circle. circle. Triangles. (12) A self-polar circle (see 1 (11) and Note) exists for every triangle. Its radius may be imaginary. (13) A circle {Hart's Circle) analogous to the N.P.C. exists for every triangle. Its centre lies on the join of the orthocentre and the cen- troid (i. e. the intersection of the medians). It touches the in- and ex- circles and has a radius tan~^ (|tan R). Quadrilaterals. (14) For any cyclic quadrilateral A + C = B + D, and for any circumcyclic quadrilateral a + c = h + d. (15) [Analogous to Ptolemy's Theorem.] If a;, y be the diagonals of a cyclic quadrilateral . a . c . b . d . X . y sin- sin 2 +sin-sm2 =sin2sm2. * Cf. Euc. iii. 21. This is really a part of Lexell's Theorem. t Cf. Euc. iii. 31. : Cf. Euc. iii. 35, 36, 37. SPHERICAL TRIGONOMETRY 91 (16) Remy's Theorem. If z be distance between midpoints of diagonals, X y z cos a + cos & + cos c + cos rf = 4 cos ^ cos ^ cos ^^ A Z Z Transversals, 4'C' (17) If in a triangle ABC a great circle DEF cuts the sides in D, E, F, then sin JBJ) sin CE sin AF _ sin 2)0 shTEA ' sinFB " ^^^^ ^^^^ ^ ^^^ (18) If AD, BE, CF be concurrent, the same relation holds (paying no respect to sign). (19) Conversely, if this relation holds, AD, BE, CF are concurrent if 1 or 3 of the points D, E, F lie tvithin the sides ; D, E, F are coUinear if or 2 points do so. AnJiarmonics. (20) The A.R. of 4 points A, B, C, D is sin AB sin CD , ^^^ smBC sin DA ^ ^ Note. Strict attention must be paid to sign here, AD = DA. (21) The row is harmonic when this A.R. = l. (22) In this case tan AB, tan AC, tan AD are in H.P. So also tan DA, tan DB, tan DC. (23) The A.R. of the pencil formed by joining to A, B, C, D is sin^lOB^ sin COD ^ r^^^^^^. ^ c^^^D}. sin BOC sin DOA ^ i i ) 2. Sphebical Mensuration. (1) Arc of small circle = corresponding arc of equator X cosine of latitude. (2) Area of lune (lA) = 2Ai^. (3) Area of triangle = {A + B+ C-'n)f^, = Er^. [See 8.] ]^ote. The angles must be measured in radians. (4) Area of small circle, radius p = 27rr2 (1 - cos p). 92 SPHERICAL TRIGONOMETRY 3. The Subsidiary Triangles. (1) Tlie Supplemental Triangle. Any theorem in Sph. Trig, remains true if we keep one side and the opposite angle unchanged and write for everything else its supplement. Note. Produce AB, AC to meet in A\ A'BC is the supplemental triangle. There are three such. (2) The Polar Triangle. Any theorem in Sph. Trig, remains true if we write for each angle the supplement of the opposite side, and for each side the supplement of the opposite angle. Note. If -4i be a pole of BC, &c. , A^B^Ci is a polar triangle. There are eight such triangles. For one of them the sides are ir A, n-^B, n O, and the angles are tt a, :r 6, tt c. Three others are supplemental to this one. The other four are antipodal to these four. 4. Relations between Sides and Angles. INote. These are most easily learnt in immediate connexion with the corresponding plane formulae, which are accordingly printed in parallel columns.] Plane. (1) f/2 = ii + c'^-2bc cos A, &c. sin A sin B sin C (2) where A 2a ahc = A/s{s-a){s-h){s-c). ,3,si|=V<^-'!:^^-l be COS be tan^= V^'"*^''~'^ s{8 a) Spherical. cos a = cos6cosc+sin6sin ccos^,&( sin A sin B sin C sin a sin b 2d sine sin a sin b sin c where 8 = v^sin s sin (s a) sin (s b) sin {s c] . A A /sin (s b) sin (s c) sm ^ = V . / ' ' 2 sm b sin c A A /sin s sin (s a) cos - = V . , : -' , 2 sin b sin c A^ ^ sin(.-l.)Bin(.-.) ,^^ 2 sin .< sm (s a) SPHERICAL TRIGONOMETRY 98 Plane. (4) JV'ofe. The spherical formu- lae giving the sides in terms of the angles, which have no useful analogues in piano, must be found from (3) when required and not remembered. They are added for the sake of reference only. (5) Napier s Analogies. B-C tan 2 .A Y, . cot TT (6 + c) 2 , B\C .A tan ^ = cot ^ (6) Gauss's Analogies C a . B- 2- ^^"-2 a B-C 2-^^-2- h-c A -2--^^2 b + c . A -2- ^^^2 &c. Spherical. .a . I cosScos(S A) sin ^ = V - -' sin B sin C tan B-C cos (S-B) cos {S-C) sinJSsinC cos Scos(S -4) cos (S-) cos (S-C) (b-c) ,&c. sin tan B+C cos COS 2 .A 2 (b-c) 2~ ,A T&T^)^"*2 &c. The Polar Transformation ( 8 (2)) of these gives two other formulae not to be remembered : B-C sin ^ tan tan ^? tan b + c B+C 2 B-C 2 a ^TC*^^2' sin rt sin -C . b-c A ^- = sin-2-C0S2, . b+c . A sin -g- sin 2 , ^-C sin 2 cos a.jB+C 6-c A cos 2 sin = cos -^ cos cos h cos -B+0 = cos b + sin 2' 2"^" 2 2 ""'2 ^otes. (1) As the plane formulae here given are not usually learnt in Plane Trig, it should be noted that the Gauss Formulae may be easily derived from the Napier Formulae by equating the * numerators ' &c. 94 SPHERICAL TRIGONOMETRY on both sides, thus sin jr . [ ] = sin ^ (6 c)eos -^ ,and filling up the [ ] by analogy, arguing, 'To B on L. H. corresponds & on R. H., A a ..to on R. H. corresponds ^ on L. H.,' using sine in tliis term in the Napier Sine Formula, cosine in the Cosine Formula. This mne- monic does not apply to the Polar Napiers which give no fresh results. (2) The Polar Napiers which are not memorized may be easily written down from the above Gauss results. Dividing the first by the third we get the first Polar Napier. (3) The Plane analogues of any Spherical formula may be deduced by putting a = Oi/r, b = 0/r, c = y/r, expanding sines, &c., in series and making r = x . (4) In Solution of Triangles it must be remembered that the case ABa is ambiguous, as well as the case a, b, A. 5. Relation connecting Four Consecutive Parts. Any four parts which are consecutive as we advance round the triangle (e.g. AcBa, cAbCy &c.) are connected by the following relation : Product of cosines of mid. parts = cot ext side . sin mid side - cot ext angle . sin mid angle. i^ofe. This important formula must be thoroughly practised. The printing indicates the mnemonic by which it is readily retained. As an example take AcBa. Here a is the * extreme ', ' exterior ' side and c is the middle side, A is the 'ext.' angle and we have cos c cos B = cot a sin c cot ^ sin B. 6. Right-angled Triangles. (1) Napier's Rules. A right-angled triangle contains 5 parts (for G is known). To write a formula connecting any three of these note that there is one of the three to which the other two are either * adjacent ' or ' opposite '. Further, mentally write/or every part its complement, except for those parts which run up to the right angle (and may therefore by way of mnemonic ^^' ' be considered to have already enough to do with ^tt), and then apply the formula sin mid. (prod.) cos opp. = {prod.) tsin sidj. SPHERICAL TRIGONOMETRY 95 Notes. (l) In full : sin middle part = product of cosines of opposite parts =^ product of tangents of adjacent parts. (2) This is one of the most remarkable mnemonics ever invented, but in applying it it is essential that no additional figure should be used (as some of the text-books recommend). The triangle must be made to furnish its ow^n figure mentally, and thorough practice with all the ten possible combinations of the sides is needed. Unless these precautions are adopted it is probably wisest to learn the formulae separately, and the student will then find that in the formula he wants he has forgotten which angle is taken as the right angle. (2) In solving right-angled triangles we have an 'Am- biguous Case ' when a and A are given (or &, B), namely one of the supplemental triangles. 7. Related Circles. Plane. (1) Circumcircle. a 2 a-) cos n a b c '2*2'2 (2) Incirde. A . B C sin -sin 2 (3) Excirde. A ri = s a B C cos - cos - 2 2 A cos- Spherical. tan tan R = cos {S-A) tan . a b . c 2 sin 2 in ^ sin ^ 8 sinS . B . C sin - sin - 2 2 tan i\ 8 sin {s a) B C cos - COS - 2 2 Notes. (1) These formulae are all so easy to find that it is a question whether to learn them is worth the risk of error. 96 SPHEKICAL TKIGONOMETRY (2) In Spherical Geometry there are excircumcircles, the circum- circles of the supplemental triangles (JJj, E^, i^s). (4) Just as in Plane Trigonometry [TBIG. 12 (5)] we have for the contacts of the incircle BZ) = s-h, CD = s-c, and that the contacts of the A excircle with AB, AC are distant s from A. Note. Hence tan r = sin (s-a) . tan ^ . [Cf. TRIG. 12 (2).] tan ri = sin s . tan -^ . [Cf. TRIG. 12 (3).] (5) If be the circumcentre Z OBC = S-A. (6) 2 tan R = cot Vi + cot r^ + cot r^ cot r. 8. Spherical Excess. (1) LexelVs (CagnoWs) Formula: sin -I E = h/(2 cos -X cos - cos -) (2) Lhuilier's Formula : s-c ~2 ' i&n^E = / tan ^ tan ^ tan -^- tan iWe. These correspond to A = ys {sa) (s h) (s c). The following may be added for reference : , E l + cosa + cos& + cosc - (3) Euler s Formula : cosjr = ^ ' 2 , a b c 4 cos -cos -cos - 9. Tetrahedra. (1) The geometry of a tetrahedron is often best dealt with by regarding it as in scribed in a parallelepiped as shown. The essential rule in drawing such a figure is that each edge of the tetrahedron must be a face-diagonal of the parallelepiped. We have the Fig. o. ,!. Piiij followmg species oi tetrahedra : SPHERICAL TRIGONOMETRY 97 Tetrahedron. Parallelepiped. Isosceles (each edge = oppo- Rectangular. site edge). Orthogonal (each edge -L^* to All edges equal, all faces opposite edge). rhombi. Regular (all edges equal). Cube. (2) The lines joining middle points of opposite edges intersect in a point, bisect each other, and in isosceles tetrahedra are mutually perpendicular. (3) Any plane through the centres of a pair of opposite edges divides the tetrahedron into two parts whose volumes are equal. (4) The volume of the tetrahedron is one-third the volume of the generating parallelepiped. (5) Isosceles Tetrahedra. The centroids of surface and volume coin- cide with one another and with the circumcentre, and lie on the joins of the centres of opposite edges. (6) Orihogonal Tetrahedra are the most directly analogous to plane triangles. In them alone an orthocentre {IT) exists ; the orthocentre, centroid, and circumcentre are collinear ; and there is a Twelve Point Sphere (analogous to the N. P. C.) which passes through the centroids of the faces, the orthocentres of the faces, and through four points of trisection of the joins of H to the four vertices. The radius of this sphere is radius of circurasphere, and the centroid and orthocentre are the centres of similitude of these spheres. (7) Regular Tetrahedra. If a be an edge, the join of the centres of opposite edges is ^a\^2, the circumradius is ^a^/Qf the inradius I a \/2> the volume -^^ *' V^- The faces intersect at Z cos~i ^. The various centres (orthocentre, circumcentre, &c.) coincide at the centre of the generating cube. 10. Parallelepipeds. If a, h, c be edges intersecting at a point and oc = Abe, /3 = ^ca, y = L ah, Diagonal through the point = Va^ + b^ + c^ + 2bccos(X + 2ca cos /3 + 2ab cos y. Volume = abc \^1 - cos^a - cos^yS - cos^y + 2 cos oc cos /^ cos y. 1372 H 98 SPHERICAL TRIGONOMETRY Proof. Let diagonal make angles A, n, v with edges. Project the diagonal and the broken line reaching from end to end of diagonal via the edges (1 ) on the diagonal, D = a cos A + & cos ju + c cos v. (2) on the edges in succession, DcosX = a + ?> cos 74-c cos j8 and two similar equations. Eliminate A, /*, r, &c. 11. POLYHEDRA. (1) Eiilefs Theorem. If S be the number of solid angles in any polyhedron, F the number of its faces, E of its edges, S + F = E-\-2. (2) Regular or Platonic Solids. There are only 5 : Tetrahedron made by 4 equilateral triangles. Cube 6 squares. * Octahedron ,, 8 equilateral triangles. Dodecahedron ,, ,,12 regular pentagons. Icosahedron ,, 20 equilateral triangles. [See MENS. 21.] (3) Archimedean Solids. These have all their angles equal, and their faces regular polygons, which are not of the same species. They are 13 in number. Truncated Tetrahedron : 4 triangles, 4 hexagons ; 2 hexagons and 1 triangle at each of 12 vertices. Cuboctahedron : 8 triangles, 6 squares ; 2 triangles and 2 squares at each of 12 vertices. Truncated Cube : 8 triangles, 6 octagons ; 2 octagons and 1 triangle at each of 24 vertices. Truncated Octahedron : 6 squares, 8 hexagons ; 2 hexagons and 1 square at each of 24 vertices. Small Rhombicuboctahedron : 8 triangles, 6 squares, 12 pentagons. Great Rhombicuboctahedron : 8 triangles, 6 squares, 12 decagons. Icosidodecahedron : 20 triangles, 12 pentagons. Truncated Icosahedron : 20 hexagons, 12 pentagons. Truncated Dodecahedron : 20 triangles, 12 decagons. Snub Cube : 32 triangles, 6 squares. Small Rhombicosidodecahe- 20 triangles, 30 squares, 12 pentagons. dron : Great Rhombicosidodecahe- 30 squares, 20 hexagons, 12 decagons. dron : Snub Dodecahedron : 80 triangles, 12 pentagons. (4) Kepler-Poinsot Solids. These correspond to the ' star-polygons ', of which the well-known ' pentagram ' is an example. They are four in number, 8 dodecahedrons and an icosahedron. GEOMETRY Point-Systems. 1. Eider's Theorem. (A, B, 0, 7) collinear.) AB . CD + AC ,BB + AD . BC ^ (i. Note. A particular case of Ptolemy's Theorem. 2. Stetvarfs Theorem, (A, B, C collinear, P any other point.) AP^. BC+ BP^ CA + CP\ AB = -BC.CA. AB. 3. The Centroid Theorem If G is the centroid of any masses m^, mg, ..., at points Ai, A^, ..., and Pis any point in their plane, 2 [mi . A^ P2] = 2 [mi . A^ G^] + [5m,] . PG\ Concurrence of Lines. 4. Ceva*s Theorem* If B, E, F are points on the sides of a triangle, AB, BE, CF concur if BD.CE.AF= BC.EA.FB, and conversely. Note. An equivalent condition is sin BAD . sin CBE. sin ACF = sin DAC sin EBA sin FCB. 5. If B, E, F are points on the sides of a triangle the -L^s at D, E, F concur if BB^ + CE^ + AF^ = BC^ + EA' + FB% and conversely. 6. If three lines through the vertices of a triangle concur so do also (a) their isogonal conjugates ; (&) their isotomic conjugates. H 2 100 GEOMETRY 7. Statical methods api^ly to many theorems involving middle points. Ex. The joins of m. pts, of opp. sides of a 4} and the join of the m. pts. of the two diags. concur. Collin EARiTY of Points. 8. Menelauss Tlieorem. If 7), E, F are points on the sides of a triangle they are coUinear if BD. CE, AF = -DC. EA , FB, and conversely. Note. The converse of this theorem (but not the direct theorem) is ti-ue for any n-gon, the sign being ( )". 9. PascaVs Tlieorem is sometimes of service [see GEOM. OOiV^. 33(1)]. Ex, ABC is a A inscd. in a circle ; D and F are m. pts. of arcs BC, BA] P is miy pt. on arc GA ; PD, BC meet in L and TE, AB in N. Prove that LN passes through the incentre. The Triangle. [Lettering of Triangle. The following lettering is adopted for points connected with A ABC-. D, E, F : feet of perpendiculars. ly, E', F' : m. pts. of sides. Di , El, F^: intersections with sides of bisectors of angles. jr,Y,Z: pts. of contact of inscribed circle. 0, I, H, G : circumcentre, incentre, orthocentre, centroid. n, n'; K, K : Brocard points, symmedian point, nine-point centre.] 10. General Properties. If the base BC is divided in 7? so that BB:CB = m:n, n AB^ + m.AC^ = n. BB^ + m . CB^ + {m + n) AB^, Notes. (1) This is a particular case of 3. (2) If i? be at m. pt. of J5C, AB^ + AC^ = 2AR^ + 2BR^. [ 12 (1).] GEOMETRY ioi 11. Isogonals. IfX, Y be any pts. on lines isogonally conjugate : (1) XM. YN = XP. Y(l and conversely. (2) MNPq is cyclic. (3) MP^ AY. (4) If three lines AX, BX, CX meet in a pt. their isogonals AY, BY, CY meet in the 'iso- gonal conj. point'. Note. The following are Isogonal Conjugate Point- Pairs : Orthocentre, Circumcentre ; Oentroid, Syinmedian Pt. ; In- centre, Incentre. Fig. 12. 27ie Medians and Oentroid (G). (1) AB'^ + AC^ = 2A1J''^ + 2BD'\ Hence AD' in terms of a, h, c. (2) D'G = iD'^, c^c. y A (8) A BGC = A CGA = /\AGB. Xi^ \ (4) ^D'E'r\\\^ABC. y /l^ \j (5) G^ is the centroid of wts. B 6 C 1, 1, 1 at A, B, C; also of Fig. 7. A D'E'F'. 1 1 (6) The distances of G from the sides are as ... ^ -' a b c (7) x^ + 2(^ + 2^ (areal co-ords. of P) is min. when P is at G. 10^ GEOMETRY Fig. 8. 13. Tlie Sifmmedianlil and Symmedian (Lemoine) Point (K). (1) The J.-symmedian bisects all antiparallels to BC. (2) If T is pole of BC w. r. O ABC, ATi^ the ^-sym- median. (3) The symmedian divides BC in ratio c^ : 1)'^. ABKC:ACKA:AAKB Note. Hence its length, by 10. (4) D'K bisects JL^- AD. Note. If A = 90,K bisects ' AD. (5) K is centroid of wts. a% h% c2 at A, B, C. (6) Distances of K from the sides are in ratio a:h :c, and K is the centroid of the feet of these distances. (7) 0(2 + y82 4 .y2 (trilinear co-ords. of P) is min. when P is at K. (8) If through K lines are drawn antiparallel to the sides of a triangle, the 6 points in which they meet the sides lie on a circle (Cosine circle) whose centre is K. (9) If through K lines are drawn || to the sides of A, the 6 pts. in which they meet the sides lie on a O (Triplicate ratio circle, or Lemoine circle), whose cent, is the m. pt. of OK. (0 is the circumcentre.) Note. The names are due to tlie fact that these circles cut off from the sides segments whose lengths are : Cosine Circle cos A : cos B : cos C. Lemoine Circle a^ : l^ : c^. 14. The Circumcircle. (1) 2R.p, = he. (2) If P is a pt. on the circumcircle and FL, P3I, PN GEOMETKY 103 J-is on the sides, L3INsive collinear (Pedal = Wallace = Sim- son Line). (3) If P and Q are on the circumcircle the Z between the pedal lines of P and Q = Z which PQ subtends on the circumcircle. (4) If H is the orthocentre the pedal line of P bisects HP in a pt. lying on N. P. C. (5) The pedal line of P is y to the isogonal conjugate of the join of P to any angular pt. [ 11 (3)]. 15. Inciicle and Excircles. (1) BD^ :CDi = c:b. Hence BD^, CD^. BDi . ODi + AD^^ - 1)C, Hence AD^. [Also by 10. J Similarly the ex- ternal bisector ^D/. (2) L is circum- centre of BICIi, M is circumcentre oiBI.fih, (3) 22)'iH=r2+r3, 2D'X = ri-r, .-. /i + r2 + r3-r=4i?. (4) l\AIZ 111 /\LB3L (5) 0P = B'^~2Rr, (6) If 2 circles with centres 0, I and radii i?, r are such that B^ - 2 Br = OP, an infinite number of triangles can be inscribed in the first circle whose sides touch the second. (7) AZ = s-a, AZ' = s. 104 GEOMETRY 16. Orthocentre, Pedal Triangle^ Polar Circle. (1) BFEC, CDFA, AEBB are cyclic. (2) EFh anti-li io BC, (3) A's AEF, BFB, CBE, ABC are similar. (4) The sides of the pedal A are a cos A, h cos B, c cos C and the angles 7r-2A, &c. (5) His the incentre, A, B, C the excentres of DEF. Fig. 10. (6) AH^^ iB^- (7) Polar Circle. H is the centre of the polar circle whose radius p is given by p^ = HA . HD. This circle (with regard to which A ABC is self-conjugate) is real only when one Z of the A is obtuse, e.g. in the figure the polar circle of BHC is real, its centre is A, its radius ^^AH,AD. (8) Taylor Circle. If LL' are the projections of EF on BC, MM' corresponding pts. on CA, NN' on AB, then LI/3IM'NN' lie on the Taylor Circle. l^-oo/. By Trig, prove BL . BL' = BN . BN' , .-. LL'NN' cyclic, .-. LL'MM'NN- cyclic ( 25 (2) Note). 17. Nine-Point Circle. (1) 0, G, H are collinear. (2) i\^ bisects 0^. (3) A circle with centre N and rad. ^ i? passes through the 9 points B,E,F; D\ E', F' ; A', B', C [the m. pts. of HA, HB, HC]. (4) G and r are the centres of similitude of this circle and the circumcircle. (5) Feuerhach's Theorem. The N. P. C. touches the in- scribed and the three escribed circles of a A. GEOMETRY 105 Notes. (1) It also touches the inscribed and three escribed circles of each of the A'aBHG, CHA, AHB, 16 circles in all. (2) To find W, the pt. of contact of N. P. C. and I. C. From D^ draw B^T tan. to I. C. Then B'T meets I. C. again in W. [For lettering see above, 10.] (3) If L is m. pt. of arc which BC cuts off from N. P. C, XL passes through W. So also YM, ZN. (4) W lies on the circumcircle of B^E^^F^, (5) With a proper choice of signs WB' Wtf WF' = 0, W being in this case the pt. of contact either with I. C. or with the excircles. (6) For proof see 38 (2). Also elegantly by Purser's Theorem [ 26 (10)]. Apply to A B'E'F'. B'X = | (&-vc), E'F' = | a, .-. 2 E'F' . B'X = 0, .'. B'E'F' touches I. C. See also AXAL. QEOM. 74, Note, and TRIG. 13 (2). (6) The circumcircle, N. P. C, and Polar Circle are coaxal. 18. Brocard Points. (1) The marked angles are all equal [ = the Brocard Angle o)]. (2) cot CO = cot u4. + cot B + cot C. (3) To construct 12 notethatOl2J. Ctouches AB at A, O ilBC touches AC at C. (4) Ail.Bil.Cil = An\Bil\ail\ (5) " are centres ^'^- H- of similitude for ABC and an equiangular inscribed A a^y, y (X B the angles equal to A, B, C being ? "^. Note. If the angles correspond in the order ocPy the circumcentre of ABC is the C. S. for any 2 pes'" of oifiy and this circumcentre is the orthocentre of ai37. See 19 (5). 106 GEOMETKY (6) mr v OK, (7) 2, 12' lie on the circle on OK as diameter [Brocard circle], which also passes through the pts. {AH., 12') (12, (7120 ((712, ^120- Two Triangles. 10. (1) Desargues" Theorem. When two triangles are in perspective (copolar, homologous) the corresponding sides intersect in three collinear points (axis of perspective, homo- logy) and conversely. (2) If ABC, A'B'C be two A's in perspective, and if B'C cuts A G, AB in the points X\ X'' respectively, &c., AT BY' CZ^_AX^ BT' Czr_ QX! ' AT ' BZ' ~ BX" ' CY'' ' AZ^'' and conversely. Note. To make the lettering clear note that JC' = (B'C', CA),r = {C'A',AB), Z' = {AlB', EC). X" = (iJV, AB\ Y" = (C^', BC\ Z" = {A'B\ CA). (3) Triangles directly similar. If ABC, A'B'C be two directly similar triangles, the centre of similitude is the other intersection of the O's BXB', CXC\ vvhere X is intersection of BC, BX\ (4) Triangles inverse- ly similar. If ABC, A^B^Ci ^ *wo tri- angles inversely simi- lar, the axis of simili- tude which bisects angles AOA^, BOB^, COCi is found by divid- ing AA^, BB^, CCi_ in the ratio of similitude, and can be easily found by finding C\ Fig. 12. GEOMETKY 107 (5) The 'Pivot Pomt\ If a APQIi be inscribed in a A ABCf the O's AQli, BEP, CPQ pass through a common point at which BC subtends an Z (J. + P), &c., QE /(tt-^), &c., so that if ABC, PQE be given in species, is a fixed pt. w. r. either A, all positions of PQE are directly similar, and is the C. S. Note. This very appropriate name, ' Pivot Point,' has been recently suggested by Mr. W. Gallatly for what other writers vaguely describe as the ' Point-0 Theorem '. PQR ill pedal triangle of 0. Harmonic Kanges and Pencils. 20. (1) Bef. If C and I) divide AB internally and ex- ternally in the same ratio, {ACBB} is a Harmonic Eange, Notes. (1) The ratio is sometimes given as {AB, CD) and sometimes, unfortunately, as (ABCD). (2) Note the particular cases (a) Z at oo , AC = CB, (6) A, B, C coincident ; D anywhere. (3) The symbol [ACBD] is also used for the numerical value of the Anharmonic Ratio. See 31 (1). (2) If be m. pt. of AB, OC. 01) = 0A\ OC:OD = AC^:A1)\ (3) If {AB, CD) is harmonic (CD, AB) is harmonic. J _ J_ 1 ^ ^1_ 1 ^ - AB AC "^ AD' BA BC ^ BD' 2 X J ^ = J_ _i- CD~ CA^" CB ' DC~ DA'^ DB' Kofe. Sign must be carefully attended to. (5) \ACBD] = (AG/CB) . [BD/DA) = - 1. (6) For any Harmonic Pencil {0 . ACBD] we have sin AOC sin BOD _ mnCOB' ^iu DO A (7) Any transversal cuts a H. P. in a H. K. 108 GEOMETKY (8) The internal and external bisectors of an angle form a H. P. with the arms of the angle, and, conversely, if two conjugate rays of a H. P. be at rt. angles they bisect the angles between the other two. (9) Given 3 rays of a H. P., to construct the fourth : through Con the middle of the 3 rays draw ACB bisected at ; a line through II to this is the required fourth ray. i^o/e. Another const, by 23 (2). 21. Involution. Def. A series of pts. PP', QQ\ ... on a line such that a point exists for which OP. OP' = Oq. OQ' = OE. OR' = ... = A;2 constitutes an involution of which is the centre and two points E and F such that OE = 0F= h the foci or douhle pts. (1) The foci, EF, of an involution constitute a H. R. with every pair of points belonging to it, e.g. [PEP'F] = - 1. (2) Given 2 point-pairs of an involution, to determine foci : Method i. Through PP' draw any O, through QQ' any O, let E. A. meet range in ; OF is the length of tan. from to either circle. Method ii. On Pq describe any A PqB, on P'Q' make P'q'R' III Pqn. RIV meets range in 0. OF is a mean pro- portional between OP and 0P\ (3) If {PP', QQ', RR'] be in involution : (a) PQ' . QR' . RP' + P'Q . Q'R . R'P = U, and conversely. , PQ-PQ' PR -PR' , , ^^^ q7Fq' = WrTFr'^ ^"^^ conversely. (4) If {PP", QQ', RR^} be a pencil in involution : (a) sin POQ' . sin QOR^ . sin ROP' + sin P'OQ . sin Q'OR . sin R'OP = 0, and conversely. , sin POQ . sin POQ' sin POR . sin POR' '^^ sin P-QQ. sin FOQ- = l inP'OR. sin POR' ' ""^ conversely. Harmonic Properties of Circle, Pole, and Polar, 22. (1) Defs. Inverse Points. P and P' on radius of circle such that OP,OP'= rad.-^ GEOMETRY 109 Conjugate Points. Polar of each passes through the other. Conjugate Lines. Pole of each lies on the other. Notes. (1) Inverse pts. are conjugate. (2) The inv. of a pt. w. r. a st. line is its image in the line. (2) If K and K^ are inverse pts. and F be any pt. on the circle, KP:K'P= const. Conversely, if K and IC are_ fixed pts. and KP : ICP = const., the locus of P is a circle (Circle of ApoUonius) the extremities of whose diameter divide KK^ in the given ratio. (3) Any circle which passes through a pair of inverse pts. of another circle cuts that circle orthogonally. (4) The line joining any pair of conjugate points is cut harmonically by the circle. (5) If the polar of P passes through Q, the polar of Q passes through P. If the pole of p lies on q, the pole of q lies on^. (6) If the polars of P and Q intersect in R, the polar of R is PQ. If the poles of j) and q lie on r, the pole of r is pq. Note. Hence if P, Q be conj. pts. A PQR is self-conjugate and its orthocentre is at the centre of the circle. (7) Salmon's Tfieorem. The distances of any two pts. P and Q from the centre of a circle are proportional to the distances of each from the polar of the other. Quadrilaterals. /i^ T^ ^ A ^ J. quadrilateral (tetragram) . ^ 23. (1) J)ef. A coniplete ^^,^^,, (tetrastigm) '' ^^"J - . lines , sides , . , meet in ^ points by 4 . , called ^. which . . , , 6 ,. points vertices are joined by lines vertices forming 3 pairs of opposite . , , , ^ , which pairs sides (connectors) ^ can be joined by _ , , lines ,, , diagonals (harmonic . . , . 3 other . , called ,. ^ , \ ,, intersect in points diagonal pts. (har- 110 GEOMETRY lines) ^ . ^ i> , , sides . , , harmonic (dia- . . which lorm the ,. or the , . , ,. , momc pts.) vertices harmomc {amgonai, gonal) , . , ^ , , quadrilateral. ^ , ,, triangle of the ^ central) quadrangle. (2) In a complete . , each pair of opposite quadrangle vertices . range .,, , o .^ . , lorms a harmonic ., with two oi the sides pencil * . , of the diagonal triangle. (3) The middle pts. of the 3 diagonals of a complete quadrilateral are collinear. Note. This Hne is sometimes called the ' diameter ' of the 4*. (4) The circumcircles of the 4 A's formed by any 4 lines pass through a common point. (5) The orthocentres of the 4 A*s formed by any 4 lines are collinear. Note. Prove by 14 (4). This line is perpendicular to the ' dia- meter ' of the 4'. (6) The circles described on the diagonals of a 4^ as diameters are coaxal and their R. A. is the line given by (5). Note. The circle on the 3rd diagl. is the Circle of Similitude of those described on the other two. (7) The three pairs of lines which join any point to the opposite vertices of a 4^ are in involution. (8) Any st. line is cut in involution by the three pairs of opposite connectors of any quadrangle. (d) li A^ JB, C, D are any 4 pts. in a plane AB.CD + AD.BO AC, BD unless A, B, C, D lie in the order ABCD on a circle or a st. line (in which case we have an equality). Notes. {1) Cf. 1. (2) A, B, C, D need not be in one plane. GEOMETRY 111 Cyclic Quadkilaterals. ^^ ,,,,, , ... , . quadrilateral circum- 24. (1) The harmonic triangle oi a j i . ^ ' ^ quadrangle in- scribed about . , . , . L I M! ^ \ ., , . a circle is seli-coniugate (seli-polar). scribed in (2) The harmonic triangle of a quadrilateral circum- scribed to a circle coincides with the harmonic triangle of the quadrangle formed by the pts. of contact. This includes the following propositions : () The internal diagls. of an escd. 4^ and of the corresponding inscd. meet in a point. {b) The third diagls. of an escd. 4^ and of the corre- sponding inscd. are collinear. (c) Any pair of opposite sides of an inscribed 4^ intersect on a diagonal of the escd. 4^ (3) The ' diameter ' [ 23 (3)] of a circumscribed 4^ passes through the centre of the circle. (4) Ftolemy's Theorem. ac+l)d = xij. [See 1 and 23 (9).] {ac + bcl) {he + ad) (5) x' = f = ah + cd {ac + bd) {ah + cd) bc + ad (6) Inscribed Quadrilateral A + C=> B + D. Escribed Quadrilateral a + c = b + d. Fig. 13. (7) For a cyclic quadrangle any straight line is cut in involution not merely by the three pairs of opposite con- nectors [ 23 (8)], but also by the circle itself. (Desargues' Theorem.) (8) For a circumcyclic quadrilateral the lines connecting any pt. with the vertices form a pencil in involution [ 23 112 GEOMETEY (7)] and the pair of tangents from the pt. also belong to the pencil. (9) If a 4^ can be inscribed in a O of radius 2?, and at the same time circumscribed to a O of radius r, and if d be the distance of their centres, (a) (22+ e?)-2 + (B-d)-'^ = r-2 [and conversely]. (6) An infinite number of such 4^^ can be described. Circles. 25. Badical Axis. (1) If the K. A. of 2 O's, centres A, B, radii a, h, cut AB in 0,AO^-BO'' = a''-b\ (2) The R A. of 3 0's, taken 2 and 2, intersect in a common point (The Radical Centre). Note. Hence if A^A^, B^B^, C^C^ be on the sides of a A and B^B.^C^C^, C^Cj^i^a, A1A2B1B2 be cyclic, then all 6 pts. lie on one and same circle. Def. The Radical Circle has centre at Rad. Centre and cuts all circles orthogonally. (3) Difference of squares of tangents from any point P = 2PJV. AB, where PN = X^ from P on RA. Note. The following are particular cases of R.A. R. A. the Line Infinity, L. P. the Circular Points. R. A. tan. at pt. of cont. with which both L. P. coincide. R. A. the line. To find L. P. use 26 (1). R. A. bisects ^ dist. between pt. and inverse, which are the L. P. R. A. the line ; L. P. the pt. and its imago in the line. Concentric Circles. Touching Circles. Line and Circle. Point and Circle. Line and Point. 26. Coaxal Circles. (1) If O's, centre A, rad. a; B,h; C, c... are coaxal, then ABC... are collinear and OA^-a^ = OB^-h'' = OC^-c^ = ... = OL^ = OL'^ where is the pt. where the K.A. cuts the line of centres, and L,L' are 2 pts. (Limiting Points) lying on opposite sides of on the line of centres. GEOMETKY 113 Note. The relation BC.a^+CA .b'^ + AB . c^ ^ -BC.CA. AB is some- times useful. (2) If a line cut one circle of the system in PP\ another in QQ\ PQ, P'Q' subtend at either L. P. angles which are equal or supplementary. (3) A common tangent to 2 circles of the system sub- tends a rt. Z at X and L\ (4) The L. P. are inverse w. r. each G of the system. (5) Any try. cuts a coaxal system in involution. l^ote. If the trv. be line of centres, the foci of the involution aie the L. P. (6) The system of O's through LV themselves form a coaxal system (whose L. P. are imaginary if LV are real) the members of which cut orthogonally every member of the given system, and have their centres on the E. A. (7) If a, p, y are 3 circles of a coaxal system, the tangents from any pt. of a to y8 and to y are in a given ratio. Conversely : If a point moves so that its tangents to 2 circles p and y are in a given ratio, its locus is a circle a coaxal with y8 and y. (8) Note the particular case of (7) in which one of tlie circles becomes a L. P., e.g. in the figure PT/PA = const, as P moves on the outer circle. (9) The polars of a fixed pt. w. r. the O's of a coaxal system ^ Fig. 14. are concurrent (10) Purser's Theorem. From the vertices of a A ABO tans. AP, BQ, CR are drawn to a circle ; if the sum of 2 of the rectangles BC . AP, CA . BQ, AB.GR is equal to the third, the given O touches the G ABC. 114 GEOMETRY Similitude. 27. (1) Defs. Centres of Similitude are 2 pts. S, S^ dividing line of centres in ratio of radii. Circle of Similitude is circle on SS' as diameter. Circle of Antisimilitude, circle centre S (extl. C. S.), rad. VST. ST\ There is also an internal centre and circle of antisimilitude. Notes. (1) Lachlan reserves name C. S, for any pt. on circle of similitude and calls SS' Homothetic Centres. (2) Particular Cases of C. S. : Two Tangent Circles. Circle and Line. Circle and Point. Line and Point. One C. S. at pt. of contact. The two ends of diam. 1.^ to line. Coincide at the pt. Coincide at the pt. (2) If in 2 circles || radii are drawn, the line joining their extremities passes through a C. S. (3) IfST'The a com. tan., and SQ'P'PQ a trv. cutting circles in Q\ P', and P, Q, SP, SP' = SQ . SQ' = ST. ST\ (4) If Sq'p'pq be a second trv., Pp, P'p' intersect on the R. A. ; so PT, P'T where STT is com. tan. (5) Every common tangent passes through a C. S. (6) The 6 C. S. of 3 O's taken in pairs lie in threes on 4 lines called Axes of Similitude. (7) If a circle touches 2 given circles, the line joining the pts. of contact passes through a C. S. of the two circles. Note. A particular case of (6). (8) Two circles are similar w. r. any pt. on their circle of similitude, and conversely any pt. w. r. which 2 circles are similar lies on the circle of similitude. (9) The tangents to 2 O's from any pt. on their circle of similitude are in the ratio of the radii. Hence : (a) The two circles subtend the same angle at any pt. on the circle of similitude. (6) The circle of similitude is coaxal with the given circles. GEOMETRY 115 (10) The three circles of similitude of three given circles are coaxal. (11) The circles of antisimilitude are also [cf. (9) (b)] coaxal with given circles. (12) The tangents to 2 circles from any pt. on their circles of antisimilitude are in the ratio of sq. roots of radii. Loci. 28. (1) A and i> are hvo fixed pts. , the locus of P, when (a) LAPB = const., is 2 arcs of O's through A and B. (h) PA^ + PB'^ = const., is O, cent. m. pt. AB. (c) PA^-PB^ = const, is st. line r AB. {(l) PA : PB = const., is O dividing AB int. and ext. in ratio. (e) m . PA^n . PB^ = const., is O. [See 3.] (2) A,B, C, three fixed collinear pts., the locus of P, when (a) /.APB = IBPC, is circle through B dividing AC ext. in ratio AB/BC. (h) l.PA^m.PB^n.PG^ = const., is O. [See 3.] (3) A, B, C, D are four fixed collinear pts., the locus of P, when (a) lAPB = Z GPD, is O on EF ii^ diam., where E,F are foci of involution (AB, BC). [See 21 (2), (1).] {h)l.PA'm,PB''-n.PG'^r.PD^== Qijxist, is O. (4) A J B, C, B are four fixed coplanar pts., the locus of P, when (a) A PABA PCD is const., is st. line. {b) m . A PABn A PCD is const,, is st. line. (c) A P^li>* : A PCD is const., is st. line through inters, of AB, CD. [d) I . PA^ m . PB- n . PC'^ r . PD^ is const., is circle. I 2 116 GEOMETRY (5) If P3I, PN be drawn in fixed directions to meet two given lines, the locus of P, when {a) PMiPN const., is st. line through the inters, of given lines. (6) m . PM\ n . PN = const., is st. line. (6) If X, y, z, ... be X^'s from P (or drawn from P in any given directions) to a series of st. lines, the locus of P is a st. line if axhi/C0... is const., where a, h, c, ... are any constants. (7) If the line joining a fixed pt. to a variable pt. on a circle be divided in P in a fixed ratio, the locus of P is a O and is the C. S. of the two. (8) Locus of P, when tans, to 2 given circles are in a fixed ratio, is O coaxal with given circles. [ 26 (7). J (9) Locus of P, when tan. to given circle is in fixed ratio to distance from a given pt., is O coaxal with O and pt. Note. Hence, easily, if variable chord of O subtend rt. Z at fixed pt., find locus of (a) its m. pt. or (&) foot of -L"^ from fixed pt. upon it. (10) Locus of P, when sq. of tan. to given circle oc J.'' from P to given line, is O coaxal with circle and line. Envelopes. 29. (1) Poncelefs Theorem: If a polygon inscd. in a G changes continuously so that each side but the last con- tinues to touch a fixed circle of a system coaxal with the given O, then the last side also continues to touch a fixed G of the system. (2) BoUllier's Theorem: If the A ABC moves so that AB, AC touch fixed G's, then BG also touches a fixed G. Note. Can be investigated by (4) (6), below. (3) An envelope of a chord of a circle can sometimes be GEOMETRY 117 determined by proving it to subtend a fixed angle at the circumference and thus to envelop a concentric circle. (4) An envelope can often in one of two ways be reduced to a locus : {a) Infinitesimals. Two consecutive positions of the moving line intersect in a point on the envelope, The locus of this point is the envelope. (b) Instantaneous Centres. By this method a problem in envelopes can often be turned into one of loci. Ex. ABj a line of fixed length, slides between two fixed trammels OA, OB. A is moving along OA, .*. turning instantaneously about some pt. / in 7^ X"" 0^. So is turning about I in JSIX"" 05. Hence AB is turning about I as ' instantaneous centre '. The envelope is the locus of N, the foot of X"" from I on AB. This is not, however, one of the simple curves. Maxima and Minima. [See also 39 (7), (8).] 30. (1) Given any 2 fixed pts. A, B, and any locus, to find the pt. G on the locus when (a) Area ABC max. or min. (h) I ABC (c) AC+BC (d) AC-BC (e) AG' + BC\, Tan. at C || AB. Draw O thr. AB touching locus. Tan. at C bisects Z ACB ext. M V J7 jj int. O centre m. p. AB touches locus ate. (2) Fermat's Problem. BA ^- PB + PC is min. when P is the pt. at which all the sides subtend equal angles. (3) For theorems involving 2(mi . A^ P^) where Wj, m^, m^, ... are any constants, ^i, A^, ... series of fixed pts., see 3 ; e.g. PA^ + PB^ + PC^ is min. when P is at centroid. (4) Of all curves of given length the circle encloses greatest area. (5) Of all curves of given area the circle has least perimeter. 118 GEOMETRY (6) Given number of sides and area of a polygon, peri- meter is min. if polygon is equilateral. (7) Given number of sides and perimeter of a polygon, area is max. if polygon is equilateral. (8) Given number of sides of a cyclic polygon, perimeter and area are max. if polygon is equilateral. (9) Given number of sides of a circumcyclic polygon, perimeter and area are min. if polygon is equilateral. (10) A convex polygon whose sides are given contains max. area when cyclic. (11) If a st. line be divided into any number of parts, the continued product of all the parts is max. and the sum of their squares is min. when the parts are all equal. Note. The Method of Infinitesimals is by far the most powerful inothnd available. Two examples are added. Exx. (1) Given any angle and any curve convex to this angle, then the ian. to the curve which forms with the sides of the angle a min. A is bisected at the point. (2) Draw a line through P meeting any 2 Q's in A and B and such that AP.PB is min. Take consecutive position, .-. AP. PB = A^P . PB', .'. AA'BB' cyclic, .*. we have to draw G touching O's in A and B and such that APB is st. line, .-. [ 27 (7)] join C. S, of G's to P meeting O's in A and B. Anharmonics. [See A. GE03I. 66.] 31. (1) Notation. A. R. of A, B, C, D= \ABCD} = (AC, BJD) = (AB/BC). (CD/DA) = A. A. R. of points arranged in other orders: 1/A, 1-A, 1/(1_A), (A-l)/A, A/(A-1). For pencil A. R. = [0. ABCB} = (sin AOB /sin BOC) . (sin OOD/sin DO A). (2) {ABCB} = {O.ABCB}. GEOMETRY 119 (3) To measure [O.ABCD] numerically, draw any line OJD cutting OA, OB, OG in X, M, N, then {0. ABCB} = -LM/MN. (5) U {O.ABCD}- = {0' .A'B'CD'} and (OA, O'A') (OB, O'B') (00, OX') lie on a line, Then (OB, O'B') also lies on it. Cor, li{O.ABCB] = {O'.A'BVB'} and OAAX' is a st. line, then (OB, O'B') (OC, O'C) (OB, O'B') lie on a st. line. (7) If a, h, c, d are fixed tans, to a circle (conic) and 2^ a variable tan. to it, then {(pa) (ph) (pc) (pd)} = const. (9) BriancJton's Theorem : If a, h, c, d, e, f are 6 tans, to a circle (conic), then the lines (ah, de) (be, ef) (cd,fa) pass through a pt. Proof. By (7) and (5) above. Notes. (1) See also G. CON. 33, and A. GEOM. 65(2). (2) Note the simplest case of Pascal's Theorem when A, B, C lie on one line and D, E, F on another. This was known to Euclid. 32. HomograpMc Manges. [See G. CON. 32, A. GEOM. 67.] (There is also a reciprocal theory of nomographic Pencils, &c., which can be readily constructed from the theory for ranges and is not given here.) (1) Befinitions. Any two ranges on the same or diff. lines are homographic when A. R of any 4 points = A. R. of coiTesponding 4 points. (4) If [ABCB] = {A'B'C'B'} and AA\ BB', CC pass through a pt., Then BB' also i)asses through it. Cor. li\ABCB} = [AB'C'B'} then BB', CC, BB', are con- current. (6) If A, B, C, B are fixed pts. on a circle (conic) and P a variable pt. on it, then {P. ABCB) = const. (8) Pascal's Theorem: If A, B, C, B, E, F are 6 pts. on a circle (conic), then the pts. (AB, BE) (BC, EF) [GB, FA) lie on a st. line. Proof. By (6) and (4) above. 120 GEOMETRY If [ABC...] (i)= {A'B'C\..}, (ii) and is pt. on (i) which corresponds to pt. at oo on (ii), 0' pt. on (ii) corre- sponding to pt. at 00 on (i), then {AB, oo ) = (A^B', co 0') ; .-. AO.A'O' = BO.B'O' = PO.P'0' and 0, 0' are called Centres (Lachlan) or Vanishing Points (Russell) of the ranges. If (i) and (ii) are on the same line, there will be 2 pts. of one range which coincide with corresponding pts. of the other. These are Double Points (Lachlan), Common Points (Russell), and are given by OS . O'S = OA . O'A'. (2) Constimction of Homographic Bangc. Given a range on one line, to construct a range homographic with it (a) on a second line, (&) on the same line. (a) Let [ABC... } be the given range. Take any three pts. A', B', C' arbitrarily on the second line. The join of {AB', A'B) {AC, A'C) which also passes through {BC, B'G) [see 31 (8) note 2] is the Homo- graphic Axis of the two ranges. Join A'D, meeting the Homograpliic Axis in 5, then A 5 meets second range in required point D'. To find the Centres, draw through A' || AB, meeting Homographic Axis in /, then AI passes through 0', &c. The construction of {h) is also available. (6) Join any pt. V outside the line to A, B, C.... Cut this pencil by any trv. whatever. Join any other pt. V' to the pts. thus obtained. The new pencil cuts the original line in a system homographic with [ABC...]. (3) Construction of Common Points. Let [ABC ...] = [A'B'C' ...], the ranges lying on the same st. line. To construct their common pts. take any O, let p be any pt. on it. Join pA, pB, pC, . . . pA' . . . meeting O in a,b, c, ...a',.... Construct Homographic Axis of [ahc.,.] {a'6V...} [see (4) below]. Let this meet O in II, H', ihen pH, pH' meet the line of the original ranges in their common points. (4) li[ABC..,} = M'JS'O^..}, the points lying on a circle (conic), then the pts. (AB\ A'B),,. all lie on a st. line, the Homographic Axis (Axis of Homology). (5) If through a pt. V outside a circle (conic) a pencil of rays be drawn cutting it in A, A' ; B, B' ; ..., then [ABC...} = [A'B'e...], the Homographic Axis being the polar of F. Proo/.^ 24 (1). GEOMETRY 121 Inversion. [See also BIFF. CALC. 21 (2). 33. Inverse Terms \ anywhere]. Term. (1) St. line through pole 0. (2) not through 0. (3) Two II lines. (4) Curves inters^ in P, Q, B. (5) touching at P. (6) inters^ at Z 0, (7) Circle through 0. (8) not through 0. (9) R.A. of 2 O's. (10) C. S. of 2 O's. Inverse. Itself. Circle through 0. Such that orig. st. line is R.A. of the circle and the circle of inv. Circles touching at 0. Curves inters^ in inv. points^, 2, r. Curves touching at inv. point x^. Curves inters^ at Z - ^. St. line J.I' diam. through 0. Together with line at oo . Circle. a C. S. of the two ; the inv. of w. r. given G inverts w. r. into cent, of inv. O. If d and r be dist. of cent, from and rad. of O, then for inv. O d' = 222d/(d!2-r2) / = i2V/(rf2_r2). Circle thr. Oand int. of G*s. Intersections of O's which touch the 2 inverses and pass through the pole. Inverse figures retain same relation. (11) A circle and a pair of points (or figures) each inv. of other w. r. O. 34. Inversion of Magnitude Relations : ._, Square of common tan. of 2 circles . i, , (2) i =r I -^ tt: IS unaltered. Product or radii 122 GEOMETRY (3) If fi, ^2 1>G tans, from ext. pt. to O's radii r-^, t\, then (^iV^i) ih^/^'i) is unaltered. (4) A. R. of 4 pts. is unaltered if be on the line of the 4. (5) If a range he inverted w. r. a pt. not on it, it becomes a homographic circular range. 35. Special Inverses, Figure. Inverse. Method. (1) Any circle. (a) Itself. R = tan. from 0. (b) Circle with cent, at inv. of pt. A. / Take at inv. of ^ w. r. O. -! See 33 (8). If R^OA i cent. inv. O is at A. (c) Any other circle. at C.S. of the two. (2) Any 2 o's. (a) Themselves. ( anywhere on R. A. , 1 R = tan. from to O's. {b) St. lines. a pt. of intersection. (c) Circles of given j anywhere on a particu- i lar O coaxal with given sizes. ( circles. See 33 (8). [Particular case of (c) ; J on either circle of anti- {d) Equal circles. similitude. (3) Any 2 o's which !| st. lines. pt. of contact. touch. (4) Any 2 O's ortho- O and its diameter. on either circumf. gonal. (5) Any 3 o's. (a) Themselves. JO at rad. cent., R = tan. ( from to any O of 3. (6) o's with centres on radical circle. collinear. (c) o*s of given size. 1 at intersections of O's [ given by 2 (c) (d) above. (d) Equal circles. (6) Coaxal circles. (a) Themselves. Cf. (2) (a) above. (b) Concurrent lines. at either inters, of O's. (c) Concentric circles. jO L. P. See 26 i (6). (d) Coaxal circles. anywhere ; the L. P. inv. into the L. P. \ J (7) Circle and 2 pts. St. line, pt., image of on the circle. See also inverse w. r. it. pt. 83(11). GEOMETRY 123 36. Inverses of Triangles and Quadrangles. Figure. Inverse. (1) Vertices of any I () Verticesofequil. A. triangle, ABC. (b) ., A given species, A'B'C Three equal circles. (2) Three sides of any triangle. (3) Vertices of any Vertices of quadrangle. \ Vertices of cyclic | quadrangle. ! 37. Linkages. (1) Peaticellier. A BCD a rhombus, OA = OC, fixed. B and D describe inverse curves. rectangle. P?. Fig. 16. Method. (Take inside A ABC; AA'B'B cyclic, &c. Do- (i\i, q, r be radii. Take any A ABC of species. Use 28 (1) {d) to determine so that OA :OB:OC--=p:q:r. Proportionate reduction. (5) To describe A ABC given in species zvith vertices on three concurrent lines, and (a) one side of given length, (b) one side through given point, (a) Triangle is given in magnitude. Use 28 (1) (a). (b) Take any A of species. Use 28 (1) (a) to draw three con- current lines through vertices similar to given lines. Through given pt. draw II to one side of A, &c. (6) To inscribe (or escribe) given A PQR to given A ABC. To inscribe, determine 'pivot-point' [ 19 (5)] for both triangles. With ^jBC-point-0 as cent., rad. OP, describe O cutting BC in P, &c. (7) To inscribe minimum A of given species in ABC. Determine < pivot-point ' [ 19 (5)]. Draw OP, OQ, OR J.' sides. (8) To circumscribe maximum A of given species about ABC. Determine ' pivot-point' [ 19 (5)]. Draw QR X"" OA thr. A, &c. (9) To inscribe in ABC a A PQR given in species and (a) one (all) side II given line (lines), (b) one side, QB, through given point X. (a) Draw any line (/R' \\ given line. Make P^Q^R' ill ABC. Join AP' meeting BC in P, &c. (h) Determine * pivot-point ' [ 19 (5)]. 0R2[ is known. Hence R [28(1) (a)]. 126 GEOMETEY (10) To inscribe in ABO a A FQIi whose sides pass through three given points a, /3, y. Through a draw any line QLQR meeting AG, AB in Q, R. Join yQ, 0R meeting BC in P/, Pj. For a series of points so determined we have {P^} = {r} = {g} = {p/}. The common points [ 32 (1)] of these homographic ranges determine P. (11) To circumscribe about ABC a A PQIi whose vertices lie on any three given lines. This is the same problem as (10). Note.^yiyi. (10), (11) may be written : ^ Given any 2 A's, to describe a A lohich is circumscribed to one and inscribed to the other.^ (12) Castillon's Problem. To inscribe in a given circle (conic) a A ivhose sides (3) pass through three given pts., {h)pass two through given pts., the third II given line. (a) If the A in solution to (10) be replaced by a O, then using 32 (5) tlie solution of (10) holds throughout. (6) Is a particular case of (a) with third pt. at 00 . Note. Elementary solutions are possible (Smith and Bryant, Euclid, p. 408, &c.). The solution given above applies to an n-gon whose sides pass through n-points. (13) To circumscribe about a given circle a A whose sides lie on three given lines. Let p, q, r be the poles of the lines ; inscribe A whose sides pass through p, q, r, &c. (14) To inscribe in ABC a A PQE whose sides subtend given angles oc, jS, y at three given points L, M, K Draw any line LQi_ to AC. Make Z QiXEj == OC, Ri on AB ; RiMPj_ = 0, Pj on BC ; and Q1NP2 = 7, P^ on BG. Prove that, for various positions of Qi, Pi and Pa describe homographic ranges, the common points of which determine P. Similar solution for polygons with n points and n angles. (15) To inscribe in a A ABC a rectangle of given area, A solution using a hyperbola is easy. Otherwise take any point Pi in AC, draw P^N^ \\ BC to AB. Find P^M^ || AB, P^ on AC, M^ on BG, and such that PiN^. P-^M^ = given area/sin B. Pi, P, describe homo- graphic ranges whose common points give P, the vertex of the rectangle which lies on AG. GEOMETEY 127 Examples (10)-(15) illustrate the Method of False Posi- tions or Method of Trial and Error, perhaps the most powerful method for geometrical problems which exists. Problems on Quadrilaterals. 40. (1) To describe a square about a given quadrilateral ABGD. Draw AX perpendicular and equal to BD, CX is the direction of a side of the square. See also (2) below. (2) To describe a quadrilateral of given shape about a given quadrilateral ABGD. Let it be KLMN, the points being in order KBLGMDNA. Let O KAB cut liM in R, I ABR = I NKM and is known, so Z BAR. Hence E is known. Similarly S by O DCM. From RS reconstruct figure. (3) To inscribe a quadrilateral of given shape in given quadrilateral ABGD. About a 4} of given shape describe a 4} similar to ABCD. Pro- portional reduction. (4) To construct a cyclic quadrilateral with given sides a, b, c, d. Suppose these sides represent AB, BC, CD, DA and that 4' is con- structed. Make I DCE = Z ACB, E on AD (prod.). Prove DCE \\\ ABC, .'. DE = ac/b and AC :CE = b :c. Hence reconstruct figure. (5) Castillon's Problem. To inscribe a quadrilateral in a given circle so that its sides should xmss through four given points. See 39 (12), Note. A solution is also possible by Inversion. (6) To circumscribe a quadrilateral about a given circle so that its angles should lie on four given lines. Cf. 39 (13). Problems on Circles. [Notation. The following problem: Describe a circle to pass through a given point (P) to touch a given circle (C), and to intersect a given line at angle a (La) is given as PCLa.] 128 GEOMETKY 41. The Tangency Problems. We have One Solution : PPP. Two Solutions: PPL, PLL, PPC. Four Solutions : LLL, PLC, PCC. Eight Solutions : LLC, LCC, CCC. PPL. Join PP and produce to meet L, draw any tan. from this pt. to any O through PP, mark oflf on L line equal to this tan., &c. PLL. Sol. i. Passes also through image of P in bisector of angle LL. Sol. ii. With centre any pt. on bisector of angle LL describe O touching LL. Join pt. (LL) to P meeting O in J. and B. Join A (or B) to cent, of O. Draw H through P, &c. PPC. Draw any O through PP meeting C in A and B. Let AB meet PP in 0. Tan. from to C, &c. PLC. Draw diam. SA of C ' L and meeting L in B. Then [ 27 (1) Note 2] S is C. S. of L and C. Join SP and take Q on it so that SP. SQ = SA. SB. Required circle passes through Q [ 27 (7)]. Problem becomes PPL. PCC. S a C. S. of CC, Q such that SP. SQ = product of tans, from S to CC, then Q lies on required circle and problem becomes PPC. LLC. Draw lines |1 LL and distance from them = radius of C. Problem becomes PLL, LCC. Cf. Sol. i to CCC. Reduces thus to PLC. CCC. Sol. i. Draw O's concentric with C3, Cg and of same radii in- creased or diminished in each case by rad. of Ci. The problem becomes PCC. Sol. ii. By Inversion from CLL. Sol. iii (Casey). Let Ci, C^, C3 be the O's ; Sj, Sg, S3 their C.S.'s forming an Axis of Similitude; Pj any pt. on C^. Join P1S3 meeting Cj in corresponding pt. Q2, Q^Si meeting C3 in R3, RsS, meeting Ci in pj. Then Pii^i describe homographic ranges and either common point P of the ranges joined similarly to the C. S.'s determines Q and R the pts. of cent, of the tan. 0. Sol. iv (Gergonne). Let be the radical centre of the CCC, and LMNhe poles of one of the 4 Axes of Similitude, then if OL meet its in Pp, &c., the O's PQRf pqr touch CCC. Koies. (1) The problem is known as Viete^s Problem, though it was proposed and perhaps solved by Apollonius. (2) Sols, iii and iv provide alternative methods for many of the previous cases LCC, LLC, &c. GEOMETEY 129 42. Ortliogonal Circles. The problem of describing circles to cut two or more circles orthogonally depends on 26 (6). Diametral Circles. The problem of describing circles to cut two or more circles diametrally (i. e. to bisect the circum- ferences) depends on the fact that if A and B be the centres and a and h the radii, AP^ + a^ = BF^ + 1^. Hence use 28(l)(c). 43. Circles cutting others at given angles. Jj^TiQ. Let OA, OB be the lines. Take any pts. A, B. Make Z OAF = 90-a, I OBP = 90-/3. Make PQ (in PB) = PA. Join AQ and produce to OB in S. Draw SR || BP to meet AP in R. OR is locus of centre. Hence easily L^jL^P (cf. PLL above), L^L^L, L^LoL^ IjjjIi^C^. Sol. i. By Inversion from L^L^Ij^. Sol. ii. Let R on the LaLp locus be required centre and let required circle cut C in M and OA in A. Join 22ilf, i?^. Then RMC = 7. Make i?Og = 7, and cut off OQ : Oi? = MC-.MR = MC:AR, .'. OQ : MC = OR :AR = const, for species of OAR is known. Hence Q is a known point. But QR:CR = QO :CM = const., .'. since Q and C are fixed, R lies on known circle. CCoC, and all other cases may now be derived by Inversion, though other constructions are possible. GEOMETRICAL CONICS THE PARABOLA 1. Linear Properties (Tan. and Normal). (1) SL = SX=-2SA. (2) SP= SG = ST= PM. (3) NT =2 AN. NG = 2AS. M < ^ \ ^ Z \ \ ^ s \ \ T X V t sj G \ \ Fig. 17. (4) PN^ = 4.AS.AN PG^ = 4: AS. SP. (5) The complete normal at P = 4 ZQ. (6) PR = SL. GEOMETRICAL CONICS 131 (7) SYM is a St. line ; TY = YP, SY = YM ; F is on tan. at vertex ; SY^ = SA . SP, {p'- = ar.) (8) The locus of iT is a coaxal parabola, vertex S, latus rectum AS. The intersections of this parabola with a circle on SO as diameter easily give the points on the original parabola, normals at which pass through 0, being any point whatever. (9) The algebraical sum of the ordinates of three conormal points is zero, and their circumcircle passes through A. Proof. -GL Note to 6 (4), and use 1 (8). 2. Angular Properties (Tan. and Normal). (1) The angles MPT, SPT, STP, SYA, NPG, PGB are all equal ; MS bisects Z TSP, .'. L MSM' is rt. Z. (2) A's 3IYP, SYP, MYZ, SYZ, MPZ, SPZ, 8YT, AYT, SAY, TPN, PNG, TPa, PEG are similar. (3) (a) Z PSZ = rt. Z = Z PZQ. {b) Tangents at end of a focal chord intersect at rt. angles on directrix. (c) SZ-' = SP.SQ. 3. Chords. (1) AT^ = AN.Air. Note. True for any diameter rp2 = pv.FV [where chord Q^Q meets diameter PT in T and QV, Q'V are ordi- nates. Cf. Fig. 19]. (2) SZ bisects exterior LPSP". (3) Semi-latus rectum is Harmonic Mean between the segments of any focal chord. Note. i.e. for a focal chord SP. SQ = iAS . PQ. 4. Diameters [marked angles are equal], (1) Locus of centres of (] chords is st. line (diameter) || to axis. (2) Tangents at end of any one of these chords intersect on the diameter. K 2 132 GEOMETRICAL CONICS (3) Tangent at P is bisected at P (between TQ, TQ'). (4) Perpendicular from S on QQ' meets PF in directrix Fig. 19. (5) Parameter [i. e. focal chord bisected by diameter] = 4:SP. (6) TP = PV. (7) QV^ = iSP.PV; QD^ = 4AS . PV. (8) LM:MN= PN:NQ\ (9) QK.KQ' = 4:SP,KM. 5. Two Tangents. (1) [Adams] SL = TMl= SL'l (2) ASTPWl ASTQ. (3) Sr^ = SP.SQ. (4) TP^:TQ'' = SPiSQ. (5) irTP=ilPSQ. Fig. 20. GEOMETKICAL CONICS 138 e. Segments of Chords. (1) li PP\ CQ' are fixed in direction, (PO,OF)/{QO.OQ') is independent of 0, and = TflT^, = Sp/Sq, = ratio of focal chords II PP", QQ\ (2) If any A has its base || to the axis, the squares of its sides are in the ratio of the || focal chords. [See also 4 (8), (9).] Fig. 22. 7. Three Tangents. (1) Circle pqr passes through 8. (2) Orthocentre of pqr lies on the directrix. (3) Rp:pq=pQ:Qr (4) If a variable tangent cut the three fixed tangents in L, M, N, then LM : MN = const. (3) TE^ = TA . TB. Note. Hence, easily, 3 (1). (4) If a circle meets a parabola in four points A, B, C, JD, the chords AB, CD are equally inclined to the axis. Note. Hence diameters of AB, CD are equidistant from axis, .-. algebraic sum of ordinates A, B, C, D = 0. Fig. 28. 134 GEOMETRICAL CONICS Fia. 24. 8. Curvature. (1) Common chord of curvature at P = 4 PT, (2) Focal chord of cur- vature at P = 4^P. (3) JRadius of Curvature atP = P6?V(2 a') = 4 SPySY = 2PK, Notes. (1) Easily recalled by P = rdr/dp from x>^ = ar. (2) To find c. of curvature atP: join SP, draw SR U SP meeting PO in R, then i? is m. pt. of radius of curvature, or use radius of curvature = 2 PK. 9. Areas. (1) ATqq'==il\PQQ\ (2) Area of parabola QPQ' = t A TQQ\ Fig. 25. 10. Drawing of Tangents. To draw the tangent at T: i. T on curve. Use SP = ST. ii. Toff curve. (1) Use Adams [5(1)]. [S cent., rad. SL = Of, describe 0. From T draw tans, to this.] (2) Tcent., TS rad., O cuts directrix in MM'. Diams. thi'ough 3f, M' meet curve in pts. of contact. (3) Or, in (2), lines bisecting SM, SM' at rt. angles meet curve in pts. of contact. (4) Circle on ST as diam. meets tan. at vertex in J, Y', &c. GEOMETRICAL CONICS 135 11. Construction of Parabola under given conditions. direction of axis = 1 pt., focus = 2 pts., Construction. Draw circles, centres P, Q, radii PS, QS. Directrix is com. tan. (Two solutions.) Find tan. at vertex by 1 (7). Draw circle through S, P touching tan. in T. Use 5 (2). (Two solutions.) Use 3 (1). Z STQ (Fig. 20) = angle which TP makes with axis. Find image of P in axis. Use 6 (3) to find pt. of contact of tan. Draw QE \\ axis meeting PR in E. Produce EQ, making TQ:EQ= PE: ER. By 4 (8) T is on tan. at P. Similarly, or by 4 (2), find tan. at Q. Then focus. Use 5 (2) or 7 (2). Use 6 (3) to find p, pt. of contact of the tan. Bisect PQ in V. Draw VV^ || axis meeting tan. in t, and pV' \\ PQ. The m. pt. tV^ is on curve. Draw tan. at this pt. Find S. (Two solutions.) Determine direction of S as in (5) above. From r, intersection of tans., draw TV || axis. Draw any line between tans, bisected by TV. (Use: diagonals of ir bisect each other.) Draw PV \\ this line. Produce PV = VP^. P' is on curve. Produce P'P to meet one of tans, and use 6 (3) to determine pt. of contact, &c. Let PQ, RS meet in 0. Find OF^ = OP. OQ, OI^^'OR. OS, F, E in PQ, RS, . . EF is direc- tion of axis [ 6 (3)]. (Two solutions.) Use 7 (1) to find S. Use 6 (3). [ Require Involution. [See 34.] [N.B. Axis - 2 pts. directrix = 2 pts.] Data. (1) P, Q, S. ' (2) 2 tans., S. .(3) P, tan., S. /(4) P,Q, axis. J (5) 2 tans., axis. (6) P, tan., axis. '(7) P, Q, R, direction of axis. (8) 3 tans., direction of axis. -< (9) P,Q, tan., direction 1 of axis. (10) P, 2 tans., direc- tion of axis. .(11) P, Q, R, S.' j (12) 4 tans. (13) P, Q, R, tan. (14) P, Q, 2 tans. (15) P, 3 tans. 136 GEOMETRICAL CONICS 12. Locus A Parabola. The following are the proposi- tions used to prove the locus of a moving point to be a parabola : 1 (2). SP = PM. 1 (4). PN^ =^4 AS. AN, 4(7). QF2= 4:AS,PV. 4 (9). If fixed points QQ' can be found and PN be drawn in a fixed direction to meet QQ' in N so that PN varies as QN. NQ\ the locus of P is a parabola. 4 (8). OA, OB are fixed lines, and A a fixed point, MN in a fixed direction {M on A, N on B), P is taken on MN such that NP : PM = OM : MA. Locus of P is a parabola. 13. Envelope a Parabola. The following are the chief propositions used in proving the envelope of a moving line to be a parabola : 1 (7). If moving line meets a fixed line in Y and pei-pendicular to it through Y passes through a fixed point. 5 (2). If moving line PT meets fixed line QT in T and there exists any point S such that Z STP is const. 7 (1). Two fixed lines intersect in 0, S is a. fixed point, the chord cut off by the fixed lines from any circle through 0, S envelops a parabola. 7 (4). If a variable line cuts^ three fixed lines in L, M, N and LM : MN = const. 7 (3). If a variable line pr moves between two fixed Lines qBj qp [Fig. 23] so that Rpipq = qr : rP, 7 (3). If a variable line pr cuts off intercepts x, y from two fixed lines (qB, qP) such that Xx + fiy = const., where A and fjL are const. This follows easily from 7 (3). Note. In all envelope problems efforts should be made to gain a preliminary knowledge of the curve by taking limiting positions of the moving line. This should also be done in locus problems. GEOMETKICAL CONICS 137 ELLIPSE. 14. Linear Properties, Principal Diameters. (1) CA = a, CB = h, e'' = l -hya'' ; CS = ae, GX = a/e, CS.CX = GA^ ; SB = a, SL ^ b^a. (2) SP= e,PM. (3) (Adams) SD = e, BQ, (4.) PN^:AN.A'N=^BG^:AG^i /^ ^ = 1^ Fn^:Bn.B'n = AG^:BG^ J W "^ 6^ )' Fig. 26. (5) SP+S'P = 2a, SP= a- ex, S'P = a + ex. The image of one focus in a tan. is dist. 2 a from the other focus. (6) SY. S'T = &2, and Y, F lie on the auxiliary circle. {7) GN'.GT=a^,Gn.Gt = h^; GG.GT=a'-h\ (8) [p2 (1Q -) 138 GEOMETRICAL CONICS (9) PF.Pa = h\ PF. Pg = aK (10) Sa = e.SP, S'G^e.S'P. (11) Projection of PG on either focal distance {PK) SL, ?> J) '^y >> ?j >> >j =tt. lPEg = 90\PE=' a. (12) Tangents at P, P' intersect on axis. (13) PN:P'N=^h:a. (14) Semi-latus rectum is a H. M. between the segments of any focal chord. (15) Fregier^s Theorem. If P is a fixed point, MN a vari- able chord such that Z 3IPN = rt. Z, MN passes through a fixed point (Fr6gier Point) on the normal at P. 15. Angle Properties. (1) K8 bisects ext. IRSQ. (2) Z KSP = 90. (3) Tangents at the end of a focal chord intersect on the directrix. (4) The joins of ends of two focal chords intersect on the directrix. (5) PG, PK are internal and external bisectors of SPS\ (6) SgS'tP is cyclic. 16. Two Tangents. (1) ITSP= I TSQ, Z TS'P = Z TS'Q. [Fig. 28.] (2) Z STP = Z S'TQ. (3) Tangents at rt. angles intersect on the director circle, rad. \/a2 + IK X I S G\ 5' ) 9 Fig. 27. GEOMETKICAL CONICS 139 (4) TP',TQ = diam. llTPrdiam. || TQ. (5) If PQ meets ,Sf-di- rectrix in K, L TSK = rt. Z. (6) GT bisects PQ and all chords || PQ. (7) Ifj9 is any point on the arc PQ, a.nd pL,pM,pN tire drawn in const, directions to meet PQ, TP, TQ, pL'^ varies si.spM,pN [a2 - h(3y]. Fig. 29. Fig. 28. 17. Interseotinq Chords. (1) If the directions of PP', QQ' are fixed, {PO.OPyiQO.OQ') is independent of 0, = Tpyiq^, = ratio of sqs. of || diams. = ratio of II focal chords. (2) If a circle intersects a conic in four points A, B, G, JD, then AB, GD are equally inclined to the axis. 18. Three Tangents. (1) The segment of a vari- able tangent between two fixed tangents subtends a constant angle at S and also at S', i.e. I MSN is const., if RP, RQ are fixed, MN variable. Fig. 30. 140 GEOMETRICAL CONICS Three Tangents, two parallel. (2) CR, CR^ are conjugate, as are diagonals of any- escribed parallelogram. (3) l\SRP III ASR'F", A S'RP III A S'R'P". Fig. 81. The dotted lines in the figure indicate the lines necessary for proof, (4) Z RSR' = ^ re-entrant Z PSP", IRS'R' = ilPS'P', (6) RPiR'P' = RQ'.R'Q'. (6) RP . R'P' = SP.S^P = SP' . S'P". (7) The figure indicates most of the groups of equal angles which occur. 19. Conjugate Diameters. (1) CD II tan. at P and bisects chords || to it, ^" >> ff -^ If >> >) ty (2) Supplemental chords || conj. diams. (3) Z P'Cjy = 90. (4) CM = P'N, CN = B'M. GEOMETRICAL CONICS 141 (5) CP'- + CD'' = a" -f &2. (6) Area A CTD = const. = -I ah, i.e. PF .CD = ah, i.e. CD = ab/p. 7) pa = a Pg = I CD. (8) SP . S'P = CDK (9) CV. GT = 0P2 ^'^- ^^ [QV,Gt= GD^]. (10) FVPr. FPi = GB'^/GP\ i.e. x'/a^ + y'^/h'^ = 1. (11) TQ . Qt = sq. of semidiam. conj. to CQ, 20. Construction of Tangent [S = focus, c? = directrix]. i. P on curve. (1) Given S, d. Use 15 (2). (2) Given 8, S\ Use 15 (5). (3) Curve only. Pascal's Theorem, 33. ii. P outside curve. (1) Given 8, d. Use 14 (3). (2) Given 8, 8\ 2a. Circle cent. 8, rad. 2a, and O cent. P, rad. P8', cut in K, L, Join fi'Z, M meeting curve in 1\ t, then PTj Pt are tangents. (3) Given 8, 8', 2 a. Circle on 8P as diameter cuts the auxiliary in Y, F ; PF, PF are tangents. (4) Join GP meeting curve in ^, take V such that GY. GP = Gp^. Through Fdraw ordinates QVQ' ; Pft P^ are tangents. 21. Construction of Curve. (See also 34.) The data liere may- be indefinitely varied between C (2 pts.), S (2), position of axis (ax = 2), length of axis (a = 1), direction of axis (1), directrix {d, 2), eccentricity {e, 1), asymptotes (4), direction of one asympt. (1), one asympt. (2), &c., and one or more up to five points (P, P,...) or tangents {t, t,...). 142 GEOMETRICAL CONICS The following cases are arranged in classes, according to order of difficulty : 1. Sde; SdP; Sdt; SS'P-, SS't ; SePP; S,e,ax,P; SPPP ; Sttt ; SaPP ; SaPt ; Satt ; dPPP ; dePP. ii. C, e, ax, P : Find (?, T, A in this order. C, e,ax,t: PN/NG is known, .. PN/CN, .-.P. C, ax, PP: Use U (4) or Pascal's Theorem [ 33]. iii. C, P, Q, R : Determine three other pts. and use Pascal. Or, Bisect PQ in M. CB \\ PQ and CM are two conj. diams. JJi2' |I CM. Hence 22' on curve. Let RR', PQ meet in E. Then EP.EQ:ER. ER' = CB^ : CA"^ [ 17 (1) CA = semidiam. along CM'], = RN^ : CA^-CN^ [ 19 (10) N is pt. where RN \\ CB meets CM]. Hence CA, CB, &c. The following are notes on the more important cases : iv. PPPPP. Use 17 (1) or see 34. V. P, it and their points of contact. Given P, TQ, TR, Q, R (Fig. 33). LP . LP' ILR^ = TQ'^/TR'^ = LM^/LR"^ ; .-. LP. LP' = LM^, ,'. P' is known, .-. diam. through Q. Fig. 33. vi. Two Conjugate Diameters CP, CD. Through P draw U CD and on it take PK = PK' = CD. Axes bisect Z KCK', trv. axis in acute Z KCK'. CK' = a + b, CK --= a-b. Or, Circle KCK' meets || through P to CD in T, t. CT, a are directions of axes. Then use 14 (7). vii. PPPPP, PPPPt, PPPtt, PPttt, Ptttt, ttttt can all be solved by Theory of Polars and Harmonics. See 34. 22. Elliptic Loci. The following are the most important propositions used in proving loci to be ellipses : (1) SP=e. PM. (2) >SP+6f'P= const. (3) PNy(AN. A'N) = const. (4) QV^iPV. VP^) = const. GEOMETRICAL CONICS 143 (5) 16 (7), i.e. if P moves so that if Oi, f3, y be its perpendiculars upon the sides of a triangle and a' = A:/3y, then P describes a conic touching AB, AC at B and C. (6) If the perpendiculars on four fixed lines be such that PiPi/P^Pi = const. [oc,3 = Jc. yh]. [Milne and Davis, p. 201.] (7) OP, OQ are two fixed lines, PQ a const, length, any point on PQ describes an ellipse. (8) More generally : OP, OQ are two fixed lines, PQB is a given A, R describes an ellipse. [IS'Lacaiilay, p. 75.] 23. Elliptic Envelopes. The following are the most important propositions employed in proving envelopes of lines to be ellipses, &c. : (1) If the foot of the -L^ on the line from a fixed point lies on a fixed circle. 14 (6). (2) If there be two fixed points, S and 8\ the product of the perpendiculars from which is const. 14 (6). (3) If X, Fmove on two fixed lines OX, OY and [a) Z XSY = const., S being a fixed point, the envelope of Zr is a conic. [18(1).] {b) Area A XOY = const., the envelope is a hyperbola with OX, OF asymptotes. (c) A . OX + fjL, OY = const., the envelope is a parabola. [See 7(3), 13.] 24. Curvature. (1) The central chord of curvature = 2GI)^/GP. (2) The focal chord of curvature = 2CD'^/CA. (3) The normal chord of curvature (2 p) = 2 CD^/p. (4) Thus p = Cnyp = CB^/aJ) = PG^SL^ = a%yp\ (5) To draw circle of curvature at P: Let normal meet axis in G. Draw GK i' PG meeting a focal distance in K. At K draw ^ to this focal distance which will meet normal in centre of curvature. 144 GEOMETRICAL CONICS HYPERBOLA [N.B. To economize space, properties given for the ellipse are not usually repeated here. Angles which are equal in the ellipse are, in the hyperhola, either equal or supplementary. ~\ 25. Metrical Peoper- TIES. (1) AE=SB^ BC. (2) CD = CA, i. e. D is on the auxiliary O to which SB is a tangent. (3) CE = OS. (4) If BK is parallel to an asymptote, SB = BK. (5) S'B-SB= AA\ (Fig. 38.) (6) BB = B'B\ Br = i? V, BT = Br. (7) BB . BB' = BC Br.Br' = AC% liBBB' and Brr' are parallel to the axes. \ Fig. 36. GEOMETRICAL CONICS 146 (8) RQ . QB' = const Jor chords, = FT^ = 0Z)2. So rQ. Q/ = CPMf rQ II CP, (9) ATCr = const, area = AC.BC, i.e. CT. OT' = const. i.e. PH.PK = const. = 1 CS^. Fig. 37. (10) One asymptote is the diagonal of the Ij^^^ completed out of CP, CD, and PD is || to the other. This parallelogram has a constant area (AC.BC). (11) To describe a hyperbola, given CP, CZ>, use (9) arid (8). See also 21 (vi). 26. Angle Pkoperties. s >>5 # .^^ y^TU } t\ ^'^^'^^-T^ /N \ Fig. 38. L 146 GEOMETKICAL CONICS (1) Cyclic Groups (Fig. 38) : (a) STS'r. (^) CrrGg. (y) SPgS't. (2) Similar Triangles (Fig. 38) : (a) A TSr III A TCS" III A rCS', and the marked angles are equal. (13) A TS'r III A TGS III A T'CS. (3) Z CnG = rt. Z = Z (7m^. Hence a good method of drawing the normal and therefore the tangent at any point on the curve. RECTANGULAR HYPERBOLA 27. Metrical Properties. (1) The eccentricity = V2. (2) Diameters which make equal angles with either axis are equal. (3) Diameters which are conjugate are equal. (4) Diameters which are at rt. angles are equal. (5) Pa= CD^ Pg, i.e. P is the centre of the O GirGg. [See26(l)(/3).] (6) PN^ = AN.A'N, QY^ = PV.VP\ 23. Angle Properties. ,. . , ( equal ) , (1) Conjugate diameters make |,pi,^^tary$ *S^^' with either f^-f*j. (2) The angle between any two diameters = Z between their conjugates. GEOMETRICAL CONICS 147 (3) LQPT^LQpP. (4) Any chord of a R. H. subtends equal (or supple- mentary) angles at the end of any diameter. (5) If a R.H. circum- scribe a A it passes through the orthocentre, and the locus of its centre is the N.P.C. ofthe A. Fig. 39. (6) All conies through the intersection of two R. H. 's are R. H.'s. (7) If PQ, PR be two chords at rt. angles, QR is ii to the normal at P. Proof. The R. H. through P, Q, B passes through orthocentre PQR, .-. X"" from P on QR is tan. at P. Note. Hence in a R. H. the Fregier point of every pt. is at oo . 29. Loci. The following proposition is of service in proving loci to be R. H.'s : If A and B are fixed points and P is such that Z PAB - L PBA = const., the locus of P is a R. H. of which AB is a diameter. Proo/. Use 28 (4). HARMONICS AND GENERAL THEOREMS 30. Pole and Polar. (1) If the polar of P passes through Q, the polar of Q passes through P. If the pole of ^ lies on g, the pole of j lies on ^. Conjugate Points : Points such that the polar of each passes through the other. Conjugate Lines ; Lines such that the pole of each lies on the other. L 2 148 GEOMETRICAL CONICS (2) If the polars of P and Q intersect in B, the polar of R isPQ. If the poles of p and q lie on r, the pole of r is pq. (3) The line joining any pair of conjugate points is cut harmonically by the conic. Particular Cases : (a) {TRVS} = - 1, (6) {PVQv} = - 1, where v is pole of RS. (4) {pRqv}^ -l,le, if two tangents be drawn to a conic any third tangent is harmonically divided by the two tangents, the curve, and the chord of contact. 31. Circumscribed and Inscribed Quadrilaterals. [See GEOM, 24.] Fig. 40. (1) The harmonic triangle of a 4^ circsd. about a conic is self-conjugate [self-polar]. (2) The harmonic triangle of a 4^ inscd. in a conic is self-conjugate [self-polar]. (3) The harmonic triangle of a 4^ circsd. to a conic coincides with the harmonic triangle of the 4^ formed by the points of contact. This includes the following proposi- tions : (a) The int. diagls. of an escd. 4^ and of the correspond- ing inscd. meet in a pt. (6) The third diagls. of these 4^^ are coUinear. (c) Any pair of opposite sides of an inscd. 4^ intersect on a diagl. of the corresponding escd. 4^. (4) The locus of the centre of a conic inscd. in a given 4^ is the * diameter ' of the 4i. (5) If a 4^ is inscd. in a conic, any st. line is cut in involution by the 3 prs. of (6) If a 4' is circsd. about a conic, the lines connecting any pt. with the vertices form GEOMETRICAL CONICS 149 opposite connectors and by the conic. (Desargues* Theo- rem. SeeGEOM. 24(7).) j^ote. By making two vertices results of (7), (8). By making two we obtain the results of (9), (10). (7) If a A is inscd. in a conic, any st. line is cut in involution by the conic, the two sides which meet at any vertex, the tan. at that vertex, and the opposite side. (9) If TP, TQ are tans, to a conic and a trv. meet TP, TQ in B, 8, the conic in M, N, and PQ in F, then F is a double pt. of the involu- tion (MN, RS). a pencil in involution and the pr. of tans, from the pt. also belong to the pencil. of the 4' coincide we obtain the pairs of opposite vertices coincide (8) If a conic is inscd. in a A, the two tans, from any ext. pt., the lines joining that pt. to the ends of any side, to the vertex opp. that side, and to its pt. of contact form a pencil in involution. (10) liTP, r^ are tans, to a conic and be any pi, OT is a double line of the in- volution (OP, OQ, tans, from 0). 32. Anharmonics and Homograph y. 32 and A. GEOM. 66.] (1) If ^,5, 0,2) are fixed pts. on a conic and P a vari- able point on it, {P.ABGD} = const. (3) Homographic Banges on Conic. [Def.) {ABC...} = {A'B'C\,.}, the pts. all lying on conic if = {Q.A'B'C\..}, P and Q being any two pts. on conic. [See GEOM. 81, (2) If a, b, c, d are fixed tans, to a conic and p a vari- able tan., {pa, pb, pc, pd] = const. (4) Homographic Tangents to Conic. (Def.){ahc..,}=^\afbV...}, the lines all touching conic if {p.abc...} = {q.aTc\..}, p and q being any two tans, to conic. 150 GEOMETRICAL CONICS (5) U{ABG...} = {A'B'C...}, the pts. all lying on conic, then the points {AB', A'B) (AC, A'C)... all lie on st. line. (Homographic Axis.) (7) If through pt. V out- side a conic a pencil of rays is drawn cutting it in AA\ BB\... then {ABC...} = {A'B'C..,}, the homographic axis being polar of P. 33. Some General Theorems (1) PascaVs Theorem. If A, B, C, D, E, F, be 6 pts. on conic, the pts. {AB, DE) [BG, EF) (CD, FA) lie on st. {ahc...} (6) If = [aW...}, the lines all touching conic, then the lines (aV, a'h) (ac\ a^c)... all pass through a pt. (Homographic Pole.) (8) If from pts. on a line p outside a conic the pairs of tangents (aa^) (hi)') ... are drawn to the conic, then {ahc...} = {aVc'...}, the homographic pole being pole of V. (2) Brianchon's Theorem. If a, h, c, d, e, / be 6 tans, to conic, the lines (ah, de) (he, ef) (cd, fa) pass through pt. line. Notes. [Vj See 36 (2) (/) ; also GEOM. 31 (8), A. GEOM. 65 (2). (2) There are 60 possible Pascal Lines which intersect 3 by 3 in 20 Steiner Points, which lie 4 by 4 on 15 st. lines. Similarly for Brianchon's Theorem. (3) Newton's Theorem. If A and B be fixed points and Q move along any st. line and /.PAQ = const., APBQ = const., the locus of P is a conic through A and 5. Cor. When Z PBQ = the theorem gives that locus of jp is a conic through A, B. Proof. {B.P} = {B.Q} (. Z PBQ is const.) = {Q} = {A.Q} = {A.P},&c. GEOMETRICAL CONICS 151 (4) Maclaurin's Theorem. The locus of the third vertex (P) of a A, two of whose vertices {Q, R) move along two fixed lines {OQ, OB) and whose sides pass through fixed pts. (A , B, C), is a conic passing through 0, B, C. >P Note. Taking Q, R at it is clear is on locus ; so also pts, {AC, OR) {AB, OQ). Proof.-{B.P} = {B.R} = {R} = {A.R} = {A.Q} = {q} = {C.Q} = {C.P}, &c. 34. Construction of Conic. (See also 11, 21, and for notation used 21.) (1) PPPPP. [Particular Cases: PPPt and its pt. cont. ; PPPP and dir. asympt. ; PPP and one asympt. ; P and both asympts., &c.] Methods : (a) The chord through any one pt. || to chord of two other pts. can be easily determined by 17 (1). Thence centre. (h) Use Pascal to determine tans, at P. Thence centre. (c) By Newton^s Theorem : A, B, C, D, E the pts. ; take ABC, BAC as the two fixed angles of Newton's Theorem, which generate conic by- rotation round B, C as poles. Determine fixed line by pts. D, E. (rf) By Maclaurin's Theorern {Note). The five given pts. are, in the fig., 0, {AC, OR), {AB, OQ), B, C. Hence A. Hence A, &c. (e) If AC, AD, AE, BG, BD^ BE meet any line in C^, D^, E^, C^, D^, E^, the common pts. of the ranges [CiD^Ei], [C^B^E^^ are on conic. Draw the line through C \\ BD. Hence centre. (/) Use 31 (2). Take any four pts. ABCD. Let F = {AC, BD\ G = {AD, BC), H = {AB, CD). Let E be fifth pt. Join B to {AE, GF). If this join meet EH in E' then E^ is on conic, &c. 152 GEOMETKICAL CONICS (2) ttttt. [Particular Case: 11(12).] Reciprocation may be applied to (1), {b\ [d), (e), (/). The readiest solution is given by Brianchon's Theorem. (3) PPPPt. [Particular Case : 11 (11).] Use 31 (5). The four pts. give the 4*, t the line. The pt. of contact of t is found as one of the double points of the involution. Case now reduces to PPPPP. (4) ttttP. [Particular Case : 11 (15).] Reciprocal of last case. {6) PPPtt. [Particular Case: 11(13).] tftPP. [Particular Case : 11 (14).] Use 31 [9) and 31 (10) respectively. 35. Reciprocation. (1) With respect to any conic : Reciprocal Terms. line Point ; coUinear ; locus ; inscribed. Join of two points. Point on conic. ^ . .., ( single) Comes with I , Y con- tact. Chord of intersection of two conies. Pole. Line at infinity. Conic. Line ; concurrent ; point ; envelope ; circumscribed. Intersection of two lines. Tangent to conic. Conies with j , , , [ con- tact. Intersection of common tangents to two conies. Polar. Centre of reciprocating conic. Conic : ellipse, parabola, or hyperbola as the tangents to the original conic from the centre of the reciprocating conic are imaginary, coin- cident, real. (2) With respect to a Circle (' Point ', * Origin '). (a) The reciprocal of a circle whose centre is G, rad. a, with respect to a circle, centre 0, radius Jc, is a conic with Focus : 0. Eccentricity : OG/a. And therefore ellipse, parabola, or hyperbola as OC < = or > a. Second Focus : Find centre and use (c) (a). GEOMETRICAL CONICS 163 Semi-latus rectum : W'/a. Directrix : Reciprocal of C This is the directrix belonging to the focus. Centre : Reciprocal of polar of w. r. circle G, Direction of Axis : 00. Noie. These results are so easily found, by polar coordinates, that it is questionable whether it is worth while to memorize them. (&) Concentric circles reciprocate into conies with common focus and directrix. Coaxal circles (a) Reciprocate into confocal conies if be a L.P. Note. The R.A. reciprocates into the other focus. (/3) Reciprocate into conies with one focus common and equal minor axes if be on rad. axis. Equal circles reciprocate into conies with one focus common and equal latera recta. (c) Magnitude Properties. The only properties easily applicable are : (a) The distance of any point P from origin varies inversely as the distance of the reciprocal line. (13) The A. R. of a range = that of the reciprocal pencil. (d) Angle Properties. The angle between any two lines is equal to the angle subtended at the origin by the line joining their reciprocals. (e) Reciprocal Terms. Centre of reciprocation, 0. Line at infinity. Line through 0. Point at infinity. Parallel lines. Points in same st. line as centre of reciprocation. Conic : anywhere. Conic, see 35 (1), also note below. 154 GEOMETKICAL CONICS Reciprocal Terms {continued). Kectangular hyperbola. Circle, see 35 (2) (a). Similar conic. Kectangular hyperbola. Coaxal circles: a limit- ing point ( 35 (2) (6)). Conies with respect to which has the same polar. Conies touching a pair of lines which meet at 0. Parabolas with parallel axes. Two points which, with the centre of reciprocation, form a self-conjugate triad with reciprocal conic. Harmonic pencil. Triangle, at orthocentre. Note. If OP, OQ be the tangents from to the conic which is to be reciprocated we have for the reciprocal conic : Asymptotes : Reciprocals of P and Q. Centre : Reciprocal of PQ, Axes : Parallel to the two bisectors of Z POQ. Eccentricity : Depends on Z POQ only. 36. Projection. (1) Properties unaltered hy Projection : Degree of curve. Tangency of curves, or of lines and curves. Kelations of poles and polars. Anharmonic ratio of ranges and pencils. Any relation connecting the distances of points in a straight line, provided that every term of the relation mentions the same points, although in different orders. Note. Under certain limitations the points need not be in a st. line, e.g. Ceva's Theorem is projective. Conic : on director circle. : at focus. : at centre. Parabola : on directrix. Confocal conies : at one of the foci. Concentric conies. Conies with same eccen- tricity. Conies touching at 0. Conjugate diameters. Harmonic range. Triangle, at orthocentre. GEOMETRICAL CONICS 155 (2) Simplifications of Figure. (a) Any line may be sent to infinity and any two angles into given angles. Send third diagonal to oo . Project one Z and Z be- tween diagonals into rt.angles. Project any two of its angles into tt/S . (6) Any quadrilateral in- to square. (c) Any triangle into equilateral A. Note. Any 4} may simultaneously be made into a H"^, or any line projected to ^ . The A may be made of given species. (d) Conic into parabola. Send any tangent to oo . Note. Any triangle may simultaneously be made equilateral. (e) Conic into circle with Send polar of point to co projection of given point as and any two angles in seg- centre. ment of conic cut off by a line through the point into rt. angles. Take the pole of the line. Draw any two angles with vertices on conic and stand- ing on chords passing through the pole. Project these into rt. angles and the line to oo . Note. Hence an easy proof of Pascal's Theorem. (/) Conic into circle and any line to oo . (g) Conies through two given points into circles. (h) Conies through four given points into coaxal circles. (i) Conies touching at two given points into con- centric circles. (j) Conic into hyperbola of given eccentricity with Project line joining points to 00 and one of the conies into a circle by (/). As in (g). Send chord of contact of tangents through to oo and 156 GEOMETRICAL CONICS centre at projection of given point G project Z between them into the given angle between the asymptotes. Note, Any pair of lines may simultaneously be projected into || lines, or any angle into a given angle. [Jc] Conic into conic with projections of given points C and S for centre and focus. Let liX be polar of S, X on OS, RP tan. to conic. S end polar of C to 00 and project PSB, SXR into rt. angles. (3) Focal Properties. To project these we must use the definition of a focus given in A, GEOM. 55 Note (2). Ex. Confocal conies project into conies inscribed in the same quadrilateral. (4) Angle Properties. These are projected by the theorem : The pencil formed by the two legs of a given angle and the imaginary lines through its vertex to the circular points at 00 (focoids) has a given anharmonic ratio. Ex. Lines at rt. angles project into lines which cut harmonically the line joining the two fixed points which are the projections of the focoids. (5) TJie Casey Projection. (Casey's A. Geom., pp. 271-7.) Casey reduces the treatment of this subject to two dimen- sions. Ox, Oy are the axes. ir\ cutting Ox in B and J. The 'projection of P'is found by joining OPj IP, the latter meeting BB' in and drawing CP' II to Ox, meeting OP in P\ Infinity Line. Any point on IT is projected to oo . ax^ ay' >'A are two lines || to Oy y = Coordinates, x = C + X' " C + X' The properties of projections can all be readily proved by this construction, which gives results identical with those of the older method. ANALYTICAL GEOMETEY RELATIONS OF POINTS 1. Distance {Xi y^ (x^ y^ is h = V(x^ - x.^f + (/i - y.^f + 2 (i^i - x^) (/i - y^ cos w. 2. (1) The coordinates of the points dividing (xiy^(x^y.^) in the ratio n-^ : n^ are : ^2 Xi + niX2 _ _ ^2 ^1 - ^1 it!2 (Internal) and (External). ^2 + ^1 ^2"^! (2) The coordinates of mid-point of (x^ y-^ {x^ y^) ^^' ^ = i(^i + ^2), / = i(/i + /2)- (3) The coordinates of centroid of (Xiyi) (x^y^^ (^3^3) are x=^^(x^ + x2 + x.^\ y = i{yi + y2 + y3)' 3. The area of the A {Xi y-^ (x^ y.^ (x^, y-s) is A = I {a^i (2/2 - ^/a) + 002 (2/3 - 2/1) + ^3 (yi - ^2)} sin o), sin CO. = ^ ^1 Vi 1 ^2 2/2 1 ^3 2/3 1 Proof. Find A OAB = | (x^yi Xj^y^) sinco, A ABC = OBC- OCA - OAB. This method applies readily to polygons. Notes. (1) For rectangular axes put qj = 90, cos cy = 0, sin w = 1. (2) The sign of the area depends on whether a man walking round it from (ajjL Vt) to {x^ 2/2) (x^ 2/3) keeps it on his left (sign + in above formula) or right ( ). This is important when two areas are added. 158 ANALYTICAL GEOMETRY 4. Polar Coordinates. X ^ Y COS ^, r^ = a;2 + y^^ y = r sin 0, tan 9 = ^//iP- iVoies. (1) The point (r,0) may also be described as ( r, ^ + Tr), &c. (2) But of these descriptions one may satisfy an equation while the other does not, e.g. the point (2,0) satisfies r= l + cos6, the point ( 2, tt) does not. 6. 52 = ri + ra^ - 2 r^ r^ cos (^i - 6 3), A = + ISrargsinC^a-^g). THE STRAIGHT LINE 6. Forms of Equation (co = 90). (1) y = mx + c. m is the tangent of the angle which the part of the line above the a;-axis makes with the + direction of that axis, c is the intercept on the ^-axis. X 11 (2) - 4- ^ = 1. a^h are intercepts on axes. This equation holds for oblique axes. ^ ' ^ ~ A- Generalized forms of the above. rx + sy = 1 ) (4) X cos a + 2/ sin a = p. (p, oc) are polar coordinates of foot of J.^ from on line. To put any equation in this form : Divide throughout by the square root of the sum of the squares of the coefficients ofx and y. (b) X = h + rcoaO] The two together constitute the y = Jc + rsinO)' equation, (h, 7c) is any point on the line, and (r, 6) the polar coordinates of the current point (x, y) with reference to (h, k). [See also 11, 13.] 7. Inclination of Straight Lines. y = Wi a; + Ci, (1) tan a = + 1 + m^ mg X cos oCj^ + y sin a^ -p^ = 0, X cos 0(2 + y sin 0^2 ~P2 = ^' 0= (0C,-0C2). ANALYTICAL GEOMETRY 159 The lines are parallel if (2) mi = mg. 0^1-0^2 = zero or a multiple of it. The lines are perpendicular if (3) 1 + nil m^ =0. (X^-OL^ = odd multiple of ^ Note. For the lines AiX + B^y-\-C^ = 0, A^x + B^y + C^ = 0, the con- ditions of parallelism and perpendicularity are ^iBj -^2-^1 = 0, AiA^ + B^B^ = 0. 8. Lines under Conditions. (1) Any line through the origin : / = mx. (^i,/i): /-/! = m{x-x^). (2) The join of {x y,) {x^, y^)'.y~y,=^ ^^ {x - x,), Xy X2 or, X y 1 X2 y^i 1 = 0. (3) Any line through the intersection of J-jiC + j^i^ + (7i = and J.2 a; + J52 / + C2 = is of form J.liC + JBi?/+Oi + A(^2^ + -^2i/+ ^^2) = ^ (4) The line through {x^^y^ if to ?/ = m + c is y-Vi = mix-x^. The line through {x^^ y^ r to / = wiu + c is The line through {x^^ y^ U to ic cos a + / sin a = p is y-y^ = tana(a;-a?i). (5) All lines || to Ax^By->rG = ^ are included in J.ic + % + i> = 0, all lines X^" to \i in Bx - Ay ^ B = 0. Thus the line through (xj, y-^ ^J to it is BxAy Bxi Ay (6) The lines (^1, B^, G^) (A^, B^, C^) (^3, -S3, O,) concur if the determinant formed by the coefficients vanishes. 160 ANALYTICAL GEOMETRY 9. Length of Perpendicular. The length of the perpendicular from (x, y) on X cos a + ^ sin a - jp = is {xcoQOi + y sin a -p). The length of the perpendicular from (re, y) on Ax+By+C Ax + By+C -=() is VA^ + B' Rule. Bring all terms to the same side, substitute co- ordinates of point, and divide by sq. root of sum of sqs. of coeffts. of X and y. Note. All points on same side of a line have perpendiculars of same sign. Hence we can determine if a point lies on same side as origin. 10. Bisectors of Angles between lines {p^, (Xj) (jpg? 0^2) ^^'^ X cos oci + y sin oc^ -Pi = (^^ cos 0^2 + ^ ^^^ ^2 ~P2)- For other lines bring to (p, OC) form first [ 6 (4)]. This leads to ^^ ^ Ax + By+G A'x + B'y+C ^ ^^^ ... _ . the form ., = + , . To settle the sign : The sign VA^ + B^ " Va'^ + B"^ for the bisector of the angle in which any pt. {h, k) lies is the sign of {Ah-^Bk+G)/{A^h + B'k + (/). 11. Polar Equations. (1) St. line through pole, 6 = a. (2) Any st. line, p = r cos (^ - a). (p, a) have the same meaning as before [ 6 (4)]. (3) The lines perpendicular to the st. line k/r = Acos{e~oC) + Bcos{d-0) are included in - = Acoa(e + ^ - Oi)+ B cos {e + - /3). 12. Two Straight Lines. (1) ax^ + 2hxy + hy'^ = represents two straight lines through which are : (a) Coincident if h^-ab = 0. (fi) At right angles if a + & = 0. (y) At an angle given by tan B = 0/ -T ANALYTICAL GEOMETRY 161 (2j The lines bisecting the angles between them are x^ - /- 2 xy a-h ^ Yh' (S) ax^ + 2hxy + by^ + 2gx+2fy + c= 0, i.e. (a, b, c, f, g, h I x, y, If = 0, represents two st. lines if A = a li y =0. a h 9 h b J y f c Proof. For determinant form assume (a, 6, c, f, g, h\x, y, zf = {px-i-qy + rz){lx+my + nz). Differentiate in x, in y, in z. In the three resulting equations choose {x, y, z) to represent the point of intersection of px + qy + rz 0, and Ix + my + nz = 0. Hence for this point ax + hy+gz = 0, hx-\-by+fz = 0, gx+fy + ez =- 0, &c. Notes. (1) The determinant is easily remembered if the order in which the letters a, b, c,/, g, h occur in it is noted. In writing it down remember that /, g, h occur with 2^s in the equation and are not coefficients. (2) Its expanded form is abc-ap~bg^-ch'^ + 2fgh = 0. (3) The angle between the lines is the Z between ax^ + 2 hxy + by"^ = 0. (4) The point where the lines intersect is {G/C, F/C), where C, F, G are the minors of c,f, g in A. Hence by (2) the lines bisecting the angles between the lines. 13. Oblique Coordinates. (1) The inclination of t/ = mx+c to the a-axis is given by msincu tan e = ^ . 1 + m cos w (2) If the perpendicular p be inclined at angles a, to the two axes a line may be written x cos Ci + ycos = p To put Ax + By+C ^ in this form multiply by sin cu and divide by y^A^ + B^^2AB cos w. (3) The angle between y rriiX+Ci and y = m^x-j-c^ is given by tan ^ = + ^- , ^ T-^ . ~ 1 + [mi + m^) cos = Z FOx is known as the Auxiliary Angle. (7) Circle on {x-^f y^ (iCg, ^2) ^^ diameter. Express that lines (x, y){x^, Vi) and (, y)(x2, y^) are at right angles. [See also Polar Equations ( 21).] 19. Tangent. [Note. It is recommended to learn at this stage the tangent for the General Equation, 52.] (1) xx^ + yy^ = a\ (2) ic cos <^ + ^ sin <|) = a. Touches the circle at the point Vi) ow the curve : tangent. 7* (*i> Vi) inside the curve : polar. (3) To determine the pole oiy = mx + c^ identify this and xx^ + yyi = a^ and thus obtain sufficient equations for x^ and y^. (4) The pole of the chord (f>i, (p^ is (5) If the point and its polar with regard to a circle of radius a be distant respectively d and p from the centre, dp = a^. 21. Polar Equations. (1) Pole at centre : r = a. (2) Pole on circumference : r = 2a cos 0. [Initial line a diameter.] r = 2a cos {6 a). [Initial line making Z a with a diameter.] (3) General equation : a^ = r'^ + f-2rlcoH (9-01) [Centre at (I, a), radius a]. (4) Tangent : should be found geometrically as required. Note. The tangent at a to r = 2a cos is 2 a cos^ = rcos (5 2 a). 22. Radical Axis, Coaxal Circles. (1) If Si = 0, S2 = be the equations to two circles (in form ic^ + ?/2 + ... = 0), S1-S2 = is their radical axis. 166 ANALYTICAL GEOMETRY (2) The radical axes of the three circles 8i, S'g, So taken in pairs pass through a radical centre given by 8^ = 82 = 8^. (3) 81 + ^82 = represents all circles coaxal with Si and 82* (4) All circles which have with Si the radical axis u = are included in Si + \u ~ 0. (5) To find the orthogonal circle of Si, S^, S3, find the radical centre (h, k), and T the length of the tangent from {h, k) to Si [ 19 (5)], then the orthogonal circle is {x h)^ + (y-k)^ = 7^. (6) A system of circles having the same radical axis and line of centres {coaxal system) is represented by /p2 ^ ^2 ^ 2 A ic + c = 0. (A. variable. ) The axis of this system is ic = 0. The limiting points are ( + Vc, 0). The orthogonal system \s x^ + y'^ + 2 ^y + c = 0. [See also GEOM. 26.] Note. In problems involving two circles it is usually best to refer the circles to their R. A. [See 15.] 23. Centre of Similitude. (1) The C. S. divides the line joining the centres internally and externally in the ratio of the radii. For coordinates see 2 (a). (2) Common Tangent. To find the common tangent to two circles : (a) (x - hi) cos a + {y- k^) sin oc = ai is a tangent to {x - 7^l)^ + (/ - ^i)*^ = i^' Express that this passes through aC.S. or, {b) Express that the U on it from (/^g, h^ = > ^^^^ thus determine a. (.3) Circle of Similitude of {h^, k^, a^) and (/laj k^, a^) is (x-hiY+iy-kiY _ (x-h,Y + (y-k^y [See also GEOM. 27.] ANALYTICAL GEOMETKY 167 THE PARABOLA 24. Forms of Equation. (1) ^2 = 4. (IX. Axes : Axis, tangent at A ; semi-Iatus rectum 2a. (2) {px + qyf = mx + ny+l. General equation : See 51 (5). (3) f/2 = 4,a'x, Oblique axes : Diameter through P, tangent at P. a' = SP = a/sin^ cy. (4) X = a/m^, / = 2 a/m. The two together constitute the parabola. Here m is the tangent of the inclination to the axis of x of the tangent at any point. Other systems '. {a) x afjfi, y = 2a/i : gives the simplest formulae. (6) X = y^/^.a, y = y,. (5) Any parabola can be represented by i ^ "i?/ - % + 1 +Ci . [See also Intercept Equation ( 29) and Polar Equation ( 30).] 25. Chord. Mnemonic: The tangent is rn'a; my + a = [x fiy + afi^ = 0'\. If the work of the m's be, so to speak, divided between Wj and m^ [or ^1 and /^a] we get the chord. Cf. 52. , Pole. ,a( + ). ai"i/^2) ^(^i + A'a)* Mnemonic : Derived similarly from 24 (4) and 24 (4) (a). 26. Tangent. (1) 2/J/i = 2a(x + x,). Cf. Mnemonic in 52. The same form for y^ = ia'x. a (2) y = mx + . The m has the same meaning as in 24 (4). The condition that y = mx + c should touch the curve is mc = a. (3) Ihco Tangents : {y^-4ax){yi^-^ax^) = {yyi'-2a(x + x^)}^ Mnemonic : (Curve) (Curve) = (Tangent)^. 168 ANALYTICAL GEOMETRY 27. Normal. (2) m^y + m^(x-2a)-a = 0. y-^-fi (x-2a)-afx^ = 0. 28. Polar. The equation yy^ = 2a{x + x^) again repre- sents the polar of {x^, y^ and has the various meanings given in 20, Note (2). The !pole of Ax +By+C= is found by comparison with yy^ = 2a{x + x^). 29. Intercept Equation. If the curve be referred to two tangents of lengths 7i, Jc making an Z oj, V ^* "^ V 7i^ (1) Curve: ,/^+ ./ | = 1- (2) Tangent at {x,, y,): -^ + -^ = 1. This is not the polar of (scj, t/i). [See 20, iVo/e (1).] (8) Focus : hx = ky = h^ k^/{h'^ + k'^ + 2 hk cos cu). (4) Directrix : jc (/i + A; cos w) + y (/f + ^ cos '2 = a'^ + t^. 170 ANALYTICAL GEOMETEY (3) x'^y' = c\ Referred to the equiconjugate diameters, c^ = |(a'^ + &'^), (w = 2 tan-i - . Many of the circle formulae hold for this form. a (4) ic = a cos , y = & sin (/>. The two together constitute the ellipse, (p is the * eccentric' angle P'CA, where P' is the point on the auxiliary circle corresponding to P. This auxiliary substitution may also be used for 33 (2) and the equations for chord, tangent, pole, polar hold, but not that for normal. [For Polar Equation, see 39.] 34. Chord. (2) Chord (!, yi)ixc^t y^) is easily remembered in form a^ "^ i>2 - 2 + j,2 ^> which reduces to g(iri + a;a) y (1/1 + 1/2) _ ^^ . VxVi , , a2 ^ 62 - 2 + 52 + ^ 35. Tangent. (2) - cos (^ + 1 sin (/) (3) y = mic+ Va^m^ + V^. The condition that ?/ = wjc + c should touch the curve is c2= a^m^H- 62. (4) a; cos a + / sin a = + Va^ cos^ a + 6^ sin'-^ a. So that if p be the central perpendicular on the tangent 292= a2cos2a + b'sin2a. (5) Two tangents from {x^ , y^) : r ^ + ^' _ 1 u^' ^' _ 1 ^ ^ ( ^1 + ^ - 1 f . Mnemonic : (Curve) (Curve) = (Tangent) ANALYTICAL GEOMETEY 171 (6) Director circle, i.e. locus of intersection of perpendi- cular tangents : x^-^-y^ = a^ + h^. 36. Normal. a^x-x,) ^ hHy-y,) Not so much learnt as written immediately from 35 (1). C0S9 sm0 (3) {y-mxy{a'- + mH-^) = (a2_t2)2^2. 37. Polar of (iCj, p^). The equation^ + ^ = 1 again represents the polar of {x-^, y-^ and has the various meanings given in 20, Note (2). The pole of Ax + By+C=0 is found by comparison with it. a cos ^-^-^ & sin - Pole of. chord 0i , 2 : ? cos^^ cos^^ 38. Conjugate Diameters (2 a', 2 V), (1) a"" + fc'2 = a2 + fe2 . ^z^,/ sin a) = aZ>. (2) If a, a' be the inclinations to the iP-axis of two con- jugate diameters, tan a . tan a' = ^ . (3) If (^, (f)' be the eccentric angles of the extremities of two conjugate diameters, (f)^ -(f) = (an odd multiple of) ^ (4) The eguiconjugate diameters are inclined to the jc-axis at an angle tan-i - . [From 38 (2) , put a' = tt- a.] a (5) If (arij^i) {x^,y^)he the extremities of any two conju- gate diameters, the properties y^ y^ overleaf belong to the determinant : "b b 172 ANALYTICAL GEOMETEY (a) Sum of squares of any row or column = 1. (0) Sum of products of rows or columns = 0. (7) Whole determinant = + 1. 39. Polar Equations. (1) Curve: r = -, [or -1. 1 + e cos t^ L 1 - e cos e^ J (2) Chord {(X + fi,(X-^): r {ecoaO + sec^ .cos(0-(X)} = I. (3) Tangent at oc: r \e cos 6 + cos{d - oc)} ^ I. /iN TIT y A . i esina . ^^ . . ^ (4) Normal at QL: - = sin(^ a) + e sm^. ^ ^ r 1 + ecosa ^ ^ Proofs. Oi {2)'. Assume r[^cos0 + Bcos (0-a)] = Z, Of (4) : See 11 (3), or it can be easily derived directly from SG = e . SP. (5) Polar of (p, a) : ( ecos^ j ( ecosa j = cos(0-a). 40. CoNCYCLic AND CoNORMAL PoiNTS. If the ecceiitric angles of four points be a, (3, y, b. (1) If a, ^, y, b are concyclic, a + /3 + y + 6 = 2mr, and conversely. Hence (i) the lines (a, /3) (7,8) are equally inclined to either axis, (ii) The point common to an ellipse and its circle of curvature. Put a = /3 = 7, &c. (2) If a, 13, y, b are conormal, 0C + l3 + y + b={2n+l)T:, but the converse is not true. (3) If a, 0, 7 are conormal, S sin {& + y) = 0, and conversely. 41. Magnitude of Eelated Lines. (1) BC=^ h.AC=a, e2 = 1-^. (2) CS = ae,CA==a, OX = ~ . (3) Semi-latus rectum (Z) = = a(l -f^). a ANALYTICAL GEOMETKY (4) SP = a + ex, S'P=a- ex, SP+ S'P = 2a, SP.S'P= CD^ (6) PG=~ GB, CG = e'x [hence NG\ (6) SY. S'Z = BC, PG bisects ISPS\ (7) If SY^p, SP = r, then K,=:~ -1. 173 (8) Up be the perpendicular from G on the tangent atP, p.CB = ab, GP^+GB^ = a^ + b'^, GP. GBsinAPGB = db. ' (9) SG =ae + e'^x = e. SP. Hence A SKG III to SNP, the ratio of linear dimensions being e : 1, .*. GK = e . PiV= e/. Hence Z >SPG^ is known. PiC = I. (10) GT=-, Gt = -. (11) ()=- GB^/p = GB^/ab = d^V^lp^, wherei>isthe central 1' on the tangent. It can be found in terms of SY, SP from d/T 174 ANALYTICAL GEOMETRY 42. Curvature. (1) Radius : p = ' -^ ^~ = ^ ^L . [Origin C] a6 06'- [Seealso 41(11).] (2) Cen^e 0/ Curvature : ( cos^ (p, sin^ . Here (j> has a geometrical interpreta- tion. Let the tan. at P meet the major axis in T, and TQ drawn A.^ to this axis meet the auxiliary circle in Q, then Z QCT = ^. {2) X = a cosh (p, y = b sinh (p. (S) X = a cos (pj y = ib sin (p. This system is on the whole recom- mended, for it admits of any formula being written from the corre- sponding ellipse formula. Great care is needed in the interpretation of results.

. Pole <^i, 03. (a) X a sec (p^y h tan . -cos ^ a 2 >! + < sin b 2 = cos^^ r sin

2 2 (6) a; = a cosh (p, y b sinli (/>. a 2 b 2 cosh />i-<^a -coshd) - sinh(/> = 1. a ^ b ^ cosh (p sinh ^ a2 + 62. a cosh (pi + (p2 6 sinh (pi + ip.^ cosh^^^ cosh^^ 45. The Conjugate Hyperbola. - &2 = -1. Note. The point conjugate to Pon the original hyperbola, i.e. such that the tangent at it is || CP and the vector to it || to tangent at P, is imaginary. A real point D is taken on a line through C || tangent at P and such that i.CB = the distance from G of the true conjugate of P. The locus of D is the conjugate hyperbola and Z) is called the point conjugate to P. The following properties are true concerning it : (1) (7P2-C2>2 = a2-62. (2) Tangents at P and D intersect on the asymptotes. (3) Area GPD is constant. [See also a. CON. 25.] Note. Thus the minor axis of the hyperbola is not really CB but i . CB. 46. Magnitude of Kelated Lines. (2) CS = ae, CA = a, CZ = a/e. 176 ANALYTICAL GEOMETKY (3) Semi-latus rectum (l) = h'^/a = a(e^- 1). (4) SF = ex-a, S'F = ex + a, S'F-SF = 2a, SF.S'F= CUP'. Fig. 46. (5) FG=^ . CD, CG = ^x. [Hence NG.] (6) SY. S'Z = BC^ and FT bisects Z >SP^'. (7) If iSr =i?, >SP = r, ^ = - + L liote. = +0 as the curve is an ellipse, parabola, or hyperbola. p2 r +1 (8) If ^ be the perpendicular from (7 on the tangent at P, i?. CZ) = a6, CF^-CB'^ = a2-62, CP. CDsinlFCD = aZ>, and the area of the A cut off from the asymptotes by any tangent is constant. (9) SG = e'x -ae= e. SF, Hence A SKG III A SNF, the ratio of linear dimensions being e : 1, . * . GK = e . FN = ey. Hence Z SFG, FK = I (10) CT = a^x, Ct = y^/y. (11) Angle between the asymptotes = 2 tan~^ h/a. ANALYTICAL GEOMETRY 177 (12) p = CD'^/p = CD^/ah = a'^h^p'^, whereiJisthe central ^ on the tangent. 47. Hyperbola referred to Asymptotes (axes oblique). (1) Curve : xy = J {a^ + h^) = c^. (2) Tangent: - + ^ = 2. Note. The polar of (Xj, t/j) is not of this form. It is xy^ + yxj^ = 2c2 and not xy^ + yx^ = 2xi y^. (3) Chord (x^,y^){x^,y^)x ^ + -^ = 1. [Cf. Mnemonic in 25.] Note. It is very useful to note that in this form any point may be denoted by {d, c/t). The following equations then hold : Tangent at i : x-^-yt"^ ^ 2ct. Chord (i, #3) : x-{-yt^t^ = citi + t^). IMnemonic, 25.] Pole of Chord : 2 d, t^/{t^ + #2), 2 c/{t^ + fg). 48. Rectangular Hyperbola (= Equilateral Hyperbola). (1) a = &, e^ = 1 + -^ = 2. Conjugate also rectangular. (2) Curve : X^-y^ = a^ (referred to axes), a xy = c^ = - (referred to asymptotes). Note. Of these equations to the curve the second is most useful and the note of 47 becomes exceedingly valuable. The following formulae may be added : Normal at t: yt-xi^ + c(t*-l) = 0. Cmtre of Curvature att: {c(Bi*+l)/2 1^, c {i* + 3)/2 1} . Radius of Curvature att: c{t* + 1)^/2 1^. The normal meets the curve again in the point l/i^. The circle of curvature meets the curve again in the point 1/i'. 1372 N 178 ANALYTICAL GEOMETRY 49. Curvature. [See also 46 (12) and 48, Note.] (1) Radius : p = ^^ . (2) Centre of Curvature : ax = (a^ + 62) sec' <{>, htj = - (a' + h"^) tan' (p. [44 (a).] ax = (a2 + 62) cosh' (p, by = - (a2 + 62) sinh' <^. [44(6).] (8) Evolute: (ax)3-(6y)l = (a2 + 62). SOME GENERAL PRINCIPLES 50. If a = 0, /3 = ... represent straight lines and Si = 0, ^^2 = 0... conies, (1) Si = 0, S2 = 0, Ss = pass through a point if it is possible to determine m and n so that Si + mS2 + nSQ = identically. e. g. Si-S^ = 0, S2-S3 = 0, S3-S1 = pass through a point because (Si ~S2)+(Sa- 53^(53 -Si) = identically. (2) Any conic through all the intersections of Si = 0, S2 = may be written Si + JCS2 = 0. (3) Any conic through the points where a = meets Si = may be written Si + (lx + my + n)a = 0. (I, m, n arbitraiy constants.) (4) Any conic circumscribing the triangle a = 0, /3 = 0, y = can be written as jp/3y + ^ya + ra/3 = 0. (5) Any conic circumscribing the quad^ a = 0, /3 = 0, y = 0, b = can be written as ay = A; . /36. X^otes. {I) If a = a;cosa + r/sina p = the circumcircle is OCy = fc/38, the upper sign being taken if is outside the quadri- lateral. (2) Any circle may be written {x + iy){x-iy) + {2gx + 2/y + c)(i0.x + 0.y+l) = and therefore passes through the points at infinity {Focoids) given by xiy = 0, O.x+O.y + 1 = 0. (6) To find the ratio in which the line {x, y) (Xi, y^ is cut by the curve f(x, y) = solve in n/fii the equation J nxi + rtiX ^ nyi 4- n^y ~\ ^ ^ ^l, n^-ni * n + rii \ ANALYTICAL GEOMETRY 179 (7) To find the equation to all the tangents from (^i, y-^ to f(x, y) = express that the equation in (6) has equal roots. (8) To find the equation of lines drawn from (Xi, y-^ to all points of intersection of two curves substitute / ., , f^ ^ for X and y in both curves and eliminate h. (9) To find the length of the line drawn from (x^, y-^) in a given direction 6 to meet the curve f{x, y) = solve /(a?! + r cos 0, y^ + r sin 6) = 0. Applications : (1) To find locus of midpoints of || chords. Make roots equal and opposite. (2) To find directions or equations of two tangents. Make roots equal. (10) Five points completely determine an ellipse or hyperbola. Four points completely determine a parabola. TJiree ,, ,, a circle. Two ,, ,, a st. line. Instead of one or more of these points may be given an equal number of tangents to the curve, but in this case there will generally be several curves satisfying the conditions. GENERAL EQUATION OF SECOND DEGREE 51. Analysis. The equation is taken as S=ax^ + 2}ixy + &/ + 2gx + 2fy + c = {a,b,c,f,g,h\x,y, 1)2= 0. (1) If h = and a b it represents a circle. [See 18(4).] If the axes are oblique h = a cos w and a = h. (2) Then determine h^ - ah. If this is - the curve is an ellipse [circle, or point]. If this is the curve is a parabola [two coincident or parallel lines]. If this is + the curve is a hyperbola [two st. lines]. N 2 180 ANALYTICAL GEOMETKY (3) If h^ -ah is + or 0, test for two st. lines by factoriza- tion. [ 12 (3).] (4) If it proves not to represent two st. lines and h^ - ab is not zero, the curve is a central conic (ellipse or hyperbola). (a) Find the Centre. Write x+Xfor x, y+Y for y, and equate coefficients of x, y to zero. Solve for (X, Y) the cenbe. Or, Solve = 0, ^ =0. hx hy Note. If either of the coordinates x, y is oo , the curve is a parabola ; if indeterminate, two parallel lines. (6) Befer to Centre ( 16). The result referred to \\ axes through the centre is found by rule ; Leave unaltered the terms oftlw second degree and the absolute term and write HALF the coordinates of the centre for x and y in the terms of the first degree. The curve is now ax^ + 2hxy + by^ = c' . Note. If two St. lines have been missed at 51 (3) they will now reveal themselves by c' = 0. (c) Position of Principal Axes. They are inclined to axis of X at angles a (acute) and + a, which satisfy the equation tan29 = a-b' Note.Csin be derived by 12 (2), the asymptotes being ax^ + 2hxy + by'^ = 0. (d) Lengths of Principal Axes. If the equation referred to them is Ax^ + By^ = c', determine A and B by AB = ab-h^ A+B = a + b. [For Oblique Coordinates, 17.] (e) Discrimination of Principal Axes. To determine whether the major or the minor axis lies along the line 6 = oc (acute) if the curve is ANALYTICAL GEOMETEY 181 A Hyperbola : Draw a rough figure, remembering that the asymptotes are ax^ + 2 hxy + hy'^ = 0. An Ellipse : It will be the major axis if g' and h have opposite signs. Or, The lines joining to the intersection of x^ + y^ = r"^ and ax^ + 2hxy-^ by"- = (/ ure (a - ;^j x^ + 2 hxy + (b - ~A y"^ = 0. These coincide if r is a semi-axis. Hence semi-axes satisfy (-,^)(-^) = -- ^ and the equation to the semi-axis of length r isl^a -^x-V^li '^ 0. This method is applicable to oblique coordinates, writing x2 + 2a;t/coscu-f-?/2 = r^, &c. (5) If it proves not to represent two st. lines and li^ - ah is zero it is a parabola. In this case it may be written (px + qyf mx + ny+l ; i. e. [px + qy + hf = (m + ^pTc) x + {n-ir 2qk) ^ + 1 + F. Choose h so that the lines px + qy + h = 0, (m + 2ph) x + (n + 2qk)y+l + W' = are at rt. angles and take them for axes of x and y respectively. By finding Y, X, the perpendiculars on these lines, the equation is written in the form Y^ = +AX. Note. The ambiguity is best determined by a rough figure, but see 9, Note. 52. Tangent at iCi, /i to ^ = is T= axxj^ + h{xyj^ + yx^) + byyi + g(x + x^)+f{y + yi) + c= 0. Mnemonic. Divide the work of the ic's in S, so to speak, between X and x^, and the work of the ya between y and y^. Cf. 25. Polar of (iCi, yi) : Same form as jf = 0. Chord whose middle point is (x^, y^): T = S^ where Si is the result of substituting Xi, y^ for x, y in S, Two tangents from (x^, y^ ; SS^ = T^. Mnemonic : (Curve) (Curve) = (Tangent)'*, 182 ANALYTICAL GEOMETRY 53. General Results. S = [a, h, c, f, g, h X ^, y, 1)^ = ^' ' a^tg j If A = hhf I and A, B, C, ... are the minors oia,h,c, ..., \9fA (1) Tangent^i(X, Y, Z = 1) ) hS bS , JS ^ Polar I'^hX^^hY^'hZ-''' Two Tangents from (Z, T, Z) : iS(x,y, ^)S(X, r, Z) = (:r + 2/ gy + ^ g^^ j (2) Centre: S^ ^0, Sy = 0. These lead to x/G = y/f' = 1/C. (3) Foci : ^^ = ^^ = 4>S. [Cf. 12 (2).] These work out to j ^(-^-^/)-2 G. + 2Fy + A-B = 0, I Cxy-Fx-Gy + H = 0. These two R.H.'s which intersect in the foci are the loci of points the two tangents from which to the conic make complementary and supplementary angles with the axis of x respectively. Proof. Express that two tangents [ 52] from foci satisfy the con- ditions for a circle [ 55, Note 2]. Note. The ellipse - + - = 1 has four foci (+ ae, 0) (0, + aei) and two eccentricities The hyperbola^ - ^ = 1 has four foci (+ ae, 0) (0, + aei) and two eccentricities, both real, Vi^^' Vi.^. (4) Axes: ^lz_ = ?%^. ^ '' a-h 2h (5) Asymptotes: aSy'^-\-bSx'^-2hSySx =^ 0, or CS = A. Conjugate Hyperbola : CS = 2A. (6) Director Circle: C{x'^ + y'^)-2 Gx~2Fy + A + B ^ 0. For the directrix of the general parabola put C ^ 0. ANALYTICAL GEOMi^TRY 183 (7) Eccentricity : e* + ^"^'^^"^Jj^- ('-l) = 0. (8) Axes{2r) : C'jHC A (a + &)r2 + A^ = 0. (9) Ix + my + n = touches S if {A,B, C,F, G, H\l, m, nf ^ 0. Note. This may also be written by equating to zero the determinant formed by ' bordering ' A with I, m, n, and zero. (10) i/ = mx, y = rufx are parallel to conjugate diameters ^^ a + h{m + rnf) + Immf = 0. 54. Invariants. I. Of a single Conic ; (1) When the axes are rectangular, to turn them through any angle whatever does not aifect the values of a&-/^^ a-vh, P^g", c, A. (2) When the axes are oblique no transformation affects ,, , ^db-h^ , a + &-2/icosa) , . , ^ni the values of ^-^ and r . Frooi : 17 . sirrii) sin^o) - ^ j II. Of two Conies : If S = 0, S' = be two conies, S + A.S' = will be a pair of straight lines if a + Aa', h + \h^, g + \g' h + Kh', h + Kh', f+Kf = 0. Sf + A/, /+A/, c + Ac' If this equation be written A + A- O + A^e' + A' A' = 0, the ratios A : : 0^ : A' are unaffected by any change of axes. = Aa' + Bh'-\-Cc'-\-2Ff-\-2Gg'-{-2mi,' ', @' = A' a + B'h + C'c + 2 F'f^- 2 G'g + 2 H'h. (1) The condition (necessary and sufficient) that the conies should touch is that the A-equation should have two equal roots. (2) The condition (necessary and sufficient) that the conies should osculate is that the A-equation should have three equal roots. (3) The condition (necessary and sufficient) that the conies should be such that a A circumscribed to S will be inscribed in S' is 2 = 4 A '. If there is one such A there is any infinite number of such. 184 ANALYTICAL GEOMETRY PARTICULAR CONICS 55. Conic ^y = h(X^ touches /3 = 0, y = 0, where a = meets them. Notes. (1) Any parabola may be written t/2 = (0 . x + .y + ia) x, .'. touches line at oo . (2) If the origin is at the focus and D = be a directrix, the conic is x'^ + y^ = \D^, .'. x + yi = OfXyi = are two tangents from focus. These pass through focoids, .*. all confocal conies have four imaginary common tangents, and two opposite vertices of 4' thus formed are foci. Application, If a 4* whose sides are L = 0, L' = 0, M = 0, ]^ = 0, in circumscribed to conic, and if R = joins the contacts of L and M, and iJ' = those of L'M', then R and R' pass through intersection of diagonals of 4'. Conic may be written LM-R'^ = 0, also L'M' R'"^ = 0, .-. LU-W- = X (L'Af'-i2'2), .-. LU-X . L'W = R^-X R"^, .'. LM-X.L'M' = is two st. lines, passing through points (L, i') (L, M') {M, L') {M, Jf'), .'. the two diagonals. It also passes through {R, R'). 66. CoNics S = hoc^, S = kcx^ (1) S = kOijS passes through the four points where a = 0, ^ = meet S = 0. (2) S = kOL^ touches S = where a = meets it. Notes. (1) From 55, Note (2), the focus may be regarded as an infinitely small circle having double contact with the conic, the chord of contact being directrix. (2) All similar and similarly situated conies have double contact on the line at go . Application. S+L^ = 0, S+M^ = 0, S+I^ = represent three conies having double contact with S = 0. Their chords of intersection are L'^-M^ = 0, M^-N^ = 0, N^-L^ = 0, and are concurrent in threes. If the conies S + L^ = 0, &c., are all line-pairs we have Brianchon's Theorem. [See 65 (3).] 57. Conic through Four Points. (1) Conic through (5, 0) {s% 0) (0, t) (0, f) is ^^ r, y^ /I 1 \ /I 1 \ ^ ANALYTICAL GEOMETRY 185 (2) Or, the points may be defined as the points where ax+hi/=l, afx ^'b'y^X meet the axes, and the conic is then {ax + &^ - 1) {a!x + Vy - 1) = ^xy. (3) Or^ taking any axes, the lines joining the points may be found and 50 (5) used. 58. Conic touching Axes at (s, 0) (0, t) is - + f - 1 = 'JTxy. s t For conic touching two given lines at given points use 55. 59. Conic touching Four Lines. Take lines as re = 0, y = 0, lx + 7}iy = 1, Vx + m'y = 1. Any conic touching X = Of y = is {ax + hy - 1)'^ = 2\xy. Determine a and b by remaining conditions of contact. Note.\ = 2(a-l)(b-m) =^ 2(a-l') (b-m'). 60. Concentric Conics may often be best treated by referring them to their common pair of conjugate diameters and writing them ax^ + hy^ = 1, a'x^ + Vy'^= 1. 61. CoNFOCAL Conics. X 11 X n (1) ^ + ^;2 = 1 ^t2i + yM,ilf+JV'=0 is M^^iLK (2) The envelope of Pcosa + Csina = B is F^+Q'^ = B\ [See DIFF. CALC 24.] CONTACT OF CONICS 64. (1) Take the conies in the form ax^ + 2hxy + l)y^ + 2gx = 0, [Contact aV + 2h'xy + hY + 2g'x = 0. First Order.] Note. This form of equation, referring the conic to a tangent (a; = 0) and normal (y = 0), is often very convenient and should be carefully noted. The line through the other two intersections P, Q is (ab' - a'b) x + 2 (W - h'b) y-2 {bg' - Vg) = 0. If lies on this line, bg' - Vg = 0. [Contact Second Order.] The equations to OPj OQ are (a/ -a'g) x^ + 2(hg' -h'g)xy + (bg' -b'g)f = 0. If OP, OQ coincide, (V - h'gf = (ag' - a'g) (&/ - b'g). [Double Contact.] If OPj OQ coincide and lies on P, 7i _ & _ /7 [Contact V ~V ~ Y' Third Order.] Hence equations to circle of curvature, centre of curva- ture, p, osculating conic, &c. [See 32, 41 (11), 42, 46 (12), 48(JVo/e), 49.] ANALYTICAL GEOMETKY 187 (2) Or thus: Let jS = be a conic, (a/, y') a point on it, and T = the tangent at (x\ y'\ Then B-\T {{y-y')-m{x-x')\ = is a conic inter- secting >S = in {x\ y') at three consecutive points, and m, A may be chosen so that this is a circle or otherwise. ^-AT2 = has Contact Third Order. SOME GENERAL THEOREMS 65. (1) Carnot's Theokem. If a conic meet the sides of a A in a, a' ; ?>,&'; c, c', Ba . Ba! Cb . Ch ' Ac. Ac' _ Ca.Ca''Ah.AV'Bc.Bc'~ ' Mnemonic B/C. C/A . A/B =1. Note. Hence describe a conic, given four points and one tangent. (2) Pascal's Theorem. If a hexagon be inscribed in a conic the crosses of opposite sides are collinear. Notes. (1) There are sixty Pascal Lines corresponding to six points on a conic. (2) By this theorem, (a) Draw a tangent at a point on conic by ruler only ; (/3) Describe a conic through five points. Proof. If the vertices be a, 6, c, d, e, /and a& = denote the line ah, then conic may be written ab .cdhc. ad = (for it circumscribes a, b, c, d) and also de.fa ef.ad = (for it circumscribes d, e, f, a) ; .*. ah. cd de .fa, = ad(bc ef). L. H. of this equation represents conic circumscribing ah, de, cd, fa. But it splits into factors, . '. is the two diagonals of this 4\ But ad is one diagonal, .*. 6c e/"= joins the vertices (ab, de) (cd^af). But it clearly passes through (6c, e/"), . &c. (3) Brianchon's Theorem. If a hexagon be described about a conic the joins of opposite angles are concurrent. [See 56 (2) Application.'] Notes. {1) There are sixty Brianchon Points corresponding to six tangents to a conic. (2) By this theorem describe a conic touching five lines. 188 ANALYTICAL GEOMETRY ANHARMONIC RATIO [See also GEOM. 20, 22, 23, 24, 31, and G. CON, 32. Some theorems are repeated here for convenience.] ee.(l){PeiJ^}=g.f = -||if5beat=c. xO. PQRS, = ^^j^Q^ sin I SOP ' ^^^^^^ if these points are on a line. Mnemonic mote order PQ, QR, RS, SP. (2) If {PQHS} = - 1, P, PjR, and PS are in H.P. and we have a Harmonic Range. (3) If the abscissae of P and B be given by ax^ + 2hx + h = 0, If the abscissae of Q and S be given by a'x^ + 2h'x+h' = 0, {PQRS\ = -1 if aV + a'h-2}iU = 0. (4) The A. R. of the pencil y = kx, y = Ix, y = mx, y = nx is 7^ r 7 z- > whatever the Z between the axes, and the same result expresses the A.R. of a = kjS, a = l^, oc = ml3j (X = n^, where a = 0, /3 = are any two lines. (5) Each of the three diagonals of a 4^ is divided har- monically by the other two diagonals. (6) The A.R. of the pencil formed by joining four fixed points on a conic to any fifth variable point is constant. (7) The A.R. of the range formed by cutting four fixed tangents to a conic by any fifth variable tangent is constant. (8) If a line through a point meet a conic in B and S and meet the polar of in P, {ORPS} = - 1. ANALYTICAL GEOMETEY 189 (9) Two tangents through any point, any other line through the point, and the line to the pole of this last line form a Harmonic Pencil. (10) In deriving theorems from these properties, especially (6) and (7) : 'The four points on the curve may be any whatever, and either one or two of them may be at oo : the fifth point to which the pencil is drawn may be either at an infinite distance, or may coincide with one of the four points, in which latter case one of the legs of the pencil will be the tangent at that point. Then again we may measure the A. K, by the segments on any line drawn across it, which we may, if we please, draw || to one of the legs of the pencil and so reduce the ratio to a simple one.' 67. HoMOGRAPHic Division. [See GEOM. 32.] (1) Points P, P'; Q, Q^; .-. in a st. line form a system in involution when OP. OP' = OQ.OQ'= ...= OK^ = OK'^. is the centre ; if, IC the double-points (foci), while P and P' are conjugate points. (2) {PQRS} = [P'Q'R'S'}. (3) {KPK'P'} = -L (4) (a) A system of points P, Q, B, S, ... on one line is homographic with a system P\ Q\ Pf, S\ ... on another line when for my four {PQRS} = {P'Q'P'S'}. (&) PP\ QQ\ PR', ... all touch the same conic, (c) nomographic systems on the same st. line are in involution. AKEAL COORDINATES 68. Cartesians into Areals. Ois {n^,n^. Areals and Trilinear. _ aoL _ ^/^ ^ ^y ^"2A^ ^~ 2A^ ^"2A' x-\-y + s = l', aa + &/3 + cy = 2A. 190 ANALYTICAL GEOMETRY 69. Relations op Points. (1) The point dividing the join of {Xi,y-^ , ^i) {x^, j/2, ^2) ^^ the ratio fii : n,2 is x= ^-^ ^ \ w=&c., ^ = &c. (2) The distance of the two points is given by (3) ^rea of Triangle (x^, y^, ^J (ajg, ^2 ^2) fe. ^3 ^3) is ^1 Vi ^1 smo). 3 /3 ^;! (4) Special Points, Coordinates of : Centroid : 1:1:1. Circumcentre : sin 2 A : sin 2 J5 : sin 2 C Orthocentre : tan A : tan B : tan (7. Incentre : a:b:c. -4-excentre : a:h:c. N. P. Centre : sin A cos B C : sin B cos C ^ : sin C cos ^ 5. Symmedian Point : a^:b^ : c^. 70. The SxRAiaHT Line. (1) The Line Infinity : x + y + ^ = 0. The more correct form is x + y + z = Lt^ -q e (Ix + my + nz). See -4sS:- W7i7;i, 265. (2) General Equation : lx + my + n0 = O. Perpendicular Form : px + qy + rz = 0, where p, q, r are X^s from A, B, C on it. To write a line in this form : Divide throughout hy ^ A/sq. of line with ^'-substitution after squaring. Note. The phrase ' S-substitution ' is used throughout this section to indicate the following : For a;2 y^ z^ yz ... Write a2 h^ c^ -6c cos -4.... ANALYTICAL GEOMETKY (3) Johi of{x^, ?/i, ^^i) (x^, y^, ^2) ' X y z H y\ ^1 H ^2 H 191 = 0. iS^^ line through [x-^, y^ , z-^ parallel to Ix + my + nz = 0. = 0. ^i ^1 -^1 7n n n l l m Proof. Forit passes through the point (Ix + my + nz ^ 0, x + y + z -r- 0). /^x ^ jw -TTi 77 7- . {Ix + my + nz = 0) (4) Condition of Parallelism of \ ^, / , ^\ ^ ^ '' '^ Wx^- mv + nz = > P = z m w Z' m' w' 1 1 1 = 0. Condition of Perpendicularity : 8 = (product of lines with ^S'-substitutions after multi- plication) = 0. (5) Angle between two lines: tan = 2AP/5. [See (4).] (6) Perpendicular Distance from (X, Y, Z) to px + qy + rz = [see (2)] is pX ^ qY + rZ. 71. Anharmonics. (1) If the lines l-^x + miy + niZ = 0, Ic^x + m^y + n^z =^ 0, l^x + m^y + n.^z = 0, ljiX + m^y + ^4^ = intersect, then their A.R. is h m. r m. h m^ h w^4 h ^2 h m^ h W3 h mi (2) The A. R. of the pencil u = 0, m + /x^ = 0, t; = 0, ^ + f/v = is fi/fi for any line coordinates whatever. 72. Choice of Triangle op Eeference. The A will usually be some triangle mentioned in the problem. For N. P. C. the A formed by midpoints of sides or the pedal triangle is often convenient. 192 ANALYTICAL GEOMETEY For a complete 4^ the A formed by the three diagonals, in which case the sides may be written lx my nz = 0, or the vertices taken as (+/, g, h). For A's ^i-Sj Oj, A2B2 C2, ... in perspective, take the centre of perspective at the centroid of the A of reference and let the vertices of this A lie on OAi, OB^, OC^, so that OJ-i i& y = z, &c., then the equations to the sides of any perspective A A^B^C^ can be taken as y-\-z + w^ = 0, z + x + w-^=^ 0, x + y + w^ = 0. Change of Triangle of Reference, Let (a^, ftgj %) (^i> i^2> ^3) (^i> ^2 J ^3) ^6 the coordinates of the vertices of the new triangle, then the old coordinates in terms of the new are given by X == a^X + hj^ Y+c^Z, y = a^X+h^Y+c^Z, z = ^x + fe;^ r+c^z. 73. General Equation of Second Degree. ={A,B,C,I),E,F\x,y,zf = ^. 1 = (f) after the /S'-substitutions have been made. A F E F B B E B C K = the discriminant ' bordered by the line infinity and zero ' = H= the discriminant A F E 1 F B B 1 E B C 1 1 1 1 (1) Tangent a.t (X, Y, Z): xcfy^ + ycpy + zcp^ = 0. Polar oi(X, Y, Z): x4^^ + y(f)y + Z(f)^ = 0, provided (j) is of the second degree. Notes. {1) Beware of applying this formula to find the polar of a^/x+b'^/y + c'^/s = 0, or of \/tc + \/my + \/ns = 0. It will give the tan- gent but not the polar [see 20, Note (1)]. (2) The polar form is also that for the clwrd of contact of tangents from (X, Y,Z). 0. ANALYTICAL GEOMETKY 193 (2) Condition of Tangency of Ix + my + nz = 0: Equate to zero the discriminant bordered by the coefficients of the line and zero. (3) Fair of Tangents from {X, Y, Z) : 4(/) (X, y, 0) {X, Y, Z) = [xcfy^ + yc\>y + zcl>^)\ (4) Centre : (j)^ = (f)y = (p^ provided (f) is of the second degree. Note. Beware of following the Cartesian analogy and writing ^x = 0. ^x ^y ^z (5) Conjugate of Ix + my + w^ = is I m n 111, Conic tvith Ix + my + nz = 0, Vx + m'y + n'z = for con- jugate diameters is A {Ix +my + nzY + B {Vx + m^y + nzf = 1. (6) Asymptotes : 4*+ ^{x + y-hzf = 0. Co-asymptotic Conies : (f) + Jc{x + y + 0y^ = 0. Conies with asymptotes \\ to those of (j) : (f) + {x + y + jc!){lx + my + nz) 0. (7) Foci of Conic: Express that two tangents from (X, Y, Z\ a focus, satisfy the conditions of (8) (&) below. Directrix : Polar of Focus. Centre, see (4). Axes : For equations compare 53 (4) and 53 (3). For magnitude : If a, 6 be the semi-axes 2AH , . ,, IH ab - , a2 + &2= - t^. 3 > - - j^2 (8) Analysis, {a) Two Straight Lines : H = 0. (c) Parabola : K = 0. - 1372 194 ANALYTICAL GEOMETRY {d) liectangtdar Hyperbola : 7 = 0. (e) Hyperbola: -fiT positive. (/) Ellipse', -ff" negative. 74. Circle. (1) Circumcircle: a^/x + b^/y + c'^/^ = 0. [See 75.] (2) General Equation : c?yz vb'^zx^- c^xy = (?ic + my + nz) {x-\-y->rz) = {p^x + (f'y + r'^z) (it- + / + z\ where ^?, 2, r are the tangents from J., 5, to the circle. (3) Tangent from (X, T, Z) : r2 = (^2js(; + 22y+,.2^)(x+r+z)-a2rz-62zx-c2zr. (4) Circles concentric with <\>'. + A; (re + ^ + ^j'-^ = 0. (5) Condition (f) is circle : See 73 (8) (6). (6) Special Circles. Incircle : V (s - a) a; + y^s -b)y+ V(s -c)z = 0. A-excircle : \/sx + v (s c) t/ + v (^s &; s = 0. K.P.C: 2aV(t/ + s-a;) = 0, Self-Polar Circle : ^a^cot ^ = 0. [Centre : the orthocentre.] Notes.^l) For Feuerbach's Theorem [GEOM. 17 (5)] find [by 74 (2), pqr-form] 2 a;/(b c) = as R. A. of N. P.O. and incircle, and show that it touches incircle. The point of contact of the two is (p^cy (s-a) : (c-ay (s-b) : (a-hy (s-c). (2) The common R.A. of N. P.O., circumcircle, and self-polar circle is 2a; cot ^ = 0. 75. CiRCUMCONICS. (1) Curve : l/x + m/y + n/z = 0. Note. The form lyz + rmx + nxy = must be used to find the polar or two tangents. (2) Chord fe, y^, z{} {x.^, y.^.z^) : Ix/x^x.j^ + my/y^y^ + nz/z^z^ = 0. (3) Tangent (x^, y^, z^ : Ix/x^^ + my/y^ + nz/z^ = 0. ANALYTICAL GEOMETRY 195 (4) px + (2y + rz =^ is tangent if Vlp + \^mq + \^nr = 0. (5) Analysis : Gircle iil:m\n = a^\})^\ c^. Parabola if VI+ Vm + s/n = 0. H.B.. if it passes through the orthocentre. 76. Inconics. (1) Owrve : \/lx + \/my + \/nz = 0. Uoie. There is an ambiguity before each root, and they cannot all be taken as + . If the conic touch the sides in D, E^ t\ different signs will be needed for the ambiguities for the parts of the conic from E to F, from F to D, and from D to F. (2) px + qy + rz = i^ B. tangent if - + + - = 0. Note. Hence curve is a parabola if Z+m + n = 0. (3) Centre : m + n',n-{-l:l-\-m. 77. Self-polar Conics. (1) Curve : Ix^ + my^ + ng^ = 0; (2) Tangent at or Polar of {x', y', z') : Ixx' + myy' + nzs' = 0. (3) Any two Conics may be referred to a common self- polar triangle [6r. CON. 31 (2) J and therefore written Ix^ + my^ + w^2 = 0, Vx^ + my + n' z'^ = 0. (4) Auxiliary Angle may often be usefully employed. \^l.x = i \^n . z cos (f), Vm > y = i Vn . z sin . {p) Analysis : Circle : I: m:n = cot A : cot B : cot G. Parabola : l/l+l/m + l/7i = 0. R.H.i iaHwbHwc^ - 0. 78. Double- CONTACT Conics. (1) Curve : x^ = Jcyz. (2) Any point may be taken Sks kfx:kfx^:l, (3) Tangent at n : 2 fix ^ y + kn^e. (4) Pole of\, IX : {ix + \)/2 : fiX : 1/k. (5) Analysis : Parabola : /c = 4. R.H.: k =^-a^/bc cos A. o 2 196 ANALYTICAL GEOMETRY 79. Tangential Coordinates. (1) The tangential coordinates of a line are the perpen- diculars p, q, r from the vertices of the A upon the line. (2) Invariant Belation. p^a^ + q'^b'^ + r^c^-2qrhccosA-...= 0, or a^{p- q) {p-r) + h^ (q -p) (q-r) + c^ (r-p) (r-q) = A A^. (3) Equations : Circumconic : Vlp + Vmq + \/nr = 0. Income : l/p + m/q + n/r = 0. [For Polar Beclprocation and Projection see G. CON. 35-6. For Inversion see GEOM. 33, and DIFF. CALG. 21 (2).] DIFFEKENTIAL CALCULUS 1. Elementary Laws. ,.. d{cu) _ du ^ ' dx ~ dx (2) (3) d{u v) dx d{uv) dx dii dv dx' ~ dx dv du dx dx Learn in words : ' Diff^ product = First x diff^ sec. + Second x diff^ first.' <) w-^ = du dx dv dx Learn in words : Diffi fraction = DENR. X diffi num^ - numr. x diffi den^ , du _ du dv ^ ' dx dv dx ')S=V: dx du DENR. 2. Standard Forms. Form, c a"" logic a^ log^ a 1/x Form. sin a; cos a; tana; cotic sin~^ X tan'^ic cos a? -sinic - cosec^ X 1/(1 + a;2) 198 DIFFERENTIAL CALCULUS Notes. (1) \ogaX = logae/x. ax (2) sin-ix + cos-^a: = tt/2. Hence sin-^a; So for cot"*. Standard Forms (for Reference). (IX Form. D'ff'. Jbrw. BiffK sill X sinhic cosh a; tanho; cotha? secha; cosech X cosh X sec X cosmic cos if sinhic cp/*"l /)f sin^ic 1 sech^ X vpri~^ f 1 -1 - cosech^ X V2x- -rc2 sinhiP cosh^ X cosh a? sinh^o; Form. Equivalent. sinh" ^x = log{x+ V1 + xv cosh" ^x = log (re + ^/x^ - 1) Dir- 1 1 tanh"^ic = ^log- 1 + iC Vx^-1 1 coth ^ a; = tanh il sech "^x = cosh ^- cosech ^ X = sinh~^ l-a;2 1 x^-1 1 (^<1) (^>1) a;\/l-a;2 1 X Vx'^ + 1 DIFFERENTIAL CALCULUS 199 3. Methods of Differentiation. (1) u", mnv, &c. Take logs. (2) Note such transformations as : ^ 2X _. , ^ ,1+X TT . , tan~^z s=2tan ^x-Aan'^z = j + tan ^ or. 1-x^ 1-x 4: 4. Partial Fractions. (1) If the numerator is of equal or higher degree than the denominator divide out by the latter. (2) The partial fractions for f{x)/{x - ay {x^ -i-hx + c) are A, ^2 . 4. ^r . B,x + C^ (x-ay {x-ay-'^ '" x-a {x^ + hx + cY x'^ + hx + c* and the values of A, B, G's may always be obtained by multiplying up and equating coefficients ; but this is never the best method, though one or more of the coefficients may usually be obtained this way easily. (3) Unrepeated Linear Factors. ~X = ^ 4/ , ^ ^ ^ F(x) F\a) x-a Where f(x) is of lower degree than F (x), F(x) = [x-a^] (x-a^) ... (x-a^). This rule may be expressed thus : ' If F(x) contains a factor x-a the P. F. corresponding to ic-a is A /[x-a) J where A is the result of putting x = a in every part of the original fraction except x-a itself.' l^otes. (1) If no factors are repeated this at once gives the expression of the fraction. xc a c 1 & c 1 ix a) (x b) ~ a b x a b-^a x b (2) The text-books usually give the rule as applicable only to un- repeated factors. As a fact, however, it is true if for x a we read (x - a)'", and gives the coeflft. for the highest power of the repeated factor. 200 DIFFERENTIAL CALCULUS rr-S 1-3 Ex. ix-lf{x-2) 1-2 {X- can be determined by (2) above. {Ans. A = 1.) (3) And this method is the quickest if no factor is repeated more than twice. (4) Hepeated Linear Factors. Fraction = f{x)/{x - a)^ {x - ly' F (x). Put X ^ a + 7i in the terms which do not contain [x - a) and expand f(a + h)/{a -h + hY'F(a + h) in ascending powers of h. Contract the division so as to give the first m coefficients only, Ai, A21 ... A^, then the P. F.'s corresponding to {x - a)*" are {x-af^ [x-ay^'^ '" x-a' Then put x = h + h and work similarly for the terms corresponding to [x - &)**. Note. Application of (2) will often now complete the decom- position. (5) jRepeated Non-Linear Factors. 3? ax + h cx + d ex+j ^' {x'^+iy(x^-x+l) " (x2+l)2 "^ c^+1 "^ a;2-x+l ' Hence a^ = (ax + h) {x-^-x+l) + {cx + d) (x^ + 1) (a;2-x+ 1) + (ex+f) {x'^ + l)\ Put a;2 = 1, a? = x and reduce to x = a hx ; .-. a = 0, 6 = 1. Put x^ = x 1, a^ = 1, &c., and prove similarly e = 1,/= 0. Use (2) to deduce c = d = h But for practical applications (differential and integral calculus) it is usually necessary to employ the complex linear factors of these non-linear expressions. 5. Successive Differentiation. (1) D^ic^ = n{n-l) ... {n-r+l)x''-^. (2) D^ I ^^ . Use partial fractions if f{x), F{x) are integ. algeb. functions. DIFFERENTIAL CALCULUS 201 sin (4) D^ = - 1 when operating on x ) D'^ = - w^operat- sin ..^, sin .. ^. sin ins on mx ; fW^) mx = / ( - m^) mx. ^ cos ' -^ ^ ^ COS '' ^ ^ COS (5) Leibnitz's Theorem. ^.d^u dv d^'^u ^ \- n r + . . . . dx^ dx dx""-^ Proof given of this should be by induction. In giving this proof take care to deal with the general term. Note.B^ == D-D.,; .-. D," = (D-D^)^ ; cV*u d'^iuv) d^-'^{uv) dv " ^do^ ^ dx^ ** f7x"-i * dx + .... (6) D^ [e^ . Z] = ea^ . (D + a)^ X, and, in general, f(D) \f^ . X] = e^ . /(D + a) X. (7) sin^ X cos^ ^. Turn into multiple angles. tan-i^^^. After first diff^ split into P. F.'s and c use De Moivre. x"^ sin arr, a;^ sin*^ x cos*^ iP. Use Leibnitz. (8) [2)^2/]x = o can often be found by forming a diff^ equation. Exx, y = [sin-ix]2, [sinh-ia;]^, sin (msin-ia;), eitan-ia;, &c. 6. Expansions. (1) When f(x) changes sign with x [e. g. sin x] only odd powers of x occur in its expansion ; if it does not change sign, only even powers. dfix) d^f(x) (2) If the expansion of ^^ or of -^H is known [e.g. f[x) = e^, sinic, sin'^rr] we may obtain expansion of f{x) 202 DIFFERENTIAL CALCULUS by assuming /(a) = aQ + aiX + a^x"^ + ... and equating coeffi- cients in df{x) ^ = ai + 2a2X + Sa^x^ + ... = Known Series. (3) Taylor's Tlieorem. f(x + h) -/(rr) + V'W+ ^r(x)+...+ ^fn{x + eh). Eemainder in Taylor*s Theorem. h^ Lagrange's Form: ^/^(x+OJi). [0 a prop. frac. Cauchy'sForm: ^ /^ ' /^{x + Oh). ScMomUch's Form: /,, , ' f^(x + Oh). (w - 1) ! . (i? + 1) *^ ^ ' = Cauchy, i? = 0, = Lagrange, p = n-1. Proof. Let (p{cc + h) = {x) + hil/ (x) + ... + />-i {x) + - E. Consider the function u = (p(x-^h){y)r(y)+f-^[{cl>{y)}^riy)]+... Ex. Let X = ae. Here y = 0, (x) = e*. Let F (x) = x, F^ (x) = 1, = a + 2aV2! +...+w-i. a/n ! +..., a series which is convergent if a < 1/e. Hence, e.g., the solution of lOx = e* is x = 111832.... (9) Laplace's Theorem. If x = f {y + a tti ~z-r^ J where y is a constant which may or may not be zero. (P(x) "^ - J . /> /ft If we put / {x) = -j-T-T = a, the required problem is now to express F{x) in powers of a. This is effected by Lagrange. The name is sometimes applied to the particular case: If (p~^ (x) be the inverse function of (f) (x) so that (^(^~^ [x) = x, and if cf) (0) = 0, Note. In order that the answer may have a simple form /(a;) must be such that it may be written as {x y)/

S5 -gi + ..., secic= 1 +/S2 'pi + ^^4 "71 + ' where 6^1, 6^3, ... are 'prepared Bernoullians ', and S^^ S^, ... are Euler Numbers. (2) To evaluate the numbers : X x^ y = sec ic + tan x = 1 + 5i T-y + ^2 ' ol + > . . / cos ic = 1 + sin X. Differentiate n times and put x = 0, ^" 9I ^n-2 + Ti '^-* '" "n nil . nn + cos-2- = smY DIFFEKENTIAL CALCULUS 205 (3) Si = 1, Sa = 1, S, = 2, S4 = 5, So = 16, Sg = 61, S7 = 272, Sg = 1385, Sg = 7936, Sjo = 50521, &c. (4) The relation connecting /Sg^-i with the ordinary Bernoulli Number is (5) The Bernoulli Numbers are : B^ = B^ = Bq = ... = 0, Bq = ^, Bi = 1, -63 = 3^0, -Bg = ^, ^7 = 3^, Bq = e%, 11 = ^V -^13 = h V _3617 R _ 43867 rp .6 5"T0 -"17 798 > ^^' After g they continually increase to a limit 00 . There is un- fortunately divergence among mathematicians as to the notation. What is given above as B^n-i (following Edw^ards) is often (Chrystal) taken as B,j, sometimes as B^n, while German writers do not take them all as + . (6) The following results are added for reference : -(-|)"H5)"H-0"--Ki)"(^:- Hence the sum of 1 + (1/2)2 "+(1/3)2 "+(1/4)2 +.... 1 e^+1 1 B^x _ BsO^ B^ ^ 1 22 2* cothx= - + g-j JBiX- B^x^ + ..., 1 22 2* o cotx = BiX--BsX^+.... X 2\ 4 ! 1 2(2-1) 2(23-1) - cosecx = ^ + -^2! ^''''^ "Vi ^3^'+ .. 22(22-1) 2* (2^-1) tanx = 2! -^^^+ ^-^T"^^^ ^ *" ' "' ' - 2^(22-1) 2*(2^-l) . tanh X = ^- -' 1 X 1 i 3 x^ + . . . . (c) The following may also be expanded in Bernoullians : log (sin x/x), log (sinh x/x), log cos x, log cosh x, cosec2 x, &c. (d) l-+2'- + 3'-+...+n'- = ^ [{n + sy+^-B^+^l (1 + B)" -n + B" where after expansion B^ is written for jB^ and i"-2_i for B". 206 DIFFERENTIAL CALCULUS 8. Partial Differentiation. (1) If u = (f) [Xi^, X2, ...) where x^, x^, ... are functions of X, or of some other variable, ^x^ ^ ^X2 Notc.U ip (x, y) = and x, v, x (3) Extension of Taylor's Theorem, X X c^V To find r- use an equation containing u, x, y only. C' X DIFFERENTIAL CALCULUS 207 (2) In this work never use y" = l/^ ' for this equation involves the assumption that a relation exists between u and X which does not involve any other variable. For reference only : (3) ^ = ^ /^ . from which, by putting t = y/J ^ 1 /^ . ^ ^ dx dt/ dt' ' ^ ^ 6 if> ^^ / ^y (1) ^ _ f^^ dx _d^ d^^ /A^y ^ ^ _ ^" //"^V ^ ^ dx^~ \dt^ ' dt df ' dt) /\dt) ' dx^ dy^/\dy) (5) Polars to Cartesians : (i ^ c) sin 6 () = cos e . -- ox or r d Tie' ^ . ^ ^ c -- = sin ^ . -- + - dy Or OS r ^ ^ ^ dr dx dy {x + h, y + k) = E^KE^(i)(x,y) = e^D + kD' cfy {x, y) 208 DIFFERENTIAL CALCULUS 11. Undetermined Forms. (1) Algebraical Metliods. Suppose the form to become * undetermined ' when x = a, put x = a + Ji and expand everything in powers of h. (2) Differential Methods. , , JA4>(x)^ 0.. . . <^\a) _^,. (a) 7: If ~r{ = A its value is - If this IS of the form ^ the value = ttttt an) If = its value is -p This is necessarily also of form , but the infinite factor will often cancel. 00 (x)j find the values of x for which -f- = or 00 . For values making d<^ f. , max. .0 d(^ + to - . . 3 = we have . 11 -r- changes from , , i.e. it ax mm. dx ^ - to + ' d^4> . - , ^ , d(b , max. -7-2 IS For values making = qo we may have a . ax T ax min. under the same conditions and shall have such a value if x passes through the critical value as we travel along the function. DIFFERENTIAL CALCULUS 209 If -T^ = 0, then for . ' we must have also -~ = 0, dx^ mm. dx'^ the first diff^ coefft. which does not vanish must be of an even order and must fulfil the sign-conditions given above. (2) Between two equal values of a function at least one max. or min. value must occur. (3) Maxima and minima values occur alternately. (4) If y and x are connected by an implicit relation (f) (x, y) = 0, the formulae of 8 (1), Note, must be used, 14. (1) Maxima and Minima : Two Independent Vari- ables, w = (f){x, y). ia) For a . * value of w we must have = and mm. dx also r^ = 0. dy (&) If r = <\)y,^, s = (t)^y, t = (\>yy, thcu uukss r, 5, t all vanish when rt < s^ we have neither max. nor min. value for iv ; rt > s^ a. . value as r (or n is , ; mm. ^ ^ + ' rt = 5^, the case is indeterminate and higher diff' coeffts. must be examined. (2) Maxima and Minima : Three Independent Variables. tv = (f> (x, y, z). (a) For a . ' value of w we must have ' mm. ^x ^y OS (6) If J., J5, G, F, G, H represent (/)^^ ..., (py^ ..., we have 210 DIFFERENTIAL CALCULUS Minimum value of tv ii A, Maximum A H H B A H G H B F G F C are 4- + + Neither ,, ,. ,, ., have any other max. ' . signs. 15. Maxima and Minima : Several Dependent Variables. To find the . ' values oi iv =^ (j) (x, y, z) subject to lF(x,y,z) = OS - dw = = (l)^.dx + (f)y . dy + cf)^. dz, =f^ .dx+fy . dy+f^ ,dz, = F^.dx + Fy,dy + F^.dz. Eliminate A and /ut from \ (\)y + Kfy + ixFy = h PROPERTIES OF CURVES 16. Tangent. (1) Curve : y = f{x). Tangent : Y-y = ^{X-x). ax (2) Curve : f{x, y) = 0. Tangent : (X - ic) |^ + (Y- /) |^ = 0. (3) Curve : f{x, y, z) = 0, where z = 1 renders curve homogeneous. Tangent: X^ + Y~ +Z^ =0. dx cy oz Note. If (x, y) be used as current coordinates this equation gives the ' Polar Curve ' of {X, Y) with regard to/(a;, y) = 0. (4) At origin, if this be on the curve : Equate to zero the terms of lowest order in x, y. (5) n{n-l) tangents can be drawn from an external point to a curve of nih. degree. DIFFERENTIAL CALCULUS 211 (6) Polar Equation. The tangent at {U, a) is u = ZJcos (e-(X) + V' sin (6-01), where V = f-^j Proof. Assume it to be u = Acos{e (X) + B sin{e-~(X) and express that u and -z-r are the same for the curve and the tangent when 6 = 01. 17. Normal. (1) Curve : p = f{x). Normal : ( r - ?/) ^ +{X-x) = 0. (2) Curve : f{x, y) = 0. Normal X-x Y-y (3) n^ normals can be drawn from an external point to a curve of the nth degree. (4) Polar Equation. The normal at (C7, a) is .u= U' cos {9 - 01) -U sin (6 -01). 18. Related Lines. The figure should be committed to memory with the various lines associated with it. The following properties follow from it and should not be learnt individually, but de- duced from it as required, until the habit of doing so becomes fixed. ds2 = dx^ + chf = dr'^ + {rdOf. dr . J rdd cosc/>=-, smc/,= ^, tan (b = - dr cos\/a dx ds sin\/A = ds tan \lf = OT -Polar SubVaKgcul' Via. 47. dx x-y dx dy p = r sin (/) = p 2 ds ' 212 DIFFERENTIAL CALCULUS Leakn only Polar Suhtangent : OZ (Fig. 47) = r^ - = - ,. , 1 1 ^/dr^^ ,dw2 Perpendzcular: -, = ^ + ^(^-^) = u^-^(-) . 19. The Pedal Figure. Here again the figure should be learnt. Locus of is the Z second pedal. OYY'U coney clic, (p of curve = (/) of pedal. d and 6. (6) Cartesian Coordinates. Since the tangent is ox ox dy i>js ' j = Eliminate x and ^. , ,-2 = ^2 + ^2^ /(^,y) = 0. DIFFERENTIAL CALCULUS 213 Common Curves (Pedal Equations). St. Line : p = a. Circle (centre) : p = a, or r = a, or p = r. Circle (circumf.) : 2 ap = r^. Circle (any origin) : 2ap = r^ b\ *Parahola (focus) : p"^ = ar. ^Ellipse ) . ^' ^ ?f + 1 * Hyperbola S p^ r *Eq. Spiral : p = ar. *Epicycloid ) o _ .2 , , *Hypocycloid S Circle Involute : r^ = p'^ + a'^. Cardioid (cusp) : r' = ap"^. Also of epicycloidal form [ 28 (6)]. Lemniscate (node) : r* = a^p. Cotes' Spirals: l/p^ = a/r"^ + & [ 28]. An asterisk indicates the curves in which the p-r equation should be learnt. (2) Tangential- Polar (i. e. p-yj/ equation). Find the pedal curve [see 21 (1)]. Let it be r = f(0). Then we have | -"^ = -^ and p =r, ,'. p =f(^ - g)- Common Curves {p, \p form). Point '. p = a sin \p. St. Line -. \p = OL. Cirde (circumf.) : p =a(l + sin^). Parabola (focus) : p = a cosec \p. Parabola (vertex) : p = a cosec tp . cos^ \^. Rect. Hyp. (cent.) : p2 ^ _a2 cos 2 i//. * Ellipse (cent.) : 2)2 = a^sin'^xf^ + b^cos^ip. (3) Intrinsic Equation (i.e. s-f equation) can only rarely be determined without the Integral Calculus. Eliminate x between j tan x|/ = f{x) 1 Common Curves (Intrinsic). Circle : s = aif/. Catenary : s = c tan if/. Cycloid : s = 4 a sin ^. Epi-(Hypo-)cycloid : s = a sin 6 \p. Circle Involute : s = ^aip"^. Parabola Evolute :s = 2a (sec* i/* 1). (4) Tangential Equation (i. e. condition that Ix + my + n touches i^'l^r,^) = 0). 214 DIFFERENTIAL CALCULUS (a) Compare XI\-hYFy + ZI\ = and lX + mY+n= 0; .-. FJI = Fy/m = FJn ^ k. But F(x,y) = 0. Eliminate X, Y, A. (b) Make F(Xf y) = homogeneous by means of Ix + my + n = 0, and express condition that resulting eqn. has two equal roots. 21. Related Curves. (1) Pedal Curves : To find the Pedal Curve. (a) p, r-Form. If f{r, p) = be the original curve, p'= p^/r, r'= p; .: the pedal curve iaf(r'^/p, r) = 0. (6) Folars. Method i. / ^ /I/ TT , [Draw init. line so that 2 b' IS positive. I tan <\i =^ r -T-' dr r = r sin (/). Eliminate r, ^, <|). Method ii. Find the envelope of circles described on the radii vectores as diameters. (c) Cartesians. The condition that X cos a + Zsin a = p should touch F (ip, /) = is Eliminating ic, ^ we have an equation in ^, a ; i. e. the polar coords, of point Y. (d) The first negative pedal is the envelope of a st. line drawn through any point of the curve at right angles to the radius vector. DIFFEKENTIAL CALCULUS 215 (e) Common Pedals : Curve. Origin. Pedal. Circle. On circumference. Cardioid. M Anywhere. Lima9on. Parabola. Focus. Tangent at vertex. ) Vertex. Cissoid. Central Conic. Focus. Auxiliary Circle. ?> j> Centre. -2 = a2cos2 + 62giu2^^ Kect. Hyp. Centre. Lemniscate. Eq. Spiral (a). Pole. Eq. Spiral (a). ir/ar='''!^m9. ^ ' ^ sin Pole {r/ayn+i = ''^\x9/im+l). (2) Inverse Curves : To find the Inverse. (a) p, r-Form. FQQ'F' cyclic. . . (\) oi ciu've = 7T-4) of inverse. *. p^r^ = p/r rr' = a^ f{r,p) = Elim. ^ and p. Fm. 49. (h) Polar Form. If /(r, 6) = be the curve, /K/r, ^) = is Inverse. (c) Cartesian Form, lif(x, y) = be the curve ./ a'^x a^y \ ^ . ^ J ( -5 o ' -o ^ ) = IS Inverse. ^x^ + y^ x^ + if^ (d) Common Inverses. (See also OEOM. 33.) CMn?e. Origin. Inverse. Parabola. Focus. Cardioid. Vertex. Cissoid. Rect. Hyp. Centre. Lemniscate, Central Conic. Focus. Lima9on. ,, ,, Centre. Cassinian Oval. Note also that curves having nth contact at P invert into curves having nth contact at Q ; that the osculating circle at P inverts into 216 DIFFERENTIAL CALCULUS a circle osculating the inverse curve at Q, but if the centre of inversion is on the circle, into a st. line tangent at Q, a point of inflexion on the inverse curve. (3) Polar Reciprocal. (a) Y' is taken so that OP. OY' = a^ ^nd Y'Q is drawn , at rt. angles to OY' ; then the Polar Reciprocal is the envelope of Y^Q. Or, which is the same thing, OY is drawn perpen- YiQ, 50. ' dicular to the tangent at P and Q taken such that OY. OQ = a2, P. R. is locus of Q. (b) p, r-Form. The P. R. is clearly the inverse of Pedal. Use 21 (1) (a), (2) (a). (c) Cartesian Form. Use 21 (1) (c), then (2) (b). Result is in Polars. {d) Common Polar Reciprocals. (See also G. CON. 35.) Curve. Origin. Inverse. Eq. Spiral (a). Pole. Eq. Spiral (a). (4) Caustics. OP, PQ are drawn making equal angles with the tangent at P, the caustic is the envelope of PQ. Common Caustics. at 00 , reflecting curve a circle, caustic an epicycloid (radii 2 a, a). on reflecting circle, caustic a cardioid touching circle at 0. 22. Asymptotes, i. Cartesian Coordinates. (1) A curve of the nth degree has n asymptotes, real or imaginary. If n is odd one of these is necessarily real. (2) Asymptotes by inspection, [u^, v^ = homogeneous w-tics.] (a) (ax + by) u^_^ + v,,.^ 4- v,,.^ + . . . = 0. Asymptote is ax + by = 0. (b) (ax + by) u,,_^ + v,,^^ + v,,..^ + ... = 0. Asymptote is ax + by = - Lt. ^n-i "n-l DIFFERENTIAL CALCULUS 217 (c) {ax + hyyUn.i + v,,_-^ + Vn_2+ =0. Asymptotes are (ax+ hyf = - Lt. -- u. n-l {d) {ax + lyf w^i + (ax + hy) v^^^ + it\_^ + ^;_2 + . . . = 0. The asymptotes are ^M-l , T4- ^n-\ (ax + &y)2 + (ax + M Lt. -^^ + Lt. -^^ = 0. In all the above cases the limits (re = oo, ^ = oo) are to be X J) taken subiect to the limitation - = ^ y a (e) If F^ be a function containing terms of wth degree and lower, and break up into linear factors, no two being parallel, all these lines are asymptotic to F^ + F^., = 0. (3) Asymptotes parallel to Axes. Equate to zero the coefficient of the highest powers of x, then that of the highest power of y. The resulting lines, if any, will be the asymp- totes, and will give all asymptotes parallel to the axes. (4) Partial Fractions Method. Arrange the equation in the form ^ . /y\ , , /y\ [i.e. with terms of nth degree first, followed by terms of (w-l)th]. Then if - -r-ji: becomes, in partial fractions, - + ^ + ... t-m^ t-m2 ' the asymptotes are y = m^x + Ci^, y = m2X + C2, &g. (5) The rule of (4) is simply a convenient way of stating and developing the following : Asymptotes, not parallel to axes, may be determined by substituting y = mx + c in the equation to the curve, and equating to zero the coefficients of the two highest powers of X, thus giving two equations for m and c. 218 DIFFERENTIAL CALCULUS Note. The limitation ' not parallel to axes ' is important. Thus if we substitute y = ynx + cin y* = y'^ x'^Sind apply the rule, the resulting equations are satisfied by w = and c = 0, but j/ = is not an asymptote. (6) Curvilinear Asymptotes. Expansions of the forms y = mx + ax^ + 6 + cx~ * + ... will, by taking the first three terms, give hyperbolic or parabolic asymptotes respectively. Such expansions may be determined by substituting for y and equating successive powers of x to zero, but are usually best sought by succes- sive approximations. Notes. (1) Of the two forms given, the second must be used when there are two parallel rectilinear asymptotes. (2) Example of Successive Approximations (Edwards). Curve {y-xy (y+x) = 2ax^. Hyperbolic Asymptote. 2ax^ 2ax^ a ?/ + ^-(7Z:^-T^ = 2- (IstApp.) y + x^ = 2 + r^ ^^^^ ^^^-^ (-x+^-xy Parabolic Asymptote. \l = V = Vax. {1st App.) ^ y + x ^ x + x ^ = ... = \/^- \ . (2nd App.) Vx + ii. Polar Coordinates. Find a value of 6, say oc, which will make r = oo andj? = r^dO/dr finite. The asymptote is then the Cartesian line y = x tan a -pa sec oc. iii. Asymptotic Circle. Examine if = co gives a definite limit for r, say a. Then r = a is an asymptotic circle. 23. Singularities, &c. (1) Concavity or Convexity : (a) A curve is convex or concave to the foot of ordinate d^y . DIFFERENTIAL CALCULUS 219 [b] A curve is convex or concave to the pole as or +. t.+ ^is Note. If the sign is forgotten it can be recovered in a moment by applying the test to the known curves y = x^ or u ^ 1/a. (2) Point of Inflexion (Flex). Curve y = ^{x)\ ^"{pc) = and changes sign at the point. Polar Curve : u + j^ = and changes sign at the point. Note. The appended table is a useful summary. It is sometimes, d'^x d'^y of course, more convenient to work with -^ih^n with j^ . dy d^y d^y dx dx2 dx^ Tangent || to -axis. _ Maximum. + Minimum. fo Flex. fo + Convex downward. h ~ Concave downward. (3) Point of Undulation. Curve y = (f)(x) : f^^ . fyy for the values in question. Acnodes if f^y < f^^ . fyy. Spinodes = Cusps if /^^ = f^^^fyy, usually. But sometimes acnodes. 220 DIFFERENTIAL CALCULUS If a spinode (cusp), its species (keratoid or rhamphoid) is usually best determined by transferring the origin to the point in question, but it can sometimes be settled by noting ( keratoid } d^y V d'^x'^ , , ***' ^' i rhamphoid 1 ""^P' d^T d/J '""'' ^^"^ *'" values of < > signs very near the cusp. Nofes.~-{l) All the spinodes (cusps) lie on/|y =Ux.fyy. (2) The equation to the two tangents at a double-point (^, fc) is (^-^)'^ + 2(a:-;i)(r/-Ar) i!f +(t,-fc)2^ = 0. (3) For a triple-point at i^, k) we must have /= fh =fk =^fhh =fhk- =fkk = <> and the equation to the three tangents is +3(.-*)(v-*)^,^ +(.-;.) 5-^-0. (5) Tlie Fundamental Singularities. All singularities on a rational curve may be resolved into nodes {b), cusps (k), flexes (t), and bitangents (t). Ex. A Point of Undulation = 2 t + r. Note. The rarer singularities which do not occur on rational integral algebraical curves are : Point d^ Arret, e. g. (0, 0) on y = a^ . Branche Pointillee, e.g. in all quadrants except the first with the curve y = cc^. (6) Plucher's Equations, If there be b nodes, k cusps, t flexes, T bitangents on a curve of degree n and class m [m = no. of tang, from ext. pt.], m = w (w 1) - 2 <) - 8 K, w =m(m-l)-2r-3t, I = 3w(w-2)-6^-8k, K = 3w(w-2j-6r-8t. DIFFERENTIAL CALCULUS 221 24. Envelopes. (1) If A be a variable parameter, Envelope of Ak'^ + 2Bk+C= is B"^ = AC ; Acoa\ + Bsm\ = C is A^ + B'- = C^ (2) The envelope of (j) (x, y, X) = is the A-eliminant of 4> (x, 2/, ^) = and ^(/)/^A = 0. But this eliminant contains also as factors the locus of any nodes of {x, y, a, (3, y) = 0, where is the eliminant of a, ^, y, A, /x between (#) = 0, /i = 0, /2 = 0, Cf. 15.] r + A:- + IJL r 0^ oy oy oy 222 DIFFERENTIAL CALCULUS 25. Curvature. (1) p = ds/dyj/. (2) f. = (1 +i92)V2, where i; = S' '^ = ^J* . (3) p = rdr/dp, where i? = X^ on tangent from pole. dy\f'' (4) />=!?+ ;^, 2 (6) /J at origin = Lt. x^/2y if ?/ = be the tangent. = Lt./V2a;ifrc = Other formulae of less importance are added for reference (6) 1==- f = ^, ^ ^ p dy dx ds ds (7) 1 _ /cfxY (<^\ j^ " Vrfs2/ "^ yds') ' (12) Centre of Curvature . $ * = ^ " /^ ^^^ 'A' iy = y + pcosij/. (13) Two curves have contact of the second order at a point of intersection where -7- and ^4 have the same value dx dx^ for both curves ; contact of the third order when, in addition, ~ has the same value. dx^ DIFFERENTIAL CALCULUS 223 (14) Conic of Closest Contact. The conic through 3 consecutive points on the curve will have p the same at the points as on the curve ; through 4 consecutive pts. will have p and the same ; ds ' through 5 consecutive pts. ^P ^V xu XT Pj 1~> -3~9 the same. Now ds ds^ in any conic tan ^ = i -r'> where (p is the angle in front of CIS the normal as regards the direction in which s is measured. This gives the direction of the centre of all conies of 4-point contact (^p and being taken from the curve) or of the axis of the parabola of l-pointic contact. For the centre of the Osculating Conic (of 5-point con- tact) we measure along PO (which is drawn at the inclination <^ in front of PG) a distance r given by cos (f) 1 dcf) r p ds Here again (j), p, -= are found from the curve. UiS (15) Evolutes. To find evolute of a curve find the enve- lope of the normal. 26. Curve Tracing, Cartesian Coordinates. i. Note any Symmetry. With regard to Ox: y occurs in even powers only. 5) ?> ^y "^ 5> ?j )) ,, ,> X = y: Unaltered by interchange of x and y. ., ,, ^ = -/: Unaltered by interchange of X and y. 224 DIFFERENTIAL CALCULUS In opposite quadrants. Unaltered by writing -x, y for x^ y. If not symmetrical it may sometimes be rendered so by some obvious transformation of coordinates. Ex. j/3 = x2-4x+4, (2x-2/)2= (x + 22/~3). ii. Differentiate and find i? = ~ (XX Thu3ii (t>(x,y)=0, p = ^ci>Jcf>y. If 0x = 0, <^y = are simple curves it will be a useful confirmatory test to trace them. Wherever 0a: = intersects ^ = 0, p = 0, and wherever (Py = intersects = 0, p = oo . Wherever 4>x = intersects 2/-, .-. curve lies between y = +x and y= x in the quadrant in which ?/ = lies; also 1/2 = x2-x< = a;2(l-a;2), .-. xyy ' i? 1872 q 226 DIFFERENTIAL CALCULUS If this is not still indeterminate it will give a quadratic for the value of p at the singularity. When the position of a singularity has been determined, its nature is best examined by transferring the origin to the singularity and using vii. edition. In this quadratic for p be careful not to miss the value 23 = 00 which occurs when the quadratic reduces to a simple equation. vii. Newton's Rule, [Form of Curve at Origin and In- finity.] Let each term of the equation be represented on squared paper by a point whose coordinates are its x and y indices. Thus ^x^y^ is entered as the point (2, 3). If a straight line can be drawn joining two or more of these points, and not passing through 0, which shuts off all other points from 0, the terms of the equation represented by this line give the form of the curve at the origin. If a st. line can be drawn shutting off no point from the origin the terms represented will give the form at x, i.e. will give curves more or less asymptotic (rarely an exact asymptote). Notes. (1) Conjugate Points (Acnodes) at the origin are indicated by the * Form at the Origin ' becoming wholly imaginary. (2) Absence of Infinite Branches is indicated by the ' Form at Infinity' becoming wholly imaginary. (3) Singular Points may be studied by transferring the origin to them and using Newton's Rule. (4) Ex. Newton's Rule applied to curve y^ = x + xy + x^y"^ shows that y* = X ia the form at the origin and y^ = x^, x = cc^y^, i.e. xy = 1 is the 'form at oo'. The rule applied to y^{xl) = x^ (a^+l) gives for form at 0, x^ + j/2 = and indicates an acnode. viii. The Tentative Tracing for which sufficient details probably now exist can be examined by the following considerations : (1) By turning into polars and solving for r in terms of 6. This will usually detect loops. (2) Any straight line intersects a curve of the ^th degree in n points, real or imaginary, and imaginary roots occur in pairs. DIFFERENTIAL CALCULUS 227 (3) There cannot be single branches going to infinity. (4) In particular, if a branch approaches an asymptote, there must be a second branch approaching the same asymp- tote either at the same or at the opposite end. ix. Special Devices. (1) Auxiliary Loci. Ex. (1) yiy-'-ax) = (^Z + a^) {x-a). It is easy to see that the curve lies wholly in the regions of space marked 4- , and passes through the points marked . When the asymptote {x = y) is drawn the curve is now easily traced. Ex. (2) {y-xy = c6, .-.?/ = x2+a;i. Trace y = x^, y = x^ separately, and add and subtract their ordinates. y-0 Fig. 52. (2) Transformation of Coordinates. [Cf. Note on 26, i.J Ex. {x + y+l)^ = {y-x + S)\ Kefer to new axes j ^.q Z n* ' y ~~x -\- ii \). So also with {x + 2y+iy = (Sy + x 6)^, even although the new axes in this case are oblique. 27. Curve Tracing, Polar Coordinates. i. Note any Symmetry. If change of sign of 6 leaves equation unaltered : Initial line. If ^ + 77 for leaves equation unaltered : Opposite quad^ If-TT-^for^ ,, : Axis of y. If not symmetrical it may sometimes be rendered so by some obvious transformation. Ex. r = a (1 + sin 2 d) has at present symmetry in opposite quad*. = a (sin + cos 0)'^ = 2 a sin^ ($ + j), showing symmetry with regard to the line ^ + -7 = 0. 9a 228 DIFFERENTIAL CALCULUS ii. Note any limitations imposed on r bj^ the trigono- metrical functions of 6. Ex. r = a sin n$, .'. r 'yf>' a, .'. curve whollj'^ inside circle r = a. iii. Find the asymptotes [ 22, ii] by finding the value of dO which makes r = oo but 3- finite. du If r can be determined explicitly in terms of 6 it is now easy to trace the curve by making 6 gradually increase from up to the value (if any) when r repeats itself, and then decrease from 0. iv. In any case the values of r must be found for selected values of 6 Tparticularly 0? j > 9 > '"") > ^^^^ ^ calculated from tan cf) = [Remember that (j) and 6 are on the same side of r, and then selected points must be tabulated, 6, r, and tan (j) being entered in the table.] v. Note (1) r = 00. The curve has infinite branches. (2) r finite, oc. Asympt. circle. (3) Points of Inflexion. [See 23 (2).] (4) Concavity or Convexity. [See 23 (1).] 28. The well-kno^wn Higher Curves. (1) Chainette or Cate- nary. All the properties may be deduced from the figure, s being measured from r. Cartesian Equation : y = c cosh (x/c). Note: ^ Catenary : p = - normal. ) ( Circle : p = + normal. ) Fio. 53 DIFFERENTIAL CALCULUS 229 (2) Equiangular or Logarithmic Spiral. PC = />, PT = s. Equation : Fig. 54. (3) Cycloid. Tangent at P is II to OQ,. Arc OP = 2 chord OQ,, Arc oq = pq. pp = 2 chord AQ. Area = 3 x area of gene- rating circle. The Cartesian Equation is easily written in terms of ^ = 2 with Y. 2. Kesultants. (1) R'- = P^+Q^ + 2PQcosa, Note. The direction of R is inclined to that of P at an angle 6 given by tan 6 = Q ain a/(P+ Q cos a). Obtained by resolving Q into Q cos along P, Q sin Oi \ ELEMENTARY STATICS 233 (3) 7^2 = 2P2 + 2 ^P cos Z PQ, for any number of forces, whether coplanar or not. [Cf. SPH. TRIG. 10.] (4) Resultant of OA, OB is 20^, where G lies in AB so that AG = BG, (5) Resultant of m . OA, n . OB is {m -f n) OG, where G lies in AB so that m . ^(r = w . j5(t. (6) Resultant of OA^, OA^ ... OA,. is r . OG, where 6^ is the centroid of JLj J-g ... ^,.. (7) Resultant of Wj.O^i, m.^. OA^ ... m^. . OA,. is (2m) OG^, where 6^ is the centroid of m^ at J-i, Wg at ^2> w^ at A^, (8) Lami's Theorem. If P, Q, i2 equilibrate at a point, P_ ^ Q_ Ji sin Z QR sin Z E/^ sin IPQ' 3. Theorem of Moments. The algebraical sum of the coplanar forces . , point moments of any number oi ^ about a ,. ^ lorces line is equal to the moment of their resultant about the same, and is zero if the forces are in equilibrium. 4. Conditions of Equilibrium. (Concurrent coplanar forces.) A system of concurrent coplanar forces is in equi- librium if (1) The sum of the resolutes in each of two directions is zero. OR (2) The sum of the moments about each of two points is zero provided that the join of these points does not pass through the point at which the forces act. 5. Elastic Strings. (1) Hoohe's Law. Tension of an elastic string / Extension \ ^ Length ^ A is the 'modulus' for that particular string = tension required to double its length. 234 ELEMENTARY STATICS Note. The value of A is independent of the length of the string and varies directly with the cross-section. A for a string (wire) 1 cm.2 in section is known as * Young's Modulus ' for the material of the string. (2) Work done in stretching an elastic string = (Mean of Tensions) x (Extension). 6. Paballel Forces. (1) LiJce. The resultant of P and Q is P+Q, acting so that if ACB be any transversal cutting the forces in A and B and the resultant inC, P.AC= Q.BG. (2) Unlike. The resultant of P and Q is P- Q, acting nearer to the greater force and so that P-Q / P.AG= Q.BG. Fm. 57. 7. Couples. (1) Equimomental couples in the same or in parallel planes are equivalent. (2) The resultant of any number of couples in parallel planes is a couple whose moment is equal to the sum of the moments of the couples. (3) Axes of couples follow the Parallelogram Law. (4) Any system of forces acting on a rigid body can be reduced to a single force acting at any arbitrary point and a couple. ELEMENTARY STATICS 235 (5) Any system of forces acting on a rigid body may be replaced by a single force and a couple whose axis is along the direction of that force (Poinsot's Central Axis). This central axis is the line about which the sum of the moments of all the forces is a minimum. 8. Conditions of Equilibrium or a Kigid Body. (1) If three forces maintain equilibrium they must be coplanar, and either all concurrent or all parallel. Note. In the application of this result the following trigonometrical results are of great value : (1) (m 4- n) cot ^ = m cot a - n cot /3. (2) = ncot A mcoiB. The general form only of these equations should be remembered, and the equations found more exactly when required. (2) Any number of coplanar forces will be in equilibrium if, and only if : I Moment about any one point is zero. 1 Components in any two directions are zero. ( Moments about any two points are zero. or (&) j Components in any one direction (not at rt. angles I to the join of the two points) are zero. or (c) Moments about any 3 non-collinear points are zero. or {d) If the forces are (Xj, Z,) (Zg, ^2) ... acting at (x^, y^) (^2,^2) -, then :X = 0, 2r= 0, ^{xY-yX) = 0. (5) Any number of forces will be in equilibrium if, and only if : IResolutes in any three non-coplanar directions are zero. Moments about these or about any three non- coplanar axes are zero. 236 ELEMENTARY STATICS (5) When the forces are {X,. Y^, Z,) [X.^, Y^, Z^) acting at (x^, y^, z^) fe, ^2, ^2) . 1 ^(yZ-zY) = 2(^Z-^Z) = :S.(xY-yX) = 0. |a 1 w Fig. 59. 9. Actions at Connecting Points. i. Bigid Connexion, If the con- nexion be rigid (as at any point of a rod) the action across this connexion is equal and opposite to the resultant of all the forces on one side (either side) of the connexion and consists of two parts : (a) A force acting through the point. TV. f [ ~ component of this force along rod. ' Shear '= component of this force X^ to rod. (h) A couple whose moment is equal and opposite to the moment of all the forces on either side of the point. This couple is known as the ' tendency to bend ', ' bending stress ', ' shear-couple ', ' moment of flexure '. The rod will break, if at all, at the point where this couple is a maxi- mum, provided it is a rod of uniform thickness. Note. Thus in Fig. 59 we have for the action at C : Tension -> =0. Shear f = W+w .BC{w = wt. of unit length). Shear-couple % - W.BC+^w.BC^. ii. Smooth Joint In this case the shear-couple is zero and the force is generally resolved in two convenient direc- tions at rt. angles. Such resolutions are shown in the figures. In the second figure we think of both rods as jointed to a pivot to which the string is tied. If the string in this case be replaced by a rigid bar the actions at the hinge will usually be indeterminate. ELEMENTARY STATICS 237 tyi^N'y^'^-v-^ij^^^i^-^'^v^n y Fig. 60. Fig. 61. v^A^/y///yy////A> /y/y/////A '^yy^yyyy^yyyJyy^ Fig. 62. Fig. 63. l^oies. (1) Indeterminate Reactions. It may be said, generally, that the reaction along a I'od or bar of a connected system (framework, &c.) is determinate whenever the length of that rod can be increased without undoing the frame. (2) The ' tendency to break ' at any point of a rod is, as usual, the moment of all the forces on one side of the point, e. g. in Fig. 61 for the upper rod it is the moment of y^ only, if the rod is light. (3) Virtual Work ( 15) is very useful for these problems. 10. Centre of Gravity. i. Lines : (1) Uniform St. Line. Middle point (2) Uniform Triangular Wire. Incentre. (3) Arc of Circle. Distance from centre r sin 4 angle"] ,. ' ^ o Is. radius, y I jangle Deduce Semicircular Wire, distance = 2a/7r. ii. Areas : (1) Triangle. Centroid, i. e. intersection 238 ELEMENTARY STATICS medians, one-third up median. Coincides with C. Gr. of three equal particles at angles. (2) Parallelogram. Intersection of diagonals. (3) Sector. Distance from centre = f distance for arc. Deduce Semicircle^ distance = ^- Deduce (by Projection) Semi-ellipse (any diameter), distance = ^r- , on conjugate. OTT (4) Cone, Curved Surface. One-third up axis from base. (5) IsJrJ^ce 1 ^-^ ^P^^^^^^ cut off by parallel planes. Centre of median line. Deduce Hollow Hemisphere. iii. Volumes : (1) Pyramid on any Base. One-quarter up join of centroid of base to vertex. (2) Gone on any Base. One-quarter up join of centroid of base to vertex. (3) Hemisphere. Distance from centre = f n (4) Sector of Sphere. [Reduce to ii (5).] Deduce Segment of Sphere. [Use iii (2).] Also Zone (Vol.) of Sphere. [Diff. of 2 segments. ] (5) Paraboloid of Bevolution. Distance = %h from vertex. Note. For a tetrahedron the volume-centroid corresponds with that of four equal particles at the angles. 11. Determination of C. Gr. : General Theorems. (1) 0? = SmiiCi/Swi, y = l^m^yj^mi, [z = ^mi^i/Smj]. Notes. (1) r ^ 2 ^irj/S ?Wi, for the r's are not parallel. (2) For the solution of problems involving varying density the result V+2^+,..+n'' = + lower powers of n is needed. ELEMENTARY STATICS 239 (2) Given C. Gr. of the whole of a mass m (at G) and of a part nii (at O^), to determine that of the remainder : Intro- duce the conception of negative mass and find the C. Gr. of + m at G, and -m^ at O^. 12. Properties of C. Gr. : General Theorems. (1) If masses m^ ^2 ... m^ be placed at points J.^ ^.2-..^,^ (not necessarily coplanar) and G be their centroid, any- other point, 2m . AO^ = "^m . AG^ + OG^ ^2 m. Note. This result is exceedingly useful geometrically. (2) Guldinus' Theorem. See MENS. 31. Note. Hence mental calculation gives 0. Gr. of area and circum- ference of a semicircle, &c. 13. Laws of Friction (Experimental). i. General : (1) The direction of friction between two bodies is opposite to that in which their point of contact is urged to move. (2) The magnitude of friction is (when there is equi- librium) just sufficient to prevent motion. (3) No more than a certain amount ('Limiting Fric- tion ') can be called into play. ii. Limiting Friction : (1) The magnitude of L.F. bears for the same bodies a constant ratio {fx = ' CoeflBLcient of Friction ') to the normal pressure between the bodies, i.e. F= fxB. (2) fji is independent of the area and shape of the bodies in contact. Note. The actual value of fi for metal-metal is about 0-2 ; for wood- wood, 0-3 to 0-5. It varies greatly according to circumstances. iii. Dynamical Friction : (1) The friction called into play to prevent motion is greater than that called out to resist its continuance. 240 ELEMENTAKY STATICS (2) For the same substances the friction is approxi- mately independent of the rate of their relative motion. iv. Rolling Friction : (1) When a body is on the point of rolling upon another body the action at the point of contact ( (a) Normal Eeaction. consists of -j (/3) Tangential Friction Force. I (y) Friction Couple. (2) The moment of the friction couple is independent of the curvature and proportional to the normal pressure. Note. It is commonly stated that ' Rolling Friction ' is less than ' Sliding Friction', which means, presumably, that the force (0) of iv(l) is less than the force i^ of ii. This statement is only true with wide limitations. Recent tramway accidents have shown that brakes which nearly but not completely lock the wheels are more effective than those that do so completely. 14. Friction (Miscellaneous). (1) Problems in Friction are usually best treated by the use of the total reaction (which in the case of Limiting Friction makes an angle X = tan" ^/x with the normal). They can frequently be thus reduced to three-force problems. Note. \ is the angle of the inclined plane on which the body just rests under the action of its weight and of the reactions of the plane only. (2) For a string passing round a rough surface and in limiting equilibrium Fig. 64. Tj = Tg . C'^'^. (3) When equilibrium can be broken either by rolling or by sliding, rolling will occur unless it should involve a reaction at the point of rolling lying outside the cone of friction. ELEMENTARY STATICS 241 15. Virtual Work. [Sometimes called ' Virtual Moment '. The old name, * Virtual Velocity ', should never be used.] If a force P act at a point A of a body along AN, and the point A y\ move into a position A' very near xl9 I . r, to A, A^N >-P P. AN = P. AA^ cos (j) = virtual flg. 65. work done by P in the displacement. i. (1) If a number of forces act at a point A, the sum of the V. W. owing to the displacement of ^ = V. W. of the resultant. (2) If a particle be in equilibrium under any number of forces acting at a point A, the sum of the V. W. for any dis- placement of A is zero. (3) If a system be in equilibrium under the action of any number of forces and receive a small displacement consistent with the geometrical connexions between the bodies of the system, the sum of the V. W. for such a displacement is zero. Note. More strictly not zero but of the second order of small quantities. ii. (1) The V. W. of the tension of an elastic string, whether straight or deflected (by smooth rings or pegs), is -T.bl. (2) Provided that the geometrical conditions be not violated, we may consider the V. W. of the following to be zero : Tensions of Inelastic Strings. Mutual Attractions. Interactions at Smooth Joints. The Reaction (but not the Friction) when a body is virtually displaced by sliding along a rough surface. U2 ELEMENTARY STATICS The Keaction when a body rolls without sliding on {a) a fixed surface, or (b) a movable surface belonging to the same system. 16. Stability of Equilibrium. (1) If a body be placed at rest but not in equilibrium under the action of external forces it will begin so to move that the work of the forces in the initial displacement is positive, i. e. so that the potential energy decreases. (2) The equilibrium of a body, or system, is | i. i.i [ ,. ,1 . . 1 i. fminimum) according as the potential energy is a true ] . [ ^ ^ -^ Imaxmiuml (3) The equilibrium of a system of bodies under the influence of their own weights and such constraints and connexions as are named in 15, ii is j . \ when the height of the C. Gr. of the system above any fixed horizontal minimum ] maximum) plane is \ (4) Stable and unstable positions occur alternately for any system which moves with but one degree of freedom. (5) If a body whose radius is II rests on another body of radius r, its C. Gr. being at height h vertically above the point of contact, the equilibrium is stable or unstable as l/h^\/R+l/r. ELEMENTAKY DYNAMICS 1. (1) Parallelogram of Velocities. If a moving point possesses simultaneously velocities which are represented in magnitude, direction, (and sense), by the two sides of a parallelogram drawn from a point, its actual motion is with a velocity which is represented by the diagonal of the parallelogram through the point. (2) Triangle of Velocities. If a point possesses simul- taneously velocities represented in magnitude and direction by the sides of a triangle taken in order, the point is at rest. Hence velocities represented by AB, BG are equivalent to velocity represented hy AG. (3) Polygon of Velocities. If a point possesses simul- taneously velocities represented in magnitude and direction by the sides of a closed polygon taken in order, the point is at rest. (4) Parallelepiped of Velocities. Cf. 1 (1). (5) Components. Any velocity u may be resolved into u cos 6 along a direction making an Z ^ with u and u sin at rt. angles to that direction. 2. Kesultants. (1) w'^ = u^ + v^ + 2uv cos oc. The direction of w is inclined to that of u at an angle given by tan 6 = V sin 01/ (ju + v cos a). (2) w^ = Iiu^ + 2"^ U1U2 cos I u^u^' (3) See STAT. 2 (4)-(8), all of which theorems hold mutatis mutandis. 3. Accelerations. The results of 1, 2 above are tiue in every case if for the word ' velocity ' we substitute r2 244 ELEMENTARY DYNAMICS ' acceleration ', and for the words * is at rest ' the words * has no actual acceleration '. 4. Relative Velocity. (1) The velocities of two bodies relative to one another are unaltered by impressing on both any common velocity. (2) The velocity of A relative to B is the velocity com- pounded of the velocity of A and a velocity equal and 02>i>osite to that of B. (3) The same laws hold for Relative Accelerations. 5. Angular Velocity. (1) If a point describes a circle of radius a with uniform velocity v and angular velocity oo about the centre, v = aco. Note. For a wheel rolling uniformly along the ground without slipping, if the velocity of the centre be r, that of the highest point is 2 tj, that of the lowest point 0. (2) If the velocity of a point whose polar coordinates are (r, 0) is at any instant v, its angular velocity about the pole at that instant is given by rco = -y sin (f), where (p is the Z between r and the tangent at (/, 6) to the curve described by the point. This is also written r'^co = vp. 6. (1) Uniform Motion : s = ut. (2) Uniformly Accelerated Motion : V = u + ft, s = ut + ift\ v^ = u'^ + 2fs. Notes. ^1) The space described during the wth second is s= w + |/(2w-l). (2j For Retardations / is negative. ds dv d^s (3) tj =-- - ' / = t: = -n? for variable accelerations, &c. ^ at at at^ (3) Uniformly Accelerated Motion under Gravity. For / writer = 32 (F.P.S. system), or 981 (C.aS. system). ELEMENTARY DYNAMICS 246 Notes. (1) It is of the utmost importance to pay special attention to the directions of velocities and accelerations. Thus, if a body is projected upwards and this is chosen as the positive direction, g is negative and the formulae become v = a gt, &c. (2) The more accurate values of g are 32'19, 981-2 (London). It varies according to position on the earth from 32-09 to 32-25 (978 to 983) . Its value in latitude A and at a height of h metres above the sea is (Helmert) 980-632 - 2-593 cos 2 A - 0-007 cos^ 2 A - 0-0003086 h. 7. Motion on Smooth Inclined Plane. For / write g sin Oi in the formulae of 6 (2), where oc is the inclination of the plane to the horizon, and see 6 (3), JSfote (1). Notes. (1) This gives motion in the line of greatest slope and assumes that the plane is fixed and not free to move. (2) The result of 18 (3) is often useful. Chords of a Circle. The time of sliding down all chords of a vertical circle which pass through either the highest or the lowest points of the circle is the same. Notes. (1) The time of descent is t = v2 a/g, where a is the radius of the circle. (2) The theorem and Note (1) are true if we substitute * sphere' for * circle', and the theorem, but not Note (1), is also true if the plane of the circle is inclined to the vertical. (3) The theorem is a particular case of the following : If is any point on the upper half of a vertical circle the time of descent from to the circle is the same for all chords through 0. The theorem is also true if the chords are rough and if the motion takes place in a medium whose resistance varies as the velocity. Clvords of Quickest Descent (Brachistoclirones). (1) To find the chord of quickest descent, PQ> ^^'^ni a point P to a curve, describe a circle with P as its highest point, and touching the curve. (2) To find the chord of quickest descent, PQ, from a curve to a point P, describe a circle with P as its lowest point, and touching the curve. (3) The chord of quickest descent from one curve to 246 ELEMENTARY DYNAMICS another bisects at each end the Z between the normal to the curve and the vertical. Note. These constructions may give the line of Slowest Descent. 8. The Lav^s of Motion (Newton). Law i. Every body continues in its state of rest or of uniform motion in a straight line except in so far as it be compelled by impressed force to change that state. Law ii. The rate of change of momentum is proportional to the impressed force and takes place in the direction of the straight line in which that force acts. Law iii. To every action there is an equal and opposite reaction. 9. Measurement op Force. (1) P= Mf, W=Mg. Units : M in lb., /in F.S. units, P in poundals. or M in grams, / in C. S. units, P in dynes. Note. A poundal is roughly the weight of | oz., a dyne is roughly the weight of a milligram. These are absolule units. The weight of a pound (= 32'19 poundals) or of a gram (= 981 dynes) is sometimes taken {gravitation units). The dyne is so small that the megadyne (10^ dynes) is often employed. (2) Since P = Mf, the Parallelogram of Forces (STAT. 1) follows from the Parallelogram of Accelerations (BYN. 3). Note. This is the most strictly logical proof of the Parallelogram of Forces. All other proofs (Duchayla's, &c.) depend on so-called axioms. (3) Astronomical Unit of Mass is that mass which would attract an equal mass at unit distance with the unit of force. Notes. (1) Its value is approximately 1-543 x 10'' grams. (2) Its dimensions are L^ T~^, assuming that the gravitation constant (A of Note 3), of whose nature we know nothing, is of zero dimensions. (3) Two small spherical masses m^, m^ (in grams), placed at a distance d cm. apart, attract one another with a force Km^^mjd'^ dynes, where K = 6-66x10-8. (4) The mean density of the earth is 5-53. ELEMENTAKY DYNAMICS 247 10. Particular Cases of Motion. (1) Railway Trains. The fiictional resistance offered to railway trains depends on the velocity, and the retardation produced is not therefore constant, so that no exact solution to such problems can usually be found by elementary methods. Problems involving the work (see 12) done by the engine in maintaining a uniform speed on the level or on a gradient are capable of solution. The law of resistance for the engine is not the same as for the train, but an approximate law is, for an ordinary passenger train at the usual speeds, that the frictional resistance in pounds weight per ton is given ^y "^3^ + ale ^^ where V is the velocity in m. p. h. Wind may double this resistance, especially cross-winds which increase the flange- friction. (2) Masses m^, m.^ connected by string over smooth pulley / = ^ ^ a ; T = - - g (poundals, dynes). ^ ^i + mg m^-fma ^ Note. In applying these results to Atwood's machine there are many errors : (a) Rigidity of String (not serious, if silk thread is used). (b) Weight of String (often important, allowed for in one form of machine). (c) Friction of Air (velocities are too small for this to be important). (d) Friction of Pulley (can be approximately allowed for, if assumed uniform, by a counterweight which just overcomes it). (e) Mass of Pulley (the work of the falling weights is partly taken up in energy of rotation of the wheelwork. To allow for this take /= (wi m2)s'/(rWi + m2 + Ai), where /x is a constant experimentally determined for each machine. /* will generally lie between | M and M where M is the mass of the pulley, and is accurately equal to MJc^/a^ where k is the ' radius of gyration ' and a is the radius of the pulley). (3) Mass mi on smooth table pulled by rn^ hanging vertically, ^ m^ + m^' Wi + m^ 11. Impulse [= Force x Time during which it acts], (1) Change of momentum of a particle in a given time is equal to impulse of force which produces it and is in the same direction. 248 ELEMENTARY DYNAMICS (2) An impulsive force 'is measured by its 'impulse', i. e. by the change of momentum produced. Note. An ' impulsive force ' is an infinitely great force acting for an infinitely short time. See 16 (3) Note (4). 12. Work and Energy. (1) The measure of the ivorh done by a force = (Force) x (Distance through which the point of application is moved in the direction of the force) = P. d cos a = {P cos oc) . d, where d = actual displacement of point of application of P, and OL = IPd. (2) Power = rate of doing work. (3) Units. P.P. a C.G.S. Work Foot-poundal Erg [ = centimetre-dynej Practical Foot-pound Joule = 10^ ergs Unit [32 ft.-poundals] = f ft. -pound nearly Power Foot-poundal- second [Erg-second] Practical Horse-power Watt = 1 joule-second Unit = 550 foot-pound - = 10^ erg-seconds seconds Kilowatt = 1000 watts = 746 watts Notes. (1) The C. G. S. horse-power is defined as 75 kilogram- metre-seconds, and is about 735 watts, its precise equivalent depending on the value taken for g. (2) The actual working power of a good horse is about f H. P., and the maximum working power of an average man is about f H. P. The H. P. of a locomotive is 600-1000, and the H. P. of the Mauretania is 68,000. (3) Determination of TT.P. A rope passed once completely round fly- wheel (radius r) of engine carries at one end small weight iv, at the other is attached to spring balance fixed to floor, reading W. If fly- wheel can turn w times per second the H.P. at which engine is working is 2 rrnr(W-io) /550. ELEMENTAEY DYNAMICS 249 (4) The work done in stretching an elastic string = (Mean of Tensions) x (Extension). (5) Kinetic Energy of body of mass m moving with velocity vis^mv"^. Potential Energy of body of mass m at height h above the earth's surface (regarded as surface of zero potential) 13. Certain General Laws. (1) Motion of Centre of Inertia. If particles m^, W2 ... move with velocities (in same direction) %, U2 ..., and with accelerations /i , f^ ..., the Velocity of Centre of Inertia in that direction is 2mw/2m, and its acceleration is 2m//2m. (2) The motion of the C.I. of a system of particles is unaffected by any mutual action (collisions, attraction, &c.) between the particles. (3) The motion of the C. I. of a system of particles is the same as if all the masses were collected at the C. I. and all the external forces applied there parallel to their original directions. (4) Law of Conservation of Momentum. If the sum of the external forces acting on any system, resolved in any given direction, always vanishes, the total momentum of the system in that direction remains the same throughout the motion. (5) Law of Conservation of Energy. If a body or system of bodies is in motion under a conservative system of forces (i.e. forces depending only on position or configuration of system and not on velocity or direction of motion) the sum of its kinetic and potential energies is constant. Note. Non-conservative forces are Friction, Resistance of Air, Im- pulses of Collision, &c. (6) B'Alemherfs Principle. If a force is applied at each part of a moving system equal to the mass of that part 248 ELEMENTARY DYNAMICS (2) An ' impulsive force ' is measured by its ' impulse ', i. e. by the change of momentum produced. Note, An ' impulsive force ' is an infinitely great force acting for an infinitely short time. See 16 (3) Note (4). 12. Work and Energy. (1) The measure of the tvo^k done by a force = (Force) x (Distance through which the point of application is moved in the direction of the force) = P. dcosoc = {P cos oc) . d, where d = actual displacement of point of application of P, and a = Z Pd. (2) Power = rate of doing work. (3) Units. F.P.S. C.G.S. Work Foot-poundal Erg [ = centimetre-dynej Practical Foot-pound Joule = 10^ ergs Unit [32 ft.-poundals] = f ft. -pound nearly Power Foot-poundal- second [Erg-second] Practical Horse-power Watt = 1 joule-second Unit = 550 foot-pound- = 10^ erg-seconds seconds Kilowatt = 1000 watts = 746 watts Notes. (1) The C. G. S. horse-power is defined as 75 kilogram- metre-seconds, and is about 735 watts, its precise equivalent depending on the value taken for g. (2) The actual working power of a good horse is about | H. P., and the maximum working power of an average man is about f H. P. The H. P. of a locomotive is 600-1000, and the H. P. of the Mauretania is 68,000. (3) Determination of TJ. P. A rope passed once completely round fly- wheel (radius r) of engine carries at one end small weight iv, at the other is attached to spring balance fixed to floor, reading W. If fly- wheel can turn n times per second the H.P. at which engine is working is2nnr{W-io)/560. ELEMENTAEY DYNAMICS 249 (4) The work done in stretching an elastic string = (Mean of Tensions) x (Extension). (5) Kinetic Energy of body of mass m moving with velocity viB^mv^, Potential Energy of body of mass m at height h above the earth's surface (regarded as surface of zero potential) 13. Certain General Laws. (1) Motion of Centre of Inertia. If particles m^, W2 ... move with velocities (in same direction) %, u.2 ..., and with accelerations /i , f^ ..., the Velocity of Centre of Inertia in that direction is 2mw/2)m, and its acceleration is 2w//2m. (2) The motion of the C.I. of a system of particles is unaffected by any mutual action (collisions, attraction, &c.) between the particles. (3) The motion of the C. I. of a system of particles is the same as if all the masses were collected at the 0. 1, and all the external forces applied there parallel to their original directions. (4) Law of Conservation of Momentum. If the sum of the external forces acting on any system, resolved in any given direction, always vanishes, the total momentum of the system in that direction remains the same throughout the motion. (5) Law of Conservation of Energy. If a body or system of bodies is in motion under a conservative system of forces (i.e. forces depending only on position or configuration of system and not on velocity or direction of motion) the sum of its kinetic and potential energies is constant. Note. Non-conservative forces are Friction, Resistance of Air, Im- pulses of Collision, &c. (6) D'Alemherfs Principle. If a force is applied at each part of a moving system equal to the mass of that part 250 ELEMENTARY DYNAMICS multiplied by its acceleration, and in the direction of that acceleration reversed, these forces form an equilibrating sj^stem with the external impressed forces which act. (7) D'AlemherVs Principle (Impulses). If an impulse (or simultaneous impulses) act upon a S5^stem of bodies, then this impulse is in statical equilibrium with the impulses formed by multiplying the mass of each part by its velocity before the action of the impulse together with the impulses formed by multiplying the mass of each part by its reversed velocity after the action. 14. Dimensions OF Units in Mass(l, 0, 0), Length (0, 1, 0), Time (0, 0, 1) : Angular Velocity (0, 0, - 1), Velocity (0, 1, - 1), Acceleration (0, 1, -2), Volume Density (1, -3,0). Force (1,1,- 2), Momentum or Impulse (1, 1, - 1). Work or Energy (1, 2, -2), Power (1, 2, -3). 15. Projectiles. (1) The trajectory is a parabola with axis vertical. (2) The restdlant velocity at any point on the trajectory is that due to a vertical fall from the directrix to the point. Note. So that the vertical height of the directrix above the point of projection and the distance of the focus from the point of projection are both uY2 g. (3) Motion in any direction is independent of motion in a perpendicular direction. Hence. Vertical Velocity after time t is u sin (X-gr/. Horizontal Velocity after time t is u cos (X, Inclination of motion to horizon is tan 6 = {u sin gt)/u cos Of. Vertical Height after time t 'ib h = u s\n OL , t \ gt^. Horizontal Distance after time t is d ii cos a . f. (4) Time of Flight : 2 m sin Oi/g. Range on Horizontal Plane : u^ sin 2 QL/g. Proof, h = 0, .*. i = or 2 M sin QL/g. But (Z = w cos a . /. Maximum Range (pi =^ 45'') : u'^/g. [See (9) below.] ELEMENTARY DYNAMICS 251 (5) Range on Inclined Plane (Angle + j8): 2 w2 cos a sin {OL-&)/g cos^ ^. Proof. For t consider motion perpendicular to plane. Range = (hor. dist. in t) X sec /3. Maximum Range. The angle of projection bisects the angle between the plane and the vertical (whether /3 be + or -). (6) Greatest Height: 11^^11x^01/2 g. Proof. Vert. vel. =0, .. t = u^inOL/g. Hence h. (7) Latus Rectum of Path : 2 m^ cos^ QL/g. Equation to Path : y = x tan (gf/2 u^ cos'^ a) x^. (8) Envelope of Paths (m const., a variable) is a parabola with focus at the point of projection and common directrix of the paths for tangent at vertex. Proof. Geometrical or by second result of (7) 0. (9) Application to Guns, &c. With comparatively low velocities, heavy shot, and high angles of fire the parabolic theory gives good approximate results. Thus with v = 98 metres p. s. and a = 38 a range of 928 m. was obtained {Enc. Math. iv. 3, p. 202) which is 98 % of the parabolic range. With higher velocities, lighter shot, and lower angles the parabolic theory is worthless. Thus for German infantry musket with v = 640 m. p. s., a = 4, the actual range of 1,612 m. is only 28 % of the parabolic range (ref. cit., supra). The resistance of the air follows no simple law. It is proportional, roughly, to the square of the velocity for low (< 750 f. p. s.) and high (> 1,300) velocities, but is proportional to the cube or fifth power for intermediate velocities (ref. cit., p. 195). Its general effect is to make the path much steeper in descent than in ascent, and the final velocity much less than the initial, and to diminish the time of flight. Thus for V = 640 m. p. s., = 6 28'.3, the range was 2,000 m. (parabolic theory 9,346), the angle of fall 13 45', and the final velocity 159 (ref. cit., p. 216), The actual maximum range is stated (W. M. Roberts, Dynamics, p. 230) to be attained at 33, but this figure must depend on the velocity, and according to Greenhill (Encyc. Brit. iii. 275) the longest recorded range is 12 miles [y = 2,375 f. p. s., a = 40, time of flight 64'^]. According to the parabolic theory the range is 33 miles, and the time of flight 95''. The parabolic theory probably gives fair results for cricket-balls, but experiment on this subject is to be desired and would interest schoolboys. 16. Collision. (1) Newton's Laiv : (a) If two bodies impinge directly, the 252 ELEMENTARY DYNAMICS relative velocity after impact = -e (relative velocity before impact). {b) If two bodies impinge obliquely, the relative normal velocity after impact = - e (relative normal velocity before impact). Notes. (1) e = Coefficient of Restitution (Elasticity), depends only on substances of bodies, is independent of masses. Values (Hodgkinson), 0-94 (glass, glass), 0-81 (ivcry, ivory), 0-20 (lead, lead), 0-13 (cast iron, lead). (2) Fair results may be obtained by dropping balls and noting height to which they rise [e = \////^]. The heights may be fixed by rings of a retort stand. (3) The law is only approximate. The ratio of relative velocities decreases slightly as the velocities increase. (2) Direct Collision. The results of the collision of two smooth spheres, or of a smooth sphere against a smooth fixed surface, may be obtained by Neivton's Law and the principle of the Conservation of Momentum [ 13 (4)]. The following are added for reference : a. Particle on Fixed Smooth Plane, velocities before and after impact u and V making angles a, 6 with the normal : v^ = u^ (sin2 a + e^ cos^ a), cot B = e cot a. Loss of energy = |(1 e^j tjZ qq^i a, j9. Masses m, mf moving along same line with velocities u,u^ : (in + m')v={m em'') u + m^ (! + <*) ^^t (w + m')i/ = m{l+e)u + {mf em) u' . Loss of energy = J (1 - e^) (w n')^ mm' /{m + m'). Note that if m = m' , e = 1, the particles exchange velocities on collision. (3) Collision of Constrained Bodies. The impact must be divided into two parts, a ' period of compression ' at the end of which the colliding bodies will have zero relative normal velocity, and a 'period of restitution ' at the end of which they are just separating. If I and /' be the (normal) impulses during these periods, j^ = gj. ELEMENTARY DYNAMICS 253 Notes. (1) This law, which can be proved true for the cases con- sidered in (2), is assumed as the real interpretation of Newton's Law for all cases, and is sometimes given as Newton's Law. (2) It is important for the beginner to notice that it is only the velocities along the normal which are the same at the end of the period of compression. The velocities must not be equated along any other line of resolution. Further, it is only the velocities at the end of the period of compression, and not at the end of the period of restitution. (3) The Law of Conservation of Momentum [13 (4)] is always applicable. That of Conservation of Energy is not. Some problems which really belong to Rigid Dynamics may require D'Alembert's Principle [ 13 (7)]. Others may be indeterminate. (4) The time of contact of two steel spheres, rad. 2-5 cm,, meeting with relative velocity 1 cm. p. s, has been shown (Hertz) to be 0-00038 sec, the radius of the surface of contact 0-013 cm., and the maximum total pressure 2-47 kilograms. 17. The Hodograph. (1) If a particle P is moving in any manner, and if from any fixed point a line OQ be drawn || and : i^ to the velocity of P at any moment, the locus of Q is the hodograph of P. (2) The velocity of Q represents in magnitude and direc- tion the acceleration of P. (3) Special Hodographs. (a) P describes a st. line with uniform acceleration ; Q describes a parallel line with uniform velocity. (6) P describes a circle with uniform velocity ; Q also describes a circle with uniform velocity. (c) P a projectile ; Q describes a vertical st. line with uniform velocity. Proof. Either directly from definition (velocity at any pt. = velocity at P compounded with gt downwards), or v* = 2 gr . SPa SY^, ,'. vOi SY. But Y lies on tangent at vertex. {d) P describes a central orbit ; Q describes the polar reciprocal [6r. CON. 35] of the orbit turned through a right angle. Proof. For vp = constant. 254 ELEMENTARY DYNAMICS (e) P describes a S. H. M. ; Q describes an equal S. H. M. with phase a quarter period in advance. 18. Motion in a Circle or CuRve. (1) A point moving uniformly in a circle of radius r has a normal (i. e. central) acceleration v'^/r = rco^ inwards. (2) If a point is moving in a curve and its velocity at any moment be v, its normal acceleration at that moment is v^/p inwards, where p is the radius of curvature at the point. (3) If a point slides down the arc of any smooth curve or moves under gravity under constraints always at right angles to the direction of motion, and v and u be its velocity at points of which the former is a vertical height h below the latter, ^2 _ ,^2 + 2gh. Note. As examples of ' constraints ' a ball swinging freely attached to a string, a particle moving in any path on the surface of a smooth sphere, or in a smooth tube, &c. (4) Conical Pendulum. A body tied to a point by a string of length I, and moving so that the string remains stretched and sweeps out a right circular cone with axis vertical and vertical angle 2 a, will rotate once round the axis in time t = 2n \/lco3(H/g. 19. Simple Harmonic Motion. (1) If P describes a circle, centre 0, radius a, uniformly with angular velocity w, and PN be -L^* from P to a diameter, N describes a S. H. M. Displacement : x = ON = a cos cot. Velocity of N = dx/dt = - aw sin oit. Acceleration ofN = dv/dt = - aco^ cos o:>t = - oy^x. Periodic Time = 277/0). Notes. (1) The above assumes that t = when x = a. More gene- rally X = a cos {Q)t + ) where is the epoch. (2) The composition of two S. H. M.'s in the same direction is effected hy X = a cos {(tit + e) + a' cos (oj't + e'), and if cy ^ w' (i.e. if the periods are the same) the resultant is a S. H. M. (3) The composition of two S. H. M.'s in directions at right angles ELEMENTARY DYNAMICS 255 gives rise to Lissajou''s Figures, whose equations are found by elimi- nating t between x = a cos (cy^+e). y = &cos (a/ t + f^). (4) The essential conditions for a force to produce S.H.M. are (a) the force must be continuously in the direction of the displacement, (b) it must be proportional to the displacement and in the opposite direction. These are satisfied, approximately, by the great majority of forces resisting displacement in nature. If a body distant x from a position of equilibrium (measured along a straight path or a curved path) be continually acted on by a force, per unit of mass, of \x in the tangent to the path it will oscillate in a S.H.M. of period 2Tr/\/A, since \ = oo^. It is assumed throughout this Note that the displace- ments are small. (2) Pendulum. The periodic time of a simple pendukmi is 277 \^l/g and is approximately independent of the ampli- tude of swing (Isochronism). Notes. ~(1) More exactly, if 2 a be the total angle of swing, t = 27^^/l/'g(l+ksin^^ Periodic Time : 27r/\//>i. Velocity: Vfi.CD. Proof. Resolve force into fix, fiy and use 19 (1). {h) Force = fx/r^ (per unit mass). Orbit : Conic with as focus. Periodic Time: 27r Va^/fji, if orbit is elliptic. Velocity : v^ = fi (2/r + 1/a) if hyperbolic, = fjL (2/r) if parabolic, = fx{2/r-l/a) if elliptic. Note. Hence orbit is hyperbolic, parabolic, or elliptic, as square of velocity of projection >, =, or <2/x/r. (4) Kepler s Law of Planetary Motion. i. The planets describe ellipses having the sun in one focus. Note. The actual fact is that the earth (for instance) and the sun both describe ellipses round their common C. Gr. as focus, but as this C. Gr. is situated only about 280 miles from the sun's centre the law may be taken as true. ii. The radius vector from the sun to any planet describes equal areas in equal times. iii. The squares of the periodic times of all the planets ELEMENTAKY DYNAMICS 257 are proportional to the cubes of the major axes of their orbits. Notes. (1) From Law ii we deduce [(2) above] that the planets are acted on by a force towards the sun, from Law i [by (3) (&)] that this force = ft/r^, from Law iii that fx is the same for all planets. (2) The laws hold also if for 'planet' we read 'satellite', and for * sun ' we read 'planet'. 1S72 260 ELEMENTARY HYDROSTATICS (6) The pressure on the base of a vessel containing liquid depends only on the area of the base and the depth of the liquid and is independent of the slope or shape of the sides. 3. The Thrusts. [Horizontal (H.T.), Vertical (V.T.), Resultant (R.T.).] Name. Magnitude. Point of Aclion. RT. on planes. swhS ; S is area of plane, C. P. of plane. [See h depth C.Gr. 4.] V.T. on surfaces. Wt. of liquid vertically above S. C.Gr. of this liquid. V.T. on immersed body. Wt. of liquid displaced. C.Gr. of this liquid. H.T. on surface. H.T. on S in any direc- C.P. of projection. tion is R. T. upon pro- [See 4.] jection of S upon a vertical plane per- pendicular to that direction. R.T. on surface. For planes : see above. For other surfaces : re- Intersection of V. T. sultant of V.T. and and H.T. H.T. Notes. (1) The phrase ' Whole Pressure ' unfortunately occurs still. Its value is sichS. But it has no meaning whatever unless S is plane, and then it is the same as Resultant Thrust. (2) The V.T. may act downwards (liquid above S, as when S is part of the base of a bottle filled with water), or upwards (liquid below S, as when S is part of the shoulder of the bottle). (3) The V. T. for immersed bodies is found by same rule whether body is wholly or partially immersed. (4) H.T. The projection should be taken parallel to a (vertical) plane of symmetry of S. If there is no such plane the H. T. in two directions at right angles must be taken and their resultant found. (5) The R. T. on a curved surface may be found more easily if the surface is bounded by a plane curve, from the fact that R.T. on surface and R.T. on plane bounded by curve are statically equivalent to weight of liquid between surface and plane (see V.T. on immersed body). (6) This consideration will often give the C.P. in a difficult case, e.g. for C.P. of a circle wholly immersed (plane not necessarily ELEMENTARY HYDROSTATICS 261 vertical) consider a hemisphere bounded by circle, R. T. on hemisphere must pass through centre of sphere ; take moments about centre to express equivalence of R. T. on circle and R. T. on hemisphere (moment zero) to wt. of hemisphere of liquid. (7) Hence we can deal with pressure on (a) curved surface of right circular cylinder with plane ends, (6) curved surface of a right circular cone, (c) curved surface of frustum of cone with plane circular ends. 4. Centre of Pressure (C. P.). Methods of JDefermination. i. C. P. of any plane area [S] is the point of the area vertically below the C. Gr. of the solid formed by drawing vertical lines through the boundary of the area to meet the surface of the liquid. ii. To determine C. P. of any plane area draw horizontal lines through the boundary of the area, each line being in length ::1 to the pressure at the boundary-point through which it is drawn ; determine C. Gr. of solid thus formed ; a horizontal line through this C. Gr. meets the area in its C.P. iii. See 3, Note (6). iv. If a line of symmetry Ox of S meets the surface of the liquid in and meets S in ^ and B, and if S be so placed that the ordinates of S at right angles to Ox are all horizontal, the distance of the C. P. from along Ox is V. Split into rectangles and triangles and see below. Particular Cases. (1) Eectangle, one edge in surface: C.P. f down median. (2) Triangle^ one side in surface : C. P. J down mc dian. (3) Triangle, vertex in surface, opposite side horizontal : C. P. f down median. Note. With a little practice these are instantaneously obtained mentally by Method i, and there is danger in memorizing them. 262 ELEMENTAKY HYDROSTATICS (4) Triangle, anyhow immersed : C. P. is centre of II forces acting at mid-points of sides and of magnitudes : } to depths. Note. If a, j8, 7 be depths of angular points, C. P. is at depth where h is depth of its C. Gr. (5) Quadtilaterals, Polygons : Method v, above. Note. If a, )3, 7, 8 be depths of vertices of a 4* and h the depth of its C. Gr., the depth of its C. P. is | 2a - ScXiS. (6) Circle, wholly immersed (plane vertical, depth of centre /<, radius a). Depth of C. P. /i + "V^ h. [ 3, Note (6) .] ^ofe. If the plane of the Q be inclined at Z a to horizon the C. P. is ^ (a^/h) sin a from the centre, measured along the median line down - wards, h being still the depth of the centre. The two results may be learnt, if desired, as one : If d be the distance of the centre from the surface measured along the median line, then the distance of the C.P. is always d+a^/id. 6. Equilibrium or Floating Bodies. (1) The forces acting are : (a) Upward pressure of liquid displaced through C. Gr. of this liquid (Centre of Buoyancy). (^) Weight of body, downwards, through its C. Gr. (y) Any external impressed forces such as tension of string, reaction at hinge, &c. Apply ordinary statical methods to these. (2) A body floating freely is in equilibrium if a = /^ [ 5 (1)] and if the point of action of a is in the same vertical line as the point of action of /3. (3) The equilibrium of a body floating freely is stable or unstaUe according as the Metacentre ( 6) is above or below the centre of gravity. If the immersed portion is spherical the metacentre is the centre of the sphere. For other cases see 6. ELEMENTARY HYDROSTATICS 263 Notes. (1) If the metacentre and tJie centre of gravity coincide the equilibrium is neutral. (2) For submarines the condition of stability is that the C. Gr. should be below the C.B. 6. Metacentre. (1) Centre of Buoyancy (C. B.) is the 0. Gr. of the liquid displaced. Surface of Buoyancy: If a floating body is displaced so that the volume of the part immersed remains constant, the locus (in the body) of the C. B. is the surface of buoyancy. Curve of Buoyancy is the locus of the C. B. for displace- ments such that the C. B. remains in one plane. (2) The tangent plane at any point to the surface of buoyancy is II to the plane of floatation (P.F.). (3) The Positions of Equilibrium are found by drawing normals from the C. Gr. to the surface of buoyancy and taking P. F. X^' to these normals. (4) The Metacentre for planar displacements of the C. B. is the centre of curvature of the curve of buoyancy at the C. B. for the position considered. (5) Position of Metacentre. If the body is symmetrical and if the displacements be about an axis of symmetry in the P. F., the distance of the metacentre above the C.B. = 1/ V, where I = ' Moment of Inertia ' of the P. F. about this axis of symmetry, V = volume of the fluid displaced. Special cases : P. F. a circle : I = tt a^/L P. F. a rectangle (sides a, &) : I = a' 6/12, the displacements being in a plane at right angles to edge b. Lamina (floating vertically) : I = a'/12, where a = length of section by liquid and V = area immersed. Practical Determination of Metacentre. The 'Metacentric Height' (i.e. height of metacentre above the centre of gravity) is worked out before the ship is built from the working 264 ELEMENTARY HYDROSTATICS diagrams, but as the position of the C.Gr., which depends on the loads, can only be roughly ascertained, a further 'inclining experiment' is made when the ship is completed by moving weight w a distance h from one side of the deck to the other. This causes the ship to tilt through an angle 6, which is determined by a plumb-line, and it is quite easy to prove that if W be the weight (' displacement ') of the ship, the metacentric height is hw cot 0/W, The actual metacentric heights are greatest in shallow-draught river gunboats, in which they vary from 8 to 20 feet, and least in large mail steamers, in which they vary from 6 in. to 2 feet. For men-of-war they are between 3| and 5 feet. A small value diminishes the safety, a large value causes excessive and rapid rolling (Encyc. Brit. xxiv. 924). (6) Curves of Buoyancy. Triangle {Vertex Immersed) : A hyperbola with immersed sides as asymptotes. Bectangle : A parabola with axis the vertical line of symmetry of the rectangle, concavity upwards, and tangent at the vertex the horizontal through the C. B. Notes. (1) For rolling displacement of an ordinary ship the C. B. is approximately an arc of a hyperbola, but more nearly a parabolic arc if the sides are almost perpendicular to the water. (2) It is an admirable exercise to trace out completely the buoyancy curves and their evolutes (the metacentric locus) in these cases. In the case of the rectangle the buoyancy curve consists of a closed curve made up of four parabolic arcs united by four arcs of rectangular hyperbolas, which latter vanish if the rectangle floats with exactly half of it immersed. 7. Rotating Liquids. (1) If a liquid, whether confined or free, rotate about a vertical axis with angular velocity co, the surfaces of equal pressure are equal paraboloids of revolution, whose axes are the axis of revolution and whose equation is (2) The free surface in such a case takes the form of a paraboloid of revolution, and the pressure at any point is the pressure up to the free surface. (8) To find the pressure at any point in a confined rotating liquid : (a) If the liquid does not quite fill the containing vessel ELEMENTARY HYDROSTATICS 265 the position of the free surface must be determined by the consideration that the pai'aboloid generated by x^ = (2^/a)2) y has for volume the volume of space left empty in the vessel. This gives the free surface of zero pressure, and thus the pressure at any point. (6) If the liquid just fills the vessel, the surface of zero pressure is the paraboloid with axis the axis of revolution and touching the rotating vessel at the point of zero pressure. 8. Surface Tension. Cylinder. If the surface tension (perpendicular to length of cylinder) at any point of a thin long cylinder (rad. r), where the hydrostatic pressure is j?, be t per unit of length, t = pr. Sphere. If the surface tension at any point of a thin sphere (rad. r) be t in any direction, 2t = pr. Note. If the cylinder be of finite uniform thickness, c and T be the tension per unit area of the substance, t = cT. 9. Gases. (1) Boyle's Law. The pressure of a given mass of gas varies inversely as its volume, provided that the temperature is unaltered, i. e. pv = constant, if t is constant. (2) Charles's Laiv : Vi = Vq(1 + oct). (^constant, a = ^^g nearly.) (3) Deductions : (a) Pi = Pq/{1 + oct). {p constant, a = -^i-g- nearly.) (/3) pv/T = constant. (T = absolute temperature.) (y) p/p is always the same for a given gas {t constant). (4) Height by Barometer. The difference of height at two stations at which the barometric readings are h^^, h is, approximately, 1-84 logio \.K/h] X 10* metres. INDEX IThe references are to pages.'] Acceleration : 243, 244. Acnode : 219. Adams's Property : 132, 137. Addition Formulae (Trig.) : 71. Algebra: 11-37. Ambiguous Case : 75. Analytical Geometry : 167-96. Anchor Ring : 10. Angular Velocity : 244, 256. Auharmonics : 118-20, 149-50, 188-9, 191 ; (Spherical) : 91. Antisimilitude : 114. Apollonius : 128. Arbogast's Theorem : 203. Arc of Circle ": Length, 6 ; C. Gr. 237. Archimedean Solids : 98. Archimedes, Spiral of: 231. Areal Coordinates : 189-95. Areas : (A. G.) 157, 190. See also Mensuration. Arithmetic : 1-5. Arithmetical Progression : 20. Astronomical details: 5, 256-7. Asymptotes : Cartesian, 216-18 ; Circular, 218 ; Polar, 218. See also Hyperlola. Atwood's Machine : 247. Auxiliary Angle : 164, 167, 170, 174-5, 195. Axis : s. V. Central, HomograpMc, Radical, Similitude. Ballistics: 251. Barometer : 259, 265. Bernoulli: Numbers, 204-5 ; Sol. Eqns. 54. Bezout, Elimination : 62. Binomial-Theorem, 22 ; Equa- tions, 55-6. Bobillier's Theorem : 116. Boyle's Law : 265. Brachistochrones : 245. Branche Pointillee : 220. 1873 Brianchon's Theorem : 119, 150, 184, 187. Brocard Points : 105-6. Budan's Theorem : 42. Buoyancy : Centre of, 262, 263 ; Curves of, 264. Burmann's Theorem : 204. Cagnoli's Formula : 94. Cardan's Solution : 59. Cardioid : 213, 215, 216, 230. Carnot's Theorem : 1 87. Cartesian Ovals : 230. Casey's Theorem, 124; Projec- tion, 156. See also 128. Cassinian Ovals : 215, 230. Castillon's Problem : 126, 127. Catenary : 228. Cauchy's Remainder (Taylor) : 202. Caustics : 216. Central-Axis (Poinsot), 235 ; Orbits, 256. Centre of Buoyancy : 262-3 ; Pressure, 261-2. Centroid : 77, 99, 101, 190, 237-8. Ceva's Theorem : 99. Chainette: 228. Change of Variable (D. C.) : 206-7. Charles's Law : 265. Chords of Circle, Motion down : 245. Circles: Anal. Geom. of, 164-6, 193-5; Antisimilitude, 114; C. Gr. 237, 238 ; C. P. 262 ; C. S. 114, 166 ; Chords of. Mo- tion down, 245; Coaxal, 112-13, 166; Cotes's Properties, 88 ; De Moivre's Properties, 88 ; Dia- metral, 129; Harmonic Pro- perties, 108-9 ; Hart's, 90 ; Mensuration, 6, 7 ; Motion in, 254; Orthogonal, 129, 160; INDEX 267 p-r eqn. 213 ; p-\p equ. 213 ; Problems on, 127-9; Kadical Axis, 112 ; Kadical Centre, 112; s-xp eqn. 213; Similitude, 114; Touching other circles, 128. See also Ourvatwe, Excirde, In- circle, Nine Point Circle, &c. Circular-Cubics, 230; Motion, 254. Circumcircle : Trig. 75-6 ; Geom. 102-8; S. Trig. 95-6; A. G. 178, 194. Circumconics : 178, 184, 191-5. Cissoid : 215, 230. Clausen on tt : 81. Closest Contact, Conic of: 223. Coaxal Circles : 112-13, 166 ; In Sph. Trig. 90. Coefficient : Elasticity, 252 ; Fric- tion, 239 ; Kestitution, 252. Collinearity of Points : 100, 159. Collision: 251-3. Combinations : 21. Commensurable Roots : 44-5. Common Tangent : 166. Complex Quantities : 19, 81. Components : 232, 243. Concavity, Test of : 218-19. Concentric Conies : 185. Conchoid : 231. Cone : Centroid, 239 ; Frustum, 10 ; Mensuration, 9, 10. Confocal Conies, 185. Conic : Closest Contact, 223 ; Through 4 points, 178, 184-5 ; Touching 4 lines, 185 ; Touching 2 lines, 184, 185, 195. Conjugate Lines, 109, 147 ; Points, 109, 147 ; Diameters, 140, 142, 146, 171, 183, 193; Hyperbola, 175. Conormal Points : 172. Conservation : Energy, 249 ; Mo- mentum, 249. Contact of Conies, 186-7, 223 ; of Curves, 222. Contraparallelogram : 123. Convergence of Series : 32-3. Convergents : 25-8 ; Intermediate, 25. Convexity, Test of: 218-19. cos 6 : Expansion, 80 ; Exp. Value, 81 ; Factors, 86, 87. cosn^: Expansion, 80, 83-4; . Factors, 86, 87. cos "5: Expansion, 83. cosec 6: Expansion, 81, 205. cosh n

/;r= 1-77245385 71 2 _ 9-86960440 \^^= 1-46459189 7r/i8o = 0^01745329 logarithm 0*497150 I96i20 1*718999 0-622089 0-248575 0-994300 0-165717 2^241877 logarithm I -i- 77 = 0*31830989 1-502850 I -^ 4 7r = 0-07957747 2-900790 v^6T n^ = I -24070098 0-093667 ^3 H- 477 = 0-6203 5049 1-792637 ^i ^ 77 = 0-56418958 1-751425 I -^ 77*= 0-10132118 1-005700 ^2= 2-14502940 0-331433 180/77= 57-29577951 I-758123 Naperian (or Natural) Logarithms c = 2-7182182 log^oc == -43429448 logg 10 = 2-30258509 logio N = logg N X logio e. lege iV = log^o N x log^ 10 LOGARITHMS OF NUMBERS Mean Differences 1 2 3 4 I 2 3 4 5 6 7 8 9 10 ooooo 00432 00860 01284 01703 42 '8i 127 170 212 254 297 339 381 II 04139 04532 04922 05308 05690 39 77 116 155 193 232 270 3(>9 348 12 07918 08279 08636 08991 09342 35 71 106 142 177 213 248 284 319 13 II394 II727 12057 12385 .12710 33 66 98 131 164 197 229 262 295 14 I46I3 14922 15229 15534 15836 ?o 61 91 122 152 183 213 244 274 15 17609 17898 18184 18469 18752 28 57 85 114 142 171 199 228 256 i6 20412 .20683 .20952 21219 .21484 27 53 80 107 134 160 187 214 240 17 23045 23300 23553 23805 24055 25 50 76 lOI 126 151 176 201 227 i8 25527 .25768 .26007 26245 .26482 24 48 71 95 119 143 167 190 214 19 27875 .28103 28330 28556 .28780 23 45 68 90 113 ^35 158 180 203 20 30103 30320 30535 30750 30963 21 43 64 86 107 128 150 172 193 21 32222 .32428 32634 32838 33041 20 41 61 82 102 123 143 164 184 22 34242 34439 34635 34830 35025 20 39 59 78 98 117 137 158 176 23 36173 36361 36549 36736 .36922 19 37 56 75 94 112 131 149 168 24 38021 38202 38382 38561 38739 18 36 54 72 90 108 125 143 161 25 39794 39967 40140 40312 40483 17 34 52 69 86 103 120 138 155 26 41497 41664 41830 41996 42160 17 33 50 66 S3 99 116 132 149 27 43136 43297 43457 43616 43775 16 32 48 64 80 96 112 128 143 28 44716 .44871 45025 45179 45332 15 31 46 61 77 92 108 123 138 29 46240 46389 46538 46687 46835 15 30 45 59 74 89 104 119 134 30 47712 47857 48001 48144 48287 14 29 43 57 72 86 lOI lis 129 31 49136 49276 49415 49554 49693 14 28 42 56 70 83 97 III 125 32 50515 50651 50786 50920 51055 13 27 40 54 67 81 94 108 121 33 5I85I 51983 52114 52244 52375 13 26 39 52 65 78 91 105 118 34 53148 53275 53403 53529 53656 13 25 38 51 63 76 89 lOI 114 35 54407 5453 1 54654 54777 54900 12 25 37 49 62 74 86 99 III 36 55630 55751 55871 55991 56110 12 24 36 48 60 72 84 96 108 37 56820 56937 57054 57171 57287 12 23 35 47 58 70 82 93 105 38 57978 58093 58206 58320 58433 II 23 34 45 57 68 80 91 102 39 59106 59218 59329 59439 59550 II 22 33 44 55 66 78 89 100 40 60206 60314 60423 60531 60638 II 22 32 43 54 65 76 86 97 41 61278 61384 61490 61595 61700 II 21 32 42 53 63 74 84 95 42 62325 62428 62531 62634 62737 10 21 31 41 51 62 72 82 93 43 63347 63448 63548 63649 63749 10 20 30 40 50 60 70 80 90 44 64345 64444 64542 1 64640 64738 10 20 29 39 49 59 69 79 88 45 65321 65418 65514 65610 1 65706 10 19 29 38 48 58 67 77 86 46 66276 66370 66464 66558 66652 9 19 28 .38 47 56 66 75 84 47 67210 '67302 67394 67486 67578 9 18 28 37 46 55 64 73 83 48 68124 68215 68305 68395 68485 9 18 27 36 45 54 63 72 81 49 G9020 69108 69197 69285 '69373 9 18 26 35 44 53 62 71 79 SO 69897 69984 70070 70157 70243 9 17 26 35 43 52 60 69 78 SI 70757 70842 70927 71012 71096 8 17 25 34 42 51 59 68 76 52 71600 71684 71767 71850 71933 8 17 25 33 42 50 58 67 75 S3 72428 72509 .72591 72673 72754 8 16 24 33 41 49 57 65 73 54 73239 73320 *7340o| 73480 73560 8 16 1 24 32 1 40 48 56 64 72 LOGARITHMS OF NUMBERS 5 6 7 8 9 I 2 3 4 5 6 7 8 9 lO 02119 02531 02938 03342 03743 40 8? 121 162 202 242 283 323 364 II 06070 06446 06819 07188 07555 n 74 III 148 185 222 259 296 III 12 09691 10037 10380 10721 11059 34 68 102 137 170 204 238 272 307 13 13033 13354 13672 13988 14301 32 ^l 95 126 158 190 221 253 284 14 16137 16435 16732 17026 17319 29 59 88 118 147 177 206 236 26s 15 19033 19312 19590 19866 20140 28 55 83 no 138 165 193 221 248 i6 21748 2201 1 22272 22531 22789 26 52 78 104 130 156 182 208 233 17 24304 24551 24797 25042 25285 24 49 71 98 123 147 171 196 220 i8 '26T17 26951 27184 27416 27646 ^i 46 70 93 116 139 162 185 208 19 29003 29226 29447 29667 29885 22 44 66 88 no 132 IJ4 176 198 20 31175 31387 31597 31806 32015 21 42 63 84 105 126 147 168 188 21 ^3244 33445 33646 33846 34044 20 40 60 80 100 120 140 160 180 22 35218 35411 35603 35793 35984 19 38 57 77 96 115 134 153 172 23 37107 37291 37475 37658 37840 18 Z7 55 71 91 no 128 146 i6c 24 38917 39094 39270 39445 39620 18 35 53 70 88 105 123 140 158 25 40654 40824 40993 41 162 41330 17 34 51 67 84 lOI 118 135 152 26 42325 42488 42651 42813 42975 16 32 49 65 81 97 114 130 146 27 43933 44091 44248 44404 44560 16 31 47 63 78 94 no 125 141 28 45484 45637 45788 45939 46090 15 30 45 61 7^ 91 106 121 136 29 46982 47129 47276 47422 47567 14 29 44 58 71 87 102 117 131 30 48430 48572 48714 48855 48996 14 28 42 56 71 85 99 113 127 31 49831 49969 50106 50243 50379 14 27 41 55 68 82 96 109 123 32 51188 51322 51455 51587 51720 13 27 40 53 66 80 93 106 119 33 52504 52634 52763 52892 53020 13 26 39 51 64 77 90 103 116 34 53782 53908 54033 54158 54283 13 25 38 50 ^l 75 88 100 113 35 55023 55145 55267 55388 55509 12 24 36 49 61 73 85 97 109 36 56229 56348 56467 56585 56703 12 24 35 47 59 71 83 95 106 37 57403 57519 57634 57749 57864 12 23 35 46 58 69 81 92 104 38 58546 58659 58771 58883 58995 II 22 34 45 56 67 78 90 lOI 39 59660 59770 59879 59988 60097 II 22 33 44 55 66 76 87 98 40 60746 60853 60959 61066 61172 II 21 32 43 S3 64 74 85 96 41 61805 61909 62014 62118 62221 10 21 31 42 52 62 71 83 94 42 62839 62941 63043 63144 63246 10 20 30 41 51 61 71 81 91 43 63849 63949 64048 64147 64246 10 20 30 40 50 60 70 79 89 44 64836 64933 65031 65128 65225 10 19 29 39 49 58 68 78 87 45 65801 65986 65992 66087 66181 10 19 29 38 48 57 67 76 86 46 66745 66839 66932 67025 67117 9 19 28 17 47 56 65 74 84 47 67669 67761 67852 67943 68034 9 18 27 36 46 55 64 71 82 48 68574 68664 68753 68842 68931 9 18 27 Z^ 45 54 62 71 80 49 69461 69548 69636 69723 69810 9 17 26 35 44 52 61 70 78 SO 70329 70415 70501 70586 70672 9 17 26 34 43 51 60 68 77 51 71181 71265 71349 71433 71517 8 17 25 34 42 50 59 67 75 52 72016 72099 72181 72263 72346 8 16 25 33 41 49 58 66 74 53 72835 72916 .72997 73078 73159 8 16 24 Z^ 40 48 57 65 73 54 73640 73719 73799 73878 73957 8 16 24 32 40 48 55 63 71 Mean Differences LOGARITHMS OF NUMBERS Mean Differences 1 2 3 4 I 2 3 4 5 6 7 8 9 55 74036 74115 74194 74273 74351 8 l6 24 Ji 39 47 55 6j n 56 74819 74896 74974 75051 75128 8 15 23 31 39 46 54 62 69 57 75587 75664 75740 75815 75891 8 15 23 30 38 46 53 61 68 58 76343 76418 76492 76567 76641 7 15 22 30 Z7 45 52 60 67 59 77085 77159 77232 77305 77379 7 15 22 29 Z7 44 51 59 66 60 77815 77887 77960 78032 78104 7 14 22 29 36 43 51 58 65 61 78533 78604 78675 78746 78817 7 14 21 28 36 43 SO 57 64 62 79239 79309 79379 79449 79518 7 14 21 28 35 42 49 56 63 63 79934 80003 80072 80140 80209 7 14 21 27 34 41 48 55 62 64 80618 80686 80754 80821 80889 7 14 20 27 34 41 47 54 61 65 81291 81358 81425 81491 81558 7 ' 13 20 27 33 40 47 53 60 66 81954 82020 82086 82151 82217 7 13 20 26 ZZ 39 46 52 59 67 82607 82672 82737 82802 82866 e > 13 19 26 32 2,9 45 52 58 68 83251 83315 83378 83442 83506 t > 13 19 25 32 38 45 SI 57 69 83885 83948 8401 1 84073 84136 t > 13 19 25 31 38 44 50 56 70 84510 84572 84634 84696 84757 <^ ) 12 19 25 31 37 43 49 56 71 85126 85187 85248 85309 85370 C ) 12 18 24 31 2>7 43 49 55 72 85733 85794 85854 85914 85974 ^ ) 12 18 24 30 36 42 48 54 73 86332 86392 86451 86510 86570 t ) 12 18 24 30 36 42 48 53 74 86923 86982 87040 87099 87157 t ) 12 18 23 29 35 41 47 53 75 87506 87564 87622 87679 87737 ^ ) II 17 23 29 35 40 46 52 76 88081 88138 88191; 88252 88309 ( ) II 17 23 29 34 40 46 51 77 88649 88705 88762 88818 88874 t ) II 17 22 28 34 39 45 51 78 89209 89261; 89321 89376 89432 t ) II 17 22 28 33 39 44 50 79 89763 89818 89873 89927 89982 ) II 16 22 27 33 38 44 49 80 90309 90363 90417 90472 90526 ! 5 II 16 22 27 33 38 43 49 81 90849 90902 90956 91009 91062 ) II 16 21 27 32 37 43 48 82 91381 91434 91487 91540 91593 ^ 5 II 16 21 27 32 37 42 48 83 91908 91960 92012 92065 92117 - 5 10 16 21 26 31 37 42 47 84 92428 92480 92531 92583 92634 . 5 10 15 21 26 31 36 41 46 85 92942 92993 93044 93095 93146 5 10 15 20 26 31 36 41 46 86 93450 93500 93551 93601 93651 . ) 10 15 20 25 30 35 40 45 87 93952 94002 94052 94101 941 5 1 . ) 10 15 20 25 30 35 40 45 88 94448 94498 94547 94596 94645 . 5 10 15 20 25 30 34 39 44 89 94939 94988 95036 95085 95134 . 5 10 15 19 24 29 34 39 44 90 95424 95472 95521 95569 95617 . 5 10 14 19 24 29 34 39 43 91 95904 95952 99599 96047 96095 . ) 10 14 19 24 29 33 38 43 92 96379 96426 96473 96520 96567 . 5 9 14 19 24 28 33 38 42 93 96848 96895 96942 96988 97035 . 5 9 14 19 23 28 33 37 42 94 97313 97359 97405 97451 97497 . 5 9 14 18 23 28 32 37 41 95 97772 97818 97864 97909 97955 . 5 9 14 18 23 27 32 36 41 96 98227 98272 98318 98363 98408 5 9 14 18 23 27 32 36 41 97 98677 98722 98767 988 1 1 98856 i ^ 9 13 18 22 27 31 36 40 98 .99123 99167 9921 1 99255 .99300 . ^ 9 13 18 22 27 31 35 40 99 99564 99607 99651 99695 99739 ^ \ 9 13 17 22 6 31 35 39 LOGARITHMS OF NUMBERS 5 6 7 8 9 I 8 2 16 3 23 4 3! 5 39 6 47 7 55 8 62 9 55 74429 74507 74586 74663 74741 70 S6 75205 75282 75358 75435 75511 8 15 23 31 38 46 53 61 69 57 75967 76042 761 18 .76193 76268 8 15 ^Z 30 38 45 53 60 68 58 76716 76790 76864 .76938 77012 7 15 22 30 n 44 52 59 66 59 77452 77525 77597 '^^6^Q 777M 7 15 22 29 36 44 51 58 65 60 78176 78247 78319 78390 78462 7 14 21 29 36 43 50 57 64 61 78888 78958 79029 .79099 79169 7 14 21 28 35 42 49 56 63 62 79588 .79657 79727 79796 79865 7 14 21 28 35 42 48 55 62 63 80277 80346 .80414 .80482 80550 7 14 20 27 34 41 48 55 61 64 80956 81023 81090 .81158 81224 7 13 20 27 34 40 47 54 60 65 81624 81690 81757 81823 81889 7 13 20 26 33 40 46 53 59 66 82282 82347 82413 82478 82543 7 13 20 26 11 39 46 52 59 67 82930 82995 83059- 83123 83187 6 13 19 26 32 39 45 51 58 68 83569 83632 83696 83759 83822 6 13 19 25 32 38 44 51 57 69 84198 84261 84323 84386 84448 6 12 19 25 31 n 44 50 56 70 84819 84880 84942 85003 85065 6 12 18 25 31 37 43 49 55 71 85431 85491 85552 85612 85673 6 12 18 24 30 36 42 48 54 72 86034 86094 86153 86213 86273 6 12 18 24 30 36 42 48 54 73 86629 86688 86747 86806 86864 6 12 18 24 29 35 41 47 53 74 87216 87274 87332 87390 87448 6 12 17 23 29 35 41 46 52 75 87795 87852 87910 87967 88024 6 II 17 23 29 34 40 46 SI 76 88366 88423 88480 88536 88593 6 II 17 23 28 34 40 45 51 77 88930 88986 89042 89098 89154 6 II 17 22 28 Zl 39 45 50 78 89487 89542 89597 89653 89708 6 II 17 22 28 11 39 44 50 79 90037 90091 90146 90200 90255 5 II 16 22 27 Zl 38 44 49 80 90580 90634 90687 90741 9079s 5 II 16 22 27 32 38 43 48 81 91116 91 169 91222 91275 91328 II 16 21 27 32 Z7 42 48 82 91646 91698 91751 91803 91855 10 16 21 26 31 Z7 42 47 83 92169 92221 .92273 92324 92376 10 16 21 26 31 Z^ 41 47 84 92686 .92737 92788 92840 92891 10 15 20 26 31 36 41 46 85 93197 93247 93298 93349 93399 10 15 20 25 30 35 40 46 86 93702 93752 93802 93852 93902 10 15 20 25 30 35 40 45 87 94201 94250 94300 94349 94399 10 15 20 25 30 35 40 44 88 94694 94743 94792 94841 94890 10 15 20 25 29 34 39 44 89 95182 95231 .95279 95328 95376 10 15 19 24 29 34 39 44 90 95665 95713 95761 95809 95856 10 14 19 24 29 33 38 43 91 96142 96190 96237 96284 96332 9 14 19 24 28 ZZ 38 43 92 96614 .96661 96708 96755 .96802 9 14 19 23 28 33 37 42 93 97081 97128 97174 97220 .97267 9 14 19 23 28 Z2 Z7 42 94 97543 97589 97635 97681 97727 9 14 18 23 27 32 17 41 95 98000 .98046 98091 98137 98182 9 14 18 23 27 32 36 41 96 98453 98498 98543 98588 98632 9 13 18 22 27 31 36 40 97 98900 98945 98989 99034 99078 9 13 18 22 27 31 36 40 98 99344 .99388 99432 99476 99520 9 13 18 22 26 31 35 40 991 99782 99826 99870 99913 99957 9 13 17 22 26 31 35 39 Mean Differences ANTILOGARITHMS Mean Differences 1 2 3 4 I 2 3 4 5 6 7 8 9 oo lOOOO 10023 10046 10069 10093 1. 5 ~7 ~9 12 14 i6 ^ 21 01 10233 10257 10280 10304 10328 2 5 7 9 12 14 17 19 21 02 1 047 1 10495 10520 10544 10568 2 5 7 10 12 15 17 20 22 03 I07I5 10740 10765 10789 10814 2 5 7 10 12 15 17 20 22 04 10965 10990 11015 11041 1 1066 3 5 8 10 13 15 18 20 23 OS II220 1 1 246 11272 1 1 298 1 1324 3 5 8 10 13 16 18 21 23 06 1 1482 1 1 508 11535 11561 11588 3 5 8 II 13 16 18 21 24 07 1 1 749 1 1776 1 1803 11830 11858 3 5 8 II 14 16 19 22 24 08 12023 12050 12078 12106 12134 3 6 8 II 14 17 19 22 25 09 12303 12331 12359 12388 12417 3 6 9 II 14 17 20 23 26 10 12589 12618 12647 12677 12706 3 6 9 12 15 i8 20 23 26 II 12882 1 29 1 2 12942 12972 13002 3 6 9 12 15 18 20 24 27 12 13183 13213 13243 13274 13305 3 6 9 12 15 18 21" 24 27 13 13490 13521 13552 13583 13614 3 6 9 12 16 19 22 25 28 14 13804 13836 13868 13900 13932 3 6 10 13 16 19 22 26 29 IS 14125 141S8 14191 14223 14256 3 7 10 13 16 20 23 26 30 16 14454 14488 145 2 1 14555 14588 3 7 10 13 17 20 24 27 30 17 1 479 1 14825 14859 14894 14928 3 7 10 14 17 21 24 27 31 18 15136 15171 15205 15241 15276 4 7 II 14 18 21 25 28 l^ 19 15488 15524 15560 15596 15631 4 7 II 14 18 22 25 29 32 20 15849 15885 15922 15959 15996 4 7 II 15 18 22 26 29 33 21 16218 16255 16293 16331 16368 4 8 II 15 19 23 26 30 34 22 16596 16634 16672 16711 16749 4 8 II 15 19 23 27 31 34 23 16982 17022 1 706 1 17100 1 7 140 4 8 12 16 20 24 28 32 35 24 17378 17418 17458 17498 17539 4 8 12 16 20 24 28 32 36 2S 17783 17824 17865 17906 17947 4 8 12 16 21 25 29 33 37 26 18197 18239 18281 18323 18365 4 8 13 17 21 25 30 34 38 27 1 862 1 18664 18707 18750 18793 4 9 13 17 22 26 30 34 39 28 1905 s 19099 19143 19187 19231 4 9 13 18 22 26 31 35 40 29 19498 19543 19588 19634 19679 5 9 14 18 23 27 32 36 41 30 19953 19999 20045 20091 20137 5 9 14 18 23 28 32 37 42 31 20417 20464 20512 20559 20606 5 9 14 19 24 28 11 38 43 32 20893 20941 20989 21038 21086 5 10 15 19 24 29 34 39 44 33 21380 21429 21478 21528 21577 5 10 15 20 25 30 35 40 44 34 21878 21928 21979 22029 22080 5 10 15 20 25 30 35 40 46 35 22387 22439 22491 22542 22S94 5 10 16 21 26 31 36 41 47 36 22909 22961 23014 23067 23I2I 5 II 16 21 27 32 Z7 42 48 37 23442 23496 23550 23605 23659 5 II 16 22 27 33 38 44 49 38 23988 24044 24099 24155 24210 6 II 17 22 28 Zl 39 44 50 39 24547 24604 24660 24717 24774 6 II 17 23 28 34 40 45 51 40 25119 25177 2523s 25293 25351 6 12 17 23 29 35 41 47 52 41 25704 25763 25823 25882 25942 6 12 18 24 30 36 42 48 54 42 26303 26363 26424 26485 26546 6 12 18 24 30 36 43 49 55 43 26915 26977 27040 27102 27164 6 12 19 25 31 37 44 50 56 44 27542 27606 27669 ^7711 27797 6 13 19 26 32 38 45 51 57 45 28184 28249 28314 28379 28445 7 13 20 26 33 39 46 52 59 46 28840 28907 28973 29040 29107 7 13 20 27 33 40 47 53 60 47 29512 29580 29648 29717 29785 7 14 21 2y 34 41 48 55 62 48 30200 30269 30339 30409 30479 7 14 21 28 35 42 49 56 63 49 30903 30974 31046 31117 3II89 7 14 21 2. 36 43 1 50 57 64 10 ANTILOGARITHMS 5 6 7 8 9 I 2 3 4 5 6 7 8 9 00 10116 10139 10162 10186 10209 2 ~5 "7 "9 12 U 16 ^ 21 01 10351 10375 10399 10423 10447 2 5 7 10 12 14 17 19 22 02 10593 10617 1 064 1 10666 1 069 1 2 5 7 10 12 15 17 20 22 03 10839 10864 10889 10914 10940 3 5 8 10 13 15 18 20 23 04 1 1092 11117 11143 1 1 169 11194 3 5 8 10 13 15 18 20 23 OS 1 1350 1 1376 1 1402 1 1429 1 1455 3 5 8 II 13 16 18 21 24 o6 11614 11641 1 1668 1 1695 11722 3 5 8 1 1 14 16 19 22 24 07 11885 11912 1 1940 1 1967 1 1995 3 6 8 II 14 17 19 22 25 o8 12162 12190 12218 12246 12274 3 6 8 II 14 17 20 23 25 09 12445 12474 12503 12531 12560 3 6 9 12 14 17 20 23 26 10 12735 12764 12794 12823 12853 3 6 9 12 15 18 21 24 26 II 13032 13062 13092 13122 13152 3 6 9 12 15 18 21 24 27 12 13335 13366 13397 13428 13459 3 6 9 12 16 19 22 25 28 13 13646 13677 13709 13740 13772 3 6 9 13 16 19 22 25 28 14 13964 13996 14028 14060 14093 3 6 10 13 16 19 23 26 29 15 14289 14322 14355 14388 1442 1 3 7 10 13 17 20 23 26 30 i6 14622 14655 14689 14723 14757 3 7 10 14 17 20 24 27 30 17 14962 14997 15031 15066 15101 3 7 10 14 17 21 24 28 31 i8 15311 15346 15382 15417 15453 4 7 II 14 18 21 25 28 32 19 15668 15704 15740 15776 15812 4 7 II H 18 22 25 29 33 20 16032 16069 16106 16144 16181 4 7 II 15 19 22 26 30 33 21 16406 16444 16482 16520 16558 4 8 II 15 19 23 27 30 34 22 16788 16827 16866 16904 16943 4 8 12 16 19 23 27 31 35 23 17179 17219 17258 17298 17338 4 8 12 16 20 24 28 32 36 24 17579 17620 17660 17701 17742 4 8 12 16 20 24 29 33 37 25 17989 18030 18072 18113 18155 4 8 12 17 21 25 29 33 37 26 18408 18450 18493 18535 18578 4 9 13 17 21 26 30 34 38 27 18836 18880 18923 18967 19011 4 9 13 18 22 26 31 35 39 28 19275 19320 19364 19409 19454 4 9 13 18 22 27 31 36 40 29 19724 19770 19815 19861 19907 5 9 14 18 23 27 32 37 41 30 20184 20230 20277 20324 20370 5 9 14 19 23 28 33 37 42 31 20654 20701 20749 20797 20845 5 10 14 19 24 29 33 38 43 32 21135 21184 21232 21281 21330 5 10 15 20 25 29 34 39 44 33 21627 21677 21727 21777 21827 5 10 15 20 25 30 35 40 45 34 22131 22182 22233 22284 22336 5 10 15 20 26 31 36 41 46 35 22646 22699 22751 22803 22856 5 II 16 21 26 32 37 42 47 36 23174 23227 23281 23335 23388 5 II 16 21 27 32 38 43 48 37 23714 23768 23823 23878 23933 5 II 16 22 27 33 38 44 49 38 24266 24322 24378 24434 24491 6 II 17 22 28 34 39 45 51 39 24831 24889 24946 25003 25061 6 12 17 23 28 35 40 46 52 40 25410 25468 25527 25586 25645 6 12 18 24 29 35 41 47 53 41 26002 26062 26122 26182 26242 6 12 18 24 30 36 42 48 54 42 26607 26669 26730 26792 26853 6 12 18 25 31 37 43 49 55 43 27227 27290 27353 27416 27479 6 13 19 25 32 38 44 50 57 44 27861 27925 27990 28054 28119 6 13 19 26 32 39 45 52 58 45 28510 28576 28642 28708 28774 7 13 20 26 33 40 46 53 59 46 29174 29242 29309 29376 29444 7 14 20 27 34 41 47 54 61 47 29854 29923 29992 30061 30130 7 14 21 28 35 42 48 55 62 48 30549 30620 30690 30761 30832 7 14 21 28 35 42 50 57 64 49 31261 31333 31405 31477 31550 7 14 21 29 36 43 51 IL 65 Mean Differences II ANTILOGARITHMS Mean Differences 1 31696 2 3 4 1]^ 3 4 5 6 7 8 9 so 31623 31769 31842 3191S 7 IS 22 29 37 44 SI S9 66 SI 32359 32434 32509 32584 32659 8 15 23 30 3S 45 53 60 68 S2 33113 33189 33266 33343 33420 8 15 23 31 38 46 53 61 69 S3 33884 33963 34041 34119 34198 8 16 24 31 39 47 55 63 71 S4 34674 j 34754 34834 34914 34995 8 16 24 32 40 48 56 64 72 '5S 3S481 3SS63 3S64S 3S727 3S8io 8 16 2S 33 41 49 S8 66 74 S6 36308 36392 36475 36559 36644 8 17 25 34 42 50 59 67 76 S7 37154 37239 37325 3741 1 37497 9 17 26 34 43 52 60 69 78 S8 38019138107138194 38282 38371 9 18 26 35 44 53 62 70 79 S9 38905 38994 39084 39174 39264 9 18 27 36 45 54 63 72 81 60 3981 1 39902 39994 40087 40179 9 18 28 37 46 SS 6S 74 83 61 40738 40832 40926 41020 41115 9 19 28 38 47 57 66 76 85 62 41687 41783 41879 41976 42073 10 19 29 39 48 58 67 77 87 63 42658 42756 42855 42954 43053 10 20 30 40 49 59 69 79 89 64 43652 43752 43853 43954 44055 10 20 30 40 51 61 71 81 91 65 44668 44771 4487s 44978 4S082 10 21 31 41 52 62 73 83 93 66 45709 45814 45920 46026 46132 II 21 32 42 53 63 74 85 95 67 46774 46881 46989147098 47206 II 22 32 43 54 65 76 86 97 68 47^^3 47973 48084 48195 48306 II 22 33 44 55 66 78 89 100 .69 48978 49091 49204 49317 49431 II 23 34 45 57 68 79 91 102 70 SOI 19 S0234 S03S0 S0466 SOS82 12 23 35 46 S8 70 81 93 104 71 51286 51404 51523 51642 51761 12 24 36 48 59 71 83 95 107 72 52481 52602 52723 52845 52966 12 24 36 49 61 73 85 97 109 73 53703 53827 53951 54075 54200 12 25 37 50 62 75 87 100 112 74 54954 55081 55208 55335 55463 13 25 38 51 64 76 89 102 114 75 S6234 S6364 S6494 S6624 S67S4 13 26 39 S2 6s 78 91 104 117 76 57544 57677 57810 57943 58076 13 27 40 53 67 80 93 107 120 77 58884 59020 59156 59293 59429 14 27 41 55 68 82 95 109 123 .78 60256 60395 60534 60674 60814 14 28 42 56 70 84 98 112 126 79 61660 61802 61944 62087 62230 14 29 43 57 71 86 100 114 128 80 63096 63241 63387 63S33 63680 IS 29 44 S8 73 88 102 117 131 8i 64565 64714 64863 65013 65163 15 30 45 60 75 90 105 120 135 82 66069 66222 66374 66527 66681 15 31 46 61 77 92 107 122 138 83 67608 67764 67920 68077 68234 16 31 47 63 78 94 no 125 141 84 69183 69343 69502 69663 69823 16 32 48 64 80 96 112 128 144 85 7079s 709S8 71121 7128s 7I4SO 16 33 49 66 82 98 IIS 131 147 86 72444 7261 1 72778 72946 73114 17 34 50 67 84 lOI 117 134 151 87 74131 74302 74473 74645 74817 17 34 51 69 86 103 120 137 154 88 75858 76033 76208 76384 76560 18 35 53 70 88 105 123 140 158 .89 77625 77804 77983 78163 78343 18 36 54 72 90 108 126 144 162 90 79433 79616 79799 79983 80168 18 37 SS 74 92 no 129 147 166 91 81283 81470 81658 81846 82035 19 38 56 75 94 113 132 151 169 92 83176 83368 83560 83753 83946 19 39 58 77 96 116 135 154 174 93 85114 85310 85507 85704 85901 20 39 59 79 99 118 138 158 177 94 87096 87297 87498 87700 87902 20 40 61 81 lOI 121 141 161 182 9S 8912s 89331 89S36 89743 899SO 21 41 62 83 103 124 144 i6s 186 .96 91201 91411 91622 91833 92045 21 42 63 84 106 127 148 169 190 97 93325 93541 93756 93972 94189 22 43 65 86 108 130 151 173 195 98 95499 95719 95940 96161 96383 22 44 66 88 III 133 155 177 199 99 97724 97949 98175 98401 98628 IL AL 68 90 113 136 158 181 204 12 ANTILOGARITHMS 5 6 7 8 9 I 2 3 4 5 6 7 8 9 50 31989 32063 32137 32211 3228s "7 IS 22 30 37 44 52 59 67 51 32734 32810 32885 32961 lion 8 IS 23 30 38 45 53 61 68 52 33497 33574 33651 33729 33806 8 15 23 31 39 46 54 62 70 53 34277 34356 34435 34514 34594 8 16 24 32 40 48 55 63 71 54 35075 35156 35237 35318 35400 8 16 24 32 41 49 57 65 73 55 35892 35975 360S8 36141 36224 8 17 25 33 42 50 58 67 75 56 36728 36813 36898 36983 37068 9 17 26 34 43 SI 60 68 77 57 37584 Z7^7o m^^i 37844 37931 9 17 26 35 44 52 61 70 78 58 38459 38548 38637 38726 38815 9 18 27 36 45 54 62 71 80 59 39355 39446 39537 39628 39719 9 18 27 36 46 55 64 73 82 .60 40272 40365 40458 40551 40644 9 19 28 37 47 S6 65 75 84 61 41210 41305 41400 41495 41591 10 19 29 38 48 57 ^7 76 86 62 42170 42267 42364 42462 42560 10 20 29 39 49 59 68 78 88 63 43152 43251 43351 43451 43551 10 20 30 40 50 60 70 80 90 .64 44157 44259 44361 44463 44566 10 20 31 41 51 61 72 82 92 6s 45186 45290 45394 4S499 45604 10 21 31 42 52 63 73 84 94 66 46238 46345 46452 46559 46666 II 21 32 43 54 64 75 86 96 .67 47315 47424 47534 47643 47753 II 22 }>l 44 55 66 77 88 99 68 48417 48529 48641 48753 48865 II 22 34 45 56 67 79 90 lOI .69 49545 49659 49774 49888 50003 II 23 34 46 57 69 80 92 103 70 50699 S0816 50933 Siosi 51 168 12 23 35 47 59 70 82 94 106 .71 51880 52000 52119 52240 52360 12 24 36 48 60 72 84 96 108 .72 53088 53211 Sl?>ZZ 53456 53580 12 25 n 49 62 74 86 98 III 73 54325 54450 54576 54702 54828 13 25 38 50 ^l 75 88 100 113 74 55590 55719 55847 55976 56105 13 26 39 52 64 77 90 103 116 75 5688s S7016 57148 57280 57412 13 26 40 S3 66 79 92 105 119 .76 58210 58345 58479 58614 58749 13 27 40 54 67 81 94 108 121 77 59566 59704 59841 59979 601 17 14 28 41 55 69 82 97 no 124 .78 60954 61094 61235 61376 61518 14 28 42 56 71 85 99 113 127 79 62373 62517 62661 62806 62951 14 29 43 58 72 87 lOI 116 130 .80 63826 63973 641 2 1 64269 64417 IS 30 44 59 74 88 104 118 133 81 65313 65464 65615 65766 65917 15 30 45 60 76 91 106 121 136 82 66834 66988 67143 67298 67453 15 31 46 62 71 93 108 124 139 83 68391 68549 68707 68865 69024 16 32 48 63 79 95 III 127 143 .84 69984 70146 70307 70469 70632 16 32 49 65 81 97 114 130 146 8s 71614 71779 71945 721 1 1 72277 17 33 50 66 83 100 116 133 149 86 73282 73451 73621 73790 73961 17 34 51 68 84 102 119 136 153 87 74989 75162 75336 75509 75683 17 35 52 70 87 104 122 139 156 88 76736 76913 77090 77268 77446 18 36 53 71 89 107 124 142 160 .89 78524 78705 78886 79068 79250 18 36 55 73 91 109 127 145 164 .90 80353 80538 80724 80910 81096 19 37 56 74 93 112 130 149 167 .91 82224 82414 82604 82794 82985 19 38 57 76 95 114 133 152 171 .92 84140 84333 84528 84723 84918 19 39 58 78 97 117 136 156 175 93 86099 86298 86497 86696 86896 20 40 60 80 100 120 140 160 179 94 88105 88308 88512 88716 88920 20 41 6j 82 102 122 143 163 184 95 90157 9036s 90573 90782 90991 21 42 63 84 104 125 146 167 188 .96 92257 92470 92683 92897 93111 21 43 64 85 107 128 150 171 192 97 94406 94624 94842 95060 95280 22 44 66 87 109 131 153 175 197 .98 96605 96828 97051 97275 97499 22 45 ^1 90 112 134 157 179 201 99 98855 99083 99312 99541 99770 23 46 |69 92 115 137 160 183 206 I riean Diff eren( ;es 13 NATURAL FUNCTIONS Differences are given for every lo'. Intermediate values can be found by th method of proportional parts ; e. g. : To find tan 43 5^' and cos 37 34' tan 43 50' = -96008 + diff. for 6'= 337 .-. tan 43 5^'= -96345 cos 37"30 = -79335 diff. for 4'= 71 cos 37 34'= 79264 When there is no entry in the difference column, the value of the function change too rapidly for correct interpolation by proportional parts. Greater accuracy is thei obtained by expressing the function in terms of the sine and cosine. To find tan 6^'' 23' tan 67 20'= 2-39449 by proportional parts, diff. for 3'= 592 This gives tan 6f 23'= 2-40041. Diff. for 10' = 1972 (The correct value is 2 40038.) Subtract differences when dealing with co-functions 10' 20' 30' 40' } 10' 20' 30' 40' 50' 2* 10' 20' 30' 40' 3" 00291 00582 00873 01 164 01454 01745 02036 02327 02618 02908 03199 03490 03781 04071 04362 04653 04943 05234 291 291 291 291 290 291 291 291 291 290 291 291 291 290 291 291 290 291 00 34378 171-89 114-59 85-946 68^757 57-299 49-114 42-976 38^202 34-382 31-258 28-654 26-451 24-562 22-926 21-494 20^230 19-107 tangent 00000 00291 00582 00873 01 164 01455 01746 02037 02328 02619 02910 03201 03492 03783 04075 04366 04658 04949 05241 D cotangent 291 291 291 291 291 291 291 291 291 291 291 291 291 292 291 292 291 292 cotangent 00 343-77 171-89 114-59 85-940 68-750 57290 49-104 42-964 38-188 34-368 31-242 28636 26-432 24-542 22^904 21-470 20-206 19081 tangent secant I -00000 I -00000 I -00002 I -00004 I -00007 I-OOOII I 00015 I -0002 1 I ^00027 1-00034 I -00042 I -0005 I I 00061 1-00072 1-00083 1-00095 1-00108 I 001 22 I -00137 cosecant 000 002 002 003 004 004 005 006 007 008 009 010 on on 012 013 014 015 I 00000 I -00000 -99996 99993 -99989 -9998s 99979 -99973 -99966 99958 -99949 -99939 .99929 99917 -99905 -99892 99878 99863 000 002 002 003 004 004 006 006 oo^ 008 009 010 010 012 012 013 014 015 90^ D 87" 14 LOGARITHMIC FUNCTIONS The values given here are the true logarithms ; the characteristic is not increased by lo as in many tables. Differences are given for every lo'. Intermediate values can be found by the method of proportional parts. The differences for the logarithm of a function and of the reciprocal of the fimction are the same in magnitude but opposite in sign. When there is no entry in the difference column, the rate of change of the logarithm changes too rapidly for correct interpolation by proportional parts. The following rules may be used when the angle is small : Log sine. Add 6-68557 to the log of the angle expressed in seconds and subtract ^ of^the log secant. Log tan. Add 6-68557 to the log of the angle expressed in seconds and add of the log secant. When the log sine is given, the angle is found in seconds by adding 5-31443 to the log sine and ^ of the corresponding log secant (found in the ordinary way). When the log tan is given, the angle is found in seconds by adding 5*31443 to the log tan and subtracting of the corresponding log secant (found in the ordinary way). Subtract differences when dealing with co-functions log sin D log cosec log tan D log cotan log sec D log COS 00 00 00 00 00000 00000 90 10' 3-46373 2-53627 3-46373 2-53627 o-ooooo on T O-OOOOO 50' 20' 376475 2-23525 3-76476 2-23524 o-ooooi 001 001 1-99999 40' 30' 3-94084 2-05916 3-94086 2-05914 0-00002 1-99998 30' 40' 2-06578 1-93422 2-06581 I-93419 0-00003 002 002 1-99997 20' 50' 2-16268 1-83732 2-16273 1-83727 0-00005 1-99995 10' 1 2-24186 1-75814 2-24192 175808 000007 002 1-99993 89 10' 2-30879 I-69121 2-30888 I -691 12 0-00009 I -9999 1 50' 20' 2.36678 1-63322 2-36689 1-63311 0-00012 003 003 003 004 004 004 1-99988 40' 30' 2-41792 1-58208 2-41807 1-58193 0-00015 1-99985 30' 40' 2-46366 1-53634 2-46385 1-53615 0-00018 1-99982 20' so' 2 2-50504 2-54282 1-49496 1-45718 2-50527 2-54308 1-49473 1-45692 -0002 2 000026 1-99978 i -99974 10' 10' 2-577S7 1-42243 2-57788 1-42212 0-00031 1-99969 50' 20' 2-60973 1-39027 2-61009 I-3899I 0-00036 005 005 006 006 1-99964 40' 30' 2-63968 1-36032 2-64009 I-3599I 0-00041 1-99959 30' 40' 2-66769 1-33231 2-66816 I-33184 0-00047 1-99953 20' 50' 2-69400 1-30600 2-69453 1-30547 0-00053 T-99947 10' 3 2-71880 I-2812O 2-71940 1-28060 0-00060 (JvJU 1-99940 87"' log^cos D log sec log cotan U log tan log cosec D log sin 87" 15 3 NATURAL FUNCTIONS lO 20' 30' 40' 10 05234 1 05524 05814 06105 06395 06685 06976 07266 07556 07846 08136 08426 08716 09005 09295 09585 09874 10164 10453 10742 11031 11320 I 1609 I 1898 12187 12476 12764 13052 13341 13629 13917 14205 14493 14781 15069 15356 15643 15931 16218 16505 16792 17078 17365 290 290 291 290 290 291 290 290 290 290 290 290 289 290 290 289 290 289 289 289 289 289 289 289 289 288 288 289 288 288 288 288 288 288 287 287 288 287 287 287 286 287 19-1073 i8'i026 I7-I984 16-3804 15-6368 14-9579 14-3356 13-7631 13-2347 12-7455 12-2913 ii^8684 11-4737 ii^i046 10-7585 IO-4334 io^i275 9^839i2 956677 9-30917 9-06515 8-83367 8-61379 8 -40466 820551 8-01565 7-83443 7-66130 7-49571 7-337^9 7-18530 7-03962 6-89979 6-76547 6-63633 6-51208 639245 6-27719 6-16607 6-05886 5-95536 5-85539 5-75877 tangent D 05241 05533 05824 061 16 06408 06700 06993 07285 07578 07870 08163 -08456 08749 09042 09335 09629 09923 IO216 IO51O 10805 I 1099 I1394 I1688 II983 12278 12574 12869 13165 I 3461 -13758 -14054 -14351 14648 -14945 15243 15540 15838 16137 16435 16734 -17033 -17333 17633 cotangent 292 291 292 292 292 293 292 293 292 293 293 293 293 293 294 294 293 294 295 294 295 294 295 295 296 295 296 296 297 296 297 297 297 298 297 298 299 298 299 299 300 300 I9081I 18-0750 17-1693 16-3499 15^6048 14-9244 14-3007 13-7267 13-1969 12-7062 12-2505 11-8262 11-4301 11-0594 10-7119 10-3854 10-0780 9-78817 9-31436 9-25530 9-00983 8^77689 8-55554 8^34496 8-14435 7-95302 7-77035 7-59575 7-42871 7-26873 7-IIS37 6-96823 6-82694 6-69116 6-56055 6^43484 63137s 6^19703 6-08444 5-97576 5-87080 5-76937 567128 cotangent D tangent I 00137 1-00153 1-00169 1-00187 1-00205 1-00224 1-00244 1-00265 1-00287 1-00309 I-00333 1-00357 I 00382 1 -00408 I -0043 5 1 -00463 1-00491 1-00521 I 00551 1-00582 I -00614 1 -00647 1 -0068 1 1 -007 1 5 I -00751 1-00788 1-00825 I -00863 1 -00902 I -00942 I 00983 1-01024 1-01067 I-OIIII 1-01155 1 -01200 I -01247 1-01294 1-01342 1-01391 1-01440 1-01491 101543 016 016 018 018 019 020 021 022 022 024 024 025 026 027 028 028 030 030 031 032 033 034 034 036 037 037 038 039 040 041 041 043 044 044 045 047 047 048 049 049 051 052 cosecant 99863 99847 99831 99813 99795 99776 99756 99736 99714 99692 99668 99644 99619 99594 99567 99540 -995 1 1 -99482 99452 -99421 99390 -99357 99324 99290 99255 99219 99182 99144 -99106 99067 99027 -98986 98944 98902 98858 98814 98769 98723 -98676 -98629 -98580 98531 98481 016 016 018 018 019 020 020 022 022 024 024 025 025 027 027 029 029 030 031 031 033 033 034 035 036 037 038 038 039 040 041 042 042 044 044 045 046 047 047 049 049 050 87 50' 40' 30' 20' 10' 86 50' 40' 30' 20' 10' 85 50' 40' 30' 20' 10' 84^ 50' 40' 30' 20' 10' 83^ 50' 40' 30' 20' 10' 82 50' 40' 30' 20' 10' 81 50' 40' 30' 20' 10' 80 80= 16 3- LOGARITHMIC FUNCTIONS log sin D log cosec log tan D log cotan log sec D log cos 3<^ 271880 1-28120 2-71940 1-28060 0-00060 006 i-99940 87" lo' 274226 1-25774 2-74292 1-25708 0-00066 008 1-99934 50' 20' 276541 1-23549 2-76525 1-23475 0-00074 007 008 008 1-99926 40' 30' 278568 1-21432 2-78649 1-21351 -0008 1 1-99919 30' 40' 2-80585 1-19415 2-80674 1-19326 0-00089 I -999 1 1 20' 50' 2-82513 1-17487 2-82610 1-17390 0-00097 009 009 009 010 1-99903 10' -4^' 2-84358 1-15642 284464 I -15536 000106 1-99894 86^ 10' 2-86128 1-13872 2-86243 1-13757 O'ootis 1-99885 50' 20' 2-87829 1-12171 2-87953 I - 1 2047 0-00124 1-99876 40' 30' 2-89464 1-10536 2-89598 I -10402 0-00134 010 r> T T 1-99866 30' 40' 2-91040 I -08960 2-91185 1-08815 0-00144 1-99856 20' 50' 2-92561 1-07439 2-92716 1-07284 0-00155 U i 1 Oil 1-99845 10' 5 2-94030 I 05970 294195 1-05805 000166 Oil 1-99834 85^ 10' 2-95450 1-04550 2-95627 1-04373 0-00177 Oil 012 013 012 014 013 1-99823 50' 20' 2-96825 1-03175 2-97013 1-02987 0-00188 I -998 1 2 40' 30' 2-98157 1-01843 2-98358 I -01642 -00200 1-99800 30' 40' 2-99450 1-00550 2-99662 1-00338 0-00213 1-99787 20' 50' I -00704 0-99296 1-00930 0-99070 0-00225 1-99775 10' 6 1-01923 098077 i-02162 0-97838 0-00239 1-99761 84" 10' 1-03109 0-96891 1-03361 0-96639 0-00252 1-99748 50' 20' I -04262 0-95738 1-04528 0-95472 0-00266 014 1-99734 40' 30' 1-05386 0-94614 1-05666 0-94334 0-00280 014 015 015 oiS 016 1-99720 30' 40' I -0648 1 0-93519 1-06775 0-93225 0-00295 1-99705 20' 50' 1-07548 0-92452 1-07858 0-92142 0-00310 I -99690 10' r I 08589 0-91411 i-08914 0-91086 0-00325 199675 83" 10' 1-09606 993 971 949 928 909 889 871 854 837 821 805 790 0-90394 1-09947 0-90053 0-00341 016 016 017 1-99659 50' 20' I-I0599 0-89401 I-IO956 987 966 945 926 908 889 873 856 840 825 811 0-89044 0-00357 1-99643 40' 30' 1-11570 0-88430 I-II943 0-88057 0-00373 1-99627 30' 40' 1-12519 0-87481 I -1 2909 0-87091 0-00390 I -99610 20' 50' 10' I -1 3447 1-14356 1-15245 0-86553 085644 0-84755 I-13854 1-14780 I-15688 0-86146 0-85220 0-84312 0-00407 000425 0-00443 017 018 018 018 1-99593 1-99575 1-99557 10' 82" 50' 20' i-i6ii6 0-83884 I-16577 0-83423 0-00461 T-99539 40' 30' 1-16970 0-83030 1-17450 0-82550 0-00480 019 1-99520 30' 40' 1-17807 0-82193 1-18306 0-81694 0-00499 019 I -99501 20' 50' 9 1-18628 1-19433 0-81372 080567 I-19146 I-19971 0-80854 0-80029 0-00518 000538 019 020 020 1-99482 1-99462 10' 81" 10' 20' 30' 40' 1-20223 1-20999 1-21761 1-22509 776 762 748 735 723 0-79777 0-79001 078239 077491 1-20782 I-21578 I-22361 I-23130 796 783 769 0-79218 0-78422 077639 0-76870 0-00558 0-00579 0-00600 0-00621 021 021 021 I -99442 I -9942 1 T -99400 1-99379 50' 40' 30' 20' 50' 1-23244 0-76756 1-23887 757 745 0-76113 0-00643 022 1-99357 10' 10" 1-23967 076033 1-24632 0-75368 000665 022 1-99335 80 log COS D log sec log cotan D log tan log cosec D log sin 80" 17 10- NATURAL FUNCTIONS 1736s 17651 '17937 18224 18509 18795 19081 19366 '19652 19937 20222 20507 20791 21076 21360 21644 21928 22212 22495 22778 23062 23345 23627 23910 24192 24474 24756 25038 25320 25601 25882 26163 26443 26724 27004 27284 27564 27843 28123 28402 28680 28959 29237 286 286 287 285 286 286 28s 286 285 285 285 284 285 284 284 284 284 283 283 284 283 282 283 282 282 282 282 282 281 281 281 280 281 280 280 280 279 280 279 278 279 278 cosecant D 5-75877 5-66533 5-57493 5-48740 5^40263 5-32049 524084 5-16359 5^08863 5-01585 4-94517 4-87649 4-80973 4-74482 4^68i67 4^62023 4^56041 4^50216 4-44S4I 4^39012 4-33622 4^28366 4^23239 4-18238 4-13357 4-08591 4-03938 3-99393 3-94952 3-90613 386370 3-82223 3-78166 3-74198 3-70315 3-66515 362796 3-59154 3-55587 3-52094 3-48671 3-45317 342030 I u tangent 17633 17933 18233 18534 18835 19136 19438 19740 20042 20345 20648 20952 21256 21560 21864 22169 22475 22781 23087 23700 24008 24316 24624 24933 25242 25552 25862 26172 26483 26795 27107 27419 27732 28046 28360 28675 28990 29305 29621 29938 30255 30573 cotangent 300 300 301 301 301 302 302 302 303 303 304 304 304 304 305 306 306 306 306 307 308 308 308 309 309 310 310 310 311 312 312 312 313 314 314 315 315 315 316 317 317 318 cotangent 5^67128 5-57638 5-48451 5-39552 5^30928 5^22566 S-I44S5 5^06584 4-98940 4-91516 4-84300 4^77286 4-70463 4-63825 4-57363 4-51071 4-44942 4-38969 4-33148 4-27471 4-21933 4-16530 4-11256 4^06107 401078 3-96165 3-91364 3-86671 3-82083 3-77595 3-73205 3-68909 3-64705 360588 3-56557 3-52609 3-48741 3-44951 3-41236 3-37594 3-34023 3-30521 327085 tangent I-OI543 I -01595 1-01649 i^oi703 1-01758 1-01815 I 01872 I -01930 1-01989 I -02049 I -02 1 10 I -02 1 7 1 I 02234 I -02298 1-02362 I -02428 I -02494 1-02562 I 02630 I -02700 I -02770 I -02841 1-02914 I -02987 I 03061 I -03137 1-03213 1-03290 1-03368 1-03447 1-03528 I -03609 I -0369 1 1-03774 1-03858 I -03944 1-04030 I -041 17 I -04206 1-04295 I -04385 1-04477 I 04569 052 054 054 055 057 057 058 059 060 061 061 063 064 064 066 066 068 068 070 070 071 073 073 074 076 076 077 078 079 081 081 082 083 084 086 086 087 089 089 090 092 092 98481 98430 98378 98325 98272 98218 98163 98107 98050 97992 97934 97875 97815 97754 97692 97630 97566 .97502 97437 97371 97304 97237 97169 97100 97030 96959 96887 96815 96742 96667 96593 96517 96440 96363 96285 96206 96126 96046 95964 95882 95799 95715 95630 73= 18 10 LOGARITHMIC FUNCTIONS 10 lo' 20' 30' 40' ^o^ 10' 20' 30' 40' 50' 12 10' 20' 30' 40' 13 10' 20' 30' 40' 50' 14 10' 20' 30' 40' 50' 15 10' 20' 30' 40' so' 16^ \T log sin 1-23967 1-24677 1-25376 1-26063 1-26739 1-27405 1-28060 1-28705 1-29340 1-29966 1-30582 I-31189 1-31788 1-32378 1-32960 1-33534 1-34100 1-34658 1-35209 35752 1-36289 1-36819 I-3734I 1-37858 1-38368 1-38871 39369 1-39860 1-40346 1-40825 1-41300 1-41768 1-42232 1-42690 1-43143 1-43591 1-44034 1-44472 1-44905 -45334 1-45758 T-46178 1-46594 log COS 710 699 687 676 666 655 645 635 625 616 607 599 590 582 574 566 558 551 543 537 529 522 517 510 S03 498 491 486 479 475 468 464 458 453 448 443 438 433 429 424 420 416 log cosec o- 76033 0-75323 0-74624 0-73937 0-73261 0-72595 071940 0-71295 0-70660 0-70034 0-69418 0-6881 1 0-68212 0-67622 0-67040 0-66466 0-65900 0-65342 0-64791 0-64248 0-63711 0-63181 0-62659 0-62142 061632 0-61129 0-60631 0-60140 0-59654 0-59175 0-58700 0-58232 0-57768 0-57310 0-56857 0-56409 0-55966 0-55528 0-55095 0-54666 0-54242 0-53822 0-53406 log sec log tan 1-24632 -25365 1-26086 1-26797 1-27496 I-28186 1-28865 T-29535 -30195 1-30846 T-31489 1-32122 i-32747 1-33365 1-33974 34576 1-35170 1-35757 1-36336 -36909 1-38035 1-38589 1-39136 1-39677 1-40212 1-40742 1-41266 1-41784 T-42297 1-42805 1-43308 1-43806 1-44299 1-44787 45271 1^45750 1-46224 1-46694 T -47 1 60 1-47622 T-48080 148534 log cotan 733 721 711 699 690 679 670 660 651 643 633 625 618 609 601 595 587 579 573 567 559 554 547 541 535 530 524 518 513 508 503 498 493 488 484 479 474 470 466 462 458 454 log cotan 0-75368 0-74635 0-73914 0-73203 0-72504 0-71814 O-71135 0-70465 0-69805 0-69154 O-68511 0-67878 067253 0-66635 0-66026 0-65424 0-64830 0-64243 o 63664 0-63091 0-62524 0-61965 O-61411 0-60864 0-60323 0-59788 0-59258 0-58734 0-58216 0-57703 0-57195 0-56692 0-56194 0-55701 0-55213 0-54729 054250 0-53776 0-53306 0-52840 0-52378 0-51920 0-51466 log tan log sec 000665 0-00687 0-00710 0-00733 0-00757 0-00781 o 00805 0-00830 0-00855 0-00881 0-00907 0-00933 o 00960 0-00987 0-01014 0-01042 0-01070 0-01099 0-OII28 0-OII57 0-OII87 0-OI2I7 0-01247 0-01278 0-OI3IO 0-OI34I 0-OI373 0-01406 0-01439 0-01472 001506 0-01540 0-01574 0-01609 0-01644 0-01680 001716 0-01752 0-01789 0-01826 0-01864 0-01902 00 1940 022 023 023 024 024 024 025 025 026 026 026 027 027 027 028 028 029 029 029 030 030 030 031 032 031 032 033 033 033 034 034 034 035 035 036 036 036 037 037 038 038 038 log COS 1-99335 99313 1-99290 -99267 99243 1-99219 1-99195 1-99170 1-99145 1-99119 99093 I -99067 1-99040 1-99013 T-98986 1-98958 -98930 I -98901 1-98872 1-98843 I -988 1 3 1-98783 1-98753 1-98722 1-98690 1-98659 1-98627 1-98594 1-98561 1-98528 1-98494 1-98460 1-98426 -98391 1-98356 1-98320 I 98284 1-98248 I-98211 T-98174 1-98136 1-98098 1-98060 log cosec I D I log sin 80 50' 40' 30' 20' 10' 79 50' 40' 30' 20' 10' 78 50' 40' 30' 20' 10' 77 50' 40' 30' 20' 10' 76^ 50' 40' 30' 20' 10' 75_ 50' 40' 30' 20' 10' 74 50' 40' 30' 20' 10' 73" 73* 19 ir NATURAL FUNCTIONS 29237 29515 29793 30071 30348 30625 30902 31178 31454 31730 32006 32282 32557 32832 33106 33381 33655 33929 34202 34475 34748 35021 35293 35565 35837 36108 36379 36650 36921 37191 37461 '37730 37999 38268 38537 38805 39073 39341 39608 39875 40142 40408 40674 278 278 278 277 277 277 276 276 276 276 276 275 275 274 275 274 274 273 273 273 273 272 272 272 271 271 271 271 270 270 269 269 269 269 268 268 268 267 267 267 266 266 cosecant 3 42030 3-38808 3^35649 3^32551 3-29512 3-26531 323607 3-20737 3-17920 3-15155 3-12440 3-09774 3-07155 3-04584 3-02057 3-99574 2-97135 2^94737 292380 2-90063 2-87785 2-85545' 2-83342 2-81175 279043 2-76945 2-74881 2-72850 2-70851 2-68884 2-66947 2^65040 2-63162 2-61313 2-59491 2-57698 255930 2-54190 2-52474 2-50784 2-491 19 2-47477 2-45859 tangent D 30573 -30891 31210 -31530 -3185 -3217 32492 328 331. 3346c 33783 34108 34433 -34758 -35085 318 .319 1320 320 '321 ,322 ,60 i 3^4 1323 -' I ?2i; J325 325 327 35412 -35740 327 328 -36068 3^^ ^^20 36397^3^ 36727 37057^:^, -37388 ^^^ -37720 ^:f -38053 il,i t^^ 335 '^^^^' 334 -39055 11^ 39391 l'^ 39727 ^^o 40065 III 40403 338 Tiotl 340 4i42i 340 41763 III -42105 ^'^ 342 '^'^' 344 -42791 ... -43136 345 .43481 345 43828 347 -44175 III 44523 ^^' cotangent cotangent D 3-27085 3-23714 3-20406 3-I7159 3-13972 3-10842 307768 3-04749 3-01783 2-98869 2-96004 2-93189 2-90421 2-87700 2-85023 2-82391 2-79802 2-77254 274748 2^72281 2-69853 2-67462 2-65109 2-62791 2-60509 2-58261 2-56046 2-53865 2-51715 2-49597 2-47509 2^45451 2-43422 2-41421 2^39449 2-37504 2-35585 2-33693 2-31826 2-29984 2-28167 2-26374 2 24604 tangent 094 094 1 096 I 04569 1-04663 1-04757 ^4853j I -04950 i^; 1-05047 099 ^^5146 1^00 1-052461 I -05 347 1 I -05449 1 1-05552 1-05657 1-05762 1-05869 1-05976 1-06085 I -06195 1-06306 I 06418 I -065 3 1 1-06645 I 06761 i^o6878 1-06995 107115 1-07235 1^07356 1-07479 I -07602 1-07727 107853 I -0798 1 1-08109 1-08239 1-08370 1-08503 1-08636 I -08771 1-08907 I -09044 1-09183 1-09323 1-09464 lOI 102 103 105 105 107 107 109 no III 112 113 114 116 117 117 120 120 121 123 123 125 126 128 128 130 131 133 133 135 136 137 139 140 141 085 086 95630 95545 95195 !^^ 95106! 95015! -94924 -94832 94740 94646 94552 -94457 94361 94264 94167 94068 93969 93869 93769 .93667 93565 93462 93358 93253 -93148 -93042 .92935 -92827 92718 -92609 -92499 -92388 -92276 -92164 -92050 -91936 -91822 -91706 91590 -91472 91355 66^ 20 170 LOGARITHMIC FUNCTIONS I log sin 1-46594 1-47005 I -4741 1 1-47814 1-48213 1-48607 1-48998 1-49385 1-49768 1-50148 1-50523 1-50896 1-31264 1-51629 1-51991 1-52350 1-52705 1-53057 1-53405 53751 54093 -54433 -54769 1-55102 i -55433 1-55761 1-56085 1-56408 1-56727 1-57044 1-57358 1-57669 1-57978 1-58284 1-58588 1-59188 1-59484 1-59778 I -60070 1-60359 1-60646 1-60931 log^ COS 401 406 403 399 394 391 387 383 380 375 Z7Z 368 365 362 359 355 352 348 346 342 340 336 333 328 324 323 319 317 314 311 309 306 304 301 299 296 294 292 289 287 285 log cosec 0-53406 0-52995 0-52589 0-52186 0-51787 0-51393 0-51002 0-50615 0-50232 0-49852 0-49477 0-49104 0-48736 0-48371 0-48009 0-47650 0-47295 0-46944 o -4659s 0-46249 0-45907 0-45567 0-45231 0-44898 331 ^ :fo 0-44567 0-44239 0-43915 0-43592 0-43273 0-42956 042642 0-42331 0-42022 0-41716 0-41412 0-41111 0*40812 0-40516 0-40222 0-39930 0-39641 0-39354 0-39069 log tan -489841 6 49430!^;, -49872,44 -503111^^3^ 1-507461^^^ 1-51606!^ log cotan log sec 52031 j -52452! I-52870I 1-53285 j 1-53697! I-54106 54512 -54915 55315 i^557i2 1-56107 1-56498 1-56887 1-57274 57658 1-58039 1-58418 58794 1-59168 -59540 1-59909 1-60276 1-60641 I -61004 1-61364 1-61722 T-62079 1-62433 1-62785 -63135 -63484 1-63830 -64175 I -645 1 7 i -64858 loo' cotan 421 418 415 412 409 406 403 400 397 395 391 389 387 384 381 379 376 374 372 369 367 365 363 360 358 357 354 352 350 349 346 345 342 341 0-51466 0-51016 0-50570 0-50128 0-49689 0-49254 0-48822 0-48394 0-47969 0-47548 0-47130 0-46715 046303 0-45894 0-45488 0-45085 0-44685 0-44288 0-43893 0-43502 0-43113 0-42726 0-42342 0-41961 0-41582 0-41206 0-40832 0-40460 0-40091 0-39724 0-39359 0-38996 0-38636 0-38278 0-37921 0-37567 0-37215 0-36865 0-36516 0-36170 0-35825 0-35483 0-35142 log sec log cos 001940 0-01979! 0-02018 I 0-02058 0-02098 0-02139 0-02179 0-0222I 0-02262 0-02304 0-02347 0-02390 0-02433 0-02477 I 0-02521 ' 0-02565 0-02610 0-02656 0-02701 0-02748 0-02794 0-02841 0-02889 0-02937 002985 0-03034 0-03083 0-03132 0-03182 0-03233 0-03283 0-03335 0-03386 0-03438 0-03491 0-03544 0-03597 0-03651 0-03706 0-03760 0-03815 0-03871 0-03927 log tan |log( 039! 039 j 040 ! 040 041 040 042 041 042 043 043 043 044 044 044 j 045 I 0461 045 I 047' 046 047 048 048 048 049 049 049 050 051 050 052 051 052 053 053 053 054 055 054 055 056 056 1-98060 T-9802I T-97982 -97942 -97902 I -97861 I -9782 1 1-97779 97738 I -97696 -97653 1-97610 1-97567 97523 1-97479 -97435 1-97390 1-97344 1-97299 1-97252 1-97206 -97159 1-97111 1-97063 1-97015 1-96966 I -969 1 7 1-96868 I -968 1 8 1-96767 i -967 1 7 1-96665 1-96614 1-96562 1-96509 1-96456 i -96403 -96349 I -96294 1-96240 1-96185 1-96129 i -96073 log sin 50' 40' 30' 20' 10' 72^ 50' 40' 30' 20' 10' Zi! 50' 40' 30' 20' 10' 70" 50' 40' 30' 20' 10' 69 50' 40' 30' 20' 10' 68" 50' 40' 30' 20' 10' 67^ 50' 40' 30' 20' 10' 66 21 24 NATURAL FUNCTIONS 40674 40939 41204 41469 41734 41998 42262 42525 42788 43051 43313 43575 43837 44098 44359 44620 44880 45140 45399 45658 45917 46175 46433 46690 46947 47204 47460 47716 47971 48226 48481 48735 48989 49242 49495 49748 50000 50252 50503 50754 51004 51254 51S04 D 265 265 265 265 264 264 263 263 263 262 262 262 261 261 261 260 260 259 259 259 258 258 257 257 257 256 256 255 255 255 254 254 253 253 253 252 252 251 251 250 250 250 245859 2-44264 2-42692 2^41 142 2-39614 2^38107 2 36620 2^35154 2-33708 2-32282 2-30875 2-29487 2-28117 2^26766 2-25432 2-24116 2-22817 2-21535 2-20269 2-19019 2-17786 2-16568 2-15366 2-14178 2-13005 2-1 1847 2-10704 2-09574 2-08458 2-07356 2 06267 2-05191 2 04128 2-03077 2-02039 2-01014 2 00000 1-98998 1-98008 1-97029 I -96062 1-95106 1-94160 990 979 967 956 946 tangent 44523 -44872 45222 45573 -45924 -46277 -46631 -46985 47341 47698 48055 48414 48773 49134 49495 -49858 -50222 50587 50953 -51320 -51688 52057 52427 52798 53171 53545 53920 -54296 54673 55051 55431 55812 56194 56577 56962 57348 57735 58124 58513 58905 59297 -59691 -60086 D cotangent cotangent 349 350 351 351 353 354 354 356 357 357 359 359 361 361 364 365 366 367 368 369 370 371 Z72> 374 375 376 Z77 378 380 381 382 383 385 386 387 389 389 392 392 394 395 2 24604 2-22857 2-21132 2-19430 2-17749 2-16090 2^i44Si 2^12832 2-11233 2-09654 2-08094 2-06553 2 05030 2^03526 2^02039 2^00569 1-99116 I -9768 1 I -96261 1-94858 I -93470 1-92098 I -90741 I -89400 I 88073 1^86760 1-85462 1-84177 1-82906 I -8 1649 I 80405 I -79 1 74 1-77955 1-76749 1-75556 1-74375 I -7320s I -72047 I -70901 I -69766 I -68643 1-67530 1-66428 tangent 09464 09606 09750 09895 1 004 1 IOI89 10338 10488 10640 10793 10947 III03 1 1 260 II419 1 1 579 1 1 740 1 1903 12067 12233 12400 12568 12738 12910 13083 13257 13433 13610 13789 13970 14152 14335 14521 14707 14896 15085 15277 15470 15665 15861 16059 16259 16460 16663 142 144 145 146 148 149 150 152 153 154 156 157 159 160 161 163 164 166 167 168 170 172 173 174 176 177 179 181 182 183 186 186 189 189 192 193 195 196 198 200 201 203 D 91355 91236 91I16 90996 90875 90753 90631 90507 90383 90259 90133 90007 89879 89752 89623 89493 89363 89232 89IOI 88968 88835 88701 88566 88431 88295 88158 88020 87882 87743 87603 87462 87321 87178 87036 86892 86748 86603 86457 86310 86163 86015 85866 85717 59' 2a 24'= LOGARITHMIC FUNCTIONS log sin D log cosec log tan log cotan log sec log cos 24 lO 20 30 40 50' 25 10' 20' 30' 40 26 10' 20' 30' 40' 27 10' 20' 30' 40' 28 10' 20' 30' 40' 29 10' 20' 30' 40' 30 10' 20' 30' 40' 50' 31 1-60931 1-61214 1-61494 61773 I -62049 1-62323 1-62595 1-62865 63133 63398 1-63662 1-63924 1-64184 1-64442 1-64698 1-64953 T-65205 1-65456 1-65705 1-65952 1-66197 1-66441 -66682 1-66923 1-67161 67398 1-67633 1-67866 I 68098 1-68328 1-68557 68784 1^690 10 69234 1^69456 1-69677 1-69897 1-701 15 1-70332 i^70547 i^7076i 1-70973 1-71184 283 280 279 276 274 272 270 268 265 264 262 260 258 256 255 252 251 249 247 245 244 241 241 238 237 235 233 232 230 229 227 226 224 222 221 220 218 217 215 214 212 211 0-39069 0-38786 0-38506 0*38227 0-37951 0-17677 0-37405 0-37135 0-36867 0-36602 0-36338 0-36076 0-35816 0-35558 0-35302 0-35047 0-34795 0-34544 0-34295 0-34048 0-33803 0-33559 0-33318 0-33078 0-32839 0-32602 0-32367 0-32134 0-31902 0-31672 0-31443 0-31216 0-30990 0-30766 0-30544 0-30323 0-30103 0-29885 0-29668 0-29453 0-29239 0-29027 0-28816 i -64858 1-65197 65535 1-65870 T -66204 1-66537 i-66867 T-67I96 67524 67850 1-68174 1-68497 168818 69138 1-69457 -69774 1-70089 1-70404 1-70717 1-71028 71339 I -71648 71955 1-72262 i -72567 1-72872 73175 73476 l'7Z777 1-74077 1-74375 -74673 -74969 75264 -75558 1-75852 I -76144 76435 1-76726 -77015 IJ77Z02 I-7759I 1-77877 339 338 335 334 333 330 329 328 326 324 323 321 320 319 317 315 315 313 311 311 309 307 307 305 305 303 301 301 300 298 298 296 295 294 294 292 291 290 290 288 288 286 0-35142 0-34803 0-34465 0-34130 0-33796 0-33463 0-33133 0-32804 0-32476 0-32150 0-31826 0-31503 O-31182 0-30862 0-30543 0-30226 0-29911 0-29596 0-29283 0-28972 0-28661 0-28352 0-28045 0-27738 0-27433 0-27128 0-26825 0-26524 0-26223 0-25923 025625 0-25327 0-25031 0-24736 0-24442 0-24148 0-23856 0-23565 0-23275 0-22985 0-22697 0-22409 0-22123 o 03927 0-03983 0-04040 0-04098 0-04156 0-04214 004272 0-04332 0-04391 0-0445 I 0-04512 0-04573 0-04634 0-04696 0-04758 0-04821 0-04884 0-04948 0-05012 0-05077 0-05142 0^05207 0-05273 0-05340 o 05407 0-05474 0-05542 0-05610 0-05679 0-05748 005818 0^05888 0-05959 0^06030 0^061 02 0^06174 o 06247 0^06320 0-06394 0-06468 0-06543 0-06618 0-06693 056 057 058 057 059 058 060 059 060 061 061 061 062 062 063 063 064 064 065 065 065 066 067 067 067 068 068 069 069 070 070 071 071 072 072 073 073 074 074 075 075 075 1-96073 1-96017 I ^9 5 960 1-95902 95845 1-95786 i -95728 -95668 -95609 1-95549 1-95488 1-95427 1*95366 95304 95242 95179 1-95116 1-95052 1-94988 1-94923 1-94858 -94793 -94727 I ^94660 1-94593 94526 -94458 -94390 -94321 1-94252 1-94182 I -941 1 2 1-94041 1-93970 93898 1^93826 1-93753 i^9368o i^936o6 -93532 -93457 1-93382 i -93307 66^ 50' 40' 30' 20' 10' 65 50' 40' 30' 20' 10' 64 50' 40' 30' 20' 10' 63^ 50' 40' 30' 20' 10' 62 50' 40' 30' 20' 10' 6r 50' 40' 30' 20' 10' 60^ 50' 40' 30' 20' 10' 59 log COS D log sec lo" cotan log tan log cosec log sin 89' 3 31 NATURAL FUNCTIONS S1504 51753 52002 52250 52498 52745 52992 53238 53484 53730 53975 54220 54464 54708 54951 55194 55436 55678 55919 56160 56401 56641 56880 57119 57358 57596 58070 58307 58543 58779 59014 59248 59482 59716 59949 60182 60414 60645 60876 61 107 61337 61566 249 249 248 248 247 247 246 246 246 245 245 244 244 243 243 242 242 241 241 241 240 239 239 239 238 237 237 237 236 236 235 234 234 234 233 233 232 231 231 231 230 229 D 94160 93226 92302 91388 90485 89591 88708 87834 86970 86116 85271 84435 83608 82790 81981 81180 79604 78829 78062 773OZ 7^SS^ 75808 75073 74345 73624 7291 1 72205 71506 70815 70130 69452 68782 68117 67460 66809 66164 65526 64894 64268 63648 63035 62427 934 924 914 903 894 883 874 864 854 845 836 827 818 809 801 792 784 775 767 759 751 744 735 728 721 713 706 699 691 685 678 670 665 657 651 645 638 632 626 620 613 608 tangent D 60086 60483 60881 61280 61681 62083 62487 62892 63299 63707 641 17 64528 64941 65355 65771 66189 66608 67028 67451 67875 68301 68728 69157 69588 70021 70455 70891 71329 71769 7221I 72654 73100 73547 73996 74447 74900 75355 75812 76272 76733 77196 7y66i 78129 cotangent 397 398 399 401 402 404 405 407 408 410 411 413 414 416 418 419 420 423 424 426 427 429 431 433 434 436 438 440 442 443 446 447 449 451 453 455 457 460 461 463 465 468 D cotangent 1-66428 1-65337 1-64256 1-63185 1-62125 I -61074 I 60033 1-59002 1-57981 1-56969 1-55966 1-54972 I 53987 1-53010 1-52043 1-51084 I-50133 I -49 1 90 1-48256 1-47330 1-46411 I-45501 1-44598 1-43703 1-42815 1-41934 I-41061 I -40195 1-39336 1-38484 1-37638 1-36800 1-35968 I-35142 1-34323 I -335 1 1 1-32704 1-31904 1-31110 1-30323 I -29541 1-28764 1-27994 tangent 994 985 977 967 959 951 943 934 926 919 910 903 895 888 881 873 866 859 852 846 838 832 826 819 812 807 800 794 787 782 777 770 -16C63 -16868 -17075 -17283 17493 -17704 -17918 18133 18350 -18569 -18790 -19012 -19236 -19463 -19691 -19920 20152 20386 -20622 -20859 -21099 -21341 -21584 -21830 -22077 -22327 -22579 -22833 -23089 '233A7 -23607 23869 -24134 -24400 -24669 -24940 -25214 -25489 25767 -26047 26330 -26615 26902 cosecant 205 207 208 210 211 214 215 217 219 221 222 224 227 228 229 232 234 236 237 240 242 243 246 247 250 252 254 256 258 260 262 265 266 269 271 274 275 278 280 283 285 287 85717 85567 85416 -85264 -85II2 84959 -84805 84650 84495 84339 84182 84025 83867 83708 83549 83389 83228 -83066 82904 82741 82577 82413 82248 82082 819IS 81748 81580 81412 81242 81072 80902 80730 80558 80386 80212 80038 79864 79688 79512 79335 79158 78980 78801 24 31< LOGARITHMIC FUNCTIONS 31 lO' 20' 30' 8i 32 ^ 10' 30' 40' 50' 10' 20' 30' 40' 50' 34^ 10' 20' 30' 40' 50' 10' 20' 30' 40' 50' '36 10' 20' 30' 40' rro 10' 20' 30' 40' SO' 38 lo? sin 1-71184 I71393 171602 I71809 172OI4 I72218 1-72421 1-72622 1-72823 1-73022 I-73219 173416 i-73611 173805 173997 I-74189 174379 1-74568 1-74756 174943 175128 175313 175496 175678 1-75859 1-76039 1-76218 176395 1-76572 1-76747 1-76922 1-77095 1-77268 177439 T-77609 1-77778 1-77946 1-78113 1-78280 1-78445 1-78609 T-78772 1-78934 logf COS 209 209 207 205 204 203 201 201 199 197 197 195 194 192 192 190 189 188 187 185 185 183 182 181 180 179 177 177 17s 175 173 173 171 170 169 168 167 167 165 164 163 162 log cosec 0-28816 0-28607 0-28398 0-28191 0-27986 0-27782 0-27579 0-27378 0-27177 0-26978 0-26781 0-26584 o- 263 89 0-26195 0-26003 0-25811 0-25621 0-25432 0-2S244 0-25057 0-24872 0-24687 0-24504 0-24322 0-24I4I 0-23961 0-23782 0-23605 0-23428 0-23253 0*23078 0-22905 0-22732 0-22561 0-22391 0-22222 0-220S4 0-21887 0-21720 0-2I555 0-21391 0-21228 0-21066 log tan 1-77877 1-78163 1-78448 T78732 I -7901 5 1-79297 I -79579 1-79860 1-80140 1-80419 1-80697 1-80975 1-81252 1-81528 T-81803 1-82078 1-82352 1-82626 1-82899 1-83171 1-83442 1-83713 1-83984 1-84254 1-84523 1-84791 1-85059 1-85327 1-85594 1-85860 1-86126 1-86392 1-86656 I -8692 1 87185 1-87448 i-87711 1-87974 1-88236 1-88759 1-89020 1-89281 286 285 284 283 282 282 281 280 279 278 278 277 276 275 275 274 274 273 272 271 271 271 270 269 268 268 268 267 266 266 266 264 265 264 263 263 263 262 262 261 261 261 log sec I log cotan log- cotan 0-22123 0-21837 0-21552 0'2I268 0-20985 0-20703 0-20421 0-20140 0-19860 0-I958I 0-19303 0-19025 0-18748 0-18472 0-18197 0-17922 0-17648 0-17374 0-17101 0-16829 0-16558 0-16287 0-I60I6 0-15746 0-15477 0-15209 0-I494I 0-14673 0-14406 0-I4I40 0-13874 0-13608 o- 1 3344 0-13079 0-12815 0-12552 0-12289 0-12026 0-11764 0-11502 0-11241 0-10980 0-10719 log tan log sec o- 06693 0-06770 0-06846 0-06923 0-07001 0-07079 007158 0-07237 0-07317 0-07397 0-07478 0-07559 007641 0-07723 0-07806 0-07889 0-07973 0-08058 o 08 143 0-08228 0-08314 0-08401 0-08488 0-08575 008664 0-08752 0-08842 0-08931 0-09022 0-09113 009204 0-09296 0-09389 0-09482 0-09576 0-09670 o 09765 0-09861 0-09957 0-10053 0-IOI5I 0-10248 0-10347 log cosec 077 076 077 078 078 079 079 080 080 081 081 082 082 083 083 084 085 085 085 086 087 087 087 089 088 090 089 091 091 091 092 093 093 094 094 095 096 096 096 098 097 099 log cos 1-93307 1-93230 1-93154 1-93077 1-92999 I -9292 1 I 92842 1-92763 1-92683 1-92603 1-92522 1-92441 1-92359 1-92277 I -92 1 94 I -921 1 1 1-92027 I-9I942 1-91857 I.9I772 I-9I686 91599 91512 1-91425 I -91336 1-91248 -91158 I -91069 1-90978 1-90887 i -90796 90704 I -906 1 1 1-90518 1-90424 1-90330 1-90235 -90139 -90043 1-89947 1-89849 1-89752 1-89653 log sin 59 50' 40' 30' 20' 10' 58 50' 40' 30' 20' 10' 57 50' 40' 30' 20' 10' 56 50' 40' 30' 20' 10' 55 50' 40' 30' 20' 10' 54 50' 40' 30' 20' 10' 53 50' 40' 30' 20' 10' 52 52 25 38 NATURAL FUNCTIONS 6is66 61795 62024 '62251 62479 62706 62932 63158 63383 63608 63832 64056 64279 64501 64723 64945 65166 65386 65606 65825 66044 66262 66480 66697 66913 67129 67344 67559 67773 67987 68200 68412 68624 68835 69046 69256 69466 69675 69883 70091 70298 70505 7071 1 229 229 227 228 227 226 226 225 225 224 224 223 222 222 222 221 220 220 219 219 218 218 217 216 216 215 215 214 214 213 212 212 211 211 210 210 206 208 208 207 207 206 I 62427 1-61825 1-61229 1-60639 1-60054 1-59475 1-58902 1-58333 1-57771 1-57213 I -56661 1-56114 155572 1-55036 1-54504 1-53977 1-53455 I 602 596 590 585 579 573 569 562 558 552 547 542 536 532 527 522 5393815;^ :^ -^ 1-51415 1-50916 1-50422 1-49933 1-49448 1*48967 1-48491 I -48019 I -475 5 1 1-47087 1-46628 I 46173 I-4572I 1-45274 1-44831 I -44391 1-43956 1-43524 I ^43096 1-42672 1-42251 1-41835 1-41421 503 499 494 489 485 481 476 472 468 464 459 455 452 447 443 440 435 432 428 424 421 416 414 tangent D cotangent 78129 .78598 79070 79544 80020 -80498 -80978 -81461 81946 82434 82923 83415 -83910 84407 84906 85408 85912 86419 86929 87441 87955 88473 88992 89515 90040 90569 91099 91633 5^4 469 472 474 476 478 480 483 485 488 489 492 495 497 499 502 504 507 510 512 514 518 519 523 525 529 530 92170 92709 93252 -93797 -94345 94896 -95451 96008 96569 -97133 97700 98270 98843 99420 I 00000 D cotangent 537 539 543 545 548 551 555 557 561 564 567 570 573 577 580 1-27994 1-27230 I -2647 1 I -25717 I -24969 1-24227 I 23490 1-22758 I -2203 1 1-21310 1-20593 1-19882 1-19175 I -18474 1-17777 1-17085 1-16398 1-15715 I -15037 1-14363 1-13694 i^i3029 1-12369 1-11713 i-iio6i I'i04i4 1-09770 1-09131 I -08496 1-07864 I 07237 I -066 1 3 1-05994 1-05378 I -04766 1-04158 1-03553 1-02952 I-02355 i^oi76i i^oii7o 1-00583 I 00000 764 759 754 748 742 737 732 727 721 717 711 707 701 697 692 687 683 678 674 669 665 660 656 652 647 644 639 635 632 627 624 619 616 612 608 605 601 597 594 591 587 583 tangent 26902 27191 27483 27778 28075 28374 28676 28980 29287 29597 29909 30223 30541 30861 31183 31509 31837 32168 32501 32838 '33177 33519 33864 34212 34563 -34917 -35274 -35634 -35997 -36363 -36733 37105 -37481 -37860 -38242 -38628 -39016 39409 -39804 40203 40606 4IOI2 4 142 1 289 292 295 297 299 302 304 307 310 312 314 318 320 322 326 328 331 333 337 339 342 345 348 351 354 357 360 363 366 370 372 376 379 382 386 388 393 395 399 403 406 409 78801 78622 -78442 78261 78079 -77897 77715 77531 77347 77162 -76977 -76791 76604 76417 76229 76041 -75851 75661 75471 -75280 75088 74896 74703 -74509 74314 -74120 -73924 -73728 73531 73333 73135 .72937 '72737 72537 72337 72136 -71934 71732 71529 -71325 71121 70916 7071 I z6 38= LOGARITHMIC FUNCTIONS log^ sin 78934 79095 79256 79415 79573 79731 79887 80043 80197 80351 80504 80656 80807 80957 8 1 106 81254 81402 81549 81694 81839 81983 82126 82269 82410 82551 82691 82830 82968 83106 83242 83378 83513 83648 83781 83914 84046 84177 84308 84437 84566 84694 84822 84949 log COS 161 161 159 158 158 156 156 154 154 153 152 151 ISO 149 148 148 147 145 14s 144 143 143 141 141 140 139 138 138 136 136 136 135 133 133 132 131 131 129 129 128 128 127 log^ CO sec 2 1066 0-20905 0-20744 0-20585 0-20427 0-20269 O-2OII3 0-I9957 0-19803 0-19649 0-19496 0-I9344 0-19193 0-19043 0-18894 0-18746 0-18598 0-18451 0-18306 o^i8i6i 0-18017 0-17874 0-17731 0-17590 0-17449 0-17309 0-17170 0-17032 0-16894 0-16758 0-16622 0-16487 0-16352 0-16219 0-16086 15954 15823 15692 15563 15434 15306 15178 15052 log; sec log tan 1^8928 1 1-89541 i^898oi i^90o6i i^90320 1-90578 190837 i^9io95 91353 i^9i6io i^9i868 I^92I25 192381 i^92638 i^92894 1-93150 1-93406 i^9366i 193916 1-94171 I ^94426 i^9468i 1-94935 1-95190 1-95444 1-95698 1-95952 1-96205 1-96459 1-96712 i -96966 T-97219 1-97472 1-97725 1-97978 1-98231 I 98484 1-98737 1^98989 1-99242 1-99495 1-99747 000000 log cotan D 260 260 260 259 258 259 258 258 257 258 257 256 257 256 256 256 255 255 255 255 255 254 255 254 254 254 253 254 253 254 253 253 253 253 253 253 253 252 253 253 252 253 log cotan I07I9 0-I0459 0-10199 0-09939 0-09680 0-09422 0-09163 0-08905 0-08647 08390 o^o8i32 0-07875 007619 0-07362 0-07106 0-06850 0^06594 06339 006084 05829 0-05574 0-05319 0-05065 04810 o 04556 04302 0-04048 -3795 0-03541 -3288 o 03034 -2781 0-02528 -2275 -222 -1769 O-OIS16 1263 -III -758 o55 -253 0-00000 log tan log sec D 0-10347 I 446 0-I0545 I 646 i746 i848 0-10950 o-ii52 0-11156 0-11259 0-II364 -II469 0-II57S -11681 -11788 11895 0-I24 -I2II3 0-12222 0-12332 0-12443 0-12554 0-12666 0-12779 0-12893 o-i37 0-13121 0-13237 0-I3353 0-13470 0-13587 0-13705 0-13824 0-I3944 0-14064 0-14185 0-14307 0-14429 0-14552 0-14676 -1480 - 14926 0-15052 log cosec 099 99 ii I I 2 I2 102 I 4 103 105 105 I 6 106 i7 I 7 I 9 I 9 I 9 no III III 112 113 114 114 114 116 116 117 117 118 119 12 12 121 122 122 123 124 124 126 126 log cos 89653 89554 89455 89354 89254 89152 89050 88948 88844 88741 88636 88531 88425 88319 88212 88i5 87996 87887 87778 87668 87557 87446 87334 87221 87107 86993 86879 86763 86647 8653 86413 86295 86176 8656 85936 85815 85693 85571 85448 85324 852 85074 84949 log sin 52^ 50' 40' 30' 2' I' 50' 40' 30' 2' I' 50 50' 40' 30' 2' I' 49'' 50' 40' 30' 20' 10' 48 50' 40' 30' 20' 10' 47 50' 40' 30' 20' 10' 46 so' 40' 30' 20' 10' 45 45^ 97 FUUR-FIGURE TRIGUNUMETRICAL TABLES Radians De- giees Sine Cosec. Tangent Cotan. Secant Cosine OOOOO 0000 00 -0000 00 1 .0000 I -OOOO 90 I-570S0 01745 I 0175 57-2986 -0175 57-2899 I -0002 -9998 89 1-55334 03491 2 0349 28-6537 0349 28-6362 I -0006 9994 88 1-53589 05236 3 0523 19-1073 -0524 19-0811 1-0014 9986 87 I -5 1 844 06981 4 0698 14-3356 0699 14-3006 1 .0024 -9976 86 1-50098 08727 5 0872 11-4737 -0875 11-4301 1.0038 .9962 85 1-48353 10472 6 1045 9-5668 -1051 9-5144 1-0055 9945 84 1 .46608 12217 7 1219 8-2055 1228 8-1443 1-0075 9925 83 1 .44862 13963 8 1392 7-1853 1405 7-1154 I -0098 -9903 82 1-43117 15708 9 1564 6-3925 1584 6-3138 I-OI25 9877 81 1-41372 17453 10 1736 5-7588 1763 5-6713 I-OI54 9848 80 1-39626 19199 II 1908 5-2408 1944 5 -1446 1-0187 .9816 79 1-37881 20944 12 2079 4-8097 2126 4-7046 1-0223 9781 78 1-36136 22689 13 2250 4*4454 2309 4-3315 1-0263 9744 77 I -34390 24435 14 2419 4-1336 2493 4-0108 1 .0306 9703 76 1-32645 26180 IS 2588 3-8637 2679 3-7321 I-0353 -9659 75 1-30900 27925 16 2756 3-6280 2867 3-4874 1-0403 9613 74 1-29154 29671 17 2924 3-4203 3057 3-2709 1-0457 9563 73 I -27409 31416 18 3090 3-2361 3249 3.0777 1-0515 .9511 72 1-25664 33161 19 3256 3-0716 3443 2-9042 1-0576 9455 71 1. 23918 34907 20 3420 2-9238 3640 2-7475 I -0642 9397 70 1.22173 36652 21 3584 2.7904 3839 2-6051 1-0711 -9336 69 1.20428 38397 22 3746 2-6695 4040 2-4751 1-0785 .9272 68 I. 18682 40143 23 3907 2-5593 4245 2-3559 1.0864 9205 67 I. 16937 41888 24 4067 2-4586 -4452 2-2460 I -0946 9135 66 1-15192 43633 25 4226 2-3662 4663 2-1445 1-1034 9063 65 I -1 3446 45379 26 4384 2-2812 4877 2-0503 1-1126 -8988 64 1.11701 47124 27 4540 2-2027 5095 1-9626 1-1223 -8910 63 1-09956 48869 28 4695 2-1301 5317 1-8807 1-1326 -8829 62 1-08210 50615 29 4848 2-0627 5543 1 .8040 I -1434 .8746 61 1-06465 52360 30 5000 2-0000 5774 I-732I I -1 547 8660 60 1-04720 54105 31 5150 1-9416 6009 1-6643 I -1666 8572 59 1-02974 .55851 32 5299 1-8871 6249 1-6003 1-1792 -8480 58 1-01229 '57596 33 5446 1-8361 -6494 1-5399 1-1924 8387 57 -99484 59341 34 5592 1-7883 6745 1-4826 1-2062 -8290 56 97738 '61087 35 5736 1-7434 7002 1-4281 1-2208 8192 55 -95993 62832 36 5878 1-7013 7265 1-3764 1-2361 8090 54 .94248 64577 37 6018 I -6616 7536 1.3270 1-2521 7986 53 .92502 66323 38 6157 1-6243 7813 1-2799 1-2690 7880 52 .90757 68068 39 6293 1-5890 8098 1-2349 1-2868 7771 51 .89012 69813 40 6428 1-5557 8391 1-1918 1-3054 7660 50 .87266 71559 41 6561 1-5243 8693 1-1504 1-3250 7547 49 -85521 73304 42 6691 1-4945 9004 I '1106 1-3456 7431 48 -83776 75049 43 6820 1-4663 9325 1-0724 1-3673 -7314 47 82030 .76794 44 6947 1-4396 9657 I-0355 I -3902 7193 46 -80285 78540 45 7071 1-4142 I -000 3 I -oooo 1-4142 7071 45 78540 Cosine Secant Cotan. Tangent Cosec. Sine De- jjrees Radians 28 THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $t.OO ON THE SEVENTH DAY OVERDUE. APfi ^^^^'^^ MAI? > f^ 1 ""^n j.s ?-;f:<^. OCT 15 , 'v\J. r