SB Sh3 3m 
 
MECHANICS 
 
 MOLECULAR PHYSICS AND HEAT 
 
 A TWELVE WEEKS' COLLEdE COURSE 
 
 BY 
 
 ROBERT ANDREWS MILLIKAN, Pn.D. 
 
 M 
 
 ASSISTANT PROFESSOR OF PHYSICS IN THK 
 UNIVERSITY OF CHICAGO 
 
 BOSTON, U.S.A. 
 
 GINN & COMPANY, PUBLISHERS 
 1903 
 
COPYRIGHT, 1902 
 BY SCOTT, FORESMAN & COMPANY 
 
 COPYRIGHT, 1903 
 BY GINN & COMPANY 
 
 ALL RIGHTS RESERVED 
 
 !.** c .' 
 
 ;%** " 
 
PEEFACE 
 
 This book is neither a laboratory manual in the ordinary sense 
 of the term, nor yet is it simply a class-room text. It is intended 
 to take the place of both. It represents the first portion of a 
 college course in General Physics in which the primary object has 
 been to establish an immediate and vital connection between theory 
 and experiment. Of course such connection always exists in the 
 mind of the teacher; but the use in class-room and laboratory of 
 separate texts, separate courses, and separate instructors is on the 
 whole unfavorable to making it clear to the student. The stu- 
 dent who takes an experimental course which is out of imme- 
 diate connection with class-room discussion, who is provided in 
 the laboratory with an isolated set of directions, or with a labora- 
 tory manual which is essentially a compendium of directions for 
 all conceivable experiments, may perhaps in some cases obtain, 
 with the aid of references to text-books, a comprehensive grasp of 
 the theory and bearings of his experiment; but it is safe to say 
 that in a great majority of cases he does not do so. The most 
 serious criticism which can be urged against modern laboratory 
 work in Physics is that it often degenerates into a servile following 
 of directions, and thus loses all save a purely manipulative value. 
 Important as is dexterity in the handling and adjustment of 
 apparatus, it can not be too strongly emphasized that it is grasp of 
 principles, not skill in manipulation which should be the primary 
 object of General Physics courses. 
 
 Furthermore, an intimate connection between lecture and 
 laboratory work is no less important from the standpoint of the 
 former than of the latter. Without the fixing power of laboratory 
 applications, a thorough grasp of physical principles is seldom, or 
 never, gained. This is particularly true in Mechanics, the most 
 fundamental of all the branches of Physics, for it is only through 
 it that the door is opened to insight into the theories of Heat, 
 Sound, Light, and Electricity. 
 
 3 
 
4 PREFACE 
 
 In the second place, this book represents an attempt to teach 
 thoroughly a few fundamental principles rather than to present 
 superficially a large mass of facts. Most of the general texts 
 which combine a full presentation of facts, with a satisfactory 
 discussion of their relation to theory, have grown too bulky for 
 general class-room use. On the other hand, other texts have 
 appeared in which the necessity for condensation has resulted in 
 such abridgment of discussion that there is little left but a 
 skeleton of experimental and theoretical results. 
 
 In attempting to avoid both of these extremes, a selection lias 
 been made for the present course of such principles only as can be 
 most effectively presented in connection with laboratory demon- 
 strations. This course is presented in the first third (twelve 
 weeks) of a year of General Physics in the Junior College at the 
 University of Chicago. The time is divided nearly equally between 
 class-room and laboratory work ; but the former is wholly occupied 
 with the discussion and application to practical problems, of the 
 twenty-three principles presented in the text. No demonstration 
 lectures whatever are given. The second third of the college year 
 is occupied with the presentation, by the same general plan, of 
 those parts of Electricity, Light, and Sound which can be most 
 profitably studied in connection with laboratory instruments and 
 methods. This course is embodied in a second volume. The 
 last third of the year is devoted wholly to demonstration lectures 
 upon subjects which have been omitted from the preceding courses 
 because they are more suitable to lecture than to laboratory 
 methods of presentation. Such are, for example, Static Elec- 
 tricity, Electric Radiation, the Discharge of Electricity through 
 Gases, the Radiation, Absorption, Polarization, and Interfer- 
 ence of Light, Physiological Optics and Acoustics; in a word, 
 all phenomena the presentation of which requires primarily quali- 
 tative rather than quantitative experiment. This demonstration 
 lecture course is given last instead of first because a thorough 
 grounding in the fundamental principles of physical measurement 
 is deemed absolutely essential to the intelligent following of lec- 
 tures of college grade in any branch of Physics. 
 
 A third aim has been to prepare a book which should represent 
 a continuous course in the study of the principles of experimental 
 Physics, rather than a compendium either of facto or experiments. 
 
PREFACE 5 
 
 Hence, in the first place, all purely manipulative experiments 
 have been altogether omitted. A student is not required to make 
 a useless measurement for the sake of learning to use an instru- 
 ment ; he rather learns to use the instrument for the sake of making 
 a needed measurement. The inversion of this order has invariably 
 weakened interest in laboratory work. In the second place, all 
 needless repetition of slight variations of the same experiment has 
 been avoided. For example, in the subject of Heat, there is but 
 one general principle involved in the method of mixtures, whether 
 it be applied to the determination of the specific heat of a solid, or 
 of a liquid, the latent heat of fusion, or of vaporization. Hence 
 it has been illustrated in this course by but one laboratory exercise. 
 Similarly, the usual half-dozen or more experiments upon the den- 
 sities of liquids and solids have been reduced to two. In a word, 
 experiments have been made incidental to the study of principles, 
 not principles incidental to the study of experiments. 
 
 Fourthly, an especial effort has been made to present Physics 
 as a science of exact measurement. Much harm is often done by 
 attempting to teach a course in an exact science with the aid of 
 instruments capable of giving results which can be called quanti- 
 tative by courtesy only, and which therefore foster the impression 
 that science is after all very inexact. The apparatus which is 
 used in this course has therefore been selected and designed with 
 a special reference to its ability to yield accurate results in the 
 hands of average students. All of the new pieces, such as the 
 acceleration machine (p. 11), the model balance (p. 36), the bal- 
 listic pendulums (pp. 54 and 62), the Young's modulus (p. 67), 
 the torsion machines (p. 75), the inertia disc (p. 82), the 
 pendulum arrangement (p. 97), the centripetal machine (p. 102), 
 the pressure apparatus (p. 117), the form of air thermometer 
 (p. 131), the vapor pressure arrangement (p. 158), have been 
 designed in this laboratory, slight modifications having in some 
 instances been introduced by the instrument maker, William 
 Gaertner, from whom any of the special pieces used in the course 
 may be obtained. Furthermore, in nearly all of the exercises, the 
 quantity sought is obtained by two distinct methods and the 
 results compared. 
 
 Finally, since the book represents a college, not a high school, 
 course, the aim has been, not so much to acquaint the student 
 
6 PREFACE 
 
 with interesting and striking phenomena, as to give him an insight 
 into the real significance of physical things to introduce him to 
 the very heart of the subject by putting him in touch with the 
 methods and instruments of modern physical investigation, and by 
 carrying him through the processes of close reasoning by which 
 the present science of Physics has been developed. Students who 
 enter upon the course are expected both to have completed a year 
 of secondary school Physics, and to have gained some familiarity 
 with the principles of Trigonometry. 
 
 The author's justification for the publication of this, the first 
 book in the series, is the hope that the presentation of a method 
 of instruction which has been found most satisfactory in the Uni- 
 versity of Chicago may not be altogether without usefulness, or at 
 least suggestiveness, to teachers of Physics in other institutions. 
 The custom usually followed is to hold lectures and quizzes upon 
 a group of eight experiments, before taking up the laboratory 
 work. The problems are of course invaluable aids in the fixing 
 of principles. In the laboratory not more than twenty students 
 are ever permitted in a single section. No more than two dupli- 
 cate pieces of apparatus are ever used. Duplicates of the forms of 
 record which have been inserted in the manual are filled out by 
 the student and handed to the instructor as each experiment is 
 completed. In addition to filling out these record slips each 
 student keeps a systematic note-book, in which are entered in 
 the laboratory, not at home, all observations and all calcula- 
 tions of whatever kind. This note-book is a complete record of 
 the student's work, and should of course be so arranged as to 
 be easily intelligible to anyona, even though he be unfamiliar with 
 the course. 
 
 This book is the successor of A College Course of Laboratory 
 Experiments in General Physics, published by the present director 
 of the National Bureau of Standards, Professor S. W. Stratton, 
 and the author. It is to Professor Stratton that the design of 
 much of the apparatus is due. In the preparation of the present 
 course the author has had the invaluable assistance not only of 
 Professor Stratton, but also of Mr. G. M. Hobbs and Dr. H. G 
 Gale, both instructors in Physics at the University of Chicago, 
 
 UNIVERSITY OF CHICAGO, 
 August 27, 1902. 
 
CONTENTS 
 
 MECHANICS 
 
 PAGE 
 
 I. UNIFORMLY ACCELERATED MOTION ..,'. 9 
 II. FORCE PROPORTIONAL TO RATE OF CHANGE OF MOMENTUM 
 
 (f = ma] . . . '; ... . \ ." . 15 
 
 III. COMPOSITION AND RESOLUTION OF FORCES . . . 21 
 
 IV. THE PRINCIPLE OF WORK . . . . . .*'. 29 
 
 V. ENERGY AND EFFICIENCY . . . . . . . 42 
 
 VI. THE LAWS OF IMPACT . . , . , . . . 52 
 
 VII. ELASTIC IMPACT. COEFFICIENT OF RESTITUTION . . 58 
 
 VIII. ELASTICITY. HOOKE'S LAW: YOUNG'S MODULUS . . 65 
 
 IX. THE COEFFICIENT OF RIGIDITY . . . . 71 
 
 X. MOMENT OF INERTIA .- . . . . . . .78 
 
 XL SIMPLE HARMONIC MOTION . . . . :; . . 87 
 
 XII. DETERMINATION OF "g" . 95 
 
 XIII. THE LAW OF CENTRIPETAL FORCE . . . 100 
 
 MOLECULAR PHYSICS AND HEAT 
 
 XIV. BOYLE'S LAW . . . . . . . . . > 105 
 
 XV. DENSITY OF AIR. , .[ ' . . . . . .. . 114 
 
 XVI. THE MEASUREMENT OF TEMPERATURE . ... . 122 
 
 XVII. LAW OF AVOGADRO DENSITIES OF GASES AND VAPORS 138 
 XVIII. THE PRESSURE-TEMPERATURE CURVE OF A SATURATED 
 
 VAPOR . . ... . . . . . . 152 
 
 XIX. HYGROMETRY . . . . . . " ( . . 164 
 
 XX. ARCHIMEDES' PRINCIPLE . . . . . . .. 173 
 
 XXI. CAPILLARITY . ... . . . . . 181 
 
 XXII. CALORIMETRY . . . . . . . ."- , . 193 
 
 XXIII. EXPANSION . . . . . . . . ^ . 215 
 
 APPENDIX . . . . . . . . . ' . . 223 
 
 INDEX 
 
 7 
 
MECHANICS ; * \ 
 
 I 
 
 UNIFORMLY ACCELERATED MOTION 
 
 Theory 
 
 Conceive of a body moving in a straight line with continually 
 
 changing velocity. Its motion is said to be uniformly accelerated 
 
 when it makes equal gains of velocity in equal intervals 
 
 acceleration of time. The rate of change of velocitv, or, for the 
 
 defined. . . . , . 
 
 present case, the gain in velocity per unit 01 time is 
 called the acceleration of the body. 
 
 LAWS OF UNIFORMLY ACCELERATED MOTION. The following 
 laws are derived at once from the above definition : 
 
 1. If v represent the velocity of the body at the end of t units 
 of time, a its acceleration and V Q its velocity at the beginning of 
 
 the t units, then 
 Velocity in 
 
 terms of *. _ n f , , . /i \ 
 
 acceleration " V > V J > 
 
 or, if the body start from rest (i.e. if v = 0) 
 
 v = at. (2) 
 
 This law is nothing more than the mathematical statement of the 
 definition. 
 
 2. If s t , s 2 , s 3y 5 n , and s n + 1 represent the distances traversed 
 AcceUra- ^v the body during the 1st, 2d, 3d, nth and ( 
 o/8w?ce? ms un ^ s f ^ me respectively, then 
 
 sive apace /o \ 
 
 intervals. . = S 2 5i = S 3 5 2 = S n + i S a (O) 
 
 : 
 
 FIGURE 1 
 
10 MECHANICS 
 
 Proof. Let the straight line (Fig. 1) be the path of a body 
 moving with uniform acceleration, and let v^ v z , etc. , be the veloc- 
 ities at the ends of the units of time 1, 2, etc., and s^ s 2 > etc., the 
 spaces traversed during these units. The 'space passed over during 
 any,ineVval Of time must always be the mean velocity multiplied 
 by 'the number, of units of time in the interval. In case the 
 velocity. increases uniformly this mean velocity is evidently the 
 half sum of the velocities at the beginning and at the end of the 
 interval. Hence, e.g. (See Fig. 1), 
 
 . (4) 
 
 Similarly s 3 = v z + -r- .'. s s 3 = v 3 - v z . But v 3 - v z is by defi- 
 
 /o 
 
 nition a. .*. s 4 s 3 = a. Similarly for s 3 s a , etc. Q. E. D. 
 
 The acceleration can therefore be most directly determined by 
 measuring distances traversed in successive units of time. 
 
 3. If & represent the total space traversed during t units of 
 
 Space in time > then 
 
 %%$** tf-iof + t,.*; (5) 
 
 or, if the body start from rest 
 
 S=.%at\ (6) 
 
 Proof. Total space = mean velocity x time = 4 (initial velocity 
 
 + final velocity) x time = y + (^ + g O x t = at * + ^. Q. E. D. 
 
 Z 
 
 4. If v represent the velocity of a body after it has moved 
 over a space 8 with an acceleration , and if v represent the 
 
 velocity which the body had at the point from which 
 X?o/ in S is measured, then 
 
 space and . - - - - 
 
 accelera- V=V%aS+V<f. (7) 
 
 or, if the body start from rest, (i.e., if V Q - 0) 
 
 v = N/2otf. (8) 
 
 Proof. From (1), v = at + v . '. t = Now if v be used to 
 represent the average velocity with which the body traverses the 
 
 a , , - V + VQ , a V + VQ V - VQ V* V<? _ 
 
 space 8, then v = - but S = vt = ^~ x - = - Hence 
 
 Z A OL Ztt 
 
 solving for v, 
 
 v = v/2a + vf ; or if v =0,v= \/2aS. Q. E. D. 
 
UNIFORMLY ACCELERATED MOTION 
 
 11 
 
 Experiment 
 
 The object of this experiment is to study the motion of a 
 freely falling body; in particular, (1) to ascertain whether a freely 
 falling body moves with uniformly accelerated motion, 
 and (2) to determine the acceleration of this motion 
 in centimeters per second. 
 
 The falling body is a brass frame , (Fig. 2), which weighs 
 about 1 kgm. and falls a dis- 
 tance of some 120 cm. through 
 guides which offer very 
 
 srssusx. little fricti n - Tbe 
 
 cord, which is shown 
 in the figure passing over the pul- 
 ley and supporting balancing 
 weights, is in this experiment de- 
 tached from the frame , so that 
 the latter falls freely. The fall- 
 ing frame carries with it a tuning 
 fork #, one prong of which is pro- 
 vided with a light stylus which, 
 during the descent, traces a wavy 
 line upon the blackened glass 
 plate M (see Fig. 3). The frame 
 is first raised to the top of the 
 ways where it is held in place by 
 an eccentric catch, which keeps 
 the prongs slightly spread. Turn- 
 ing the lever I draws the eccentric 
 up into the crossbar o, releases 
 the prongs, and allows the frame 
 and vibrating fork to fall. The 
 plate M can be shifted sidewise so 
 that a number of traces can te 
 obtained. Two dash-pots at the 
 base of the instrument catch the 
 falling frame and take up the jar. 
 DIRECTIONS. Hang a plumb- 
 bob from r and adjust the level- 
 FIGUBE 2 ing screws in the base until a FIGURE 3 
 
12 MECHANICS 
 
 strictly vertical fall is assured. Remove the glass plate M and 
 smoke it evenly from top to bottom in a gum-camphor flame. 
 
 Replace it and adjust the stylus so that it will follow 
 ondmeoStre- smo thty behind the falling fork, pressing very lightly 
 tface f the a g ainst tne P^te. Then release the fork. Having 
 
 obtained at least two good traces, remove the plate, set 
 it in the horizontal wooden frame provided (see Fig. 4), and count 
 off the waves by tens from top to bottom of each trace, marking 
 
 FIGURE 4 
 
 as accurately as possible the crest of ea-ch tenth wave by means of 
 a pin-scratch (see Fig. 3). Next turn the plate over, smoked side 
 down, and hold a meter stick on edge upon the unsmoked side of 
 the plate just over the trace. It will then be easy to read upon 
 the scale the distances between successive pin-marks. 
 
 Although the scale and the trace are the thickness of the glass 
 apart, the error of parallax, i. e., the error in reading arising 
 because the line of sight is not perpendicular to the 
 P^Q, may be avoided entirely. For, since the 
 blackened plate acts as a mirror, it is only necessary, 
 in taking a reading, to place the eye so that the images of the 
 scale divisions and the scale divisions themselves appear in the 
 same straight line. In taking the readings estimate in each case 
 to tenths mm. 
 
 By subtracting Si from % #2 from s 3 , s 3 from s 4 , etc. (see 
 
 Fig. 3), obtain from each .trace as many values of the acceleration 
 
 as is possible, when the unit of time chosen is the time 
 
 Acceleration , , 
 
 for different of 10 vibrations of the fork. Repeat the same opera- 
 tion when the unit is the time of 20 vibrations; then 
 when it is the time of 30 vibrations ; then of 40. If the motion 
 is uniformly accelerated the values of a will not change as you 
 move from the top to the bottom of the plate. 
 
UNIFORMLY ACCELERATED MOTION 
 
 13 
 
 By comparison of the mean values of the accelerations for differ- 
 ent units of time find the law which connects the 
 numerical value of the acceleration with the length of 
 'value of g. the unit of time. From this law and the rate of the 
 fork, as furnished by the instructor, determine #, the 
 acceleration per second of the falling body. 
 
 The true value of g is 980 cm. On account of the slight fric- 
 tion in the guides in addition to the air resistance, the value here 
 found will fall a trifle short. From the total length of 
 resistance the fall and the correct value of g 'calculate what should 
 be the total time of descent of the falling body. Com- 
 pare this with the observed time obtained from the rate of the fork 
 and the total number of waves upon the plate. The difference 
 represents the retardation due to the friction of the air and of the 
 guides. The difference between 980 and the observed value of g 
 represents of course the negative acceleration due to friction. 
 Since, when force is measured in dynes, force = mass x accelera- 
 tion (see Ex. II), it is possible from an observation of the mass of 
 the frame and fork to express the value of the retarding frictional 
 force in dynes. 
 
 Record 
 
 FIRST TRACE 
 
 10 20 30 40 
 vib's. vib's. vib's. vib's. 
 
 Mean 
 
 Means reduced * 
 Mean of means 
 Rate of fork = 
 
 vib. per sec. .. g = 
 
 SECOND TRACE 
 
 10 20 30 40 
 vib's. vib's. vib's. vib's. 
 
 Mean of means 
 
 ' 9 ~ 
 
 % of difference between the two values of g (observational error) = - 
 % of departure of final mean from correct value of g = . . . - 
 Total time of fall, calc'd value = obs'd value = % dif. = - 
 Weight of frame and fork = .'. No. dynes friction = - 
 
 * Divide mean of 20's by 4, of 30's by 9, of 40's by 16. 
 
14 MECHANICS 
 
 Problems 
 
 1. Show that the law discovered experimentally above, viz., 
 that the numerical value of the acceleration varies directly as the 
 square of the unit of time, can also be obtained theoretically from 
 the definition of acceleration, viz., the velocity (measured in cm. 
 per second) which is gained during one second. 
 
 2. If the earth's period of rotation were to be suddenly 
 doubled, and if a second of time were still to be defined as the 
 same fraction of that period as at present, what effect would be 
 observed in the numerical value of g 9 
 
 Disregard all centrifugal tendencies. 
 
 3. If the earth's period were slowly changing, how could the 
 change be detected? 
 
 4. Show that a body shot from the earth upward has at any 
 given height the same velocity in the ascent as in the descent. 
 
 5. The Eiffel Tower is 335 meters high. How many seconds 
 will elapse before an arrow, shot upward from the tower with a 
 velocity of 4000 cm. per sec. reaches the earth? 
 

 II 
 
 FORCE PROPORTIONAL TO RATE OF CHANGE 
 OF MOMENTUM (f=ma) 
 
 Theory 
 
 NEWTON'S LAWS OF MOTION. The whole of Mechanics is 
 based upon three great experimental principles first clearly stated 
 in Newton's laws of motion. As will presently appear, 
 o}iaws nt the First and Second Laws contain but one experimental 
 principle, the Third with the subjoined scholium con- 
 tains two. Newton's statement of these laws is -as follows : 
 
 /. Every body continues in its state of rest or of uniform 
 motion in a straight line except in so far as it is compelled by force 
 to change that state. 
 
 II. (Rate of) change of (quantity of) motion is proportional 
 to force and takes place in the straight line in which the force acts. 
 
 III. To every action there is ahvays an equal and contrary 
 reaction; or the mutual actions of any two bodies are always equal 
 and oppositely directed. 
 
 The First Law is completely contained in the Galilean defini- 
 tion of force, viz. : "Force is that which changes the 
 tf e Fir*tLaw s ^ e f res ^ or uniform motion of a body."* It is 
 therefore essentially a definition of force, and hence 
 requires no proof. 
 
 The Second Law asserts that force is proportional to, and there- 
 fore may be measured by, the rate of change of quantity of motion. 
 But "quantity of motion,"" commonly called momentum, 
 wmd*Law. ' IS ^ e P r duct of two factors, mass and velocity. Rate 
 of change of momentum is therefore mass x rate of 
 
 * Since the existence of forces in equilibrium is denied by this defini- 
 tion of force it is often found thus modified. "Force is that which 
 changes or tends to change," etc. Newton, however, (and in this most 
 modern writers follow him) regarded equilibrium as the balancing of 
 motions, not of forces. Thus according to the Newtonian view each force 
 actually produces its proper motion which is neutralized by the equal and 
 opposite motion produced by the opposing force. 
 
 IS 
 
16 MECHANICS 
 
 change of velocity, or mass x acceleration. The mathematical 
 statement of the Second Law is therefore/ oc ma. This law is a 
 definition of the measure of a force; but a little con- 
 sideration will show that it is a definition which 
 presupposes experimental knowledge, and conse- 
 quently finds its justification in experiment only. 
 
 Thus, suppose two bodies whose masses are m 1 
 and m 2 (see Fig. 5) are dropped together toward the 
 
 earth. The fact that they fall shows that 
 Second^Law. certain forces act upon them (see First 
 
 Law). Let these forces be denoted by/ 
 and / 2 . Now observation shows that the two bodies 
 fall at the same rate, i. e., that they have 
 a common acceleration. Call this accel- ^ [ - ] | | f~ 
 eration g. Then the assertion made in the m 2 m 2 m 3 m 3 
 statement of the Second Law is that FIGURE 5i 
 
 / 2 
 
 Next suppose that the two bodies are brought back to their 
 original positions, where, necessarily, the same two original forces, 
 /! and/g, act upon them. Let now each body descend again, but 
 this time let each be obliged to drag along with it some third mass 
 which would otherwise remain at rest. Thus let m l be placed upon 
 either one of the two equal masses m^ which hang over a pulley 
 of negligible weight and friction (see Fig. 5). In this case the 
 moving force is evidently /, the mass moved is m l + 2w 3 , and 
 the acceleration is some measurable quantity a^. Then let the 
 masses w 3 be brought back to their original positions and m% 
 placed upon one of them. The moving force is now / 2 , the mass 
 moved m z + 2w 3 , and the acceleration some measurable quantity, 
 # 2 . The assertion made by the Second Law in regard to these two 
 motions is 
 
 /' 
 
 . (10 ) 
 
 / 2 
 
 Now (9) and (10) cannot both be true unless 
 
 fl t , 
 
 Since there is no sort of a priori ground for asserting the correct- 
 
NEWTON'S SECOND LAW 17 
 
 ness of this relation, it is only the experimental proof of it which 
 furnishes the justification for the Second Law. This proof can be 
 made in any laboratory. But the most convincing evidence of the 
 absolute correctness of the Second and Third Laws is furnished 
 by astronomical observations. Predictions of eclipses made years 
 in advance are wholly based upon these laws taken in connection 
 with the law of gravitation. 
 
 UNITS OF FORCE. If we arbitrarily choose as the absolute 
 unit of force the force which is required to impart to unit mass 
 one unit of acceleration the expression / a ma changes 
 *o/ = ma 5 ^ or ^ ^y definition, /= 1 when m = 1 and 
 a = 1, then evidently /= 4 when m = 2 and a = 2, etc. 
 The equation f=ma is then merely the statement of Newton's 
 Second Law. Note, however, that it holds in this form only when 
 the force is expressed in the units defined above, i. e., in absolute 
 units. If the unit of acceleration be taken as 1 cm. per sec., and 
 the unit of mass as the mass of 1 cc. of water at 4 C. (i. e., 1 
 gram) the absolute unit of force is called a "dyne." A dyne of 
 force is then defined as the force required to impart to 1 gram of 
 mass an acceleration of 1 centimeter per second. 
 
 There is another unit of force in common use, which is called 
 
 the gram of force. It must not be confused with the gram of 
 
 mass. The gram of force is defined as the force of the 
 
 Number of 3 . 
 
 dynes in a earth's attraction upon a gram of mass. Since this 
 force is able to impart to the gram of mass an accelera- 
 tion of 980 cm. per second, it is evident that one gram of force is 
 equivalent to 980 dynes of force. 
 
 Experiment 
 
 Object. To prove that/ oc ma. 
 
 The apparatus is the same as that used in Ex. I, save that the 
 
 cord c (Fig. 2) is attached to the frame and fork, carried over the 
 
 pulley f) and fastened at the other end to the weights 
 
 Method. i i j. , -.1 T/. AT. T.J. 
 
 wz, one of which is adjustable. If the weights m just 
 balance the frame and fork, the removal of any weight m l from m 
 will cause the frame and fork to descend with an acceleration 
 which may be measured as in Ex. I. The moving force will be m l 
 grams, the mass moved will be the mass of frame and fork plus 
 
18 
 
 MECHANICS 
 
 After leveling the instrument as in Ex. I, adjust the mass m until 
 the fork, when given a slight downward movement, will just con- 
 tinue to move, but without acceleration. This elimi- 
 nates the friction of the pulley and guides. Now 
 remove a mass mi (say 400 gm.) from m and obtain two traces 
 precisely as in Ex. I. Repeat when m has been diminished by a 
 second 400 gm. so that the moving force is now, say, 800 gm. 
 (= m 2 ). Measure the accelerations in the two cases according to 
 the method given in Ex. I, using, however, but one unit of time,! 
 say that of 20 vibrations of the fork. The mass of the frame and 
 fork* as well as the masses m, m l7 and m* are to be determined 
 by weighing upon a pair of trip scales. The test here made of the 
 Second Law consists in verifying equation (11). 
 
 Record 
 
 Weight of frame and fork ( W) = . Balancing weight (m) = 
 
 1st 
 spaces 
 
 trace 
 
 2d trace 
 spaces a l 
 
 1st trace 3d trace 
 spaces tt 2 spaces 0*1 
 
 S2 
 S3~ 
 
 \ 
 
 \ 
 
 I 
 
 \ 1 ; 
 
 1 
 
 1 \ 
 
 f 
 
 1 
 
 f 
 
 \ \ 
 
 
 \ \ 
 
 I 
 
 f 
 
 \ \ 
 
 Si 
 
 I 
 
 I 
 
 \ \ 
 
 f 
 
 f 
 
 \ f 
 
 
 I 
 
 I 
 
 1 1 
 
 f 
 
 f 
 
 \ \ 
 
 S 
 
 Means 
 Final m 
 
 fi_ m i 
 
 .- *f 
 
 
 
 
 
 
 [W+^-m^Xa, ^ ^ ofcrro 
 
 ft >2 
 
 
 ^ W +(m-m 2 )]Xa 2 
 
 *The mass which, at the circumference of the wheel, would have 
 a moment of inertia (see Ex. X) equal to that of the wheel should theo-j 
 retically be included in this weight. This is usually a negligible quantity] 
 
 f Do not simply indicate the division. This blank is for the result 
 the division. 
 
NEWTON'S SECOND LAW 19 
 
 Problems 
 
 1. Over a weightless and frictionless pulley (see Fig. 5) are 
 suspended masses of 200 gm. each. A weight of 100 gm. is 
 added to one side. Find the acceleration thus imparted to the 
 weights. 
 
 Take g, the acceleration of free fall, as 980 ; the acting force is then 
 100 X 980 dynes. 
 
 2. To the ends of a rope passing over a weightless pulley are 
 attached two bodies of unequal masses, one of 200 gm. , the other 
 of something more than 200 gm. The acceleration imparted to the 
 masses is observed to be 245 cm. per second. Find (1) the exact 
 mass of the larger body; (2) the tension in the string. Express 
 the tension both in dynes and grams. 
 
 SUGGESTION (1). If #=the difference between the two masses, 
 then the acting force is xg dynes and the mass moved is 400 -f- & 
 
 SUGGESTION (2). Since all freely falling bodies have the acceleration 
 g, the force of gravity acting upon any mass of m grams is mg dynes. If, 
 because of some retarding force (e.g. tension in a string), the downward 
 acceleration is not g but some smaller quantity a, it is at once clear from 
 the statement of the Second Law that the value of the upward or retard- 
 ing force is m(g a) dynes. Thus, if the body is at rest, the upward force 
 is mg dynes. If it has an upward acceleration amounting to a units the 
 upward force is m(g -f- a) dynes. 
 
 3. A 10 gm. mass is moving with a velocity of 40 meters per 
 second. A force of 2000 dynes opposes its motion. How soon will 
 it be brought to rest? 
 
 Find a from Second Law, then t from Ex. I. 
 
 4. A 900 kgm. projectile struck an embankment with a velocity 
 of 400 meters per second. It penetrated 4 meters. Find the 
 resistance in kgms. which the embankment opposed to its motion. 
 
 Find a from Ex. I. 
 
 5. A 50-lb. mass hangs from a spring balance in an elevator. 
 How much does it appear to weigh at the instant at which the 
 elevator begins to descend with an acceleration of 400 cm. per 
 second? (1 kilo = 2. 2 Ib.) 
 
 See problem 2, suggestion (2). 
 
 6. A man pushes steadily upon a car weighing 1000 kilos. 
 After 5 seconds it is moving with a velocity of .5 meters per 
 second. Find the man's force in kilos. 
 
 Find a from Ex. I. 
 
20 MECHANICS 
 
 7. A man who can jump three feet high on the earth could 
 jump how high on the moon? The mass of the earth is 80 times 
 that of the moon. The diameter of the earth is 3f times that of 
 the moon. 
 
 SUGGESTION. The mathematical statement of Newton's law of grav- 
 itation is foe -, in which/ is the force acting between any two bodies, 
 
 ra and M their respective masses and r the distance between their centers 
 of gravity. By the application of this law first find the relative values 
 of \ the acceleration of a body at the surfaces of the earth and moon 
 respectively, then note that the initial velocity produced by the spring 
 is always the same. 
 
Ill 
 
 COMPOSITION AND RESOLUTION OF FORCES 
 
 Theory 
 
 COMPOSITION OF FORCES RESULTANT. From the statement 
 that "change of motion is proportional to, and in the direction of 
 
 the impressed force," it follows, by implication at 
 r ^condLaw f l eas t? that a given force always produces the same 
 
 change of motion in the direction of its action, whether 
 the body upon which it acts is at rest or in motion, whether it is 
 acted upon by other forces at the same time or not. 
 
 Consider, for example, a very short interval of time, say one- 
 millionth of a second, during which a force / 15 acting in the 
 
 direction AB (see Fig. 6), is accelerating a body at A. 
 I force. At .the end of the interval the body will have acquired 
 Iratwn a ve l c ity by virtue of which, during the second which 
 
 begins at the close of this interval, it will move uni- 
 formly and in a straight line (see First Law) to some point B. If 
 the interval of time be taken suffi- 
 ciently small the acceleration 
 during the interval may always 
 be considered constant; and if, 
 in all cases considered, the in- 
 terval chosen be of the same 
 length, then even for a variable 
 force the acceleration will be sim- FIGURE 6 
 
 ply the velocity acquired in one 
 
 of these intervals divided by the length of the interval. Since, 
 then, the velocity acquired in an interval is always proportional to 
 the acceleration, the line AB may be taken as representing the 
 acceleration a^ due to the force/!. 
 
 If, instead of receiving an acceleration represented in magnitude 
 and direction by a^ the body were acted upon by a force / 2 acting 
 in the direction AD, it would receive, in the interval considered, 
 
22 MECHANICS 
 
 a velocity which would carry it in one second to some point D. 
 AD would then represent the acceleration due to f z . 
 
 If, now, the two forces / t and / 8 act simultaneously, the 
 Second Law asserts (by implication) that the effect of each force is 
 the same as though the other did not act ; i.e., that in the short 
 interval considered, the body will receive a velocity which will 
 carry it at one and the same time a distance AB in the direction 
 of (i. e., parallel to) AB and a distance AD in the 
 oftwo e f&rces direction of AD. This is merely saying that the veloc- 
 ity acquired in the interval will carry the body in one 
 second to C. Further, the path between A and C must be a 
 straight line, because a body can move, by virtue of an acquired 
 velocity, only in a straight line. Hence the line AC represents 
 the joint, or resultant, acceleration due to the joint action of the 
 two forces / x and / 2 . 
 
 But one single force / 3 can be found which, acting in the 
 
 direction AC, would produce this acceleration 3 . Such a single 
 
 force is called the resultant of the two given forces. 
 
 Resultant Thus the resultant of any number of forces is defined 
 
 force defined. 
 
 as that single force whose action would produce the same 
 '"''change of motion" as is produced by the joint action of the several 
 forces. 
 
 It is evident from the above, since, for a given mass, forces are 
 proportional to accelerations, that the resultant of any two forces 
 
 acting upon a particle is represented 
 
 Resultant of . . 6 r . 1 
 
 any number in intensity ana direction by the 
 diagonal of the parallelogram the 
 sides of which represent the two forces. By 
 the successive application of this rule to the 
 case of the simultaneous action, of three or more 
 forces, such as , #, c (see Fig. 7), the follow- 
 ing general rule is obtained: The resultant 
 FIGURE 7 C^) f an y num ^ er f forces- is represented in 
 
 magnitude and direction by the line which closes 
 the polygon whose sides represent the several forces when the latter 
 are conceived as acting successively. 
 
 The problem, then, of finding the magnitude and direction 
 of the resultant of two forces a and b which include an angle 
 (see Fig. S) resolves itself into the trigonometrical problem 
 
NEWTON'S SECOND LAW 
 
 23 
 
 "of finding the length of 
 
 the line .c and the value of 
 
 the angle 9 in 
 
 Calculation . , 
 
 of resultant terms o i the 
 
 of two forces. 
 
 given magni- 
 tudes a, b, and <. 
 
 To find the length of c 
 in terms of a, 5, and <, drop 
 a perpendicular pr upon a. 
 
 Then 
 Hence 
 Now 
 Hence 
 
 But 
 
 FIGURE 8 
 
 <? = or + pr . But pr = 
 c 2 = or rs + b 2 . 
 or = (a rs) = a z %ars+rs. 
 c z = a z + b z %a rs. 
 
 
 Hence 
 
 -r- = cos \lt = cos 
 o 
 
 c* = 2 + # 2 + 
 
 i. e. 
 
 cos 
 
 Again to find the angle in terms of a, b, and 
 
 = sin and - = sin \b = sin 
 c o 
 
 Hence 
 
 sn 
 
 sin 
 
 , 
 
 or sin 6 = sin 
 c 
 
 (12) 
 
 (13) 
 
 The resultant of three or more forces may be found by com- 
 bining the resultant of the first and second with the third, this 
 resultant with the fourth, etc. A better method, however, is given 
 in a following paragraph. 
 
 RESOLUTION OF FORCES COMPONENT. The component of a 
 force in any specified direction is defined as that force which, act- 
 ing in the direction specified, would produce the same 
 effect, so far as motion in this direction is concerned, as 
 is produced by the action of the given force. Thus if 
 N A C (Fig. 9) represents the distance over which a body would move in 
 one second by virtue of a velocity acquired 
 through the action of the given force during 
 one of the infinitesimal intervals above consid- 
 ered, AB evidently represents the distance 
 moved over toward the east during this second ; 
 FIGURE 9 i.e., if A C represents the actual acceleration, 
 
24 MECHANICS 
 
 AB represents the eastward acceleration. Or, finally, a force 
 represented by AB is equivalent, so far as motion toward the east 
 is concerned, to the given force A C. AB is therefore the com- 
 ponent of A C in the easterly direction. 
 
 Thus, if the body which is acted upon by the force A C is free 
 to move only in the direction AB (conceive, for example, of a car 
 on overhead rails pulled forward with the aid of a rope by a man 
 on the ground), its motion under the action of the force A C must 
 be precisely the same as though the force AB were acting instead 
 of AC. 
 
 The process of finding the component of a given force in any 
 direction consists, then, in finding the projection in the required 
 direction of the line which represents the given force. 
 Thus AM * ( Fi &- 10 ) is the component in the direction 
 AE of the force AF. But AM = AF cos 6. Hence, 
 in general, the component in any specified direction of a given force 
 is the product of the given force into the cosine of the angle included 
 between the specified direction and the direction of the given force. 
 Just as AM is the component in the direction AE ot the force 
 AF (see Fig. 10) so A G is the component of AF in the direction 
 
 AH. Now, if AE and AH are at 
 
 ty resolution right angles to each other, then the 
 
 figure AGFM is a parallelogram of 
 which A Fis the diagonal and AM and AG the 
 sides. Hence, (see above under RESULTANT) the 
 single force AF might entirely replace two forces 
 represented by AM and AG. Conversely the 
 single force AF must be in every respect replace- 
 FIGUBE 10 a ^ e by the two forces A M and A G. This process 
 of replacing a given force by its components in 
 any two directions at right angles to each other is called the resolu- 
 tion of force. 
 
 With the aid of the resolution of forces into their components 
 in any two rectangular directions, the resultant of any number of 
 forces may be easily obtained. Thus : Replace all of 
 the forces ^i, F* ^3 etc. (see Fig. 11), by their com- 
 P onen ts along any two rectangular axes X and Y. 
 Find the. algebraic sum x, of all the x components x^ 
 x 3 , etc. Similarly let Y represent the algebraic sum of all of 
 
NEWTON'S SECOND LAW 
 
 25 
 
 the y components y,, ?/ 2 , y^ etc. The resultant of x and Y (see 
 Fig. 11) will then be the resultant of the forces F^ F z , F^ etc. 
 The direction and the intensity R of this resultant may be obtained 
 from any two of the following simple relations : 
 
 Y Y/x = tan 6 Y/R = sin 6 \ ^ 
 
 If a particle 
 which is subjected 
 to the action of 
 forces is at rest, the 
 .jgi total effectiveness 
 of the forces in pro- 
 ducing motion along any particular line 
 must be zero, i.e. , the algebraic sum of the 
 components of the forces in 
 condition of each particular direction must 
 
 equilibrium . _ , , . . . _ J . 
 
 of a particle, be zero. If the position of the 
 particle be taken as the origin 
 of a system of rectangular coordinates this 
 condition is evidently fulfilled when the 
 sum of the x components is zero and the 
 sum of the y components is zero. Sym- 
 FIGURE 11 bolically the condition of equilibrium of 
 
 a particle is therefore 
 
 ^x = and % = (15) 
 
 in which 2 is the sign of summation. 
 
 I 
 
 Experiment 
 
 To verify the laws of composition and resolution of 
 force. 
 
 The apparatus consists of a horizontal force table (see Fig. 12) 
 about the circumference of which may be set at any desired angles 
 the four pulleys , , c, fZ, over which any desired 
 weights may be hung. A pin holds the intersection of 
 the cords in position at the center. When a test for equilibrium 
 is to be made the pin is removed. 
 
 Object. 
 
 Method. 
 
26 
 
 MECHANICS 
 
 1. Set two of the pul- 
 leys at the angles marked 
 
 30 and 
 
 FIGURE 12 
 
 Resultant of 
 
 two forces. an( J apply 
 
 weights of 100 gm. and 
 150 gm. respectively, re- 
 membering not to over- 
 look the weights of the 
 pans themselves. Then 
 calculate, with the aid of 
 equations 12 and 13, the 
 direction and magnitude 
 of the resultant. Set a 
 third pulley 180 from 
 the calculated angle, ap- 
 ply the calculated weight 
 
 and remove the center-pin. To show that the equilibrium is not 
 due to friction, displace the junction of the cords and tap the 
 table. The junction should return to the center. 
 
 2. Resolve each of the three forces into its x and y corn- 
 Condition of P onen ts and verify the condition of equilibrium ex- 
 
 equilibrium. pressed in (15). 
 
 3. Insert the center-pin, set three of the pulleys at any angles, 
 Resultant of and a PPty an y weights. Calculate by means of a 
 many forces, resolution of all of the forces into their x and y com- 
 ponents the direction and magnitude of 
 
 the force requisite to produce equilibrium. 
 Test as in 1. 
 
 4. Fasten one end of a spring balance 
 to the hook in the wall at o (see Fig. 13) 
 
 and the other end to the ring 
 
 at the extremity of the stick 
 force. ' mHf From the ring hang 
 weights, and adjust the other end of the 
 stick until it is perpendicular to the wall 
 (see Fig. 13). Calculate the force in the 
 spring balance and compare with the ob- 
 served value. Do not overlook the weight 
 of the stick itself. FIGURE is 
 
 Vertical 
 
SECOND LAW 
 
 27 
 
 1. Forces 
 
 1st 
 3d 
 . . Resultant 
 
 Record 
 Direction Magnitude 3. Forces 
 
 x Comp. y Comp. 
 
 2. Forces 
 1st 
 2d 
 3d 
 Sum 
 
 x Comp. y Comp. 
 
 1st 
 
 3d 
 
 3d 
 
 Sum 
 
 . \ Resultant magnitude = 
 
 .-. Resultant direction =. 
 
 4. Force in spring bal. cal'd 
 Force in spring bal. obs'd 
 
 Problems 
 
 1. Show that the acceleration of a body sliding without friction 
 down an inclined plane of length I and height li is g x j- 
 
 2. Show that the velocity acquired in sliding down the plane is 
 the same as that acquired in falling through the vertical height 
 
 , vz., v = 
 
 3. By dividing the arc ab (Fig. 14) of a vertical 
 circle into a large number of small inclined planes 
 and applying the result of Problem 2, show that a 
 body sliding without friction down the arc ab acquires 
 the same velocity as though it fell vertically from 
 b to c.. 
 
 4. A 150-lb. man standing in the middle of a tight rope 60 ft. 
 long depresses the middle 5 feet below the ends. Find the addi- 
 tional tension in the rope which is caused by his weight. 
 
 See Fig. 13 for suggestion as to method. 
 
 5. A bridge span is 5 meters high and 20 long (see Fig. 15). 
 Find the vertical pressure and 
 
 the horizontal thrust upon each 
 of the piers per ton of weight at o. 
 
 6. A bullet is fired from a 
 gun with a velocity of 200 meters 
 per second at an angle of 30 
 with the horizontal. How high 
 
 will it rise? If the gun is on the FIGURE 15 
 
28 MECHANICS 
 
 surface of the earth at what distance from the gun would the 
 bullet strike the earth if there were no air resistance? 
 
 7. If the gun mentioned in 6 were on the top of the Eiffel 
 tower how far from the base would the bullet strike the earth? 
 (Height of tower = 335 meters.) 
 
 8. Show that the time of descent down all chords which start 
 from the top of a vertical circle is the same. 
 
 

 IV 
 
 THE PRINCIPLE OF WORK 
 THE LAW OF MOMENTS 
 
 Theory 
 
 SCHOLIUM TO THIRD LAW.* The Third Law asserts that force 
 is dual in its nature, that a change in the momentum of a body 
 never takes place without an equal and opposite change 
 in the momentum of some other body or bodies. The 
 reasons for this assertion and some of its consequences 
 are discussed in Ex. VI. 
 
 An additional and wholly distinct assertion is made in the 
 scholium to the Third Law. It is that the work done by the force 
 which produces the motion is always equal to the work 
 done against the resistance to the motion. The 
 scholium reads: "If the action (activity) of an agent be 
 measured by its amount and its velocity conjointly, and 
 if similarly the reaction (counter-activity) of the resistance be 
 measured by the velocities of its several parts and their several 
 amounts conjointly, whether these arise from friction, molecular 
 force, weight, or acceleration, action and reaction in all combina- 
 tions of machines will be equal and opposite." In other words, 
 the acting force F times the velocity v of its point of application 
 equals the resisting force R times the velocity v f of its point of 
 application. Symbolically Fv = Rv' . 
 
 Now the "work" W of a force is defined as the product of the 
 force acting F and the distance s through which its point of 
 The definition application moves during its action. Thus the equation 
 ofwrii. W = Fs . (16) 
 
 is merely the definition of the quantity which is called work. 
 
 *See Encyclopedia Britannica, Article on "Mechanics." 
 
 29 
 
30 
 
 MECHANICS 
 
 Since velocity is merely rate at which space is traversed, i. e., 
 space per second, it is seen that the assertion of the scholium is 
 then simply that in all machines, during any given time, 
 the work of the acting forces is equal to the work of the 
 rces. Symbolically, 
 
 The "wor/c 
 principle. 
 
 Fs = Rs'. 
 
 (17) 
 
 This equation represents, then, the briefest possible statement of 
 the "work principle," which is the essence of the scholium to the 
 Third Law. The principle is a generalization from experiments 
 upon all sorts of machines. It is, however, to be observed, that in 
 all experiments in which there is acceleration the resistance to 
 acceleration which the body offers because of its inertia must be 
 included among the reacting forces, and must be put equal and 
 opposite to that force whose single action would produce the 
 observed acceleration, i. e., equal to ma, m being the mass of the 
 body, and a its actual acceleration. 
 
 THE PRINCIPLE OF MOMENTS. The great principle above 
 asserted can of course be tested only for particular cases. One of 
 the simplest of these is that of a rigid but weightless 
 ^pie^appued l ever > ^ ree t rotate about a fulcrum. Fj,g. 16 repre- 
 uver aiQht sen ^s the force diagram for the case of such a simple 
 lever, ^represents a force which is producing a clock- 
 wise rotation . W represents a force which is resisting this rotation. 
 
 P is the upward force exerted by 
 the fulcrum against the lever. 
 If the fulcrum be a knife-edge", 
 and the bar inflexible, there are 
 no frictional or molecular forces. 
 (Molecular force is such as is 
 called into play by bending a bar, 
 stretching a spring, etc.) If the 
 motion be made so slowly that 
 the acceleration may be consid- 
 ered zero, then the weight W is 
 the only resistance, and the work 
 principle gives Fs = Ws' (see Fig. 
 16). The force P does not appear in this equation because its 
 
 FIGURE 16 
 
THE SCHOLIUM TO NEWTON'S THIRD LAW 
 
 31 
 
 point of application does not move; hence the work of this force 
 
 I 
 , 
 I 
 
 s I 
 is zero. Now, from Geometry, = , Hence 
 
 FI= m. 
 
 (18) 
 
 Worn 
 principle 
 applied to 
 rotation of 
 body acted 
 upon by any 
 number of 
 forces in a 
 plane. 
 
 This equation is rigorously true only when the acceleration is reduced 
 to zero; that is, when the bar is at rest or is moving uniformly. It 
 represents, therefore, the condition of equilibrium of the bar. 
 
 Next consider any rigid body which is pivoted at o and acted 
 upon by the forces F, }\\, W 2 and P (see Fig. 17). ^is produc- 
 ing clock- 
 wise rota- 
 tion, W l 
 and W z are 
 resisting 
 this rota- 
 tion, and P, the pres- 
 sure of the pivot, has 
 no influence upon the 
 rotation, since its point 
 of application does not 
 move. Since the direc- 
 tion in which the force 
 F acts is not parallel 
 to the direction in 
 which its point of ap- 
 plication moves, the 
 force which is effective 
 in producing the mo- 
 tion along s, the direc- 
 tion in which this point 
 of application of F 
 must move, is not .Pbut the component of F in the direction of 
 5, viz., F cos 0. Similarly the effective resistances, i. e., the 
 resistances in the directions Si and s 2 in which the points of appli- 
 cation of W l and TF 2 must move, are not W l and TF 2 , but W l cos 15 
 and W z cos 6 Z . The work principle then asserts, provided there 
 be no acceleration, that 
 
 (F cos 0) s=( W l cos 00 s l + ( W z cos 2 ) s z . (19) 
 
 FIGURE 17 
 
32 MECHANICS 
 
 But from similar triangles, siSii s 2 = h : h\ : h 2 . Hence (19) becomes 
 
 (F cos 6) h = ( W l cos ft) h, + ( W z cos 0,) h 2 . (20) 
 But again, if perpendiculars /, ? 1? 4 be dropped from o upon 
 
 the lines of direction of the forces (see Fig.), then cos = ^-, 
 
 /L 
 
 cos ft = ^- cos 2 = Y' Hence (20) becomes 
 A! / 2 
 
 *7 = TFA 4- TF 2 / 2 . (21) 
 
 This is therefore, according to the "work" principle (the scholium 
 to the Third Law), the condition of no acceleration about o; i.e., 
 it is the condition of rest or uniform motion of rotation (equilib- 
 rium) of the body. 
 
 Now the lever arm of a force is arbitrarily denned as the per- 
 pendicular distance from the line of direction of the force 
 ?f e iSoera!rm * ^ ie ax * s f Dotation. Thus , ^ and 4, are by definition 
 the lever arms of the forces F, }}\ and JF 2 , respectively. 
 The "moment" of any force about any axis is defined as the 
 Definition Product of the force and its lever arm. Thus Fl is by 
 of moment, definition the moment of the force F about o. 
 
 Hence the condition of rotational equilibrium (21) expressed in 
 words is "The sum of the moments of the forces producing clockwise 
 
 rotation must be equal to the sum of the moments pro- 
 Condition of * 1 
 
 rotational ducing counter-clockwise rotation; or, if clockwise rota- 
 
 equilibrium. , . , , . , , , . 
 
 tion be called negative, and counter-clockwise positive, 
 the algebraic sum of the moments of all the forces acting must be 
 zero. Symbolically 
 
 3*7 = 0. (22) 
 
 Although equation (22) has been developed only with reference 
 
 to moments about the axis passing through o, nevertheless, if the 
 
 pressure P be numbered among the acting: forces the 
 
 Equation 22 . 
 
 holds for equation also holds tor moments taken about any ideal 
 
 all axes. . _ . .. . . , . 
 
 axis conceived as passing parallel to the original axis 
 through any point whatever in the plane of the forces. Thus if 
 (see Fig. 18) the equation 
 
 Fl + W^ + W& + P1 = (in which 1 = 0) 
 
 be true for a body pivoted at o, then it is also true that, if o' be any 
 point whatever in the plane, 
 
 PL = 0. 
 
THE SCHOLIUM TO NEWTON'S THIRD LAW 
 
 33 
 
 \*7 
 \ 
 
 I^CH"""! 
 ''''l' \ 
 
 l~, 
 
 \ . 
 1 
 
 I 
 
 \ 
 \ 
 \ 
 \ 
 \ 
 \ 
 \ 
 
 This conclusion be- 
 comes self-evident 
 
 from the consideration 
 
 that, if the body be in 
 
 equilibrium, P is the 
 
 e<quilibrant,of the 
 
 three forces F, TFi, and 
 
 W 2 ' y i. e., so far as 
 
 any effects produced 
 
 by jF, Wi arid TF 2 , are 
 
 concerned, these three 
 
 forces may be replaced 
 
 by one single force 
 
 equal and opposite to 
 
 P applied at o. The 
 
 moment, then, of this 
 
 resultant force about 
 
 any axis whatever must 
 
 be equal and opposite 
 
 to the moment of P; 
 
 hence the sum of the 
 
 moments of the forces 
 
 which it replaces, viz., F, 
 
 to the moment of P. 
 
 If a body of mass M be conceived as made up of equal infini- 
 tesimal particles each of mass m, then the center of mass of the 
 
 body is defined as 
 that point whose 
 distance from each 
 of three coordinate 
 planes is the mean of the dis- 
 tances of all of the particles 
 from each of these planes. 
 Thus if n represent the total 
 number of particles, and x^ x% 
 
 x n the distances of these 
 
 particles from the YZ plane, 
 
 then, from the above definition, the center of mass must lie in the 
 
 plane whose distance x c from the YZ plane is such that 
 
 FIGURE 18 
 
 and JF 2 , must be equal and opposite 
 
 FIGURE 19 
 
34 
 
 MECHANICS 
 
 05. 
 
 -HET- ( 23 ) 
 
 ?i ?i w/t . M 
 
 Similarly it must lie in planes whose respective distances y c and z t 
 from the XZ and XY planes are such that 
 
 and z a = . TT- 
 
 The one point which can lie in all three of these planes is, by defini- 
 tion, the center of mass of the body. If the three coordinate 
 planes be so chosen as to pass through this point, x c = 0, y c = 0, 
 z c = 0. Hence for this case 
 = 0, 
 
 FIGURE 20 
 
 = 0, 2mz = 0. (24) 
 
 Or, if r be a general symbol rep- 
 resenting the distance of a par- 
 ticle of the body from any plane 
 whatever, the center of mass of 
 the body is that point which is so 
 situated that a plane may be 
 passed through it in any direction 
 whatever (see Fig. 20), and yet 
 the following condition be always 
 satisfied : 
 
 0. (25) 
 
 Of course distances to one side of the plane are reckoned as posi- 
 tive, to the other side as negative. 
 
 If a body be placed in a uniform field of force, * equal forces 
 will act upon all the equal elements of mass. Hence the point 
 which satisfies the condition 5/w = must also satisfy the condition 
 3/r =0 (see Fig. 21). 
 Now the center of 
 gravity of 
 
 Definition , -, . 
 
 of center of a body IS 
 gravity. 
 
 defined as 
 
 that point which is 
 so situated that, what- 
 ever the position of 
 the body, a plane 
 
 * A uniform field is defined as one in which the force acting upon a 
 particle does not change as the particle moves about in the field. 
 
 FIGURE 21 
 
THE SCHOLIUM TO NEWTON'S THIRD LAW 
 
 35 
 
 which passes through this point and is parallel to the direction of 
 the force always satisfies the condition 
 
 S/r = 0. (26) 
 
 It is evident that if a body has any center of gravity at all this 
 point must always coincide with its center of mass; and yet a 
 body possesses a center of gravity only when it lies in a uniform 
 field of force. Its center of mass, however, is a perfectly definite 
 point which depends in no way upon the location of the body. 
 
 Now a rigid body placed in a uniform field must move without 
 
 rotation, for every element tends to move at every instant with 
 
 the same velocity. But the body would also move with- 
 
 Property of , . ... . ... . , . , 
 
 the center of out rotation if a single force (of any intensity or 
 direction) were applied to its center of inertia. For 
 let $!, < 2 etc. (see Fig. 22) represent the resistances to accelera- 
 tion due to the inertias of the masses m^ 
 ?/?2 etc. Since for the center of inertia 
 ^mr = 0, and since inertias are propor- 
 tional to masses, it follows that 3<f>r = ; 
 i. e., the moments of the resistances due 
 to inertia are always balanced about the 
 center of inertia. Hence a single force 
 applied at the center of inertia can 
 never cause rotation. It is evident, then, 
 
 that a single force applied at the center of inertia will cause exactly 
 the same motion as is produced by the uniform field. This force 
 must, of course, be equal in intensity to the sum of the forces ex- 
 erted by the field upon all of the particles of the body. Hence, so 
 far as all effects produced are concerned, it must be possible to replace 
 the action of the field by the action of such a single force. It is 
 therefore customary to treat a body which is acted upon by gravity as 
 though it were under the action of but one force, that force being ap- 
 plied at its center of gravity and being equal, in intensity to its 
 weight. Thus the center of gravity is sometimes defined as that point 
 at which the weight of the body may be conceived as concentrated. 
 
 Experiment 
 
 To find the center of gravity of a body; to verify the prin- 
 ciple of moments ; and to study the theory of weigh- 
 
 Object. J J 
 
 ing. 
 
 FIGURE 22 
 
36 
 
 MECHANICS 
 
 The apparatus consists of a meter bar furnished with three 
 knife-edges a, , and c (see Fig. 23). a and c are attached to a 
 
 metal frame through which the beam may be slipped 
 balance* 61 an( ^ ^ wn i n ^ may be clamped by means of the 
 
 screw s. c is adjustable in a vertical plane; b is fixed 
 to the bar. The beam is also provided with notches m and m' ', and 
 
 FIGURE 23 
 
 with pans P and P' which are supported from the top of the bar 
 by knife-edges n and ri '. A small hole passes vertically through 
 each of the knife-edges c, n, and n', so as to permit plumb lines to 
 be dropped from these points. 
 
 DIRECTIONS. 1. Set the knife-edge c about 1 cm. above a. 
 Support the beam from c and adjust the bar, by slipping it through 
 the frame, until it balances in a horizontal position. 
 Then hang P and P' from the bar, well out toward the 
 ends, and slide one of them along until the bar is again 
 horizontal. Eead off the lever arms upon the graduated bar and 
 write down the equation of moments. Now add 100 gm. to 
 P, and, without altering the position of P', slip P toward the ful- 
 crum until a balance is again found. Write again the equation of 
 moments. The solution of the two equations thus obtained will 
 give the weights P and P '. Check the results by weighing on the 
 trip scales. 
 
 To find 
 weights of 
 pans. 
 
THE SCHOLIUM TO 
 
 THIRD LAW 
 
 37 
 
 2. Remove the pans, support the beam this time from # , and 
 adjust the movable knife-edge and the bar until the whole system 
 
 is in neutral equilibrium about a\ i. e., until the sys- 
 
 Toflndcen- *. 
 
 terof gravity tern shows no tendency to move out of any position in 
 
 of system. . . . . . * . . 
 
 which it is placed. This amounts to bringing the cen- 
 ter of gravity of the system into coincidence with a. A small bit 
 of soft wax, which may be attached to the bar at any desired point, 
 greatly facilitates this adjustment. 
 
 3. (a) Having brought the center of gravity of the system into 
 coincidence with , 
 
 support the beam from 
 c, hang the pans from 
 
 FIGURE 24 
 
 uf beam. a n ^ place 
 
 such weights upon 
 them that the beam 
 assumes a position in- 
 clined about 45 to the 
 horizontal (Fig. 24). 
 The equation of mo- 
 ments now involves 
 the three forces TPi, 
 JF 2 , and the weight of 
 the beam PF 3 , and the 
 three corresponding 
 
 lever arms ? x , 7 8 , 1 3 . These latter are to be measured by bringing 
 a half meter rule, which rests horizontally upon some firm sup- 
 port, up to the plane of the plumb lines which hang from the 
 knife-edges. The equation of moments contains but one unknown 
 
 quantity, viz., the weight of the 
 beam. Solve for this unknown 
 quantity. 
 
 (b) Check the weight thus ob- 
 tained by supporting the bar upon 
 the knife-edge b and producing 
 a balance by means of a known 
 weight W* (see Fig. 25). 
 
 * The weights of the pans and plumb bobs should be included in W,, 
 W., and W : as they occur. 
 
 FIGURE 25 
 
38 MECHANICS 
 
 (c) As a final check weigh the beam upon the trip scales. 
 
 4. (a) When the knife-edges n, ri ', and c are not in the same 
 straight line. The " sensitiveness" of a balance is defined as the 
 
 displacement produced when some arbitrarily chosen 
 
 Sensitiveness %. . , . , -, , T . J 
 
 for different small weight is added to one pan. it should decrease 
 as the load upon the pans increases, provided the sup- 
 porting knife-edge c is above the line connecting the pan knife- 
 edges n and n' (see Problems 2 and 3, pp. 39, 40). Set the knife-edge 
 c three or four centimeters above the line connecting n and n'. 
 Support the system from c. Hang the pans from the beam, but 
 not in the notches (a little friction at the knife-edges n and n' 
 vitiates completely the results of this experiment), and bring the 
 beam into the horizontal position. Set up a vertical meter stick 
 behind one end of the beam and read off upon it the exact deflec- 
 tion produced by adding very carefully one gram to one of the 
 pans. Then add 300 gm. to each pan. If the equilibrium is 
 destroyed, reestablish it by sliding along the pan to which the 
 small weight was not added, and again take the sensitiveness. 
 Repeat both observations to make sure that the results can be 
 duplicated. 
 
 (b) When the knife-edges n, n', and c are in the same straight 
 line. In this case the sensitiveness should be independent of the 
 load (see Problem 2, p. 39). Lower the knife edge c until it is in 
 line with n and n'. Observe as in 4 (a). (The bending of the 
 beam may still .cause** slight dependence of sensitiveness upon the 
 load.) 
 
 5. Hang unequal pans P and P' in the notches n and n' ', and 
 slip the bar through the frame until a balance is obtained. The 
 
 arms of the balance are now unequal, the pans are of 
 arrrecfweiah- une( l ua l weight, and the center of gravity of the beam 
 with false j s no t beneath the point of support. Nevertheless, if 
 an unknown weight w be placed in the pan P and the 
 
 li I* 
 
THE SCHOLIUM TO NEWTON'S THIRD LAW 39 
 
 beam brought back to its original position by the addition of a known 
 weight a to the other pan, the following equation must hold: 
 wk = al z (see Fig. 26). Now place win P' and balance by means 
 of a known weight b placed in P. The equation which now holds 
 I is wl z = bli. The solution of these two equations for w gives 
 w = */ab. If a and ~b have nearly the same value, it is sufficiently 
 
 correct to write w= Hence all the errors of a balance are 
 eliminated by a double weighing. 
 
 Record 
 
 1. Reading at a in condition of balance 
 
 Read'gforP = forP' = forP+100 = .*. P= P' = 
 
 By trip scales P= P' = % error in Ps = in P's = 
 
 (b) W = lever arm of W = - - of W 3 = .-. W 3 = 
 
 (c) By trip scales W 3 = - % error in (a) = in (b) = 
 4. No load, (a) Reading before adding 1 gm. after dif. 
 
 With load, (a) 
 
 " " (b) " " " " " " 
 
 5. a= b= .'. w= . By scales w % error = 
 
 Problems 
 
 1. Explain why a balance beam returns to a horizontal position 
 when displaced therefrom. 
 
 2. Fig. 27 represents the case in which the three knife-edges 
 ft, ri, and c are in the same straight line. Show from this figure 
 
 that if in the hori- 
 zontal position the 
 
 1 2 "-._ N c ; It J t center of gravity g of 
 
 the beam be directly 
 beneath c, i. e.> if Ph 
 = P7 2 , then in the dis- 
 placed position due to 
 the addition of a small 
 
 w *^, . weight p to P, the 
 
 FIGURE 27 new equation of equi- 
 
40 
 
 MECHANICS 
 
 FIGURE 8 
 
 librium, viz/, (P + p}l\ = P'l f z + Wl^ reduces to pl\ = W1 3 . Hence 
 show that the sensitiveness, which is the displacement of the 
 beam produced by p (a quantity of which Z 3 may be taken as the 
 measure), is independent of the load, i. e., of P and P ' . 
 
 3. Show from 
 Fig. 28 that when 
 n, ri ', and c are not 
 in the same straight 
 line, then, after dis- 
 placement, Pl\ is 
 not equal to P'l\, 
 and hence that the 
 sensitiveness, i. e., 
 ? 3 , depends upon the 
 load, decreasing 
 with the load if c be above the line nri ', increasing with the load if 
 c be below the line nri . 
 
 4. Does the period of vibration of a balance vary with the load? 
 If so, how and why? 
 
 5. With a given load how should the period be aifected by a 
 diminution in the distance between the knife-edge c and the center 
 of gravity g? 
 
 6. A circular ring weighing 5 Ib. rests hor- 
 izontally upon three points of support 120 apart. 
 What is the least downward force, applied to the 
 ring in a direction perpendicular to its plane, 
 which will cause it to leave one of the points of 
 support? v oeeFig. 29.) 
 
 7. A uniform bar weighing 10 Ib. is 
 pivoted at one end (see Fig. 30). A hori- 
 zontal force of 5 Ib. is applied to the free 
 end. When a condition of equilibrium is 
 reached, what angle does the bar make with 
 the horizontal? 
 
 8. The axle of a wheel carries a load of . 500 
 kilos. What horizontal force must be applied to 
 the axle to raise the wheel" over an obstacle 12 cm. 
 high, the radius of the wheel being 50 cm.? 
 FIGURE 31 (See Fig. 31.) 
 
 FIGURE 29 
 
 FIGURE 
 
THE SCHOLIUM TO NEWTON 5 S THIRD LAW 41 
 
 9. A uniform board 3 feet square and weighing 25 
 Ib. rests on a block at a (see Fig. 32), and is kept 
 from falling by a horizontal force at B. Find the 
 force at B and the vertical and horizontal pressures 
 
 upon the block at a. FIGURE ca 
 
 10. Two men carry between them a weight of 50 
 
 kilos, supported upon a uniform bar weighing 30 kilos. Where 
 must the load be placed in order that one man may carry twice as 
 much as the other? 
 
 Choose some convenient point as axis and apply the equation 2FI = 0. 
 
 11. A man wishes to overturn a cubical block which weighs 
 100 kilos and has a 5 ft. edge. In what direction and with what 
 force must he push in order that he may accomplish his object 
 most easily? After the block has once started will the required 
 force increase or decrease? Why? 
 
V 
 
 ENERGY AND EFFICIENCY 
 
 Theory 
 
 In the preceding experiment work was defined as the prod- 
 uct of the force acting and the distance through which it 
 moves the point to which it is applied. Symbolically, 
 W=Fs. 
 
 The energy of an agent is defined as its capacity for doing 
 work. Energy and work must of course be measured in the 
 same units ; yet it is obvious that they are not synonymous terms, 
 for a body may possess energy and yet never apply it to the pro- 
 duction of work. Work is done only when energy is expended. 
 
 Since work is a product of force and space, the work unit must of 
 course involve a force unit and a space unit. The absolute centi- 
 meter-gram- second (c. g. s.) unit of energy or of work is 
 energy and the dyne-centimeter, also called the erg. An erg of 
 work is done when a dyne of force moves the point upon 
 which it acts through a distance of one centimeter. Other work 
 units are the gram-centimeter, kilogram-meter, foot-pound, etc., 
 the definitions of which are evident from their names. Since a 
 gram of force is 980 dynes, it is evident that a gram-centimeter is 
 equal to 980 ergs. 
 
 It was shown in Ex. IV that Newton's interpretation of the 
 
 Third Law as given in the scholium is equivalent to the statement 
 
 that in all mechanical operations the energy expended 
 
 ergy. Resist- by the agent is equivalent to the work done against the 
 
 ance to motion J & * . 
 
 is inertia four resistances, inction, molecular force, gravity, 
 and inertia. If the first three resistances are absent 
 the only effect of the force is to impart velocity to the body. By 
 virtue of this velocity the body itself becomes possessed of the 
 capacity for doing work, for it can now move itself against a fric- 
 
 * Electrical and magnetic forces are here classed as molecular. 
 
 42 
 
THE SCHOLIUM TO NEWTON 'S THIRD LAW 43 
 
 tional resistance, compress a spring, raise itself against gravity, or 
 by impact overcome the inertia of some other body. This energy 
 which a body possesses because of the motion which has been com- 
 municated to it is called "kinetic energy." 
 
 That the kinetic energy imparted to a body by the action of a 
 
 force is exactly equal to the work done upon it, and 
 
 PJjCflSfai that this is equal, in the absolute system of units, to 
 
 one-half the mass of the body into the square of its 
 
 velocity, may be shown as follows: 
 
 Let a body of mass m acquire a velocity v under the action of 
 a constant force ^acting for a time t, and in that time moving the 
 body a distance s. The work done upon the body is then by defi- 
 nition Fs. Now let the body be brought to rest by being subjected 
 to the action of an oppositely directed constant force F' which 
 requires a time t' and a space s' in order to destroy the velocity v. 
 The ''kinetic energy" of the body, i. e., the work which it is 
 capable of doing because of its velocity, is -by definition F's'. But 
 since every force is measured by the rate of change of momentum 
 which it produces, the first force F is measured by the rate at 
 which it imparts momentum, and the second force F' by the rate 
 at which it destroys momentum, i. e., 
 
 F '= ma and F' = ma' . (27) 
 
 Now, since .Fand F' are both constant forces, i. e., forces which 
 produce uniformly accelerated motion, by Ex. I 
 
 s = %at z and s' = a't'*. (28) 
 
 Therefore from (27) and (28) 
 
 Fs = $ma*t* and F's' = %ma'*t'*. (29) 
 
 But by hypothesis the velocity imparted by F and the velocity 
 destroyed by F' are one and the same velocity. Hence 
 
 v = at and v = a't'. (30) 
 
 It follows from (29) and (30) that 
 
 Fs = iwv 8 and F's' = %mv*. 
 Hence 
 
 Fs = F's' = %mv\ (31) 
 
 Q. E. D. 
 
 If F and F' are variable forces it is only necessary to conceive 
 them as made up each of the same number of very small elements, 
 
44 MECHANICS 
 
 each element being a constant force, v will then be the velocity 
 gained under the action of one of these constant elements of .Fand 
 destroyed under the action of the corresponding element of F'. 
 Hence the above demonstration is perfectly general. 
 
 Next suppose that the resistance which the working force 
 experiences is gravity alone, as when a body is taken from position 
 Potential a ( see ^ ] 'S' ^)> and placed upon a hook in position b. 
 sManceis 6 ' ^ ne wor ^ done upon the body is the force of the pull 
 molecular times the distance ab. The pull is a variable force, 
 force alone, being a little greater than the weight of the body when 
 the motion is starting, and a little less when it is stopping. The 
 kinetic energy which is imparted to the body during the first 
 instants, when the velocity is being acquired, is all 
 given back during the last instants when the velocity 
 is being lost. When the operation is taken as a 
 whole no velocity is imparted; hence the only re- 
 sistance is gravity. Then by Newton's interpre- 
 tation of the Third Law the work which has been 
 done upon the body is equal to the work done against 
 gravity, viz., the force of gravity upon the body 
 times the distance ab. But, in this case, as in the 
 FIGURE 33 preceding in which the resistance was inertia alone, 
 the work may all be regained without the expendi- 
 ture of any more work on the part of the agent. For, in returning 
 to position a, the body is capable of lifting through the height ab 
 any other body, e.g. body 2 (see Fig. 33), whose weight does not 
 exceed its own. The ability to do work which the body possesses 
 by virtue of its position at b is called its potential energy. 
 
 If the work be done against molecular force alone, as when a 
 perfectly elastic spring is compressed, the work done can all be 
 regained by releasing the spring, which, when compressed, is pos- 
 sessed of potential energy. Potential energy is in general any 
 energy which is put into a system by a change in the position of its 
 parts. Thus when the resistance is gravity or molecular force 
 alone, an amount of potential energy equal to the work done is 
 stored up; when the resistance is inertia alone, kinetic energy 
 equal in amount to the work done appears. 
 
 But when the resistance is friction, the work done by the 
 agent can not be regained. In Newton's day it was supposed to 
 
THE SCHOLIUM TO NEWTON'S THIRD LAW 45 
 
 have disappeared altogether; but about the middle of the nine- 
 
 teenth century it was proved by Joule that for every 
 
 Resistance fe erg of work which so disappears there always appears a 
 
 friction alone. 6 _ . . ., . 
 
 perfectly definite quantity of heat; hence it is now 
 customary to. say that the work expended has been transformed 
 into heat energy. 
 
 The experiments of Joule, whereby the principle of the equiv- 
 alence of heat and work was established, consisted in transforming 
 
 heat into work in as large a variety of ways as possible, 
 principle of e ' " ^ means of the friction of different sorts of sub- 
 
 S ^ ances 5 D J percussion, by compression, by the generation 
 of electric currents the energy of which was finally dis- 
 sipated in heat, etc. When the experiments were so arranged that 
 the heat generated was taken up by a given quantity of water, it 
 was observed that a given expenditure of mechanical energy always 
 produced the same rise of temperature in the water. These experi- 
 ments had much to do with securing general acceptance for the 
 principle of conservation of energy, a principle blindly grasped at 
 by philosophers from the earliest times; first stated, in the 
 scholium to the Third Law, by Newton in 1687, but with respect to 
 mechanical operations only ; first asserted as a principle of universal 
 applicability by the German physician J. R. Mayer in 1845; first 
 generally accepted and universally recognized as the most funda- 
 mental and most fruitful principle, in all physical science, after 
 Joule, by his series of experiments extending from 1843 to 1878, 
 had demonstrated the equivalence of heat and ivork. The present 
 accepted value of the mechanical equivalent of heat, i. e., the 
 number of ergs of work required to raise 1 gm. of water at 15 
 C. through 1 C. is 
 
 4.19xl0 7 . 
 
 But the principle of conservation of energy is more than the 
 assertion of the equivalence of heat and work. It may be stated 
 thus: Every physical (or chemical) change of condition 
 rf uas a fixed mechanical equivalent, i. e., can be equated, 
 un( ^ er a ^ circumstances, to one and the same amount of 
 mechanical work. In other words, whenever a change 
 takes place in the condition of a body because of the expenditure 
 upon it of mechanical energy (kinetic or potential), the change is 
 equivalent to the work done, in the sense that if the body can be 
 
46 MECHANICS 
 
 brought back to its original condition the whole of the energy 
 expended may be regained either in the form of work or the equiv- 
 alent heat. 
 
 Thus, applying the principle to a mechanical problem, it asserts 
 illustrations at once that the kinetic energy of a moving body is 
 of principle. e q ua j ^ foe work done in setting it in motion. 
 
 Applying it to a chemical problem it asserts, since the burning 
 of 1 gram of carbon, i. e. , the formation of carbon dioxide 'from 
 carbon and oxygen, generates enough heat to raise 97,000 gm. of 
 water through 1 C., that, if it were possible to directly pull apart 
 the united carbon and oxygen atoms, 97,000 x 4.19 x 10 7 ergs of 
 work would be required to secure 1 gram of carbon from this 
 compound. Applied to an electrical problem the principle asserts 
 that if it requires 1000 kilogram-meters of work per second to 
 drive a dynamo, then the work which this dynamo does per second 
 in the motors which it runs, plus the mechanical equivalent of the 
 heat developed in all of the machines, and in all the connecting 
 wires must be exactly equal to 1000 kilogram-meters. 
 
 The principle is perhaps the most important generalization 
 
 which has ever been made. It is merely an extension of the 
 
 scholium to the Third Law, and, like it, rests upon uni- 
 
 Basisfor . ' 
 
 assertion of versal experience rather than upon any one particular 
 experiment. The only kind of direct test of which it 
 is capable is of the kind which Joule made, and consists in 
 mechanically producing a given change of condition (e. g., a given 
 rise in the temperature of water) in as large a variety of ways as 
 possible, and observing whether the work required always cornea 
 out the same. Of course, such an experiment tests the law only 
 for one particular -kind of physical change. 
 
 In all mechanical devices the work which the machine accom- 
 plishes is inevitably less than the work which is put into it, for 
 the reason that there is always some friction and hence 
 a part of the applied work disappears as heat. Effi- 
 ciency is defined as the ratio of the work done by the machine in 
 any given time to the energy expended upon it in the same time. 
 
 Experiment 
 
 To determine the efficiency cufve (1) of a system of 
 pulleys, (2) of a water motor. 
 
THE SCHOLIUM TO NEWTON'S THIRD LAW 47 
 
 1. First weigh the movable block of pulleys and each of 
 
 the pans (see Fig 34). Then give to JF, which includes /- 
 the weight of pan and movable block, succes- 
 sive values of about 300, 600, 1000, 1400, 
 1800 grams, and find the corresponding values which 
 must be given to P, including pan, in order to produce 
 unaccelerated downward motion of P. Calculate the 
 efficiency corresponding to each case, and plot a curve 
 with efficiencies as ordinates and loads as abscissae. 
 Since the efficiency is the ratio of the ivork of the force 
 P and the work of the force TF, a determination of 
 efficiency must involve a determination of the ratio of 
 the distances through which the points of application of P 
 and W move. This can be obtained without a measure- 
 ment, as a little consideration will show, from the number yy 
 of strands between which the weight of W is divided. FIGURE 34 
 
 2. In order to determine the energy expended upon a 
 
 water motor in any time, it is necessary to know, first, the pres- 
 sure p under which the water issues from the orifice o 
 expended (see Fig. 35) ; second, the volume of water V which 
 issues during this time. The energy expended upon 
 the motor is then p V. This will be evident from the following 
 considerations: 
 
 Proof No. 1. Suppose a column of water of cross-section o 
 be issuing from the orifice with a velocity v. Since "pressure" 
 means force per unit area, the force driving the water forward 
 is po. This force carries the water forward a distance v in one 
 second ; hence the work done per second by the force is pov, and if 
 the experiment last t seconds the total work done is povt. But 
 ovt = V. Therefore the total energy expended upon the machine 
 during the experiment ispV. Q. E. D. 
 
 Proof No. 2. Suppose the pressure p to be due to a column of 
 liquid of height 7^, and suppose a mass of M grams of liquid to 
 have issued from the orifice o. In order to restore the condi- 
 tions existing before the mass M had passed through the orifice, 
 it is necessary to raisf^f throug^Jie height li and to return it to 
 the liquid in the reservoir, i. e.^Rs necessary to do a quantity 
 of work Mil. ^jThis therefore represents the energy which has 
 been expended in the passage of the mass M from the orifice. But 
 
48 
 
 MECHANICS 
 
 if d be the density of the liquid, p = lid. 
 
 ^- But 
 
 M 
 
 d 
 
 ^-= F. 
 
 Mh=pV. 
 
 d 
 Q. E. D. 
 
 In this experiment friction is applied to the axle by turning 
 down the thumbscrew s (see Fig. 35) of a Prony brake until the 
 
 tension in the spring S is just balanced, i. e., until the 
 fcy motw ne lever arm cn C = r ] f tne brake rests midway between 
 
 the stops ^, f. It is generally impossible to eliminate 
 completely oscillation of the lever arm, but its mean position can 
 be estimated with sufficient accuracy. It is evident that the ten- 
 
 FIGURE 35 
 
 sion F in the spring 8 represents the constant pull which the 
 machine exerts at a distance cn from the axle. If, therefore, cn 
 were the radius of a pulley upon which a cord were being wound 
 up, the constant pull which the cord would exert upon any load 
 which it were moving would be equal to F. One revolution of the 
 water wheel would cause this load to move a distance %-n-r. It is 
 evident, then, that the work accomplished by the machine in N 
 
THE SCHOLIUM TO NEWTON 'S THIRD LAW 49 
 
 revolutions is %-n-rNF. In order to determine the value of 
 lower end of the spring is detached from the lever arm, after the 
 run has been made, and known weights added to the spring until 
 it is stretched to the length which it had during the run. 
 
 The motor is attached to the regular water supply of the room, 
 but irregularities in the pressure of the latter are equalized by 
 constant introducing before the motor a large air-tight tank T 
 pressure (200 liters), the entrance and exit both being at the 
 
 bottom. It is thus the air in the upper part of the 
 tank, compressed by the water-works pressure, which is the imme- 
 diate source of the pressure applied to the machine. 
 
 The pressure under which the water issues from the orifice o is 
 obtained from a reading of the mercury manometer AB. If there 
 
 were no water in either arm, this pressure, measured in 
 measurement cen ti me ters of mercury,would evidently be the difference 
 
 between the mercury levels in the two arms of the 
 manometer. This could be reduced to grams by multiplying 
 by the density of mercury. However, since the arm A of the 
 gauge fills with water, and since it is the pressure at the level of 
 o which is sought, the pressure indicated by the mercury height 
 must be diminished by that due to a water column of height equal 
 to the difference between the level of o and the mercury level in 
 A . In solving for efficiency, it is of course essential that p be 
 expressed in the same units as F. 
 
 Starting with cock K' closed, partially open cock K and allow 
 a considerable pressure to be produced in tank T. Then slowly 
 
 open K' till the difference in the levels of the mercury 
 
 in the pressure gauge is, e. g., 100 cm. Then, while one 
 observer holds the gauge-reading constant by continually adjusting 
 K, let another adjust the screw s till the lever arm en maintains in 
 the mean a horizontal position. These adjustments made, at an 
 accurately observed time deflect the discharge-water, by means of 
 the flexible rubber tube <?, from tank E into the empty tank /), 
 and al the same time let a third observer begin to count the revo- 
 lutions of the speed counter (7, which is attached to the axle by 
 means of a flexible rubber connection. When tank D is about 
 two-thirds full, stop the run at an accurately observed time by 
 closing cock K' ' . Determine V by weighing the water in tank D 
 upon the platform scales. Determine Floy stretching the spring 
 
50 MECHANICS 
 
 S by means of known weights to the length which it had during 
 the run. If both levels of the mercury in the gauge were not read 
 during the experiment, reestablish the pressure and read the lower 
 level. Measure the lever arm en [= r], and compute the efficiency. 
 Next vary the tension in the spring S by raising or lowering the 
 support to which the upper end of this spring is attached, and 
 again determine the efficiency for this new "load," the pressure 
 being kept the same as before. In this way make five different 
 runs with loads which vary between zero and the maximum which 
 the machine is able to carry without stopping altogether. The 
 speed will then vary between "racing" speed and the slowest pos- 
 sible speed. Plot a curve with speeds as abscissae and efficiencies 
 as ordinates, and thus determine the speed for which, for the 
 given pressure, the machine is most efficient. If the time of each 
 run be made of exactly the same length the speeds will of course 
 be proportional to the total numbers of revolutions N. In any 
 case, if T represent the number of minutes of duration of the 
 
 experiment and n the speed, n = In this experiment it is 
 
 recommended that one observer regulate and read pressure, that 
 another attend entirely to the adjustment of the Prony brake 
 (it may need to be continually regulated throughout the run), and 
 that a third observe the time and the number of revolutions of 
 the speed-counter. 
 
 Record 
 
 1. Wt. of movable block = of W pan = of P pan = - 
 
 1st value of W = of P = . . efficiency = 
 
 3d value of W = of P = ,.-. efficiency = 
 
 3d value of W = of P = .: efficiency = 
 
 4th value of W = ofP= . . efficiency = 
 
 2. Mean Hg read 'g in B in A Water oor'n. cm. .-.p gm. 
 
 Wt. of D of D+ water- - \V F T N .-. eff. - 
 
 Second run " " " - - " '* " " 
 
 Third run " " " - -" " "- 
 
 Fourthrun " " " - -" " "- 
 
 Fifth run " " " " " " " 
 
THE SCHOLIUM TO NEWTON *S THIRD LAW 51 
 
 Problems 
 
 1. If the friction between two surfaces be proportional tx the 
 pressure existing between them, and if, in the experiment with the 
 pulleys, the friction due to the bending of the cord in passing over 
 the pulley be a wholly negligible quantity, how ought the efficiency 
 of the system of pulleys to vary with the load? 
 
 2. Solve, Problems 2 and 3, page 27, from a consideration 
 of the potential and kinetic energies of the body at the top and 
 bottom of the plane and arc. 
 
 3. Taking the distance of the sun as 90,000,000 miles, the 
 density of the earth as 5.50, its radius as 4000 miles, find the 
 kinetic energy in kgm. -(meters) 2 , which the earth possesses by 
 virtue of its orbital motion. Find how many kilograms of water 
 it could raise from C. to 100 C. if its energy were suddenly to 
 be transformed into heat by a collision. Take 1 kilometer as 
 equal to .62 miles. 
 
 Carry to three significant figures only and use throughout the expo- 
 nential notation, thus 7.14 X 10 21 . 
 
 4. What mean force has been applied to a kilogram weight if it 
 has been raised 5 meters and at the same time given an upward 
 velocity of 2 meters per second? 
 
 5. A bullot enters a target with a velocity of 120 meters per 
 second and penetrates 10 cm. What velocity should it have to 
 penetrate 18 cm.? 
 
 6. A Joule is 10 7 ergs. A Watt is an "activity" of 1 Joule 
 per second. A horse-power is an "activity" of 746 Watts. Find 
 the horse-power of a steam pump which lifts 100,000 liters of 
 water per hour from a well 30 meters deep. 
 
 7. How many Joules of work are expended by a 200-lb. man in 
 climbing stairs 60 ft. high? How much heat would he develop if 
 he were to fall to the ground, i. e., how many grams of water 
 would the heat generated by the fall raise through 1 C. (1 inch 
 = 2.54 cm., 1 kilo. = 2.2046 Ib.) 
 
VI 
 THE LAWS OF IMPACT 
 
 Theory 
 
 If two bodies be connected by a stretched or a compressed 
 
 spring, then if the spring be released, common observation teaches 
 
 that both bodies are set into motion. According to the Second 
 
 Law the forces which produce these two motions are 
 
 Meaning and i 
 
 proof of measured, by the rates 01 change 01 momentum of the 
 
 Third Law. ,- v. ,-, m i . ^ T 
 
 two bodies. Hence the Third Law in asserting the 
 equality of the two forces asserts the equality at every instant of 
 these two rates of change of momentum. Experiment alone can 
 furnish proof of the correctness of the assertion. But the Third 
 Law is much more than the assertion of this equality in this par- 
 ticular case. It asserts that all forces are essentially like those 
 existing in stretched or compressed springs; in short, that all 
 motions in the universe arise from stresses, and that whenever 
 one body is set in motion some other body always receives the same 
 quantity of motion in the opposite direction. This assertion is 
 a generalization from a large number of experiments upon forces 
 whose effects can be observed and measured. Astronomical obser- 
 vations attest the correctness of the law for gravitational forces. 
 Laboratory experiments must be relied upon to furnish evidence 
 regarding forces arising from impact, magnetization, electrification, 
 etc. In this experiment the law is put to the test for certain cases 
 of impact. 
 
 When any moving mass m^ (see Fig. 36) makes impact with 
 any stationary mass m z , it is observed that 
 the latter is set into motion 
 
 The conser- 
 
 vationof while the former loses part of 
 
 momentum. . , . , . 
 
 its velocity. Consider any in- 
 stant of the impact and let 2 be the accel- 
 FIGURE 33 eration which the force /* (see Fig.) is 
 
 52 
 
NEWTON'S THIRD LAW 53 
 
 imparting at that instant to m 2 , and # t the negative acceleration 
 which the force fi is imparting to m^ The Second Law gives 
 
 /! = w^, and / 2 = 
 The assertion of the Third Law, then, is 
 
 /i =/*, or Wii 
 But if the rates of change of momentum ot m^ and m z are at every 
 instant equal, it follows that the total momentum imparted to m^ 
 during the whole time of impact is equal to the total (opposite) 
 momentum imparted to w 2 , or symbolically, if Wj represent the 
 velocity of m^ before impact, v l its velocity after impact, and v 2 
 the velocity imparted to w 2 , 
 
 WI(MI-VI) =m 2 v 2 ; 
 or by transposition 
 
 m^ = miVi + m 2 v 2 i (32) 
 
 an equation which asserts that the total momentum before impact is 
 equal to the total momentum after impact. 
 
 If m 2 has an initial velocity u z -> precisely the same line of 
 reasoning gives 
 
 m^Ui + m 2 u 2 = m v Vi + m 2 v 2 . (33) 
 
 Thus the Third Law asserts that momentum is conserved in all 
 impacts, be it betiveen elastic or inelastic bodies. 
 
 While thus the Third Law asserts that there is never any loss 
 
 of momentum in an impact, it does not assert that there is 
 
 no loss in kinetic energy. The mechanical energy is 
 
 Uons after always less after impact than before, and in the case of 
 
 an inelastic impact between two bodies one of which is 
 
 at rest (u 2 = 0) the loss can be theoretically calculated from a knowl- 
 
 edge of the masses alone, Thus, for this case, since in inelastic 
 
 impact the bodies remain together after the collision, Vi = v z , and 
 
 the general equation (33) becomes 
 
 m^Ui = (raj + m 2 ) v z . (34) 
 
 Substituting the value of v. z obtained from (34) in the expression 
 which represents the fractional loss / of kinetic energy, viz., 
 _ K E before K E after _ \m^.if | (m t 4- m z )v 2 z , 
 KE before 
 
 there results, after reduction, the simple formula, 
 
 (36) 
 
MECHANICS 
 
 The Third Law therefore leads to the interesting conclusion that 
 the per cent loss in kinetic energy, when one inelastic body 
 impinges upon another which is at rest, is altogether independent 
 of the velocity of the impinging body. 
 
 Experiment 
 
 To test the equality of momenta before and after 
 impact for the case of inelastic impact. 
 It is not easy to measure directly the velocities immediately 
 before and after an impact. But if the velocity of the impinging 
 body be acquired by a fall from a known height 7^, and 
 
 Method. . i i_ j j -L i -j. 
 
 if the struck body expend its velocity in rising to a 
 known height 7^ 2 , the required velocities u v and v z can be easily 
 calculated from the measured heights 7^ and 
 
 ~ t 
 
 Object. 
 
 In order to realize these condi- 
 tions a lead ball m^ rendered per- 
 fectly inelastic by means of a piece 
 of soft wax, is allowed to fall down 
 a circular arc from the position d 
 to the position d' (see Fig. 37). In 
 so doing it acquires the same veloc- 
 ity as though it had fallen through 
 the vertical distance db (see Ex. 
 Ill, Problem 3, and Ex. V, Prob- 
 lem 2). After impact the cylin- 
 der and ball move together up the 
 
 FIGURE 37a 
 
XEWTOX'S THIRD LAW 
 
 55 
 
 Let o 
 together. 
 
 Calculation 
 of 7t 2 from 
 index 
 readings. 
 
 e. , the reading cor- 
 
 arc M, which is graduated in degrees, to a point which is regis- 
 tered by the light aluminum index 1. The vertical height cor- 
 responding to this movement up the arc is calculated from the 
 index readings as follows : 
 
 be the index reading when ball and cylinder hang 
 The center of gravity of the system formed by the ball 
 and cylinder together must then be at some point c 
 (see ideal diagram, Fig. 38) which is directly under- 
 neath the point of support A. Let 0' be the index 
 reading when the cylinder hangs alone, i. 
 responding to the position 
 of the system at the instant 
 of impact. At this instant 
 the center of gravity of the 
 system is at some point c', 
 which is to the right of c a 
 distance such that c c' = o 
 o'. Now if the force of 
 the blow were zero, as soon 
 as the ball and cylinder were 
 joined, the force of gravity^ 
 alone would cause the sys- c/ 
 tern to move forward, for c, 
 not c'', is the natural posi- 
 tion of its center of gravity. The velocity acquired in falling from 
 c down to c would of course carry the center of gravity of the 
 system up again to some point c" such that c - c = c" - c. It is 
 evident, then, that only the movement over the arc c"c'" [= o"k} 
 can be attributed to the velocity imparted by the blow. Hence li z 
 is the vertical height corresponding to this portion only of the arc, 
 i. e. (see Fig. 38): 
 
 li^c"s. (37) 
 
 and uv = uA vA. (38) 
 
 But c"s = uv, 
 
 uA , vA ,. 
 
 is ow - = cos a, and = cos 0. 
 
 r r 
 
 Hence uA vA =r (cos a - cos 0) 
 
 in which is the angle whose arc is co" [= o/ 
 
 (39) 
 
 (40) 
 and a is the angle 
 
 whose arc is c"c [= cc = oo']. Since the arc is graduated in degrees, 
 
56 
 
 MECHANICS 
 
 a and 6 are obtained at once from the readings. Finally, then, 
 from (37), (38), and (40), 
 
 Ji 2 = r (cos a cos 0). (41) 
 
 The radius r should be measured from the point of support to 
 the common center of gravity. Cos a and cos 6 may be obtained 
 from any trigonometrical table. 
 
 The momentum equation which it is sought to verify is m l u l 
 = (m j -fw 2 )v 2 . But, since u^ = V^gli^ and v* 
 
 tionsto be this equation may be written in the form 
 
 verified. _ _ 
 
 mi v% = (mi + m 2 ) \/// 8 . (42) 
 
 By a similar substitution the expression for the loss of kinetic 
 
 energy (35) may be written 
 
 ' 
 
 
 Make several preliminary trials, adjusting, if need be, the posi- 
 tions of the clamp R (Fig. 37), and the suspensions of the ball and 
 cylinder by means of the thumbscrews t (Fig. 37a), 
 until, when the ball is released by burning the thread 
 /, the cylinder moves smoothly up the arc without wabbling. 
 This done, measure the height from the top of the table to the top 
 of the ball by means of a meter stick furnished with a sliding clip 
 j (Fig. 37). Then move the index up the arc to nearly the point 
 which will be reached by the cylinder. After the impact take 
 very carefully the index readings at &, o, and 0', and also the dis- 
 tance from the table or floor to the top of the ball when the latter 
 is held at the point of impact. (It must of course be provided 
 that the reference plane from which the heights ab and ad are 
 measured is accurately horizontal.) 
 
 Obtain at least two sets of readings. Weigh the ball and 
 cylinder upon the trip scales. 
 
 Record 
 
 1st Trial 
 
 ad (Fig. 37) ab - - . . h l - 
 
 Read'g at k r at o - - at o> 
 
 Mom. bef. aft. % dif . - 
 
 I from (43) from (36) 
 
 2d Trial 
 
 ad =- ab = 
 
 atk = at o = 
 
 r== = - 
 
 Mom. bef. aft. 
 
 I from (43) = from 
 
 r *i 
 
 at o' 
 
NEWTON'S THIRD LAW 57 
 
 Problems 
 
 1. What relation exists between m l and m z when -J of the K E 
 is lost in heat? When ? When f? 
 
 2. The moon moves toward the earth a distance of 15 ft. per 
 minute. Find how far the earth moves toward the moon in the 
 same time. (Ratio of masses, 1 to 81.4.) 
 
 3. A rifle bullet weighing 20 gm. was fired into a ballistic pen- 
 dulum weighing 4 kilos. The latter moved up an arc a distance 
 
 corresponding to a vertical rise of 5 cm. Find the velocity of the 
 bullet. 
 
 4. Is it true that, at the start, the wagon pulls back with the 
 same force with which the horse pulls forward? If so, how is any 
 motion produced? If not, reconcile your answer with the Third 
 Law. 
 
 5. A bullet weighing 20 gm. and having a speed of 300 meters 
 per sec., struck and imbedded itself in a bird weighing 5 kilos, 
 which was flying in the same direction as the bullet with a speed 
 of 150 kilometers per hour. Find the velocity of the bird, in 
 kilometers per hour, the instant after it was shot. 
 
 6. What becomes of the momentum of a meteorite which col- 
 lides with the earth? What becomes of its energy? 
 
 7. Two equal inelastic balls moving with equal velocities in 
 opposite directions collide. Show that in this case the momentum 
 Before impact is the same as that after impact. 
 
 Note that velocity is a directed quantity. 
 
 8. A billiard ball weighing 100 gm. and moving east with a 
 speed of 2 meters per sec. was struck by a putty ball weighing 4 
 gm. and moving south with a speed of 20 meters per sec. Find 
 the speed and direction of the ball the instant after the impact. 
 
 9. A 500 gm. bird sat on a pole 30 meters high. A boy 
 standing 20 meters from the base of the pole shot the bird with a 
 10 gm. bullet which had a speed of 150 meters per sec. How far 
 did the bird rise above the pole? How far from the base of the 
 pole did it strike the ground? 
 
 Assume that the bullet lodged in the bird. 
 
VII 
 
 ELASTIC IMPACT. COEFFICIENT OF RESTITUTION 
 
 Theory 
 
 Since force is equal to rate of change of momentum it follows 
 from the Third Law that the mean force acting between two 
 impinging bodies is the total change in the momentum 
 ^ e ither divided by the time of duration of the impact. 
 Since this time can not in general be determined, it is 
 customary to confine attention to the total change in momentum 
 which each body experiences by virtue of the impact. This 
 quantity is called the * 'impulse" of the force, and will be here- 
 after represented by the letter R. If, then, u and v l represent 
 the velocities of the first body m, before and after impact 
 respectively, u z and v z the velocities of the second body m^ 
 before and after the impact, then by definition 
 
 R = m l (u l -v 1 ), (44) 
 
 or (see Third Law), 
 
 R = m z (v z u^). (45) 
 
 In the case of elastic impact, it is convenient to divide R into 
 two parts RI and R z , of which R represents the impulse during 
 the compression, and R% the impulse from the instant of greatest 
 compression to the instant of separation. 
 
 In impacts between inelastic bodies R z = 0. If the colliding 
 bodies are perfectly elastic, it might be expected that R z would be 
 equal to R^ In point of fact this is never the case, 
 ^coefficient ^ ol there are losses due to internal friction even with 
 ti(m StHu bodies which when subjected to static tests show per- 
 fect elasticity (see definition of perfect elasticity in 
 Ex. VIII). However Newton proved experimentally that for any 
 
 two given bodies the ratio -y^ is always a constant so long as the 
 
 *+i 
 
 impact is not so violent as to produce permanent deformation. 
 
 58 
 
THIRD LAW 59 
 
 This ratio is called the coefficient of restitution , and will be here- 
 after represented by the letter e. It is always less than unity. 
 
 This coefficient of restitution e = may also be shown to be 
 
 [a 
 
 the velocity of recession, v 2 v^ of the two colliding bodies divided 
 
 by their velocity of approach, u t u 2 . For it is evident 
 
 c = y, v that at the instant of greatest deformation m l and m 2 
 
 Ui ~ u * have a common velocity. Call this velocity S. Then, 
 
 from the above definitions of R l and R^ 
 
 A = mi (ui -S)=m 2 (S- M,), (46) 
 
 R, = mi (S - Vl ) = m 2 (v 2 - S). (47) 
 Division of (47) by (46) gives 
 
 R 2 r , B-V! v*-S /AQ . 
 
 J*- (48) 
 
 From (48) come the two equations, 
 
 '8-i\ = e (?/!-), (49) 
 
 v 2 -S=e (8-u 2 ). (50) 
 
 Addition of (49) and (50) gives 
 
 v*-v l = e (wi-w). (51) 
 
 Q. E. D. 
 
 In the special case of impact against a fixed plane, u 2 = 0, 
 v 2 = 0, and v is opposite in direction to ?/ t therefore 
 
 *--2f- (52) 
 
 M, 
 
 Newton's law as to the constancy of e may therefore be very easily 
 tested by varying the velocity of approach of a body toward a 
 fixed plane and measuring the corresponding velocities of rebound. 
 If the body be dropped vertically upon the fixed plane, the 
 velocities u v and Vi can easily be determined from the heights of 
 fall and of rebound. 
 
 The actual loss L of mechanical energy upon impact (not the 
 fractional loss /) is evidently 
 
 Ims of J 
 
 L = (ii"i* + 4W) ~ (faff + ^10?) ; (53) 
 
 impact. 
 
60 MECHANICS 
 
 or from (44) and (45) 
 
 7? 72 7? 
 
 =Y (iii + v^- (u 2 + v 2 )= {(Ui-uJ-fa-vJ}. (55) 
 
 The combination of (55) with (51) gives 
 
 L = ~( Ul -u,)(l-e). (56) 
 
 But substitution in (51) of the values of v 2 and Vi obtained from 
 (44) and (45) gives 
 
 R= jnsn*_ (l + e)(u u)f (57) 
 
 mi + mt 
 
 Substitution of this value in (56) gives for the loss of kinetic 
 energy 
 
 - ta -^-,tf J55L. (58) 
 
 The fractional loss I is this expression divided by the initial energy. 
 Hence for the simple case in which ra 2 is at rest, i.e. , for which u 2 = 
 
 , (.59) 
 
 m 
 
 and for the still simpler case in which m z is not only at rest but 
 is also infinitely large, i.e., the case of impact upon a fixed plane, 
 
 l=(l-e z ). (60) 
 
 Equation (59) shows that, as in the case of inelastic impact, 
 the loss in kinetic energy is independent of the velocity of the 
 striking body provided the struck body is at rest. When e = 
 (59) reduces to (36). 
 
 It appears from (58) that the sole condition of no loss of 
 kinetic energy in an impact is e = 1. The fact that e is always 
 somewhat less than unity means then that in all impacts there is 
 some transformation of mechanical into heat energy. 
 
 It is evident from the general momentum equation (33), viz., 
 
 Velocities m ^ + m * U * = m M + ^> 
 
 after impact and from the equation (51) which determines e, viz., 
 
 = ^TT' ( 61 ) 
 
 that the velocities after impact can always be found if the 
 coefficient of restitution, the two masses, and the two initial 
 
NEWTON'S THIRD LAW 
 
 61 
 
 velocities are known. For the simple case in which e = 1, i. e., 
 
 for the case of so-called "perfectly elastic"* impact, the solution 
 of these two equations gives 
 
 ~ m ) fnt)\ 
 
 , (bxi) 
 
 ~ m *) * /AQ\ 
 - ( 63 ) 
 
 Object. 
 
 Directions. 
 
 Experiment 
 
 1. To test Newton's law as to the constancy of e; 2. to prove 
 the equality of momenta before and after impact and 
 to find the coefficient of restitution and the per cent 
 
 loss of mechanical energy in the impact 
 
 of two steel balls. 
 
 1. Drop steel and glass balls from 
 
 the clamp c (see Fig. 39) through the 
 ring 0, and slip the hori- 
 zontal rod r up or down the 
 
 vertical support, until, in the rebound 
 
 from the smooth top of the heavy steel 
 
 plate Pf, the bottom of the ball just 
 
 becomes visible above o. Make obser- 
 vations for at least three different 
 
 heights of fall which lie between, say, 
 
 30 cm. and 100 cm. In each case 
 
 make the measurement from the steel 
 
 plate to the bottom of the ball. 
 
 Since in general for falling bodies 
 
 starring from rest v = 
 
 (64) 
 
 FIGURE 39 
 
 2. The form of apparatus used in 2 (see Fig. 40) is the same as that 
 used in Ex. VI, save that all three heights, viz. , the height of fall of m l 
 
 *The term is rather unfortunate since "perfect elasticity" is often 
 used in a somewhat different sense (see Ex. VIII). 
 
 f A slab of slate or any smooth, hard, and heavy body may replace the 
 steel plate. 
 
62 
 
 MECHANICS 
 
 Apparatus. 
 
 FIGURE 4) 
 
 with the hand as 
 it swings back 
 after the impact, 
 and take the 
 readings at 5 and 
 c. Note the zero 
 reading of each 
 ball, i. e., the 
 reading when 
 each ball hangs 
 alone^ and finally 
 take the reading 
 of index 1 when 
 m l is at the point 
 
 Directions. 
 
 before impact and the heights of rise 
 of both mi and m 2 after im- 
 pact are measured upon a 
 graduated arc. 
 
 First make such adjustment of the 
 lengths of the supporting strings that, 
 when the balls hang freely, 
 the wire frame carried upon 
 the bottom of m z clears index 1 (see 
 Fig. 41), but catches index 2, while 
 the like frame on mi catches index 1. 
 Then adjust so that when the thread 
 which holds mi back is burned, m z is 
 driven straight up the arc by the im- 
 pact from mi. Take the reading of 
 index 1 when mi is tied in position a. 
 Next slide index 1 down nearly to Z, 
 the point to which m t 
 will move after the im- 
 pact (this point should be 
 approximately located by 
 a preliminary trial), and 
 place index 2 in the 
 neighborhood of c. Then 
 burn the thread, catch wi, 
 
 FIGURE 41 
 

 NEWTON'S THIRD LAW 63 
 
 of impact d. Tliis is not tlie reading when both halls hang together, 
 but the reading when m 2 hangs freely and m^ is brought down so 
 as just to touch m z . 
 
 If a represent the difference between the zero reading of index 1 
 
 and the reading at the point of impact d, the difference between the 
 
 zero of 1 and the reading at , /? the difference between 
 
 the zero of 1 and the reading at b, and to the difference 
 
 between the zero of index 2 and the reading at c, it is evident 
 
 from the discussion of Ex. VI that the height hi through which w?. t 
 
 falls before impact, the height h 9 through which m z is raised by 
 
 the impact, and the height /// through which m-i rises by virtue of 
 
 the velocity which it retains after the impact, are given by the 
 
 relations (see Fig. 41) 
 
 hi= r (cos a cos 0) , ) 
 h*= /-(l-cosco), (65) 
 
 Ui = r (cos a- cos /?). ) 
 The equation to be verified, viz., 
 
 miUi = m^v* + m^Vi, 
 may be written, since v = -v/2</A, 
 
 m^lii = m z ^lh + m^lii ; (66) 
 
 or 
 
 m^y (cos a - cos 0) = w z */ (1 - cos CD) -f mi \/(cos a - cos (3) . (67) 
 A dmilar substitution in the expression for the coefficient of resti- 
 tution [see (61)] gives 
 
 >/(! -cos w) - v/(cos a - cos )8) 
 
 e = = 
 
 ^(COS a COS 6) 
 
 The loss of energy is to be obtained from e, w^ and m z [see (59)]. 
 
 Record 
 
 Steel 
 
 Glass 
 1st 2d 3d 
 
 1. Ht. of fall 1st 3d 3d - 
 
 Ht. of rebound " - - " - 
 
 % loss of K E " - - " - - " - 
 
 2. Rdg. of index 1 at a at 6 at zero at pt. of impact 
 
 < " 2 " at c 
 
 m^ m 2 .'. o = -.'. = .'. /3 = .. w=r 
 
 Momentum before after e = % loss of K E = 
 
64: MECHANICS 
 
 Problems 
 
 1. Show from (57), (44), and (45) that when two equal balls, 
 for which e = 1, collide centrally, they simply exchange velocities, 
 i. e., the result is the same as though one had passed through the 
 other without in any way influencing it. 
 
 2. Hence explain why the only effect of a central impact of one 
 marble upon a row of marbles is to drive off the end marble. Also 
 why the impact of two marbles drives off two from the end, etc. 
 
 3. A 300 gm. ball approaches a bat with a velocity of 50 meters 
 per second; it leaves with an opposite velocity of 100 meters. 
 Find the mean force of the blow if the impact last ^ second. 
 
 4. A rapid-fire gun shoots 500 30-gm. bullets per minute. 
 Find what force is necessary to hold it in place if the velocity of 
 the bullets be 500 meters per second. 
 
 For a constant force, or for a succession of impulses so rapid that the 
 effect is the same as that of a constant force, "rate of change of mo- 
 mentum" is change of momentum per second. 
 
 5. A fire engine throws 400 liters of water per minute from a 
 pipe furnished with a nozzle of 4 cm. diameter. What force does 
 a wall experience against which the jet is directed at short range 
 (assume inelastic impact)? If each particle of water were "per- 
 fectly elastic," i. e., rebounded with the velocity of approach, what 
 would be the value of the force? 
 
 6. A bullet weighing 50 gm. is fired into a block weighing 125 gm. 
 Find the per cent loss of mechanical energy. Had ball and block 
 been elastic bodies for which e =1, what would have been the loss? 
 
 7. eh very nearly unity for equal spheres made of perfectly 
 elastic materials (steel, glass, ivory, etc.), but it is not unity for 
 unequal balls of the same substances. It may be as low as . 75 for 
 steel balls of greatly different size. Thus e is a constant of the 
 colliding bodies, not of the material. Why? 
 
 Consider vibration losses, and the conditions under which vibrations 
 will persist in the bodies after impact. 
 
 8. Find the velocities after impact of two directly impinging 
 bodies whose masses are 50 gm. and 100 gm., whose velocities 
 before impact are in the same direction and have values of 600 
 cm. and 350 cm. respectively, and for which e is unity. Ditto for 
 balls for which e = .90. 
 
 9. Explain the rise of a rocket. 
 
VIII 
 
 ELASTICITY 
 HOOKE'S LAW: YOUNG'S MODULUS 
 
 Theory 
 
 Most substances possess in greater or less degree two quite dis- 
 tinct kinds of elasticity: (1) elasticity of volume, (2) elasticity of 
 form. A body is said to have volume elasticity if it 
 ^ eil< ^ s t return to its original volume after being com- 
 pressed or dilated by the application of force; i. e., if 
 its molecules tend to maintain fixed distances with reference to 
 one another, and resist any attempt to increase or decrease these 
 distances. A body possesses form elasticity, or rigidity if its 
 molecules tend to maintain a fixed configuration, and resist any 
 attempt to produce slipping motions among themselves. 
 
 The "volume coefficient" or the "volume modulus" of elasticity 
 is a constant which measures the restoring force called 
 ^ P^ty ^v a given change in the mean distance 
 between adjoining molecules, the. configuration remain- 
 ing unchanged. 
 
 The coefficient of rigidity or "rigidity modulus" is the constant 
 which measures the restoring force called out by a given change 
 in the relative positions of the molecules, the mean distance 
 remaining unchanged. No connection whatever exists between 
 the two kinds of elasticity. Thus all liquids possess a volume 
 modulus which is enormous, a rigidity modulus which is essentially 
 zero. India-rubber has nearly the same volume modulus as 
 water, but with a very pronounced rigidity modulus. The metals 
 have very large moduluses of both volume and form. 
 
 A body is said to be perfectly elastic if it always requires the 
 same force to produce the same displacement. Thus a wire would 
 show perfect elasticity if the successive removal of a 
 number of stretching weights caused it to resume 
 exactly the lengths which it had during the successive 
 addition of the weights, and that no matter how long or how 
 
 G5 
 
66 MECHANICS 
 
 short a time the weights had been in place. If static experiments 
 only are taken as the test of perfect elasticity, all liquids are per- 
 fectly elastic, and most solids also so long as the displacements are 
 kept within certain limits. The limits of perfect elasticity, how- 
 ever, differ very widely for different substances. Jellies and rubber 
 show perfect rigidity through very wide limits, iron through very 
 small, lead through smaller still, etc. 
 
 There is no connection between the "degree" of elasticity of 
 a body and its elastic constants. The former measures the per- 
 
 fectness of the return to the initial condition, the latter 
 dSKffs ^ e ma gnitude of the force required to produce a given 
 
 displacement. Thus jelly is nearly perfectly rigid, but 
 has a very small rigidity coefficient. Lead has large coefficients, 
 but a small degree of elasticity. In popular usage a body is said 
 to be "very elastic" which possesses nearly perfect elasticity 
 through wide limits. Properly speaking, ability to rebound is a 
 measure of resilience, not of "degree of elasticity." Neverthe- 
 less, the former depends in large measure upon the latter.* 
 
 Experiment shows that within the limits of perfect elasticity 
 Hole's ^ e displacement produced is proportional to the force 
 Law. applied^ whether it be the bulk or the form elasticity 
 
 which is made the subject of test. 
 
 For the determination of the volume modulus, force must be 
 applied in such a way that it will compress or dilate the body 
 
 equally in all directions, but will not distort it. This 
 moduim me can ^ course ^ e done only by a uniform pressure or 
 
 tension applied to all points of the surface of the body. 
 The bulk modulus k is then defined as the ratio between the force 
 per unit area (the stress) and the change in volume per unit volume 
 (the strain). By Hooke's Law this ratio must be constant. 
 Thus 
 
 , , / force\ /change in vol.X 
 
 vol. mod. = I -)-(- , ), 
 
 \ area / \ vol. / 
 
 or symbolically, since by definition - = pressure, 
 
 v (69) 
 
 V 
 
 *Art. on "Elasticity" by Lord Kelvin in Encyclopaedia Britannica* 
 
COEFFICIENTS OF ELASTICITY 
 
 67 
 
 in which k stands for the volume modulus, P for pressure (i. e., 
 force per unit area) , V for volume, and v for change in volume. 
 Unfortunately the direct determination of Jc is not easy. It 
 is therefore customary to determine instead a quantity called 
 YOU 's Young's modulus, which is a cross between the volume 
 modulus. modulus and the rigidity modulus. A known weight is 
 hung from a wire and the elongation measured. Young" 1 8 modulus 
 is then defined as the ratio between the force per unit cross-section 
 of the wire and the elongation per unit length. It is evident that 
 the operation here described tends to produce a change in form as 
 well as in volume, for instead of increasing or decreasing all 
 dimensions it increases the length and decreases the diameter. 
 Analysis which is beyond the scope of this book shows that from 
 this hybrid modulus Y, and the rigidity modulus n (see Ex. IX) 
 the bulk modulus Tc can be found by the equation . ;i ' t \ 
 
 Y = 
 
 3k + n 
 
 (70) 
 
 Experiment 
 
 Object. 
 
 To test Hooke's Law and to find Young's modulus for 
 steel. 
 
 The instrument for determining the elongation produced in a 
 wire by a given stretching force is shown in Fig. 
 42. The upper end of the wire is 
 firmly clamped in a chuck at a. At b 
 the wire is gripped by a second chuck which is 
 set into a cylindrical brass piece. This cylinder 
 passes, with very little play, through a circular 
 hole in the cross piece n. n is rigidly clamped 
 
 Description. 
 
 FIGURE 42 
 
68 MECHANICS 
 
 to the upright rods R and 7?', and carries a small horizontal 
 table upon which rest the front feet of the optical lever m. In 
 order to prevent easy displacement of the lever these feet are 
 set in a groove o. The rear foot of the lever rests upon the face 
 of the chuck b. At a distance of twelve or thirteen feet from 
 the instrument a telescope T, to which is attached a vertical scale 
 $, is so placed that the image of the scale formed by the mirror 
 m is visible in the telescope. The addition of a weight to P 
 stretches the wire and lowers the rear foot of the lever a distance 
 which will be denoted by e. If I represent the distance from the 
 rear foot of the optical lever to the mid-point between its front 
 feet, it is evident that the production of the elongation e in the 
 
 wire ab has caused the mirror m to turn through an angle of 
 
 radians. This angular motion of the mirror causes some pointy 
 instead of p to come under the cross-hairs of the telescope. The 
 beam of light reflected by the mirror has thus been turned through 
 
 the angle radians. But this angle is twice that through which 
 
 the mirror turns (see Problem 1 below) ; hen-ce the distance c is 
 easily determinable in terms of the measurable quantities^/, mp t 
 and I. 
 
 DIRECTIONS. First locate the image of the scale in the tele- 
 scope. To do this move the head about near the telescope until 
 the image of the eye is seen in the mirror m. If the 
 
 Finding the J , _. , . . 
 
 scale in the eye cannot be found at the distance of the telescope, 
 move the head up toward the mirror until it is found ; 
 then move back again. Keeping the image of the eye in sight, 
 move the telescope and scale into, or near to, the position occu- 
 pied by the eye. Then adjust positions until, when the eye 
 sights over, not through, the telescope, the image of the scale is 
 seen in the mirror. The image of the scale must then fall 
 upon the objective of the telescope. Then looking through 
 the telescope, focus by means of the rack and pinion r until 
 the mirror itself is seen, then slowly push in the eye-piece by 
 means of r until the scale is brought into view. If the cross- 
 hairs are not in sharp focus, move the eye-piece alone, in or out, 
 until they appear perfectly sharp, then refocus upon the scale by 
 means of r. 
 
COEFFICIENTS OF ELASTICITY 69 
 
 Next turn the mirror m until the portion of the scale seen in 
 the telescope is that near the objective, and take a careful reading 
 of the position of the cross-hairs upon the scale, esti- 
 readings mating to tenths millimeters. If a slight change in 
 the position of the eye changes at all the reading, focus 
 again carefully until this "error of parallax" is altogether 
 removed. Add kgm. weights successively to the pan and take the 
 corresponding readings. Make similar records as the weights are 
 successively removed. If upon removal of the weights the cross- 
 hairs do not return to their original position, the limit of perfect 
 elasticity has been overstepped and the readings must be repeated 
 with the use of fewer weights. In adding or removing weights, 
 use extreme care to prevent jarring the instrument or changing 
 the position of the lever-point on chuck. A divergence in succes- 
 sive elongations of more than one per cent indicates carelessness. 
 
 In order to determine I take the imprint of the three feet of 
 
 the optical lever upon a sheet of paper. With a knife-edge draw a 
 
 line connecting the centers of the two front feet. The 
 
 mecmtre- distance from the middle point of this line to the center 
 
 of the third foot may be measured with a steel rule held 
 
 on edge. Estimate to tenths millimeters. Measure the distance 
 
 mp with a tape. 
 
 Since in this case Young's modulus involves the sectional area 
 of a very small wire, it is necessary that the diameter be measured 
 with great care. Measure with micrometer calipers, and let the 
 final result be a mean of at least a dozen observations taken at 
 equal intervals from top to bottom of the wire. In using the cali- 
 pers always take the zero reading as well as the reading when the 
 wire is between the jaws. 
 
 In the calculation of Young's modulus, the stress (force per 
 unit cross-section) must be expressed in dynes per square centi- 
 meter, the strain (elongation per unit length) in centi- 
 uw measure- meters per centimeter. It is also required to plot a 
 curve in which the total weights in the pan at each addi- 
 tion shall represent abscissae, and the corresponding total elonga- 
 tions, measured from the first reading, shall represent ordinates. 
 As a check upon the accuracy of the work, change the position 
 of the telescope and scale, and make a second complete determi- 
 nation of Y. 
 
70 
 
 MECHANICS 
 
 Record 
 
 1st Determination 
 
 2d Determination 
 
 Wts. Edgs. Difs. 
 
 Diams. 
 
 Rdgs. 
 
 Difs. 
 
 Diams. 
 
 Means 
 7 = mp = wire length = 
 
 
 ,'. F = 
 
 X 10 
 
 F= 
 
 X10 
 
 Problems 
 
 ^ 1. From the optical law angle of incidence equals angle of 
 reflection, prove that a beam of light reflected by a mirror turns 
 through twice the angle through which the mirror turns. 
 
 v0. Can a body whose bulk modulus is infinite have a finite 
 Young's modulus? 
 
 \t 3. A wire 80 cm. long and .3 cm. in diameter is stretched .3 
 mm. by a force of 2 kilo. How much force would be required to 
 stretch a wire of 180 cm. length and 8 mm. diameter through 
 1 mm.? 
 
 * 4. An iron and a brass wire have each the length of 15 cm. 
 when each is stretched by a force of 1 kgm. The length of the 
 iron wire becomes 15.4 cm. under a stress of 3 kgm. and that of 
 the brass wire becomes 15.6 cm. under a stress of 9 kgm. Com- 
 pare the modulus of iron with that of brass. 
 
 5. What force is needed to double the length of a steel rod 
 whose diam. is 2 mm.? Assume perfect elasticity. 
 
IX 
 
 Definition 
 and 
 
 illustration 
 of shear. 
 
 THE COEFFICIENT OF RIGIDITY 
 
 Theory 
 
 In order to find the coefficient of rigidity of a substance it is 
 necessary to apply a force which will cause the molecules to shift 
 their relative positions without altering at all their 
 distances apart. Such a change is called a " shear." 
 To take as simple a case as possible, imagine a 
 rigid cylindrical shell whose wall is one molecule 
 in thickness, and let the height of the cylinder be so small 
 that it contains but two rows of molecules [see Fig. 43 (1)]. 
 Let a tangential force act upon each of the molecules 1, 2, 3, 4, 
 etc., of the upper row so as 
 to bring them into the posi- 
 tions sjiown in Fig. 43 (2), 
 the molecules of the lower 
 row being held fast by equal 
 and opposite forces. The 
 change produced is evi- 
 dently a pure shear, since 
 configuration alone has been 
 
 changed, all distances re- FIGURE 43 
 
 maining exactly as at first. 
 
 The shearing force is the total force which has acted, or the sum 
 of the forces upon the individual molecules. The angle through 
 which the line connecting any two molecules which originally lay 
 in the same vertical line has been turned by the shearing force, 
 is taken as the measure of the shear. This angle is always 
 expressed in radians. 
 
 Were the cylinder three rows of moleculesjn height instead of 
 two [see Fig. 43 (3)], then, upon the application of the same shear- 
 ing force, the upper row would move twice as far as before, but 
 the angle of shear 6 would remain the same, for the case would be 
 
 71 
 
MECHANICS 
 
 precisely the same as though the middle row were clamped fast 
 and the equal and opposite forces on the upper and lower rows 
 
 produced each the same effects which were considered 
 fnde indent * n case ^* ^ rom an extension of the same line of rea- 
 of cylinder sonm g t a still longer cylinder, it is evident that the 
 
 shear produced by the application of a given shearing 
 force to the top of the cylinder is independent of the height of the 
 cylinder, for the shearing motion ultimately ceases only when the 
 restoring force due to the rigid connection between the top row and 
 the next to the top row of molecules is equal to the shearing force, 
 i. e., when there is a given angular displacement between these 
 two rows. In order that the second row from the top may be in 
 equilibrium, the same angular displacement must exist between it 
 and the third row as exists between rows 1 and 2, and so on to 
 the bottom row. Thus a given shearing force must produce a 
 given angular tilt in a row of vertical molecules whether the 
 cylinder be long or short. 
 
 Conceive now the ideal cylindrical shell to be replaced by an 
 actual thin hollow cylinder of length /, of mean radius r and of 
 
 thickness t [see Fig. 44 (1)]. Divide the upper surface 
 
 into unit areas, and let a tangential force/ be applied 
 
 rigidity n. t() each unit area |- gee ^ ^ ^j ^ coe jft c j e}l t O f 
 
 rigidity n is noiu defined as the ratio between the force per unit area 
 
 FIGURE 41 
 
 (the stress) and the shear (the strain) produced by this force. 
 Symbolically [see Fig. 44 (1)], 
 
 (71) 
 
COEFFICIENTS OF ELASTICITY 73 
 
 The total shearing force is, however,^/*. Call this force f and 
 
 shearing force n . 
 
 the area 01 the ring A . Then n = - *- angular dis- 
 
 area 
 
 placement; or 
 
 From Hooke's Law this ratio is the same for all values of/'. 
 
 It is not so easy to measure directly as it is to measure <f> 
 
 Measurement \- F[ 8' 44 WL the an le through which the end of 
 
 hollow ^e cylinder is twisted by the force/'. Since the arc 
 cylinder. a [Fig. 44 (1)] is always small in comparison with 
 the length of the. cylinder, it is possible to write without apprecia- 
 ble error, 
 
 y = 0. But also ~ = <fr. 
 
 Hence = ^f- (73) 
 
 I 
 
 Again A = 2r*. (74) 
 
 Substitution of these values of A and in (72) gives 
 
 *- . .'?' (75) 
 
 If the force/' be called into play in the manner shown in Fig. 
 44 (3), i. e., by clamping one end of the cylinder, screwing the 
 other rigidly to a grooved circular disk and applying a twisting 
 force of say F [= Wg~\ dynes to the circumference of the disk by 
 means of weights TF, then by the principle of moments [see Fig. 
 44 (3)], 
 
 Hence finally from (75) 
 
 This equation, then, expresses w, the coefficient of rigidity, in 
 terms of quantities all of which are easily measurable. 
 
 If, as is usually the case, it is found more convenient to twist 
 a solid cylinder rather than a hollow one, formula (76) can be 
 
74 MECHANICS 
 
 transformed to fit the case as follows : Imagine the solid cylinder 
 to be made up of a large number of concentric 
 hollow cylinders of equal thickness and 
 
 Measurement 
 
 of n from of radii r^ i\, r s . . . . r n (see Fig. 45). 
 
 solid cylinder. m i i 
 
 The moment of force, say Fh^ which 
 it is necessary to apply in order to twist any 
 particular hollow cylinder of radius r k through 
 FIGURE 45 the an gl e <, is found from (76), viz., 
 
 . (77) 
 
 The total moment of force Fli which must be applied in order to 
 twist the solid cylinder through the angle <f> is evidently the sum 
 of all the moments Fh k . Thus Fh = FH^ + Flt z + FJi* f . . . Fh n = 
 
 %irn<!>t , . o o x %Trn<t> i i TI .LI T 
 
 y (r* + rf + r s s + . . . Tn) = - x in which R is the radius 
 
 of the solid cylinder.* Hence, finally, for a solid cylinder, 
 
 2 Fhl 
 
 an equation which shows that the twist < produced by a given 
 moment of force, or " torque," Fli, applied to one end of a cylin- 
 drical wire the other end of which is firmly clamped, is directly 
 proportional to the length and inversely proportional to the fourth 
 power of the diameter of the wire. 
 
 It is evident from the definition of the coefficient of rigidity n 
 
 (also called the "modulus of torsion") that it is a constant which 
 
 is characteristic of the substance and is independent of 
 
 ^mmnSof the dimensions of the particular wire used. But for a 
 
 torsion" T R 
 
 E77, 
 
 ticular wire, the ratio must also be constant (Hooke's Law). 
 
 This constant of the wire is technically called its "moment of 
 torsion," and is represented by the symbol T . Thus the equa- 
 tion 
 
 *If the student is not familiar with simple integrations he may 
 take this summation for granted. Elementary integral calculus gives 
 
 C 
 I 
 
 - 
 r*dr = , the thickness t in the above expression being the same as dr. 
 
COEFFICIENTS OF ELASTICITY 75 
 
 is simply the definition of the moment of torsion. Expressed in 
 words, the moment of torsion of a wire is the constant ratio of the 
 moment of the restoring force exerted ~by a twisted wire and the angle 
 of twist. If in (79) <j> = unity (i. e., 1 radian), then T = Jli. 
 Hence the moment of torsion of a wire is sometimes denned 
 as the moment of force required to twist one end of the wire 
 through one radian. 
 
 It is evident from (78) and (79) that the modulus of torsion n 
 may be expressed in terms of the moment of torsion T and the 
 dimensions I and R. Thus, 
 
 Experiment 
 
 (1) To test Hooke's Law for torsion; (2) to determine the 
 moments of torsion of steel wires of different lengths 
 and diameters; (3) to find w, the coefficient of rigid- 
 ity of steel. 
 
 Three steel wires 1, 2, and 3 (Fig. 46) are provided, of 
 
 which 1 and 2 have the same lengths I (about 1 m.) but different 
 
 diameters, 1 and 3 the same diameters (about 2.5 mm.) 
 
 but different lengths. By means of a set screw at ^ 
 
 the wires may be clamped rigidly to the grooved and graduated 
 
 circular wheel C. Displacements produced by the weights W are 
 
 read off by means of an index attached to the frame. Ball bear- 
 
 ings virtually do away with all friction and render possible a high 
 
 degree of nicety in the readings. 
 
 FIGURE 46 
 
76 MECHANICS 
 
 Clamp wire 1 in position and take readings first as 100 gm. 
 weights are successively added to the pan, then as they are 
 removed. Repeat with wires 2 and 3. Measure the 
 Directions. di ame ters with the micrometer calipers (see Appendix), 
 taking a mean of a large number of readings at regular intervals 
 along the wire. 
 
 Use the following method in calculating the mean twist per 
 100 gm. : 
 
 If the total twist due to 6 hundred grams is, say, 4.32 
 
 U it it tt tt it g tt .it it it g Q1 
 
 tt tt ii a a a 4. a a a tt ^ 9 
 
 a a tt a a a o tt tt it a o iiv 
 
 tt tt tt a tt "9 " " tt tt -\ A A 
 
 tt it tt tt tt a ^ a tt tt n iv o 
 
 " " sums = 21 and 15.16 
 
 then the most correct value of the twist per 100 gm. which can 
 be obtained from this set of readings is 15.16 * 21 = .722. 
 
 This method gives to each observation precisely the amount of 
 consideration which it deserves. Thus it gives a weight of 6 to 
 the observed displacement for 600 gm., a weight of 2 to the 
 observed displacement for 200 gm., etc. 
 
 Record 
 
 Wire No. 1 No. 2 No. 3 Diameters 
 
 Wts. Reads. Difs? Reads. Difs. Reads. Difs. No. 1 No. 2 No. 
 
 
 
 100 
 200 
 300 
 
 - , - 1 
 
 200 
 
 - 
 
 400 -[_ 
 
 500 - -(_ 
 
 600 - f - f 
 
 500 - f ~ - f 
 
 400 - f _ 
 
 300 
 
 f ~ _ f 
 I 
 
 f 
 
 / 
 
 100 f 
 
 200 
 
 r -Means 
 
COEFFICIENTS OF ELASTICITY 77 
 
 I of No. 1 Mean twist of No. 1 
 
 _ 
 
 I of No. 3 ~ ' Mean twist of No. 3 
 
 Diam. of No. 1 /Mean twist of No. 
 
 
 . 2\} 
 . I/ 
 
 
 Diam. of No. 2 \Mean twist of No. 
 
 Radius of wheel C . . T for 1 for 2 for 8 
 
 . . n from 1 from 2 from 3 Mean % s difs. 
 
 Problems 
 
 1. If one of the wires had a real diameter of 2.513 mm., but 
 was measured as 2.501 mm., what per cent of error was thus 
 introduced into n? 
 
 2. Decide from a study of your observations which one of the 
 quantities involved in n introduces the largest error into the 
 result. Why is it needless to take great pains in measuring the 
 lengths? 
 
 3. Show from equation (78) that n would be correctly denned 
 as the moment of force required to twist a cylinder of 1 cm. 
 length and 1 sq. cm. cross-section through 360. 
 
 4. The moment of torsion of a particular wire is 7.21 x 10 6 
 absolute units; its diameter is 2.732 mm. ; its length is 50.1 cm. 
 Find the moment of force required to twist a wire of the same 
 material of 1 mm. diameter and 4 cm. length through 90. 
 
 5. A man grips upon the circumference of a bar 1,00 in. long 
 and 1 in. in diameter and twists it through 1. He applies the 
 same force (not .the same moment of force) to the circumference 
 of a bar 2 in. in diameter and 80 in. long. Find the twist. 
 
 * Blanks are for results of division. 
 
MOMENT OF INERTIA 
 
 Theory 
 
 It was experimentally shown in Ex. IV that the condition of 
 rotational equilibrium of a rigid body acted upon by the two 
 forces F and W (see Fig. 16) is Fl = Wl' . But equilib- 
 wtaS!m f rium is reached only when the two rates of rotation 
 due to the forces F and W are equal and opposite. It 
 follows, then, from Ex. IV, that the rate at which a force can 
 impart angular velocity to a rigid body is proportional to the 
 product of the force and its lever arm, i. e., to the applied 
 moment of force. Thus, while linear acceleration is propor- 
 tional to the acting force , angular acceleration is proportional to 
 the acting moment of force. Hence "moment of force" bears 
 precisely the same relation to rotary motion which force bears 
 to linear motion. 
 
 The inertia of a body is that property by virtue of which it 
 
 offers resistance to acceleration. The measure of inertia is the 
 
 resistance offered to unit acceleration; or, since this 
 
 resistance is always equal to the force producing the 
 
 acceleration (see scholium), the measure of the inertia of a body 
 
 is the force necessary to impart to it unit acceleration. This was 
 
 experimentally proved in Ex. II to be proportional to mass ; in 
 
 the absolute system of units equal to mass. Hence a gram of 
 
 mass has one unit of inertia, two grams of mass two units of 
 
 inertia, etc. 
 
 Moment of inertia is that property of a rotating body by virtue 
 of which it offers resistance to angular acceleration. It is meas- 
 ured by the "moment of force" necessary to impart to 
 inertia tof ' ^ ie ^^ un ^ an 9 u ^ ar acceleration. Thus, a rotating 
 body has unit moment of inertia if it requires the 
 application of a unit moment of force (1 dyne-centimeter) to 
 increase or decrease its angular velocity at the rate of one radian 
 per second; it has two units of moment of inertia if it requires 
 two dyne- centimeters to impart one radian of acceleration, 10 
 units if it requires 5 dyne-centimeters to impart an acceleration 
 of i radian per sec., etc. Symbolically, if /represent moment of 
 
 78 
 
MOMENT OF INERTIA 79 
 
 inertia, Fli the acting moment of force, and a the angular accel- 
 eration produced, 
 
 This equation is to be regarded merely as the definition of /. It is 
 thus seen that "moment of inertia" is a perfectly definite phys- 
 ical quantity which can be determined for any body whatever by 
 merely applying a known moment of force and measuring the 
 angular acceleration produced. The definition of / might be put 
 in the following form : Moment of inertia is that physical prop- 
 erty in which two rotating bodies agree when it requires the same 
 moment of force to give to each a given angular acceleration. 
 
 It is evident that / is not proportional to mass alone, as is 
 
 inertia, for everyday experience teaches that two rotating bodies 
 
 mav have precisely the same mass and vet offer widely 
 
 Calculation .* , . , ,-. 4.- * 
 
 of moment different amounts of resistance to the operation of 
 
 of inertia. . , , . . -, 
 
 starting or stopping; e.g., two wheels, one of which 
 has its mass concentrated near the axle, the other on the circum- 
 ference. Thus / is a function both of mass and of the distribution 
 of mass, i.e., of the distances of the elements of mass from the 
 axis. In order to calculate /, the moment of force necessary to 
 impart a radian of acceleration must 
 be found in terms of the masses of 
 the particles and their distances from 
 the axis. Take a single particle mi at 
 a distance TI [=0^] (Fig. 47) from the 
 axis and think of it as moving inde- 
 pendently of all the other particles FIGURE 47 
 
 under the action of a force /i which 
 gives it a linear acceleration a { . The second law gives 
 The moment of the force/! about the axis is/^v 
 Hence /i^ = ??? 1 a 1 r 1 . 
 
 But since - = a it follows that 
 
 Similarly the moment of the force / 2 which is necessary to give to 
 m z the angular acceleration a about the axis is 
 
 / 2 r 2 = 
 
.80 MECHANICS 
 
 The total moment of force Fli which must be applied to give all 
 the particles of the body the angular acceleration a is manifestly 
 the sum of the moments applied to the several particles. Thus : 
 
 Fh =/!/*! +/ 2 r 2 -f- etc. = a (m^* + m^r*? + etc.) = a2mr* (82) 
 or 
 
 Fli 
 
 = lmr\ (83) 
 
 i. e., the moment of force necessary to impart unit angular accel- 
 eration is Swr 2 . But this is by definition / (see 81). 
 Hence / = 2mr 2 . (84) 
 
 In order, then, to calculate / for any body, it is necessary to 
 multiply the mass of each particle in the body by the square of its 
 distance from the axis of rotation, and then to find the sum of all 
 these products. For an irregular, non-homogeneous body, this 
 would evidently be an impossible undertaking. Hence for such 
 bodies the moment of inertia can not be calculated. It can only 
 be obtained by direct experiment, i. e., by applying a known 
 moment of force and observing the angular acceleration produced 
 (or by means of some experiment which is equivalent to this). But 
 for certain regular, homogeneous bodies, it is possible to perform 
 the summation indicated and hence to check an experimental 
 value of / by means of a calculated one. This summation is done 
 with the aid of the integral calculus. Only a few results of such 
 summation will be indicated here. 
 
 The moment of inertia of a uniform cylinder of radius R and 
 mass M rotating about its own axis is 
 
 (85) 
 
 * This may be obtained as follows : If <r represent 
 the density of the cylinder and I its length, then with 
 the use of polar coordinates (see Fig. 48), an element 
 m of mass may be taken as 
 
 m = o-lr dr dd 
 
 Hence 
 
 FIGURE 48 I=2mr^ = 
 
 =ffl C* 
 
 \}o 
 
 The other integrations are somewhat more complicated and will not here 
 be attempted. 
 
MOMENT OF INERTIA 81 . 
 
 The moment of inertia of a uniform sphere of radius R and mass 
 M rotating about an axis passing through its center is 
 
 I=^MR\ (86) 
 
 o 
 
 The moment of inertia of a uniform rectangular bar of length /, 
 width a, thickness , and mass M, rotating about an axis parallel 
 to #, and passing through its center of gravity is 
 
 /= S (?+a2) - (87) 
 
 From a comparison of the equations / = ma and F/i = Ia it 
 appears that / bears precisely the same relation to rotary motion 
 which mass bears to linear motion. / is therefore some- 
 *i mes called the mass of rotary motion. Whenever m 
 a PP ears i n an expression which relates to linear motion 
 / always appears in the corresponding expression which 
 relates to rotary motion. Thus, for example, the kinetic energy of 
 linear motion is rav 2 . The kinetic energy of rotary motion is 
 ^/o> 2 , to being the angular velocity of the rotating body. This 
 may be shown as follows : The kinetic energy Jce^ of the particle 
 m l (see Fig. 47) is ^m-p*. 
 
 But Vi = 0)7*!. 
 
 Hence lce\ = \m^r^^. 
 
 Now the total kinetic energy KE of the body must be the sum of 
 the kinetic energies of its parts. 
 
 Therefore KE = 2rar 2 <o 2 = |/o> 2 . (88) 
 
 Q. E. D. 
 
 Experiment^ 
 
 To determine the moment of inertia of a circular disk by apply- 
 ing a given moment of force and measuring the corresponding 
 angular acceleration, and to compare the result with 
 
 Object. & x , ' 
 
 the theoretical value 01 1. 
 
 A disk weighing several kilograms is mounted upon ball bear- 
 ings so as to rotate with very little friction about its own axis (see 
 Fig. 49). While a weight m imparts to the disk an 
 
 Description. * ' . r . PI*. 
 
 angular acceleration a, an electrically driven fork of 
 known period writes a trace upon the blackened face of the disk. 
 The angular acceleration a is determined precisely as were the 
 linear accelerations in Exs. I and II, i. e., by subtracting succes- 
 
82 
 
 MECHANICS 
 
 L t 
 
 s' 
 
 sive angular distances traversed during successive equal intervals 
 of time. These angular intervals are obtained in degrees from 
 readings made upon a circular scale with which the unblackened 
 
 face of the disk is provided. By 
 means of the rack and pinion s, 
 the frame which carries the fork 
 may be moved through ways in a 
 direction parallel to the face of 
 the disk, so that the traces made 
 during successive revolutions of 
 the disk need not interfere. The 
 screw s r shifts the whole disk in 
 the direction of its axis, and thus 
 makes it easy to secure a suita- 
 ble pressure of the stylus against 
 the blackened face of the disk. 
 
 Wrap a fine 
 thread three, or 
 four times 
 around the cir- 
 cumference, at- 
 taching one end 
 to the disk by 
 means of a small 
 bit of wax. To 
 
 the free end attach just enough weight to equalize the friction 
 of the ball bearings and of the stylus as it bears upon the face 
 of the disk. Set the fork in vibration, adjust the 
 stylus, add 100 grams to the thread, and then suddenly 
 release the disk, at the same time moving forward the fork 
 by means of s. As soon as m touches the floor, stop the disk, 
 remove it from the frame, and carefully mark off the trace as in 
 Ex. I, taking a group of 50 waves as the unit. Eeplace the 
 disk in the frame, set the cross-hairs of the low-power microscope 
 t upon the limiting mark of the first group of waves, and take the 
 reading of the cross-hairs of the microscope t' upon the circular 
 scale graduated upon the unblackened face of the disk. Then 
 turn the disk until the second mark comes underneath the cross- 
 hairs and read again. From such readings a is easily obtained. 
 
 FIGURE 49 
 
 Directions. 
 
MOMENT OF INERTIA 
 
 83 
 
 It must, of course, be expressed in radians (see definition of 
 moment of inertia). Take at least two traces, using different 
 masses for m\ e. g., let m l = 100 gm., m z = 200 gm. 
 
 The force F which produces the rotation of the disk is evi- 
 dently the tension in the thread q. This is not the force -acting 
 upon the mass m, viz., mg, for a part of the acting 
 force mg is expended in producing the acceleration a 
 which is imparted to m. By the scholium to the Third 
 Law, mg = F+ ma, or F = m(g a). Since the acceleration of the 
 weight is the same as the acceleration at the circumference of the 
 disk, a = a 7?, R being the radius of the disk and its angular 
 acceleration in radians. Hence the acting moment of force Fh 
 may be found from the measurement of the three quantities a, 
 R, and m. Fh must of course be expressed in dyne-centimeters. 
 From Fh and a, / is at once obtained (see definition of / ), The 
 theoretical value of /, see (85), involves only the mass M and the 
 radius R of the disk. M is to be obtained by weighing upon the 
 platform scales. 
 
 Record 
 
 1ST DETERMINATION 
 
 2D DETERMINATION 
 
 Angle Angular Angular 
 Read'gs Spaces Ace's 
 
 | 
 
 Angle Angular Angular 
 Read'gs Spaces Ace's 
 
 f 
 
 I ; ( 
 
 f J 
 
 I ' 
 
 i i 
 
 f ' 
 
 f i 
 
 I f 
 
 I f 
 
 f 1 
 
 f 
 
 / j 
 
 [ 
 
 ( / 
 
 1 | 
 
 | 
 
 Mean ace 
 
 
 
 J2 / 
 
 Mean ace. 
 
 . T Mpan T 
 
 
 
 I 
 Problems 
 
 1. A constant pull of 200 kgm. acting on the circumference 
 of a wheel of 1 m. radius imparts in 30 sec. a speed of two revolu- 
 tions per second. Find / for the wheel. 
 
84 MECHANICS 
 
 2. Find what part of the kinetic energy of a rolling solid 
 cylinder is energy of translation, and what part energy of rotation. 
 
 The latter energy, viz., J/w 2 , can in this case be expressed in terms 
 of the mass M of the cylinder and its velocity of translation v. (See 85.) 
 A simple relation exists between v and w. 
 
 3. Solve Problem 2 for a rolling hoop; for a rolling solid 
 sphere. 
 
 4. Find what relation exists between the velocities acquired by 
 a solid cylinder in sliding without friction down an inclined plane 
 and in rolling without slipping down the plane. 
 
 Equate in each case potential energy at top to total kinetic energy at 
 bottom. 
 
 5. The wheel used in the falling body machine (Ex. II) has a 
 radius of 5 cm. and moment of inertia of 275 gm. cm. 2 * Find 
 what mass at the circumference would offer the same resistance 
 to an accelerating force and therefore what number of grams 
 should be added to the mass of the frame in order to allow for the 
 presence of the wheel. 
 
 6. Find as in Problem 4 the relation between the velocities of 
 solid spheres sliding and rolling down an inclined plane. 
 
 See equation (86) and the suggestion under Problem 4. 
 
 7. A bullet weighing 5 gm. and moving 
 
 with a velocity of 100 meters per second in the ^ 
 
 direction ab (see Fig. 50), strikes the pro- 
 
 jection b of a fixed wheel whose moment of 
 
 inertia is 200,000 gm. cm. 2 ,* and whose radius 
 
 is 20 cm. Find the number of revolutions per FIGURE s 
 
 second communicated to the wheel. 
 
 * In order to understand the meaning of this symbol it will be neces- 
 sary to consider briefly the origin and use of dimensional formulae. The 
 dimensional formula for any quantity is simply the symbol which repre- 
 sents the way in which the fundamental units, i.e., the units of Mass, 
 Length and Time, enter into the definition of that quantity; e. g., note 
 the following dimensional formulae: 
 
 L 
 T 
 
 Since acceleration is velocity divided by time, a = -77^ = 
 
MOMENT OF INERTIA 85 
 
 Assume inelastic impact. Then if v represent the initial velocity of 
 the bullet, w the angular velocity of bullet and wheel after impact, and 
 t the duration of the impact, the velocity lost by the bullet is evidently 
 v wR, R being the radius of the wheel. Hence the mean force acting 
 
 between bullet and wheel during the impact is / [= ma] = - 7 
 
 The moment of this force imparts to the wheel a mean angular accelera- 
 tion a such that at = w. From these equations and that which defines 
 I, w may be found. * 
 
 8. Find the number of revolutions per second which would 
 have been imparted to the wheel if the bullet had moved along 
 the dotted line (see Fig. 50) and struck the wheel at a point mid- 
 way between b and c. 
 
 9. A hoop and a solid disk of the same diameter start down a 
 hill together. Which will reach the bottom first? Find the ratio 
 of their velocities at the bottom. 
 
 Assume in each case rolling without slipping and neglect air resist- 
 ance. See Problem 4. 
 
 10. Why can a heavy man on a bicycle always coast faster than 
 a light man on the same wheel? 
 
 In answer disregard friction. 
 
 Since force is mass multiplied by acceleration, / = ^=^- = MLT~ 2 . 
 
 Since moment of force is force multiplied by length, Fh = -=j- = ML 2 T~ 2 . 
 
 Since angular accel. is linear accel. divided by length, a = ^ = ^~ 3 - 
 
 Since moment of inertia is moment of force divided by angular acceler- 
 ation 1= ML 2 . 
 
 This last result might have been seen at once from the fact that it 
 has been shown that 1= 2mr 2 . Enough has been said to show the method 
 of procedure for the derivation of dimensional formulae and the meaning 
 of such formulae. 
 
 Now when units have been defined but have been given no particu- 
 lar names it is customary to write the dimensional formula in place of a 
 name, replacing in this formula the general symbols M, L, T, by the par- 
 ticular units of mass, length, and time which have been used in the defini- 
 tion of the unit under consideration. Thus if it were desired to say that 
 the moment of inertia of a body was 275 units and to explain that in the 
 definition of these units the gram and the centimeter were taken as the 
 fundamental units of mass and length and that these were involved in such 
 a way in the definition that the dimensional formula for moment of 
 inertia was 1= M I? it would only be necessary to write "moment of 
 inertia of body = 275 gm. cm. 2 " 
 
V ^ 
 
 86 MECHANICS 
 
 11. Will increasing the weight of the tires increase or decrease 
 the coasting speed of a bicycle? What effect will increasing the 
 weight of the frame have upon the coasting speed? 
 
 12. A clay ball weighing 50 gm., and moving with a velocity of 
 30 meters per second, struck and stuck to the end of 
 
 a rectangular bar 1 meter long and 5 cm. square 
 which was pivoted at its center of gravity o (see Fig. 
 51). If the weight of the bar were 5 kilos and the 
 motion of the ball were at right angles to the length FIGTJBE 51 
 of the bar, what number of revolutions per second would be com- 
 municated to the bar? 
 See equation (87). 
 
XI 
 
 SIMPLE HARMONIC MOTION 
 
 Theory 
 
 The two very simple forms of motion thus far con- 
 s idered, viz., uniform and uniformly accelerated, belong 
 to the general class of non-periodic motions. 
 Among periodic motions the simplest and most important type 
 is so-called simple harmonic motion. This is denned as motion in 
 which the oscillating body is at every instant urged toward some 
 natural position of rest with 
 
 a force which varies directly _ 5 _ v _Jsl_ c 
 as its distance from that posi- a o^ * 5 
 
 tion. Thus suppose a particle FIGURE 52 
 
 to be moving back and forth 
 
 over the path ab (see Fig. 52), under the action of a force which 
 has its origin in o, and let the law of action of the force upon the 
 particle be expressed by the equation, 
 
 1 mA * (89) 
 
 / 2 \ maj d 2 
 
 in which/j and/ 2 represent the forces acting upon the particle 
 when it is at the distances di and d% respectively from o, and a^ and 
 a z represent the, accelerations toward o of the particle when.it is at 
 these distances. The last equation may be written in the form, 
 
 flj _ 2 _ s _ 
 
 . . 7 tJL^. , 
 
 d l d 2 d s 
 
 or, in general, if this constant ratio of the acceleration to the dis- 
 placement be denoted by the symbol k, the equation which defines 
 simple harmonic motion [see (89)] may be written in the form, 
 
 a = Ted. (90) 
 
 Since/ = ma, (90) may evidently be written in the form, 
 
 f=mkd. (91) 
 
 87 
 
88 MECHANICS 
 
 Any one of equations (89), (90), or (91) may be taken as the defi- 
 nition of simple harmonic motion. These equations express sim- 
 ply a proportionality between force and displacement. But, in 
 view of a very simple relation which exists between the character- 
 istic constant k of the motion and the period of vibration of 
 the system, it is possible to express the characteristic equation 
 of simple harmonic motion in still another form. The evalua- 
 tion of k in terms of the period will be made in three steps, as fol- 
 lows: 
 
 1. The first step will consist in finding Ic in terms of the half 
 length R of the path of the particle, the velocity v which it has at 
 
 any particular point of this path, for example at c, and 
 o/# r <mdd ^ e di s t ance d of the chosen point c from o. Since the 
 
 particle comes to rest at the end of its path, i. e., at #, 
 the velocity v is acquired while it is moving from b to c under the 
 action of the force which has its origin in o. Hence, by equation 
 (30), p. 43, the kinetic energy which the particle has acquired 
 when it reaches c, viz., %mv 9 , is equal to the work done by the 
 force in moving it from b to c. Now it maybe seen from (91) that 
 the value of the force acting upon the particle when it is at b is 
 inkR) while its value at c is mJcd. Since, by the definition of 
 simple harmonic motion, the force is always proportional to the 
 distance of the particle from o, the mean value of the force acting 
 
 .,,.,.,. . . mkR + mTcd m , ,. 
 
 upon it while it is moving from b to c is - The dis- 
 
 iC 
 
 tance be is equal to R d. 
 
 TT mJcR + mkd , 7 , 
 
 Hence fynv* = - x (R - d) ; 
 
 Z 
 
 or v*=k(R*-(F). (92) 
 
 2. The second step will consist in expressing Jc in terms of R 
 and the speed 8 with which a shadow cast h_ 
 
 by the particle moves about the 
 o/Band*s circumference of a circle drawn 
 
 upon ab as a diameter. Imag- 
 ine that a distant source of light, situ- 
 ated directly below ab (Fig. 53), casts a FIGURE 53 
 
 shadow of the moving particle upon the 
 circumference alib. If the velocity v of the particle at any point 
 
SIMPLE HARMONIC MOTION 89 
 
 c be represented by the line ce, then the velocity S of the shadow 
 upon the circumference is represented by c'e' ' . But 
 
 ce . 
 -r- f = sin 6. 
 
 c e 
 
 Hence v = S sin 0. 
 
 Or v z - S 2 sin 2 6 = S 2 (1 - cos 2 0) = Wl - ^Y 
 
 Or <- |r (&!*). (93) 
 
 From (92) and (93) there results at once 
 
 S 2 
 
 This equation shows that the shadow travels with uniform speed 
 about the circumference, for the expression for 8 is independent 
 of d. It involves only the two constants k and R. 
 
 3. Since the shadow has a constant speed, the time which it 
 requires to move over the semicircumference bha is the length of 
 
 Kin terms this path divided by the speed, viz., ^- But this 
 
 of the half- S 
 
 time is manifestly the time t which the particle con- 
 sumes in moving from 1) to , i. e., it is the half -period of the 
 vibration. Thus, 
 
 <- (95) 
 
 From (94) and (95) it follows at once that 
 
 k = ^- (96) 
 
 Characters- Hence the characteristic equation (90) of simple har- 
 
 tic equations 
 
 of S.H.M. monic motion becomes 
 
 : = * ' (97) 
 
 and (91), which simply represents the combination with (90) of 
 the equation f= ma, becomes 
 
 ^ is called the force constant of the system considered. 
 
90 MECHANICS 
 
 If it is known, t can at once be determined. Thus, if the oscilla- 
 tory motion is due to the elasticity of a spring, the 
 
 The force f 
 
 constant. force constant 4 can be determined once for all by 
 
 d 
 
 observing the force required to stretch the spring through a given 
 distance. Then the period of oscillation of the system when any 
 mass m is hung from the spring can be calculated from (98). 
 
 The enormous importance of simple harmonic motion in the 
 study of Physics arises in part from the fact that all vibrations 
 arising from the elasticity of matter are cases of simple 
 nature' * n harmonic motion. For, by Exs. VIII and IX, the restor- 
 ing forces called into play by any sort of strains in 
 material bodies are proportional to the displacements, so long as 
 the limits of perfect elasticity are not exceeded. Thus the vibra- 
 tions of weights suspended from springs, the vibrations of tuning 
 forks, of the strings of any stringed instrument, of masses vibrat- 
 ing under the torsion of a wire, are all cases of simple harmonic 
 motion. 
 
 A body oscillating about an axis has simple harmonic motion 
 of rotation when each of its particles 
 , . moves with linear simple har- 
 
 Cnaracteris- A 
 
 of C f Q j^M n o/ mon ic motion. Thus let the 
 rotation. ]i ne om (Fig. 54) oscillate be- o 
 tween the positions oc and ob, and let each 
 
 point n of the line follow the equation of 
 
 ,. -I T i- FIGURE 54 
 
 linear simple harmonic motion, viz., 
 
 a = d. Since the linear quantities d [= be] and a [= mp] are 
 t 
 
 related to the corresponding angular quantities 6 and a by tlic 
 
 J 2 
 
 equations -^ = and -^ = a [R = om], the equation a = -5- d may be 
 H It t 
 
 at once transformed into 
 
 *-*. 
 
 The combination of this simple harmonic rotational equation with 
 the general rotational relation Fli = la [see (81) p. 79], gives 
 
SIMPLE HARMONIC MOTION 91 
 
 Wh 
 
 - is the force constant of the rotational system. In the case 
 
 u 
 
 Force con- ' n wn ^ cn the Dotation was due to torsion this constant 
 
 stant of Jjri 
 
 rotation * T&t\o - z - was given the name "moment of torsion" and 
 u 
 
 was represented by the symbol T (see Ex. IX). 
 
 It is evident from (100) that the moment of inertia / of any 
 body may be experimentally determined by suspending it from a 
 iandT from wire of known moment of torsion T and observing the 
 half-period of vibration ^. Thus, 
 
 ' j m / 2 
 
 or I,*-. (101) 
 
 Or again, if the moment of torsion T of the suspending wire be 
 not known, the observation first of ^ and then of a second period 
 t% in which the system is caused to vibrate by the addition to 1 
 of a known moment of inertia / , furnishes all necessary data for 
 determining either /or T . Thus the half -period of the system 
 after the addition of I is given by 
 
 The elimination of T from: (101) and (102) gives 
 
 7 -^V ( 103 > 
 
 Or the elimination of /from (101) and (102) gives 
 
 Experiment 
 
 1. To determine the force constant of a spiral spring, and to 
 
 compare the observed and calculated values of the periods of the 
 
 spring for different loads. 2. To calculate the periods 
 
 of several "torsion pendulums" from the known T^s 
 
 of the suspending wires, and to compare with observations. 3. To 
 
 determine / and T by the addition to a vibrating system of a 
 
 known moment of inertia I . 
 
 DIRECTIONS. 1. First test Hooke's Law for the spring and 
 determine its force constant by observing the elongations produced 
 
92 
 
 MECHANICS 
 
 Force con- 
 stant of 
 spiral spring. 
 
 by the successive addition of 100 gm., 200 gm., 300 gm., 400 gm., 
 beginning with a load of 50 gm. A graduated mirror placed behind 
 
 the spring (see Fig. 55) enables 
 the position of the index for any 
 value of m to be accurately de- 
 termined. In taking a reading place the 
 eye so that the image of the tip of the index 
 is brought into line with the tip of the 
 index itself. In computing the mean elon- 
 gation per hundred grams use the method 
 outlined on page 76. In computing the 
 
 force constant -s express all forces in dynes, 
 
 all lengths in centimeters. 
 
 Now replace m by other masses, e. g., 
 150 gm., 250 gm., 350 gm.,and, from the 
 force constant just determined, 
 
 m 
 
 FIGURE 55 
 
 Periods/or . 
 
 different calculate in each case what should 
 be the period of the suspended 
 
 system when it is set into vertical oscilla- 
 tion. Compare these theoretical values of 
 
 the periods with observed values obtained by taking with a stop- 
 watch the time of 50 vibrations. 
 
 2. The torsion pendulum here used con- 
 sists of a large disk suspended as in Fig. 56 
 
 from one of the wires whose force 
 
 Periods of 
 
 torsion pen- constant (moment of torsion) was 
 found in Ex. IX. First find the 
 weight of the disk by means of the platform 
 scales, then measure its diameter and compute 
 its moment of inertia / [see (85) p. 80]. From 
 / and the value of T found in Ex. IX com- 
 pute the period. Compare this with the 
 observed period obtained by taking with a stop- 
 watch the time of 25 vibrations. In setting the 
 disk into oscillation, do not twist the wire 
 through an arc of more than 5 or 10. 
 
 3. The determination of T by the addition 
 
 of a known moment of inertia I is accomplished FIGURE 56 
 
SIMPLE HARMONIC MOTION 93 
 
 by adding to the plate P (Fig. 56) the large brass ring D, and 
 observing the new period t 2 . T is then given by (104). I can 
 T O and 1 of course be easily calculated from its mass and mean 
 moment of d diameter [see (84) p. 80]. Care must be tak^h to make 
 inertia. the wire pass through the center of the ring ; otherwise 
 the calculated value of I will be incorrect. Determine in this 
 manner T for each of the three wires used in Ex. IX, and then cal- 
 culate from each wire the coefficient of rigidity n of steel. [See 
 (80) p. 75]. 
 
 From the same observations calculate with the aid of (103) the 
 moment of inertia of the brass disk, and compare with the 
 theoretical value (see 85). 
 
 Record 
 
 Added Scale 
 W'ts Read's Dif's 
 
 | _ 
 
 100 
 
 200 
 300 
 400 
 
 mean d per 100 gm. = 
 force cons't of spring = 
 
 Masses (obs.) (calc. ) % error 
 150 
 250 
 
 2. Mass of disk P= Radius = 
 Wire 1, To (from IX) - . '. t x calc. - 1 1 obs. 
 Wire 2, T (from IX) - - . '. t x calc. - - 1 1 obs. - 
 Wire 3, T (from IX) - . . 1 1 calc. t x obs. - 
 
 3. Mass of ring D = -- Mean radius = 
 
 Wire 1, t 2 = . . To = --- . . n = -- % dif. from mean = 
 Wire 2, t. z = .-.T = -- .-. = " " " " = 
 = - -.-. T = - -.:n = - -" " " " = 
 
 / from 1 from 2 -- from 3 Mean % error -- 
 
 Problems 
 
 1. Within a solid sphere of uniform density the force varies 
 directly as the distance from the center. If the earth were such a 
 sphere, and if a hole passed completely through it along a diam- 
 eter, how long a time would be required for a body dropped 
 through the hole to reach the other side? 
 
 Take the radius of the earth as 4000 miles. 
 
 2. A horizontal wire one meter long clamped at both ends is 
 set into vibration in a vertical plane. The amplitude at the mid- 
 
94 MECHANICS 
 
 die is 4 mm. Find the shortest period which is permissible if 
 the rider at the mid-point is at no instant to lose contact with 
 the wire. 
 
 3. Show that the apparent motion of a bright point on the rim 
 of a distant wheel, rotating at uniform speed about an axis at 
 right angles to the observer's line of sight, is simple harmonic. 
 
XII 
 DETERMINATION OF "0" 
 
 Theory 
 
 Let Fig. 57 represent any irregular body which is oscillating 
 Moment of un( ^ er the action of the force 
 force restor- o f gravity about a horizontal 
 
 my a pen- <* 
 
 duium. axis at o. Let ? be the dis- 
 
 tance from o to c the center of gravity 
 of the body. 
 
 In order to find the law which gov- 
 erns the motion of the body it is neces- 
 sary to express the moment of force Fh 
 which is acting upon the body at any 
 instant in terms of the angular dis- 
 placement at that instant from the 
 position of rest. If M be the mass 
 
 of the body, the total force acting upon it is Mg dynes, and this 
 force is applied at the center of gravity c (see Ex. IV, pp. 34, 
 35). The moment of this force is therefore Mg x dc. But since 
 dc = I sin 0, it follows that 
 
 Mg 
 
 FIGURE 57 
 
 Fh = Mgl sin 0. 
 This equation shows that the motion is 
 
 The period 
 
 duium. 
 
 (105) 
 
 not simple harmonic, 
 for the restoring moment Fh is not proportional to the displace- 
 men t but to sin 0. Nevertheless, as approaches 
 zer ' s ^ n ^ approaches 0. Hence in the limit, i. e., in 
 the case of vibrations of infinitely small amplitude, 
 pendular motions follow the law of simple harmonic motion. The 
 simple harmonic formula may then be applied to pendulum problems 
 if only the arc be kept so small that the error introduced by the 
 approximation sin = is smaller than the necessary observa- 
 tional errors of the experiment. This means that in the following 
 experiment 6 should not exceed 5. Under these conditions, 
 
 95 
 
96 MECHANICS 
 
 then, the pendulum formula is 
 
 J71 7 
 
 i. e., the fcjrce constant of the motion, viz. ^-i is equal to Mgl. 
 Substitution in the S. H. M. formula (100) gives, then, 
 
 This is the general formula for the compound pendulum. If 
 Period of the pendulum be merely a particle suspended from 
 pendulum. & weightless thread of length Z, then 
 
 / (= 2wr 2 ) = M l\ (107) 
 
 Therefore, for such a pendulum, (106) becomes 
 
 i. (108) 
 
 ^7 
 
 The length of a compound pendulum is defined as 
 ^6 length of the simple pendulum which has the same 
 period. 
 
 The center of oscillation of a compound pendulum is the posi- 
 tion of the particle which is oscillating naturally, i. e. , just as it 
 would oscillate if it alone were suspended as a simple pendulum 
 from the point of support. 
 
 The radius of gyration Tc of any rotating body is the distance 
 from the axis at which the whole mass M might be concentrated 
 without changing the value of the moment of inertia /. Thus the 
 equation, 
 
 I=^mr* = Mlc* (109) 
 
 defines the radius of gyration &. 
 
 Experiment 
 
 To find g by determining the length and the period 
 of a simple pendulum. 
 
 A simple pendulum is chosen for the determination, because 
 
 the length of such a pendulum can be measured directly, while 
 
 the length of a compound pendulum is not easily 
 
 theory din obtainable. The time measurement consists in com- 
 
 paring, by the method of coincidences, the period of the 
 
 unknown simple pendulum with that of a compound pendulum 
 
 of known period. The electric circuit of the battery B (see Fig. 
 
DETERMINATION OF "</" 
 
 97 
 
 58) is completed through the electro-magnet E, the contact 
 points c and d, and the two pendulums A and C. The length of 
 the simple pendulum A is adjusted until its 
 period is nearly the same as that of the 
 known pendulum C. When both pendulums 
 strike the mercury contacts at precisely the 
 same instant, the click of the sounder (or 
 the stroke of the bell) E is heard. Since 
 the pendulums have slightly different pe- 
 riods, no further sound is heard until the 
 faster pendulum has gained one half-vibra- 
 tion upon the slower, when the conditions 
 are right for another click. Theoretically, an 
 observation of the interval elapsing between 
 these two successive coincidences is sufficient 
 for the determination of the half-period t of 
 the unknown pendulum. For, division of 
 the time interval between coincidences by the 
 half -period of C gives the number of vibra- 
 tions made by C during the interval. The 
 addition (or subtraction) of 1 to this num- 
 ber gives the number of half-periods of A 
 during the same interval. Hence the divi- 
 sion of the length of the interval by this 
 number gives the half-period sought, viz. /. 
 However, on account of the difficulty of 
 observing accurately the exact instant of a 
 coincidence, it is preferable to observe the 
 time interval between two coincidences which 
 
 area considerable distance apart, e. g., the interval between the 
 1st and the 30th coincidence. The calculation is then made pre- 
 cisely as outlined, save that the 1 is replaced by 29. 
 wac$ce n ^k* 8 re( ^ uces the Observational error to ?\th its former 
 value. It is not necessary, however, to watch the 
 pendulums through 30 coincidences. For, if the interval between 
 two successive coincidences be somewhat accurately observed, the 
 number of coincidences which have occurred within any longer 
 interval bounded by two coincidences must evidently be the 
 nearest whole number obtained by dividing the long interval by 
 
 FIGURE 58 
 
98 MECHANICS 
 
 the time between two successive coincidences. The divisor, how- 
 ever, must be determined with such exactness as to remove all 
 uncertainty as to what that whole number is. 
 
 The difficulty of determining the exact instant of a coincidence 
 is increased by the fact that, on account of the finite width of the 
 contact points c and (7, the clicks will often continue through a 
 number of swings. The mean time between the first and last 
 click is then taken as the instant of coincidence. 
 
 First very carefully adjust c and d until, when A and C are at 
 rest, each pendulum point touches the middle of its contact. 
 Then make battery connections as in the diagram. 
 Next by means of a thread tie back pendulum A a 
 distance of about 2 inches and set it into vibration by burning the 
 thread. If a click is heard at every passage, set the heavier pen- 
 dulum (7 into vibration, giving it an amplitude somewhat smaller 
 than A's. For the determination of t use an ordinary watch, but 
 not one which has an error of more than 10 sec. per day. Take, as 
 accurately as possible, the times of the first four coincidences; 
 then allow the pendulum to swing for 20 or 30 minutes and take 
 the time of another coincidence. During the interval watch the 
 pendulums very carefully to see which is the faster, observing 
 just after a coincidence and noting which reaches the end of its 
 swing first. The length of A may be taken, without appreciable 
 error, as the distance from the knife-edge n to the center of the 
 ball. Leaving the pendulum in place, measure the diameter of 
 the ball with the vernier-calipers (see Appendix), and the distance 
 from the top of the ball to the knife-edge with a meter-stick. 
 
DETERMINATION OF "0" 99 
 
 Record 
 
 h. m. s. Coinc. perfect at 
 
 1st coincidence began at h. ra. s. Interval 
 
 ended at ) m. s. 
 
 3d began at 
 
 " " ended at 
 
 3d " began at 
 
 " ended at - 
 
 4th " began at - 
 
 " ended at 
 
 Last " began at 
 
 ended at - 
 
 Interval bet. 1st and last - No. coincidences in interval = 
 
 No. vib's of C in interval = - No. vib's of A = .'. t = 
 
 Length of wire - Diam. of ball - . . length of pend. - .\g = 
 
 Problems 
 
 1. The moment of inertia of a long and thin cylindrical body, 
 oscillating about one end, is ^MD, L being 
 
 the length of the body and M its mass. Find 
 the radius of gyration of the body in terms of 
 L. Find the length of the simple pendulum 
 which has the same period. 
 
 See (109) and (106). FlGUBE 59 
 
 2. A rigid pendulum oscillating about a 
 
 horizontal axis has a period of .75 sec. Find its period when the 
 axis is inclined 45 to the horizontal. (See Fig. 59.) 
 
 3. The period of a pendulum at the sea level was 1.002 sec. 
 It was carried to the top of a mountain and the period found to 
 be 1.005 sec. Find the height of the mountain. 
 
XIII 
 THE LAW OF CENTRIPETAL FORCE 
 
 Theory 
 
 Acceleration has been defined as rate of change of velocity. But 
 
 velocity is a directed quantity (a so-called vector) and may change 
 
 either in direction, or in mere numerical value, or in 
 
 correspond- both. The term speed has been agreed upon to denote 
 
 ing to change , . .'- ,T ,. / 
 
 in speed the numerical value of the velocity. If, then, (case 1), 
 the velocity of a body at any time is represented by the 
 line 0?*! (see Fig. 60), and if its velocity 
 after a lapse of t seconds is represented 
 by 0r 3 , then there has been no change in 
 FIGURE 60 direction, but only a change in speed, 
 
 and the mean acceleration has been - The expression for the 
 
 t 
 
 acceleration at the instant at which o?'! represents the velocity is 
 evidently the limit of this quantity as t approaches zero; thus, 
 
 (110) 
 
 = 0. 
 
 But if, (case 2), after the lapse of the t seconds, the velocity is 
 
 represented not by or 3 but by or 2 , a line equal in length to or^ then 
 
 there has been no change in speed, but only a change in 
 
 Acceleration ., i * < 
 
 correspond- direction, and, bv the principle of composition of 
 
 ing to change 
 
 indirection directed quantities (see lix. Ill, p. 22), the new velocity 
 or z is equivalent to the two simultaneous velocities or x 
 and TVs, i- e -> to the old velocity oi\ plus a new velocity ?vy ?Va 
 is then the line which represents the gain in velocity during the 
 time t. Hence the mean acceleration during the interval t has been 
 
 the value of the acceleration at the instant at which or l 
 
 represents the velocity is 
 
 -(-) CUD 
 
 t /t=o. 
 100 
 
THE LAW OF CENTRIPETAL FORCE 
 
 101 
 
 Since, as t approaches zero, r x r 2 becomes more and more nearly per- 
 pendicular to or l5 it is evident that in the limit represented by 
 (111) the acceleration a is at right angles 
 to the velocity oi\. No win Ex. II force 
 was defined as that which changes the mo- 
 tion of a body, and, by the Second Law, 
 whether < the change is one of direction 
 or of speed, the measure of the force is 
 always ma, and the direction of the force 
 is the direction of 'the vector a. Thus, in 
 
 case 1 , the force ma = m ( -y- 3 \ acts 
 
 in the direction of the velocity, i. e. in the 
 direction oi\. In case 2 the force 
 
 ma\ = 
 
 acts in the direction 
 
 (iV'g^oj a direction which is rigorously perpendicular ft 
 velocity or^ 
 
 Apply these principles to the consideration of the case of a 
 
 body m moving with uniform speed S upon the circumference of 
 
 a circle. (See Fig. 61). Let oi\ represent the velocity 
 
 circular at the end of the interval t, and let o'r, or the equal and 
 
 ilini inn* ^B 
 
 parallel line or^ represent the velocity a"he beginning 
 of the interval t. Since the velocity is continually changing in direc- 
 tion, a force/ must continually act, and since the direction of this 
 force is always at right angles to the velocity (see preceding para- 
 graph), it must act continually toward the center. Its value is then 
 
 f =ma = m (p~\ (112) 
 
 But since S represents the constant speed and t the element of 
 time considered, it is evident that 
 
 o'o = St. 
 
 But 
 
 TT /\ ty *^ ^ 
 
 Hence r^\ = o ' = jr- 
 
 Substitution of this value in (112) gives 
 
 mS* 
 /- ~ R ' 
 
 (113) 
 
102 MECHANICS 
 
 Cf 
 
 But if o> represent the angular speed, then o> = 
 
 H 
 
 Hence f = = m*R, (114) 
 
 an equation which asserts that the central force which must 
 be applied to keep a body in a circular orbit is directly propor- 
 tional both to the second power of the angular velocity, and to the 
 first power of the radius of the orbit. 
 
 Experiment 
 
 Object. To verify the law of centripetal force. 
 
 The masses m x which slip along the rod ab (Fig. 62) are 
 
 attached by cords which pass over pulleys in the case p to the slid- 
 
 , . v ing collar c. The central force which is necessary to 
 
 ' kesp the weights moving in a circle is represented by 
 
 ^ the 1 tension jin the cards. For a certain critical value of the speed 
 
 this tension is equal to the weight of the collar c. In order, there- 
 
 nii p mi 
 
 FIGURE 62 
 
 fore, to verify the law stated in (114), it is only necessary to 
 measure the radius R, the masses m l and c, and to observe the speed 
 required to lift c. All forces must of course be expressed in 
 dynes, all masses in grams, all linear distances in centimeters, all 
 angular distances in radians. 
 
 First count the number of revolutions of the axle t*b one revo- 
 lution of the wheel Jc. Then, from measurements upon 
 m lt c, and 72, calculate what number of turns N of the 
 wheel per second is necessary just to lift c. 
 
THE LAW OF CENTRIPETAL FORCE 103 
 
 To obtain an experimental value of the same quantity, let one 
 experimenter maintain a constant rotation of k at such speed that 
 the collar c is either held balanced between the upper and lower 
 stops, or else continually oscillates back and forth between them 
 (the stops are so arranged that c is free to move through, only 
 about 1 mm.). Then, as soon as this constant condition is 
 attained, let a second experimenter take with a stop-watch the 
 time of fifty revolutions, repeating several times in order to test 
 the accuracy of the observations. Then change both m and R 
 and repeat. Compare in each case the observed an*d calculated 
 values of the speed. 
 
 Record 
 
 1st value of m t = R= c = ^-^ . '. N calc. = 
 
 N obs. 1st trial = - 2d = - 3d = - mean =^ ft error = 
 
 2d value of m l = R = c = . . N calc. = 
 
 N obs. 1st trial = - 2d = - 3d = mean = - % error = 
 
 3d value of m 1 = R = c = . . N calc. = 
 
 N obs. 1st trial = 2d = 3d = mean = % error = 
 
 Problems 
 
 1. Taking the radius of die earth as 6370 kilometers, find 
 how many dynes of force are required to hold a grarifof mass upon 
 the surface (1) at the equator; (2) in latitude 45. Hence, find 
 what would be the values of g at the equator and in latitude 45 
 if the earth did not rotate. Also find how many times the velocity 
 of rotation would need to be increased in order that bodies at the 
 equator might have no weight. 
 
 2. The radius of the moon's orbit is approximately 60 times 
 the radius of the earth. Calculate the force in dynes which must 
 act upon each gram of the moon's mass in order to hold it in its 
 orbit, the period of the moon's rotation being 27 days 8 hours. 
 Compare this result with the force of the earth's attraction upon a 
 gram of mass at the distance of the moon as computed from the 
 law of gravitation. It was precisely this computation which led 
 Newton to assert the law of gravitation. 
 
 3. A skater describes a circle of 10 meters radius with a speed 
 of 5 m. per second. What must be his angle of inclination to the 
 vertical in order that he may be in equilibrium? 
 
MOLECULAR PHYSICS 
 AND HEAT 
 
 XIV 
 BOYLE'S LAW 
 
 Theory 
 
 The elastic properties of gases were very early made the sub 
 ject of observation and speculation, but the first results of experi- 
 ments made for the purpose of discovering the exact 
 w re ^ a ^ on wn i cn exists between the pressure exerted by a 
 confined gas and the volume which it occupies, were 
 published by the English physicist Boyle in 1661 in a work 
 entitled "Defence of the Doctrine of the Spring and Weight of 
 Air." These experiments brought to light the law which has since 
 been called Boyle's Law; sometimes also called Mariotte's Law. 
 This law asserts that so long as the temperature remains constant, 
 the pressure which a gas exerts upon the walls of the containing 
 vessel is directly proportional to its density or inversely propor- 
 tional to the volume which it occupies ; symbolically, 
 
 F 8 
 
 , 
 
 or P! F! = P z F 2 = P 3 F 3 = etc. = constant. (116) 
 
 The French physicist Mariotte independently discovered the same 
 law fifteen years later. 
 
 Before the discovery of Boyle's Law, in fact before the begin- 
 ning of the Christian era, two theories had been advanced to 
 account for the elastic properties of "air. The first was the repul- 
 sion theory, according to which the pressure exerted by confined 
 air was attributed to repellent forces existing between the mole- 
 
 105 
 
106 MOLECULAR PHYSICS AND HEAT 
 
 cules which were assumed to be at rest. This theory was held 
 by prominent scientists even as late as the middle of the nine- 
 teenth century. In order to reconcile the theory witli 
 
 The repulsion . . . J 
 
 theory of JBoyle s Law, it is necessary to assume that the mole- 
 cules repel each other with forces which are inversely 
 proportional to the distances between them. The theory has now 
 been altogether abandoned; first, because such a law of molecular 
 force is wholly at variance with all modern views as to the nature 
 of molecular force; second, because it necessitates the conclusion 
 that the pressure which a gas exerts is a function not of its 
 density and temperature alone, but also of the shape and size of 
 the containing vessel, a conclusion which is directly contradicted 
 by experiment; third, because the fact that a gas does not 
 experience a rise in temperature when it expands into a vacuum, 
 proves that no repulsion exists between its molecules. 
 
 According to the kinetic theory, the pressure which a gas 
 exerts against the walls of a containing vessel is due to the bom- 
 bardment of the walls by rapidlv moving molecules 
 
 The kinetic ... _. J J 
 
 theory of which at ordinary pressures are so far apart that they 
 exert no forces whatever upon one another, and which 
 occupy so little space themselves that the total number of impacts 
 per second against the walls is simply the product of the number 
 of molecules present and the number of impacts which one single 
 molecule would make if it were alone in the vessel and were mov- 
 ing with the mean velocity of all the molecules. Although a 
 crude form of this theory is as old as Greek philosophy, it can not 
 be said to have taken definite shape before about 1738, when it 
 was advanced by Daniel Bernoulli i. It did not gain general 
 acceptance until the middle of the nineteenth century, when the 
 labors of Joule in England and of Clausius in Germany won for 
 it well-nigh universal support. 
 
 While the repulsion theory was unable to account for Boyle's 
 Law without the aid of a highly improbable assumption, the 
 kinetic theory furnishes an immediate explanation of this law. 
 For, manifestly, if gas pressure is due to impacts alone, its value 
 at any instant must be the product of the mean force of each 
 impact and the number of impacts taking place at that instant 
 upon a square centimeter of surface. For a given gas, the first 
 factor would depend simply upon the mean velocity of the mole- 
 
BOYLE'S LAW 107 
 
 cules. For a constant mean velocity, the second factor would be 
 proportional to the number of molecules present in a cubic centi- 
 meter; i.e., to the density. If, then, the constancy of the mean 
 molecular velocity be taken as the condition of constant temper- 
 ature, it follows at 'once from the kinetic theory, that the pressure 
 should be directly proportional to the density. This is Boyle's 
 Law. 
 
 Up to 1848 Boyle's Law was supposed to hold rigorously for 
 the so-called permanent gases. In this year, however, the French 
 
 physicist Regnault performed very careful experiments 
 frrm Boyle's which showed that for air at ten atmospheres pressure, 
 
 the product P V differs by about one-fourth of one per 
 cent from its value at one atmosphere. At higher pressures, the 
 departure is more marked, amounting at 600 atmospheres to more 
 than 25 per cent. These departures are evidence for, rather than 
 against, the correctness of the kinetic theory; for when the 
 molecules are crowded so close together that the space which they 
 themselves occupy is no longer negligible in comparison with the 
 total volume of the vessel, then the kinetic theory would require 
 that the. pressure increase more rapidly than the density; i.e., 
 that the product P V increase. This is what actually occurs in 
 the case of all gases when the pressures are very high (100 atmos- 
 pheres or more) . For moderate pressures (1 to 50 atmospheres), the 
 departures are in the opposite direction in the case of all gases 
 excepting hydrogen (and probably also helium) ; i.e., the pressure 
 increases less rapidly than the density, or, in other words, PV 
 decreases. This is due to the fact that the attractive forces 
 between the molecules are not wholly negligible in comparison 
 with the forces of impact. 
 
 Experiment 
 
 Object. To verify Boyle's Law for ordinary pressures. 
 
 The body of dry air which "is to be experimented upon is con- 
 tained in the upper part of the graduated tube a (see Fig. 63), 
 which has a diameter of about 1 cm. The lower end 
 k of this tube is beneath the mercury in the cistern AB. 
 
 The air-jacket which surrounds a serves to maintain a constant 
 temperature throughout the experiment. If it is desired to take 
 
108 
 
 MOLECULAR PHYSICS AND HEAT 
 
 4 
 
 T 
 h 
 
 LL 
 
 still further precautions, the upper part of AB may 
 be filled with water, although this is generally found 
 to be unnecessary. The tube a is graduated in cc. 
 so that the volume V may be obtained directly by 
 reading the scale u^on a. The determination of 
 the pressure P, which corresponds to any par- 
 ticular volume F, requires the observation, first, of 
 the barometer height H, and second, of the differ- 
 ence in level D between the mercury in AB and in a. 
 This observation is made by means of the cathetom- 
 eter (see below). It is evident, therefore, that if the 
 temperature of the mercury in the barometer is the 
 same as that of the mercury in AB, then P, ex- 
 pressed in centimeters of mercury, is equal to 
 (II - D). If the two temperatures are different, H 
 must be corrected by multiplying the observed height 
 by the ratio of the density of 
 mercury at the temperature 
 of the barometer and its 
 density at the temperature 
 of AB. This correction is 
 wholly negligible in this ex- 
 periment unless the differ- 
 ence in the two temperatures 
 amounts to more than 5C. 
 DIRECTIONS. To deter- 
 mine the barometric height 
 H, proceed as follows : By 
 means of the thumbscrew s 
 [see Fig. 64, (1)], raise or 
 
 lower the level of the mercury in ^hc 
 
 cistern E of the barometer until the ivory 
 
 point n just touches the mercury surface. 
 
 This setting can be made with great accu- 
 racy by observing when the 
 
 . J J f> . . 
 
 image oi the point which is 
 
 . 
 
 seen in the mercury, just ap- 
 pears to come into contact with the point 
 itself. This point n is the zero of the FIGURE ei 
 
 FIGURE 63 
 
 Reading of 
 
 eter. 
 
 (2; 
 
BOYLE'S LAW 109 
 
 scale which is attached to the upper portion of the metal case 
 surrounding the barometer tube. By turning the milled head D, 
 move the vernier c with which the scale is provided until its lower 
 end is clearly above the convex surface of the mercury. Then 
 carefully lower it until it appears to be just in 'contact with the 
 highest point of this convex surface. During this operation, 
 keep the eye in such a position that the back lower edge of the 
 vernier tube seems to coincide with the front lower edge. To 
 test the setting, move the head up and down, and see to it that 
 the white background behind the barometer never becomes visible 
 above the top point of the meniscus. Read the scale and ) vernier 
 (see Appendix for theory of vernier). The capillary correction 
 for the barometer is to be obtained from the Appendix table, 
 which is headed " Capillary Depression of Mercury." 
 
 Adjust the cathetometer (see Fig. 65) in three steps, as fol- 
 lows: 
 
 (1) To make the column vertical, 
 
 loosen the set screw s% and turn 
 
 the column until the tel- 
 
 Adjmtment 
 
 of the escope is at right angles 
 
 cathetsrmeter. 
 
 to the line connecting 
 two of the feet; e.g., A and B. 
 Then bring the bubble to the mid- 
 dle by means of the leveling screw 
 in the third foot C. Xext rotate 
 the column through 180 about its 
 own axis, and if, after rotation, the 
 bubble is displaced from the middle, 
 correct half of the angular error by 
 means of the leveling screw in the 
 foot C and the other half by means of 
 the screw s which inclines the tele- 
 scope. If the bubble is against one 
 end of the level-tube, it is , impossi- 
 ble to know when just half of the 
 angular correction has been made., 
 Hence it is necessary first to estimate- 
 roughly these half-corrections, then 
 to rotate again through 180, and FIGURE 65 
 
110 MOLECULAR PHYSICS AND HEAT 
 
 again to correct, and thus to proceed until the bubble remains, 
 upon reversal, somewhere near the middle of the tube. The half- 
 corrections can then be read off accurately upon the scale on the 
 level-tube. When this adjustment has been made to such a 
 degree of accuracy that a reversal displaces the bubble through 
 perhaps one or two divisions, turn the column this time through 
 90, i.e., until the level is parallel to the line connecting A and 
 B, and bring the bubble back to the middle by turning the level- 
 ing screws A and B equal amounts in opposite directions. The 
 column should now be approximately vertical. In order to make 
 it accurately vertical, the whole operation must be repeated 
 from the beginning with more care. After the completion of the 
 second leveling, rotation of the column into any position what- 
 ever should not cause a displacement of the bubble of more than 
 half of a division. 
 
 (2) To make the line of sight coincident with the axis of the 
 telescope, first focus the eye-piece carefully upon the cross-hairs by 
 slipping the former forward or back in the draw-tube ; then, by 
 means of the rack and pinion with which the draw-tube is pro- 
 vided, focus the telescope sharply upon the scale on tube a (see 
 Fig. 63), which should be set at a distance of about a meter from 
 the cathetometer. If moving the eye slightly from side to side 
 causes the cross-hairs to appear to move at all with reference to 
 the scale, repeat both of these focusings until this parallax effect 
 is wholly removed. Now turn the telescope in its socket until 
 one of the cross-hairs is parallel to the scale divisions, and by 
 means of the screw s 3 , which moves the whole telescope up or 
 down, set this cross-hair upon some chosen division of the scale ; 
 then rotate the- telescope in its socket, i.e., about its own axis, 
 through 180. If this operation changes at all the reading of the 
 cross-hair upon the scale, correct half of the error by means of s 3 
 and the other half by means of the small screw which is found at 
 one side of the eye-piece and which moves the cross-hair across 
 the field of view. Repeat this adjustment until rotation of the 
 telescope through 180 produces no change in the reading. The 
 line of sight then coincides with the axis of the telescope. 
 
 (3) To make the axis of the telescope horizontal, proceed in 
 either one of the following ways [method (b) is generally to be 
 recommended] : (a) Bring the bubble to the middle by means of 
 
BOYLE'S LAW 111 
 
 screw s, then lift the level carefully from the telescope-tube, turn 
 it end for end and replace. If this operation displaces the bubble, 
 correct half of the error by means of s, the other half by means 
 of the screw Sj, which adjusts the position of the level-tube in 
 its case. The telescope-tube, and hence also the line of sight, 
 will be horizontal when no displacement of the bubble is produced 
 by a reversal of the level, (b) Set the cross-hairs of the telescope 
 upon some point on a scale about a meter distant. Then take 
 the telescope out of its socket, turn it end for end and replace. 
 Next rotate the vertical column through 180 and look again at 
 the chosen point. If the cross-hairs are no longer upon it, correct 
 half the displacement by means of the screw s which inclines the 
 telescope, and half by means of the screw s s which raises it 
 vertically. The telescope-tube will be horizontal when, after 
 reversal and rotation, the same point of the scale comes under the 
 cross-hair. 
 
 When the cathetometer is in complete adjustment, carefully 
 
 loosen the set screw s^ slide the telescope up or down the column 
 
 until the cross-hair is near the top of the meniscus of 
 
 Reading the ,, ., . . . _ , -. rtrtX , 
 
 mercury the mercury in the cistern AB (see ig. 60), clamp s 4 
 and make the final setting upon the meniscus by means 
 of the fine adjustment screw s 3 . Then take the reading of the 
 vernier upon the scale of the cathetometer column. Next raise 
 the telescope, set the cross-hair upon the top of the mercury 
 meniscus in the tube , and take a second reading. The differ- 
 ence between the two readings gives the distance D. Take a 
 number of observations of this height in order to see to what 
 degree of accuracy it is obtainable. At the same time read 
 through the telescope the volume F upon the scale on tube a. 
 This reading should not be taken either at the top or at 
 the bottom of the meniscus, but at such an intermediate point 
 as would correspond to the same volume if the meniscus were 
 flat. This point can be obtained only by careful estimate. 
 Starting with a pressure which is but a trifle less than one 
 atmosphere, vary the volume by five about equal steps until it 
 is as large as can be conveniently obtained with the apparatus, 
 and take the five corresponding pressure readings. If the 
 barometric height varies during the experiment, take this fact 
 into account. 
 
112 
 
 MOLECULAR PHYSICS AND HEAT 
 
 H Read'gs 1st trial 
 
 in AS in a .-. D in AB in a .-. D 
 
 Record 
 
 Read'gs 2d trial Mean .-. P . . PV % <* 
 , A. . j\ from 
 
 FIGURE 66 
 
 Problems 
 
 1. Make estimates of the probable observational errors in both 
 P and V above, and thence deduce the maximum permissible 
 error in P V. Cpmpare with last column of the record. 
 
 2. A level is in adjustment when the line joining the points upon 
 
 which it rests is parallel to the tangent 
 drawn to the highest point of the level- 
 tube, i.e., when the line cd (see Fig. 66) 
 is parallel to the line ab. Show why in 
 ~ adjusting a level which is out of ad- 
 justment and in leveling the table upon 
 
 which it rests (see Fig.) it is necessary after reversal to correct 
 
 half at s and half at s'. Hence justify 
 
 throughout the methods used in adjusting 
 
 the cathetometer. 
 
 3. A confined body of air V is placed 
 under a pressure of P mm. of mercury 
 (see Fig. 67. P = H+ab). By means of 
 the three-way stopcock s, the connection 
 between the two arms (see Fig.) is then 
 shut off and a volume v of mercury drawn 
 from the right arm. The level of the mer- 
 cury in this arm then sinks to the point 
 a'. Connection between the arms is then 
 reestablished by means of the cock and the 
 left arm lowered until the level in the right 
 arm is again at a' . The pressure upon 
 the air in the bulb is now found to be 
 
 h mm. less than at first. Find F, the FIGURE e? 
 
BOYLE'S LAW 113 
 
 volume of the bulb down to the point , in terms of v, 
 P, and h. 
 
 4. In Problem 3, V was found to be 500 cc. A pow- 
 der was introduced into F and the mercury in the right 
 arm brought back to its original height a. The pressure 
 was then found to be 900 mm. 250 cc. of mercury were 
 drawn off at s precisely as above, and the pressure was 
 found to fall to 500 mm. Find the volume of the powder. 
 
 5. A volume of 30 cc. of air is confined in the closed 
 
 arm of a manometer (see Fig. 68) . In the open arm the mercury 
 stands 60 cm. higher than in the closed arm. What will be the 
 difference in the levels when the volume of the air is reduced to 
 10 cc.? (Bar. Ht. = 76 cm.) 
 
XV 
 
 DENSITY OF AIR 
 
 Theory 
 
 The existence of Boyle's Law makes the determination of the 
 
 density of air at a given temperature a very simple matter, at 
 
 least in theory. For, let a globe A of known volume 
 
 Density of . . J . 
 
 airfromtwo V containing air under atmospheric pressure P, be 
 
 pressures, a 
 
 weight and a balanced upon a beam 01 equal arms against some 
 
 Then let air be forced into the 
 
 Suppose that the weight 
 
 body PF (see Fig. 69). 
 globe until the pressure within it is P 2 . 
 now needed to balance the globe 
 is W + w. Since it requires the 
 , same weight to balance the glass 
 in each case, it is evident that 
 w is the weight of the air which 
 has been forced into the globe in 
 changing the pressure from P 1 
 to P 2 . If, then, V represent the 
 
 volume of the globe, and d\ and d% the densities of air corre- 
 sponding to the pressures P l and P 2 respectively, it is evident that 
 
 FIGURE 
 
 (117) 
 
 But by Boyle's Law, 
 
 dl Pl 
 
 The elimination of d z from 
 which is sought, viz., 
 
 (117) and (118) gives the density 
 
 wP, 
 
 (119) 
 
 FOP, -P.) 
 
 If the counterpoise W consists of a few small weights and a closed 
 glass globe of the same external volume as A, any errors which 
 would arise from changes in the barometric height or the tem- 
 perature during the experiment are altogether eliminated, pro- 
 
 114 
 
DENSITY OF AIR 115 
 
 vided only that the temperature at which the air was introduced 
 into the bulb under pressure P z is the same as that which corre- 
 sponds to P lt For, with this arrangement, the buoyant effect of 
 the air upon both sides of the balance is the same no matter how 
 rapidly the barometric pressure or the temperature may change. 
 This device was first used by Eegnault in his classical determina- 
 tion of the density of air. 
 
 Experiment 
 
 To find the density of dry air under existing con- 
 Object. T ' P 
 
 ditions ot temperature and pressure. 
 
 DIRECTIONS. 1. In order to obtain dry air for the experi- 
 ment, first see to it that the stopcock c is well greased, then 
 connect the bulb to an air pump through a calcium chloride drying- 
 tube in the manner shown in Fig. 70. A pump capable of produc- 
 
 FIGURE 70 
 
 ing a high vacuum is unnecessary. The water pump of Fig. 87 
 may be conveniently used. First close the pinchcock s and 
 exhaust. Then very slowly open s, and thus fill A with air whi h 
 
 has been dried by passage through the calcium chloride 
 ?nmebuU) air tube. Repeat this operation from two to six times, 
 
 according to the degree of exhaustion obtainable with 
 the pump. Then, in order that the bulb may assume accurately 
 the outside temperature, wait about five minutes before removing 
 the bulb from the drying-tube or closing the cock c. Observe the 
 temperature by means of a thermometer hung near the bulb, then 
 carefully close cock c, remove and weigh as follows : 
 
 2. First remove all dust from the bulb and the pans of the 
 analytical balance (see Fig. 71) by means of the camel's-hair 
 brush which will be found in the balance case. Then very care- 
 fully suspend the bulb from one of the hooks c and place upon 
 
116 
 
 MOLECULAR PHYSICS AND HEAT 
 
 To make 
 
 Hiefirst 
 
 weighing. 
 
 the other pan the counterpoise and a weight of a few grams from 
 the hox of weights, taking pains to touch these weights only with 
 the pincers, never with the fingers. Next release the 
 pans by pushing in and fastening the button e which 
 controls the pan-arrest ; then turn very slowly the milled 
 head a and thus lower the beam-arrest just enough to see whether 
 the pointer begins to move to 
 left or right; i.e., whether the 
 chosen weight is too heavy or too 
 light. This done, raise the beam- 
 arrest immediately, but so slowly 
 as not to endanger the knife- 
 edges by the slightest jar; re- 
 place the chosen weight by the 
 one next heavier or next lighter, 
 and try it in the same way. Take 
 the utmost care never to place a 
 weight on the pans or to take a 
 weight off except when the beam 
 is arrested. Proceeding thus, 
 make a systematic trial of the 
 
 know 
 
 gram weights until you 
 between what two consecutive numbers of grams the condition 
 of balance must lie; then try in the same way the milligram 
 weights in order of magnitude until a weight is found such that 
 when the beam-arrest is completely lowered the pointer oscillates 
 near the middle of the scale over from 3 to 6 divisions. A larger 
 swing than this indicates insufficient care in lowering the arrest. 
 The rider r may be used in place of the small milligram weights, 
 if desired, but no attempt should ever be made to add fractional 
 portions of a milligram by means of the rider. Before taking the 
 resting point, raise again the beam-arrest, taking pains to avoid a 
 jar by raising at a time when the pointer is at the middle of its 
 swing, close the face of the balance case so as to shut out all air 
 currents, and stop all swinging of the pans by alternately pressing 
 and releasing the button e which controls the pan-arrest. Then 
 carefully lower first the pan-arrest, then the beam-arrest, and 
 take the resting point R lm This is to be done by averaging the 
 mean of three successive turning points of the pointer on one side 
 with the mean of the two intervening turning points on the other 
 
DENSITY OF AIR 
 
 117 
 
 side. This use of an odd instead of an even number of turning 
 points eliminates completely the effect of damping. The following 
 example taken in connection with Fig. 72, which represents the 
 scale s of Fig. 71, will make clear the method of procedure: 
 
 TURNING POINTS 
 
 Left 
 
 15 
 
 6.8 
 7.1 
 
 7.4 
 
 Means 7.10 
 
 Right 
 11.-& 7 
 
 11.45 
 
 FIGURE 72 
 
 9.28 
 
 Having determined the resting point R^ slowly raise the beam- 
 arrest when the pointer is in the middle of a swing, then from the 
 absences in the box of weights, count the weights which have 
 been used and check by counting again as the weights are 
 replaced. Call the sum of these 
 weights Wi. Finally, close the bal- 
 ance-case, see that every weight is 
 in its proper compartment in the 
 box of weights, replace the pincers 
 in the box, and place the latter in 
 the drawer of the balance-case. 
 
 3. Fill the bulb with air under 
 pressure P 2 as follows: Attach it 
 
 securely to the pressure 
 
 a PP aratus shown in Fig. 
 
 73. This consists of a 
 bicycle pump p attached to an air- 
 tight jar J, which is furnished with 
 a mercury pressure-gauge g and a 
 calcium chloride drying-tube t. 
 Open communication between the 
 jar and the bulb through the dry- 
 ing-tube, and produce a difference 
 of level of 50 or 60 cm. in the 
 manometer arms. After waiting 
 about five minutes for the com- 
 pressed air to regain the tempera- FIGURE 73 
 
118 MOLECULAR PHYSICS AND HEAT 
 
 ture of the room, read simultaneously the two arms of the 
 manometer, and at the same instant close the stopcock c. Take 
 the temperature by means of a thermometer hung near the bulb. 
 If this temperature differs by more than half a degree from that 
 previously taken, the first weighing must be discarded and another 
 taken under the same conditions of temperature as those which 
 here exist. This need not be done until after the completion of 4. 
 4. To make the final weighing, place the bulb upon the same 
 scale-pan as before and, proceeding exactly as in 2, balance it by 
 means of such weights W t that the pointer again oscil - 
 the second lates near the middle of the scale. Then take the new 
 
 weighing. 
 
 resting point R z in precisely the manner described 
 above. If R% coincided exactly with R^ then evidently W z W l 
 would be the weight w which is sought. But, in general, 7? 2 will 
 not coincide with R^ and it is a very slovenly proceeding to 
 attempt to make it do so, as is often done, by repeatedly shifting 
 the position of the rider. The most rapid and the only correct 
 method of making an accurate weighing is to determine the 
 sensitiveness * i.e., tlie number of scale divisions which the pointer 
 is shifted ~b\j the addition of one milligram, and then to calculate 
 by interpolation the exact correction which must be applied to 
 W z in order to bring R 2 precisely into coincidence with J? x ; this is 
 done as follows : Immediately after finding R z , add to the lighter 
 side a small weight, say 2 mg. (1 mg. if the balance is very sensi- 
 tive; this may be done by means of the rider if desired), and 
 take the corresponding resting point 7? 3 . This procedure simply 
 determines the value in milligrams of the scale divisions. Thus, 
 if R 2 = 10.63, and if, upon the addition of 2 mj., R s = 7.01, then 
 
 10.63 7 01 
 the sensitiveness is - - =1.81. From the two resting 
 
 points RI and R*, and the sensitiveness S, it is a very easy matter 
 
 * Since, on account of the bending of the balance arms, the sensitive- 
 ness varies with the load, theory requires either that it be determined at 
 each weighing, or else that a table showing the variation of the sensitive- 
 ness with the load be made out once for all and kept for use with the 
 balance. Since, however, with good balances, it generally requires a very 
 considerable change in load to produce an appreciable change in the 
 sensitiveness, it is usually unnecessary to make more than one careful 
 determination of the sensitiveness so long as the loads involved are of 
 about the same magnitude. 
 
DENSITY OF AIR 119 
 
 to calculate the weight which would need to be added to or sub- 
 tracted from W z in order that the pointer might be brought 
 exactly to the original resting point E^ Thus, in this case, the 
 number of milligrams which would be required to move the pointer 
 
 from R, [= 10.63] back to E l [= 9.28] is W '^~^'^ = .74. This 
 
 number of milligrams must be added to or subtracted from W& 
 according as the point 10.63 is farther from or nearer to the 
 bulb than the point 9.28. Let W s represent the corrected value 
 of W z . Then W 3 - }\\ is the w of equations (117) and (119). 
 5. To find the volume of the 'bulb, fill it either with water or 
 with mercury, and weigh upon the trip-scales. Take the temper- 
 ature of the liquid used and obtain the density from a 
 table (see Appendix). From the weight and density of 
 the liquid calculate V. If the liquid used be mercury, 
 the filling can most easily be done by means of a funnel made by 
 drawing down one end of a piece of glass tubing so as to form a 
 capillary tube long enough to reach into the interior of the bulb. 
 When the bulb is full of mercury, it must of course be handled 
 with extreme care, the stopcock being always left open so as to 
 prevent breakage by expansion. Filling with water may be done 
 in the same way if sufficient pressure be applied to force the water 
 through the capillary tube. It may also be done by alternately 
 heating and cooling the air in the bulb, the neck being kept 
 under water during the cooling. If water be used, the bulb will 
 not be again fit for use until it has been thoroughly dried by 
 repeatedly exhausting after the temperature has been raised above 
 100 C. by carefully heating with a rapidly moving Bunsen flame. 
 The pressures which occur in the numerator and denominator of 
 (119), viz. P l and P z P 1? must of course be expressed in the same 
 units. If they are expressed in centimeters of mer- 
 
 Thebaropi- , J , , . , ,, 
 
 etercorrec- cury, the two columns oi mercury which they represent 
 must have the same temperature.* It is evident, then, 
 that since P% P l represents the difference in the readings of the 
 manometer arms at the temperature of the room, the barometric 
 reading P^ should also correspond to the temperature of the 
 
 * A difference in temperature not exceeding 5 C. leads to a wholly 
 inappreciable error. It may therefore be overlooked (see table of mercury 
 densities in Appendix). 
 
120 MOLECULAR PHYSICS AND HEAT 
 
 room; i.e., for use in (119) the observed barometer height needs 
 correction only for capillarity. But in the table of "Densities of 
 Dry Air" given in the Appendix, the pressures all represent 
 heights of mercury columns at C. Hence, in order to compare 
 your value of d with Regnault's value, which is represented by 
 this table, the observed barometric height must be reduced to 
 C. This reduction is made by multiplying the observed height 
 by the ratio of the densities of mercury at the room temperature 
 and at the zero temperature. To save labor, this correction is 
 worked out for all ordinary temperatures in the table in the 
 Appendix entitled "Reduction of the Barometer Height toO C." 
 (This table also contains a slight correction for the expansion of 
 the brass barometer scale.) 
 
 Record 
 
 Bar. Ht. obs'd = corrected for capil'ty= Reduced to 0= cm. 
 
 Temperature of air corresponding to P l = to P 2 = 
 
 First resting point [J5J = First weight [W t ] = gm. 
 
 Second " " [.] = - Second " [W 2 ] = - -gm. 
 
 Third " " [R 3 ] = Sensitiveness [S] = 
 
 . ! . correction to be applied to W 2 = mg. . . W 3 = gm. 
 
 . . weight of air w introduced into bulb (W 9 WJ gm. 
 
 Gauge reading, long arm = short arm = - . . P 2 P 1 = cm. 
 
 Wt.ofbulb + Hg .-.Wt.ofHg Tern.- .-. Den. of Hg ' 
 
 . . V= . '- dj = Regnault's value = % error = 
 
 Problems 
 
 1. A glass tube open at one end is 60 cm. long. The inside is 
 covered with a soluble pigment. After a sea sounding, in which 
 the tube was lowered vertically, open end down, the pigment was 
 found to be dissolved to within 5 cm. of the top. The density of 
 sea water being 1.03, find the depth of the sea. 
 
 2. A tube 100 cm. long is half filled with mercury. It is then 
 inverted in a cistern of mercury. How high does the mercury 
 stand in the tube above the cistern? (Bar. Ht. 76 = cm. ) 
 
 3. A barometer tube was imperfectly filled. When the space 
 above the mercury contained 10 cc., the barometer indicated 70 
 cm. pressure. When the space above was reduced to 5 cc. by 
 
DENSITY OF AIR 121 
 
 pushing the barometer down into the cistern, it indicated 69.5 
 cm. What was the true barometric height? 
 
 4. Show why the true zero of a vibration which is gradually 
 dying down because of damping, is not obtained by taking the 
 mean of two successive turning points, one on the right and one 
 on the left. 
 
 5. Show why the true zero is obtained by averaging the mean 
 of two successive turning points on one side with the intervening 
 turning point on the other. 
 
XVI 
 THE MEASUREMENT OF TEMPERATURE 
 
 Theory 
 
 It is a fact of common observation that as a body grows hot it 
 increases in volume. Quantitative measurement shows, however, 
 that as different bodies pass through the same change 
 ^ temperature, e. g. , f rom the freezing to the boiling 
 point of water, their expansions per cubic centimeter 
 are widely different. In 1787, however, a Frenchman by the 
 name of Charles announced that all gases show the same expan- 
 sions as they pass between two fixed temperatures. This result 
 was confirmed by Gay-Lussac some twenty years later, in the first 
 series of experiments which were sufficiently exact to thoroughly 
 justify the conclusion. The law is now most generally known by 
 the name of Gay-Lussac, though it is also called the law of 
 Charles. This law, like that of Boyle, has been found by later 
 experiment to be only approximately correct. For the permanent 
 gases, however, the departures are only slight, as will be seen from 
 the table on p. 126. Since the different gases obey Boyle's Law 
 with different degrees of exactness, these departures from the law 
 of Gay-Lussac were to have been expected. In fact, the kinetic 
 theory requiies the result, established by experiment, that the 
 gases which show the largest departures from Boyle's Law, show 
 also the largest deviations from the mean value of the coefficient of 
 gaseous expansion. (See C0 2 and X 2 0.) 
 
 Direct knowledge of the temperature, i.e., of the hotness, of 
 
 a body is gained only through the sense of touch. But since, as 
 
 a body grows hot to the touch, it also expands, this 
 
 Temperature. J . & . ' . 
 
 fact of expansion has been made the quantitative 
 measure of change in temperature, even when the change is too 
 slight to be perceived by the touch. Thus, for example, the vol- 
 ume of a given weight of iron is observed first at the freezing 
 point and afterward at the boiling point of water; the increase in 
 volume is then divided into 100 equal parts, and 1 degree rise of 
 
 122 
 
THE MEASUREMENT OF TEMPERATURE 123 
 
 temperature is arbitrarily defined as any temperature change 
 which will produce an expansion of the iron equal to one of these 
 parts. Thermometers constructed in this way from different 
 substances do not exactly agree with one another for temperatures 
 other than and 100, for the reason that the expansions of the 
 different substances are not generally the same functions of the 
 temperature. Hence, it becomes necessary to choose arbitrarily 
 some particular substance whose expansion shall be taken as the 
 measure of temperature change. The gases possess peculiar 
 advantages for this purpose, first, because all gas thermometers 
 agree with one another (see law of Gay-Lussac) and, second, 
 because the kinetic theory gives to a degree measured upon a gas 
 thermometer a particular physical significance (see below). For 
 these reasons the expansion of a gas has been chosen as the 
 measure of temperature change, and all correct determinations of 
 temperature are now made either by means of gas thern*0meters 
 or else by means of other instruments which have been standard- 
 ized by comparison with gas thermometers. 
 
 Gas thermometers take two forms: (1) the constant-pressure 
 form, and (2) the constant-volume form. The first consists of any 
 constant- arrangement for measuring the expansion of a gas 
 wnicn is ke P^ under a constant pressure. For example, 
 su PP ose the g as to ^ e confined within the bulb B and 
 dent <>f gases. a par t o f the stem cd (see Fig. 74) by means of a 
 small mercury index i which moves without friction forward or 
 back as the temperature of 
 the bulb rises or falls. Let 
 the stem be open at d, so 
 
 that the confined gas is FIGURE 74 
 
 always under the condition 
 
 of pressure which exists outside. One degree of change in tem- 
 perature is then defined, in the centigrade system, as any tem- 
 perature change which, starting from any temperature whatever, 
 will produce in the confined gas an increase of volume amounting to 
 
 - of the increase which takes place when the temperature of the gas 
 100 
 
 passes between the freezing and the boiling points of water. If the 
 barometric pressure never varied, the stem of such a thermometer 
 might easily be graduated so that the instrument would read tern- 
 
124 MOLECULAR PHYSICS A^D HEAT 
 
 perature directly. It would only be necessary to place it hori- 
 zontally, first in melting ice, then in the steam rising from boiling 
 water; to mark the positions of the index at the two temper- 
 atures ; and then to divide the increase into 100 equal parts. In 
 practice a gas thermometer is never treated in this way, for the 
 reason that such a graduation would give correct temperature 
 readings only for the particular pressure P (e.g., 76 cm.) at which 
 the graduation was made. In general, then, in order to make a 
 correct determination of any unknown temperature with such a 
 thermometer, it is necessary to know, not only the index reading 
 when the bulb is at the unknown temperature , but also the 
 index readings which correspond, at the existing pressure, to the 
 freezing and boiling temperatures. From these three readings the 
 temperature t is obtained without any actual determination of any 
 one of the three volumes, i.e., the volume at [=F ], the vol- 
 ume at 100 [=Fioo], or the volume at t [=FJ; for it is only 
 the volume differences ( V t F ) and ( V m F ) which need be 
 known. Thus it is at once evident from the definition of temper- 
 ature given above, that from such observations the temperature 
 on the centigrade scale is given by 
 
 (130) 
 
 oo 
 
 100 
 
 The observation at 100 can be dispensed with if the actual vol- 
 ume F be determined, and if the coefficient of expansion of the 
 gas between and 100 be known. For, the coefficient of 
 expansion a between any two temperatures ti and t z is defined as 
 the increase in volume corresponding to a rise in temperature 
 from ti to # 2 , expressed in terms of the volume at 0. Thus the 
 following equation is arbitrarily taken as the definition of a be- 
 tween ti and t z : 
 
 Vt -Vt 
 
 n 
 
 (t Z - tl) VQ 
 
 The coefficient of expansion between and 100 is then evidently 
 given by 
 
 F 100 FQ 
 
THE MEASUREMENT OF TEMPERATURE 
 
 125 
 
 Now, if this equation be combined with (120), there results 
 
 ;= Z ^- (133) 
 
 a KO 
 
 If, then, a be known, it is evident from (123) that a determina- 
 tion of the temperature t may be made by means of the constant- 
 pressure gas thermometer, by measuring the volume of the gas at 
 0, and the increase in volume between and t. Either of the 
 characteristic equations of a constant-pressure gas thermometer, 
 (120) or (123), may be taken as the definition of temperature on 
 the centigrade scale. 
 
 Now suppose that the confined air in the bulb B, after being 
 raised at constant pressure P from the freezing to the boiling 
 c<m*tant-voi- P oinfc of water, i.e., from F to V m , had been brought 
 mowietersond back a ain while at the boiling temperature, from F 100 
 to ^ ( see Fi * ^) ky **ke a Ppl ica tio n of additional out- 
 side pressure, the final value of which is represented by 
 Pj. Then Boyle's Law gives 
 
 F 100 _ P, 
 
 To = P.' 
 
 If this equation be compared with (122), which defines the coeffi- 
 cient of expansion a, it is seen that 
 
 100 FO 
 
 100 P 
 
 (124) 
 
 Now, since, in the whole operation which has been considered, 
 the gas has been brought back to its original volume, and the only 
 resultant change which has taken place is the change in temper- 
 ature from to 100, it is clear that P l is 
 the final pressure which the gas would have 
 exerted if its volume had always remained con- 
 stant at F , while its temperature was raised 
 from the freezing to the boiling point of water, 
 i.e., P! is the pressure which the gas in a 
 constant-volume thermometer (see Fig. 75) would 
 have assumed at 100, if its pressure at was 
 P . Call it henceforth P 100 and write (124), 
 
 /1OK\ 
 
MOLECULAR PHYSICS AXD HEAT 
 
 Also conceive the above operation to be replaced by that of raising 
 the arm a (Fig. 75) and holding the mercury in b always at the 
 same point F while the temperature of B is raised from to 
 100, an operation which corresponds to that usually performed 
 in the use of a constant-volume thermometer,, 'Now the expres- 
 
 P -P 
 
 sion, 1 , applied to a constant-voltfme thermometer, is 
 
 -L \J\J -L 
 
 usually called the pressure coefficient of the gas, and is denoted by 
 the letter (3. It thus appears that for a gas which follows Boyle's 
 Law the expansion coefficient a and the pressure coefficient fi are 
 identical quantities, and hence that the equation 
 
 *=AriA (126) 
 
 The present 
 
 thermometer. 
 
 applied to a constant-volume gas thermometer , gives precisely the 
 same definition of temperature as does (123) applied to a constant- 
 pressure gas thermometer. 
 
 In practice, the constant-volume thermometer is the more con- 
 venient, as well. as the more accurate instrument. It is therefore 
 almost exclusively used. But there are other reasons 
 
 . . . 
 
 aside irom those 01 convenience and accuracy, wn.cn 
 
 . 
 
 make a choice necessary. As has just been proved, the 
 pressure and expansion coefficients must be identical for gases 
 which rigorously follow Boyle's Law. But, since such gases do 
 not exist, it was to have been expected that refined measurements 
 would show slight differences between these coefficients. The 
 following table gives the results of some of the best determina- 
 tions, those of Chappuis at the International Bureau of Weights 
 and Measures being probably the most accurate yet made: 
 
 Gas C 
 Hydrogen . 
 
 Expansion 
 Joefflcient a 
 
 0036600 * 
 
 Pressure 
 Coefficient ft 
 
 0036625 
 
 Observer 
 P Chappuis 
 
 Date 
 
 1887 
 
 Hydrogen 
 
 .003661 
 
 003667 
 
 ^ Regnault 
 
 1840 
 
 Air 
 
 .003670 
 
 .003665 
 
 Regnault 
 
 1840 
 
 Air 
 
 
 003663 
 
 Kuenen and Randall 
 
 1895 
 
 Carbon dioxide (CO 2 ). 
 Nitrous oxide (N 2 O). . 
 
 .003710 
 .003719 
 
 .003688 
 003676 
 003674 
 
 Regnault 
 Regnault 
 von Jolly 
 
 1840 
 1840 
 1874 
 
 
 
 003667 
 
 
 1874 
 
 Argon 
 
 
 003668 
 
 Kuenen and Randall 
 
 1895 
 
 Helium 
 
 
 003665 
 
 Kuenen and Randall 
 
 1895 
 
 
 
 
 
 
 *See Traverse* "Experimental Study of Gases," p, 149. Macmillan & Co., 1901. 
 
THE MEASUREMENT OF TEMPERATURE 127 
 
 Because, then, of the slight difference in- the two coefficients 
 for the same gas, and because of the slight differences existing 
 between the different gases as regards the same coefficient, the 
 International Committee of Weights and Measures adopted in 1887 
 the constant -volume hydrogen thermometer as the standard of tem- 
 perature measurement. Hence, the above approximately correct 
 definition of temperature must be replaced by that embodied in (126) 
 when applied to a hydrogen thermometer. In other words, since 
 
 fi = .0036625 = --, IC. is by definition such a temperature 
 
 change as will cause the pressure of a body of hydrogen which is 
 kept at constant volume to change by ^ of its zero value. 
 
 According to the kinetic theory, the pressure exerted by a gas 
 
 which follows Boyle's Law is, at any instant, the product of the 
 
 mean force of each impact and the number of impacts 
 
 The Kinetic 
 
 theory and occurring at that instant upon each sq. cm. 01 suriace 
 
 temperature. * 
 
 (see Jix. Alv). JNow, the number 01 impacts wnicn a 
 single molecule which is moving within a given enclosure will 
 make per second upon the walls is evidently proportional to its 
 velocity. Hence, with a fixed number of molecules present in 
 the enclosure, the number of impacts occurring at any instant 
 must be directly proportional to the average velocity. But the 
 mean force of each impact is also proportional to the velocity (see 
 second paragraph below). Hence with a given kind of mole- 
 cule, i.e., a given gas, the pressure exerted against the walls is 
 proportional to the square of the molecular velocity. But also, 
 for a given kind of molecule, the kinetic energy (%mv 2 ) of each 
 molecule is proportional to the square of the velocity. Hence the 
 ratio of the two pressures which a constant volume of gas exerts 
 at different temperatures is simply the ratio of the mean molec- 
 ular kinetic energies. Since, then, 
 
 ET 
 
 (127) 
 
 it follows that the equation (126) which, when applied to a con- 
 stant volume of hydrogen, defines temperature, may also be 
 written 
 
 (KE},-(KE\ 
 ft (KE), 
 
128 MOLECULAR PHYSICS AND HEAT 
 
 Since ft = -^-, and since, when (KE) t - (KE} Q = ft (KE)^ t = 1, 
 
 A I O 
 
 this maybe expressed in words thus: 1 change in temperature 
 means merely the addition (or subtraction) of a given amount to the 
 mean kinetic energy of translation of the flying hydrogen molecules , 
 
 and this amount is ^ of the kinetic energy which these molecules 
 
 possess at C. 
 
 It is evident, then, that if, starting from 0C., this amount 
 
 were subtracted 273 times, all kinetic energy would be removed 
 
 from the hydrogen molecules; i.e.. they would come 
 
 The absolute ' . , 
 
 zero of tem- completely to rest. Further, the interpretation of the 
 law of Gay-Lussac, in the light of the kinetic theory, is 
 that the molecules of all other gases would come to rest at the 
 same temperature, viz., at 273C. This point at which all molec- 
 ular motion would cease is called the absolute zero of temperature. 
 Of course this temperature can never be attained ; but within the 
 last twenty -five years great strides have been made toward it. Up 
 to 1877, the lowest attained temperature was 1100., produced 
 by Faraday in 1845 by the rapid evaporation in vacuo of a mixture 
 of ether and solid C0 2 . The temperature of the solid C0 2 alone 
 is 80C., that of the mixture ordinarily about the same. In 1877 
 a Swiss, Pictet, and a Frenchman, Cailletet, independently lique- 
 fied oxygen, which has a boiling point of 182.5 C. But as 
 neither of these experimenters obtained the liquid oxygen in a 
 static condition, they could make no observation of its temper- 
 ature. The lowest measured temperature was 140C., obtained 
 by Pictet by the rapid evaporation of liquid N 2 0. Following these, 
 the two Poles, Wroblewski and Olszewski, and the Englishman 
 Dewar accomplished the liquefaction of air in quantity, and by its 
 evaporation in vacuum produced and measured temperature as 
 low as 210C. In 1885, Professor Olszewski liquefied hydrogen 
 and located its boiling point at 243.5. It has since been not 
 only liquefied in quantity, but also solidified by Dewar (1900), 
 according to whose most recent determinations the boiling point 
 of hydrogen, measured by means of a hydrogen thermometer, 
 is between 252 C. and 253C. Its melting point is between 
 256C. and 257C. The lowest temperature obtained by evap- 
 orating solid hydrogen is 260C., only 13 above absolute zero. 
 
THE MEASUREMENT OF TEMPERATURE 129 
 
 Thus far it has been shown that 1 rise in temperature sig- 
 nifies an increase in the mean molecular energy amounting to 
 
 Pressure 973 ^ ^he zero value of this energy, but no attempt 
 
 andmplec- *' * 
 
 uiar unetic h as ve t been made to find the actual value of this zero 
 
 energy. > 
 
 energy, or to discover whether any simple relation 
 exists between the mean kinetic energies of the molecules of 
 different gases which have the same temperature. In 1851, the 
 English physicist Joule worked out in the following way the precise 
 relation which must exist between the pressure exerted by a gas 
 and the mean kinetic energy of the individual molecules. 
 
 Let the gas be contained in the vessel mn (see Fig. 76), the 
 lengths of whose sides are , #, c, and let N represent the number 
 of molecules in the vessel and v the 
 average velocity of each molecule. The 
 particles are, of course, moving in all 
 possible directions, but, on the whole, 
 the pressure must be just the same 
 as though one-third of them were FIGURE 76 
 
 moving parallel to each of the three 
 
 edges a, Z>, c. If a single particle were alone in the vessel and 
 were moving back and forth parallel to one edge, e. g., to c, 
 
 v 
 
 it would make against the upper wall impacts per second ; for 
 
 v is the distance which it moves per second and 2c is the distance 
 moved between successive impacts against this wall. If there are 
 other molecules present in the vessel, the molecule considered 
 may, of course, collide with them in its excursions to and fro, but, 
 so long as the volume occupied by the molecules themselves is 
 negligible in comparison with the volume of the vessel, the num- 
 ber of impacts against the wall ab will be unaltered by these col- 
 lisions. For (see Ex. VII, Problem 1) when two equal elastic 
 particles collide, the effect is the same as though one particle had 
 passed through the other without influencing it.* Since, then, 
 
 1) 
 
 each particle which is moving parallel to c makes impacts per , 
 
 * This was proved only for the case of central impact and can not 
 therefore be taken as a complete justification for the preceding statement. 
 However, a more rigorous analysis than the one here given leads to pre- 
 cisely the same final result. 
 
130 MOLECULAR PHYSICS A^D HEAT 
 
 JV 7 
 
 second against the side Z>, and since there are ^ particles so mov- 
 
 o 
 
 ing, the total number of impacts per second against ab must be 
 
 N v 
 
 x Now, at each impact each molecule first loses all its 
 o lie 
 
 velocity and then gains an equal opposite velocity. Hence the 
 change in velocity which each molecule experiences at each impact 
 is %v. The change in momentum which each particle experiences 
 at impact is therefore %mv, m being the mass of each molecule. 
 The total change in momentum experienced by the total mass which 
 
 N v 
 impinges per second against ab is then 2m v x x But rate of 
 
 O /iC 
 
 change of momentum is the measure of force (see Ex. II). Hence 
 the force /", which the w&llab experiences because of the molec- 
 ular impacts, is given by 
 
 - n N v , Nmv* /int\\ 
 
 /=2)B( , x _ x _ = i _. (139) 
 
 But the pressure p is by definition the force per unit area. Hence, 
 
 But abc is merely the volume V of the vessel. Hence equation 
 (130) becomes 
 
 ^r=-J- Nmv 9 . (131) 
 
 Now, Boyle's Law asserts that the product of the pressure and 
 volume of any gas is constant so long as the temperature remains 
 constant. Hence, according to equation (131), the condition for 
 constant temperature is a constant mean velocity of molecular 
 motion. If n denote the number of molecules in unit volume, 
 
 N 
 then n = -^, and (131) becomes 
 
 p = $ nmv*. (132) 
 
 The kinetic energy of the n molecules is %nmv*. Hence the pres- 
 sure exerted by a gas is numerically equal to two-thirds of the 
 'kinetic energy of translation of the molecules in unit volume. 
 E uanit of ket ^ W0 g ases 1 an( ^ ^ be contained in different ves- 
 
 se ^ s ' ^ut let them have the same temperature and 
 exert tne same P ressure - According to equation (132) 
 temperature. fa e pressure in Vessel 1 is 
 
 ' p = -3 tiiiniv*. 
 
THE MEASUREMENT OF TEMPERATURE 
 
 131 
 
 Similarly in vessel 2, 
 
 Now, according to the law of Avogadro, the proof of which will 
 be advanced in the next section, if the two gases exert the same 
 
 pressure and are at the same 
 temperature, they contain the 
 same number of molecules per 
 unit volume, i.e., in this case 
 HI = %. Hence, 
 
 i.e., at a given temperature the 
 molecules of all gases have the 
 same average kinetic energy of 
 translation. Thus the kinetic 
 theory leads not only to the 
 conclusion that 1 rise of tem- 
 perature means an increase of 
 kinetic energy amounting to 
 
 1 
 
 energy, 
 
 of the zero 
 
 FIGURE 77 
 
 but 
 
 also that equality of tempera- 
 ture in gases means equality in 
 the average kinetic energies of 
 the molecules. 
 
 Experiment 
 
 1. To determine the pres- 
 sure coefficient ft of air. 2. 
 
 To standardize a 
 Object 
 
 "mercury in glass " 
 
 thermometer by comparing it 
 with an air thermometer. 
 
 Fig. 77 shows 
 the constant - vol- 
 ume air thermom- 
 eter which is to be 
 used. The volume 
 is kept constant by 
 
132 MOLECULAR PHYSICS AND HEAT 
 
 so adjusting the height of the right manometer-arm that the mer- 
 cury in the left arm is brought, before each reading, exactly into 
 
 coincidence with the platinum point c (see also Fig. 77a). 
 
 The fine adjustment screw e facilitates this setting. The 
 difference in the mercury levels in the two arms is read off upon the 
 central graduated scale. A mirror is attached to the back of the 
 vernier index i and slides with it so that the readings can be made 
 entirely free from the error of parallax. A three-way stopcock s 
 makes it possible to put the bulb into communication either with the 
 manometer or with the outer air. The form of this cock is shown 
 in Fig. 77b, which represents the position when it is in communica- 
 tion, through the hole o and the tube m, with the manometer. 
 Turning the cock through 180 brings the bulb into communica- 
 tion, through the hole r and the tube n, with the outside air. 
 , A rigid arm ?, which is attached to the sliding collar (7, holds the 
 bulb in place and protects it from breakage. 
 
 If PQ represent the observed pressure at 0, H the barometric 
 height and h Q the difference between the readings in the two arms 
 
 when the bulb is surrounded with melting ice and 
 
 the mercury is brought into exact contact with c, 
 then evidently 
 
 p =H+h , 
 
 Ji Q being, of course, negative if the level in the left arm is highei 
 than that in the right. Similarly 1he observed pressure at 100 
 is given by 
 
 p m = H+h m , 
 and that at any unknown temperature t, by 
 
 p t = H+ li t . 
 
 Now if the conditions assumed in the deduction of (125) and (126) 
 had all been fully realized there would result very simply 
 
 n PIOO ~ PO PIW~PO "100 ~ UQ . 
 
 100 Po lOOjOo ~ 100 (H+ h Q ) 
 
 4 - - h - h 
 
 , . 
 
 But, in point of fact, p mi p^ andjo, are all slightly different from 
 Aooj PO ? and P,, for the latter correspond to an absolutely constant 
 volume and to a condition in which all of the confined air under- 
 goes tho heating or cooling operation. Since neither of these 
 
THE MEASUREMENT OF TEMPERATURE 133 
 
 conditions is realized in practice two corrections must be applied 
 to the observed pressures. The first is the correction for the 
 expansion of the bulb. If y represent the cubical 
 coefficient of expansion of glass, (i.e., the expansion 
 P er cc - P er Degree, a quantity which is equal to .000025), 
 then the initial volume V of the bulb will have changed 
 at 100 to F+ 100yF[=F(l + 100y)]. In order to reduce this 
 volume back to F, it would be necessary to apply some pressure 
 x such that (see Boyle's Law), 
 
 F(1+10 Y) or a-p^l-HOOy). (135) 
 
 F - 
 
 ^100 
 
 Applying this correction then to both (133) and (134) there results : 
 p m (1 + IQOy) -po p m -p<> yp m . 
 
 * wo Po "- -" 
 
 and t . - = o_.* (137) 
 
 ^o Pp*-ypt 
 
 But for even moderately accurate results a still further correc- 
 
 tion must be applied to all of the observed pressures in order to 
 
 make allowance for that portion of the confined air 
 
 Correction . . 
 
 for unheated which escapes the changes in temperature which take 
 r> place within the bulb. Thus it is evident that if the 
 air in the capillary stem and in the space about c fell to the zero 
 temperature, p Q would be somewhat smaller than it is. Similarly if 
 this air were heated with the rest to 100, p^ would have a larger 
 value. The error arising from this source is eliminated by multi- 
 plying the /? and t given by (136) and (137) by a factor of the form 
 
 i | v P * 
 
 V po 1 + . 00367*' 
 
 in which v is the volume of the unheated portion of F, t' the tem- 
 
 perature of the room near e, and p is p m if the correction is applied 
 
 to /3, p t if it is applied to t. The deduction of the form of this factor 
 
 is comparatively easy but will scarcely be found profitable here (see 
 
 Kohlrausch's Leitfaden der Praktischen Physik, 8th ed., p. 112). 
 
 Of course, if the barometer height is not 760 mm., the temper- 
 
 ature of steam will not be exactly 100, and a final modification 
 
 must be made of (136) to take into account the change 
 
 Correction . . ... . \ , \ . 
 
 for boiling in boiling point with change in pressure. The correc- 
 
 temperature. .. ' , , , ... , 
 
 tion will be made by replacing the 100 of equation 
 * The last expression is obtained by solving the preceding equation for t. 
 
134 MOLECULAR PHYSICS AND HEAT 
 
 (136) by the value of the boiling point taken from table 6 in the 
 Appendix. 
 
 DIRECTIONS. 1. Lower C' (Fig. 77) until the mercury in 
 the left arm of the manometer is below the three-way stopcock s, 
 The pressure then P en communication through r and n (see 77b), 
 coefficient p. between B and the outside air. Attach to n a calcium 
 chloride drying tube and an exhaust pump precisely as in the last 
 experiment. When B has been thoroughly dried,* turn the cock so 
 as to restore communication with the manometer. Then, with the 
 aid of an ice scraper (see Fig. 78), prepare 
 several quarts of pure shaved ice, pack 
 it carefully about the bulb within any con- 
 venient vessel from which the water may be 
 FIGURE 78 drained, at the same time lowering C' so that 
 
 the contraction of the air in B may not cause 
 
 mercury to pass over into the bulb, then pour over distilled 
 water and repack. The water is added both to insure good 
 contact and to make certain that the temperature of the ice 
 is not below 0. A mixture prepared in this way from shaved 
 ice or clean snow gives a perfectly constant zero, but dry ice 
 or snow can not be depended upon. Let the ice surround the 
 whole bulb and the tube t up to the point to which the steam will 
 have access when the bulb is immersed in the steambath. After 
 a lapse of five minutes, adjust C' and e until the mercury just 
 touches the platinum point c\ then set the sliding index i succes- 
 sively upon the tops of the mercury menisci in the two arms, and 
 take the corresponding vernier readings upon the vertical scale. 
 Repeat both the setting and the observations, and see how closely 
 successive values of ho can be made to agree. Next insert an 
 asbestos screen between the bulb and manometer, replace the ice by 
 the steam jacket shown in Fig. 77, and, as soon as the expansion has 
 ceased, make again a series of settings and readings. In order to 
 preclude the possibility of a leak, it is well, during the expansion, to 
 keep the mercury always above the stopcock. After reading the 
 barometer, obtain the volume of unheated air as follows: Place a 
 small beaker underneath n and raise C" carefully until the mercury 
 rises in the capillary tube t to the point to which this tube was 
 
 * Of course this drying need not be repeated with every experiment. 
 The instructor will decide in each case whether it is necessary or not. 
 
THE MEASUREMENT OF TEMPERATURE 135 
 
 heated. Then turn the cock s and let mercury run out through n 
 until the mercury level in the left arm has exactly reached the plati- 
 num point c. From the weight of the mercury which has emerged 
 from 5, v can at once be obtained. The volume of J9, viz., F, 
 can be calculated with sufficient accuracy from a careful estimate 
 of the dimensions of the bulb. Or the bulb may be immersed in a 
 graduate and the rise of the water noted. If v is of the order 
 y 
 an error of as much as 10% in the measurement of Fwill 
 
 introduce into the determination of (3 an error of but one- tenth of 
 one per cent. 
 
 2. Having found /?, surround B with a water bath, immerse 
 also in the bath the mercury thermometer which is to be standard- 
 ized, seeing to it that the whole of the mercury in both 
 of mercury bulb and stem is as nearly immersed as possible, and 
 
 thermometer. , _ .. . , . , 
 
 find the corrections of the mercury thermometer at about 
 25C., 50C. and 75C. [see (137)]. In all of these determina- 
 tions, stir the water continually and take no readings 
 until a stationary condition has been reached. Next find (o) 
 the correction of the mercury thermometer at 0, either 
 by filling a large funnel with pure shaved ice, pouring 
 over distilled water and immersing the thermometer up to 
 its mark ; or better still by immersing it in a small vessel 
 of distilled water (see Fig. 79) which is surrounded by 
 a freezing mixture of ice and salt, and noting the point 
 at which, with continual stirring, the mercury becomes 
 stationary. In the latter case it will generally be found 
 to fall below the freezing point and then to rise sud- 
 denly to it as the ice crystals begin to appear. Lastly 
 find the correction at the boiling point by immersing the 
 thermometer up to the mark 100 in a double-walled 
 steam jacket like that shown in Fig. 77. The correc- 
 tion will be the difference between the observed boiling 
 temperature and that given in the Appendix for the exist- 
 ing pressure. It will be positive if the mercury ther- 
 mometer reads too low, negative if it reads too high. ^^ 
 In all readings of thermometers, take great pains to 
 avoid the error of parallax. This is done by placing the stem 
 carefully at right angles to the line of sight. 
 
136 MOLECULAR PHYSICS AND HEAT 
 
 From the five observed points plot a curve of thermometer 
 corrections (cf. also the two-point curve of Figure 86, Ex. XVIII). 
 
 Record 
 
 1. Determination of /3. 
 
 Melting ice 1st setting 2d Steam 1st setting 3d 
 
 Reading at c = 
 
 Reading in rt. arm = 
 
 H =; HQ = Tiioo = 
 
 Mean 7io = . '. Po = Mean hm = . '. PICO = 
 
 v = V= . . /3 = % error = 
 
 2. Correction of mercury thermometer. H= 
 
 1st 2d 25 1st 2d 50 1st 2d 75 1st 2d 
 Ate 
 Rt. arm 
 
 Means /io ht l . . t ht 2 . . ti ht 3 . . 1 3 
 
 By mercury thermometer 1 1 t 2 1 3 
 
 Correction of Hg. ther. at at 25 at 50 at 75 at 100 - 
 
 Problems 
 
 1. From equation (126), which defines tf, the measure of tem- 
 perature upon the centigrade scale, show that, if the volume of a 
 gas remain constant while the temperature changes, 
 
 p h 273 + *!. 
 
 P h ~ 273 + 4' 
 
 or, if T denote the temperature measured from a point 273 below 
 0C., i.e., from the absolute zero, show that 
 
 A T, 
 
 pf = T 9 ' (138) 
 
 2. Show also that if the pressure of a gas remain constant 
 while its temperature changes, the volume is directly, the density 
 Inversely, proportional to the absolute temperature. 
 
 '3. Since Boyle's Law gives the variation in density when the 
 pressure alone changes, the temperature remaining constant, it is 
 evident that from this law, and the rule deduced in the preceding 
 problem, the density of a gas can be calculated for all temperatures 
 
THE MEASUREMENT OF TEMPERATURE 137 
 
 and pressures if it has once been determined for one single value 
 of the temperature and pressure. Thus, given that the density of 
 air atl6C., 745 mm., is .001192; find the density at 0, 745 
 mm. ; at 0, 760 mm. ; at 120, 755 mm. 
 
 4. Find the volume occupied by 28.88 gm. of air at 0, 760 
 mm. 
 
 5. How much work is done against atmospheric pressure when 
 10 gm. of air at 0, 750 mm., are heated to 50, 750 mm.? 
 
 6. In equation (132), nm is merely the mass per unit volume, 
 i.e., the density of the gas. The density of air at 0C., 76 
 cm., is .001293; hence, find from (132) the average velocity of 
 the molecules of air at this temperature (p must, of course, be 
 expressed in dynes). 
 
 7. Air is 14.44 times as heavy as hydrogen. Find the velocity 
 of the hydrogen molecule. What relations do the densities of 
 gases bear to their molecular velocities? 
 
XVII 
 
 LAW OF AVOGADRO DENSITIES OF GASES AND 
 
 VAPORS 
 
 Theory 
 
 The speculations of the old Greek philosophers led some of 
 
 them to the assumption of the atomic theory as to the constitution 
 
 of matter; but it was not until 1803 that the English 
 
 Origin of the . ' . 
 
 atomic chemist, John Dalton, placed this theory upon an 
 
 hypothesis. . ' ' i i * 
 
 experimental rather than upon a purely speculative 
 foundation. Experiments upon the combining powers by weight 
 of different substances reveal four principles which find their most 
 simple and natural interpretation in the atomic hypothesis. 
 These are (1) the principle of constant proportions, (2) the prin- 
 ciple of equivalence, (3) the principle of multiple proportions, 
 (4) the principle of substitution. 
 
 According to the first principle, the proportions by weight in 
 which the elements enter into a given compound are absolutely 
 
 invariable. For example, it is found that hydrogen 
 
 The law of .... 
 
 constant pro- will unite with chlorine, so as to leave no free hydrogen 
 and no free chlorine, only when the number of grams 
 of hydrogen present bears to the number of grams of chlorine one 
 definite ratio. The same may be said of the combination of 
 hydrogen with bromine or of hydrogen with iodine. These 
 ratios are 
 
 1 gm. hydrogen to 35.18 gm. chlorine j 
 
 1 gm. hydrogen to 79.36 gm. bromine > (138) 
 
 1 gm. hydrogen to 125.90 gm. iodine ) 
 
 Similarly, the chlorides of potassium, sodium, and silver are 
 always found to contain exactly the following proportions by 
 weight : 
 
 38.86 gm. potassium to 35.18 gm. chlorine \ 
 22.88 gm. sodium to 35.18 gm. chlorine > (139) 
 107.12 gm. silver to 35.18 gm. chlorine ) 
 138 
 
DENSITIES OF GASES AXD VAPORS 139 
 
 It is evident that the interpretation of this law in the light of a 
 molecular hypothesis as to the constitution of matter can only be 
 that the atoms of each substance are of constant weight and that 
 the molecules of compounds are always of the same atomic com- 
 position. 
 
 But another and still more significant relation is found to 
 exist. From the examples given above, it is evident that 35.18 
 
 gm. of chlorine, 79.36 gm. of bromine, and 125.90 gm. 
 
 of iodine may be called equivalent quantities in the 
 
 sense that each one of them combines with exactly the 
 same weight of hydrogen, viz. 1 gm. Similarly, 38.86 gm. of 
 potassium, 22.88 gm. of sodium, and 107.12 gm. of silver may 
 also be called equivalents, since each of them combines with the 
 same quantity of chlorine. These latter substances can not be 
 made to combine with hydrogen directly, but since the numbers 
 given combine with just the number of grams of chlorine which 
 has been found to be the combining equivalent of 1 gm. of 
 hydrogen, these numbars may also be said to have b3en found in 
 this indirect way to be the equivalents in combining power of 1 
 gm. of hydrogen. So much for the definition of equivalent. 
 Xow the fact of peculiar significance is this: A quantitative 
 analysis of the bromides of potassium, sodium, and silver leads to 
 precisely the same numbers for the equivalent weights of these 
 substances as did a study of the chlorides. Thus the only pro- 
 portions in which these substances will combine with bromine are: 
 38.86 gm. potassium to 79.36 gm. bromine ) 
 22.88 gm. sodium to 79.36 gm. bromine > (140) 
 107.12 gm. silver to 79.36 gm. bromine ) 
 
 When, further, a study of the iodides leads again to the same 
 three numbers, thus, 
 
 38.86 gm. potassium combines with 125.90 gm. iodine \ 
 22.88 gm. sodium combines with 125.90 gm. iodine > ? (141) 
 107.12 gm. silver combines with 125.90 gm iodine ) 
 
 it becomes certain that some very definite physical significance 
 lies behind these numbers. The simplest possible interpretation 
 to put upon them is, to take a particular case, that the particle of 
 potassium which combines with chlorine to form the molecule of 
 potassium chloride is exactly liko the particle which combines with 
 
140 MOLECULAR PHYSICS AND HEAT 
 
 bromine to form the molecule of potassium bromide, and with iodine 
 to form the molecule of potassium iodide. 
 
 If, now, it is decided to adopt a mere convention and call 
 this particle an atom of potassium; if similarly the particle of 
 bromine which enters into hydrobromic acid, and into the bromides 
 of potassium, sodium, and silver be called an atom of bromine; 
 and if a similar convention be adopted with reference to all of the 
 substances thus far mentioned, then the weights of all these atoms 
 in terms of the atom of hydrogen must be simply the numbers, 
 above given, which represent the combining powers with reference 
 to hydrogen, of the elements mentioned. Further, since it has 
 been decided to regard this quantity of each element which enters 
 into the molecule of any of the above compounds as a unit rather 
 than as a combination of two or more units, the following symbols 
 will henceforth be used: hydrochloric acid = HC1, hydrobromic 
 acid = HBr, hydroiodic acid = HI, potassium chloride = KC1, 
 sodium chloride = NaCl, silver chloride = AgCl. Corresponding 
 formulae for the bromides and iodides are: KBr, NaBr, AgBr, 
 KI, Nal, Agl. Of course, these particles which enter into the 
 above combinations may themselves be aggregations of 2, 3, 10, or 
 1000 smaller particles for aught we know, but so long as no evi- 
 dence is brought forward to show that some sort of compound 
 substance exists, the molecule of which contains a smaller amount 
 of chlorine, for example, than that quantity which is found in 
 HC1, KC1, XaCl, and AgCl, this quantity will be called by com- 
 mon consent, i.e., by definition, the atom of chlorine. An atom 
 would then be defined as the smallest particle of any element 
 which is known to enter into the molecule of any compound. 
 
 The facts of equivalence which have been above presented, 
 and which constitute one of the strongest arguments for the 
 atomic hypothesis, may be summarized thus: The study of many 
 different compounds leads often to precisely the same number as the 
 combining equivalent of a given element with reference to hydrogen. 
 
 But in some cases the study of different compounds leads to 
 different numbers for the equivalent of a given element with refer- 
 ence to hydrogen. For example, Dalton found that 
 multiple olefiant gas vielded upon decomposition the two ele- 
 
 proportions. r .1. , 
 
 ments carbon and hydrogen in the proportions by 
 weight, 6 carbon to 1 hydrogen, while marsh gas yielded the same 
 
DENSITIES OF GASES AND VAPORS 141 
 
 two elements in the ratio 3 carbon to 1 hydrogen. Further study 
 of other compounds revealed the fact that whenever elements have 
 the power of combining in different proportions, these proportions 
 always bear simple ratios to one another. This is known as the 
 law of multiple proportions. The self-evident interpretation of the 
 law, in the light of a molecular hypothesis, is that it is possible in 
 some cases for two, or three, or some other small number of atoms 
 of a given element to enter into the constitution of a compound 
 molecule. It was probably the discovery of this law of multiple 
 proportions which first convinced Dalton of the truth of the 
 atomic theory. To illustrate it by a further example, there are 
 four different compounds of the elements chlorine, potassium, 
 and oxygen in which the proportions by weight of the three ele- 
 ments are as follows : 
 
 (143) 
 
 It will be observed that in all four compounds the potassium and 
 chlorine have exactly the same ratios as in KC1. The simplest 
 and most natural interpretation is that all of the compounds con- 
 tain an atom of potassium and an atom of chlorine. The smallest 
 amount of oxygen found in the molecule of any of these com- 
 pounds weighs then 15.88 times as much as the hydrogen atom. 
 If this amount be called the oxygen atom (at least until some 
 smaller amount is found to enter into some other sort of molecule), 
 then the second compound contains two atoms of oxygen, the third 
 three, the fourth four, and the formulae for the four substances 
 are KC10, KC10 2 , KC10 3 , and KC10,. 
 
 It will have been already observed that the fact of combination 
 in multiple proportions introduces an uncertainty into the deter- 
 The rind u m i na ^i n ^ the true combining equivalents, i.e., the 
 of substitution, atomic weights of some elements. For example, from 
 Dalton's experiments on olefiant gas and marsh gas, it might be 
 inferred that the atomic weight of carbon was 6 and the formula 
 for olefiant gas CH and for marsh gas CH 2 . But 3 might with 
 equal reason be taken as the atomic weight of carbon, and 2 H 
 
 CHLORINE 
 
 POTASSIUM 
 
 OXYGEN 
 
 35.18 
 
 38.86 
 
 15.88" 
 
 35.18 
 
 38.86 
 
 31.76 
 
 35.18 
 
 38.86 
 
 47.64 
 
 35.18 
 
 38.86 
 
 63.52> 
 
142 MOLECULAR PHYSICS AND HEAT 
 
 and OH as the corresponding formulae. The following experi- 
 ment, however, makes it certain that the molecule of marsh gas 
 contains at least four hydrogen atoms. It is found that, by suc- 
 cessive treatments with chlorine, marsh gas can be made to yield 
 five different compounds in which hydrogen, carbon, and chlorine 
 are combined in the following proportions by weight: 
 
 HYDROGEN CHLORINE CARBON 
 
 4 11.91 
 
 3 35.18x1 11.91 
 
 2 35.18x2 11.91 
 
 1 35.18x3 11.91 
 
 35.18x4 11.91, 
 
 The existence of this series proves that the hydrogen in marsh gas 
 is divisible into at least four parts, and that 1, 2, 3, or 4 of these 
 parts may at will be replaced by 1, 2, 3, or 4 atoms of chlorine. 
 It is certain, then, that there must be as many as four atoms of 
 hydrogen in the molecule of marsh gas. If the quantity of carbon 
 present in this molecule be provisionally assumed to be the atom 
 (an assumption which is justified by the study of the other carbon 
 compounds), the formula for marsh gas becomes CH 4 ; for olefiant 
 gas, CH 2 ; and the hydrogen equivalent of carbon, i.e., its 
 atomic weight, is fixed not at 6 or 3 but at 12 (accurately at 
 11.91). 
 
 Again, since water is found to contain hydrogen and oxygen in 
 the ratio 1 to 7.94, this might at first be taken as the ratio of the 
 weights of the atoms of hydrogen and oxygen, and the symbol for 
 water written HO. This would indeed be inconsistent with the 
 interpretation put upon the series (142), from which it was 
 inferred that the atomic weight of oxygen was 15.88. It would 
 be possible, however, to reconcile this difficulty by assuming the 
 formulae for the compounds in (142) to be KC10 2 , KC10 4 , 
 KC10 6 , KC10 8 , an assumption not very natural, it is true, since 
 the new formulae at once suggest that it is 2 rather than 0, 
 which is the oxygen unit. But the principle of substitution leaves 
 no doubt as to which of the above possibilities must be chosen. 
 For if potassium be allowed to act on water, a portion of the 
 hydrogen is drawn off and replaced by potassium ; if it be allowed 
 to act again, all of the hydrogen is replaced, the proportions by 
 weight in the three compounds being as follows: 
 
DENSITIES OF GASES AND VAPORS 143 
 
 HYDROGEN 
 
 POTASSIUM 
 
 OXYGEN 
 
 2 
 
 
 
 15.88 
 
 1 
 
 38..S6 
 
 15.88 
 
 
 
 77.72 
 
 15.88 
 
 This series makes it certain that there are at least two atoms of 
 
 hydrogen in the water molecule, and therefore supports the first 
 
 conclusion that the atomic weight of oxygen is 15.88, and the 
 
 constitution of water H 2 0. This gradual replacement of a given 
 
 element of a compound by successive reactions is called substitution. 
 
 After the atomic weights of carbon and oxygen have been fixed 
 
 at 11.91 and 15.88, the discovery that carbon monoxide contains 
 
 these elements in exactly these proportions, and that 
 
 Some other J . ' 
 
 atomic carbon dioxide contains the same elements in the pro- 
 
 cjie'micni portions 11.91 to 31.76, leads at once to the formulae 
 CO and C0 2 to represent the chemical constituents of 
 these gases. Again, when carbon and nitrogen are found to 
 unite in the ratio 11.91 to 13.93, and carbon, nitrogen, and 
 hydrogen in the ratio 
 
 CARBON NITROGEN HYDROGEN 
 
 11.91 : 13.93 : 1 
 
 it is natural to take 13.93 as the probable value of the weight 
 of the nitrogen atom. Then the formula for nitric oxide, which 
 contains 13.93 gm. of nitrogen to 15.88 gm. of oxygen, becomes 
 NO, and that for nitrous oxide, in which nitrogen and oxygen are 
 found in the ratio 13.93 to 7,94, becomes N 2 0. Furthermore, 
 the fact that sulphur combines with chlorine in the ratio 31.83 
 to 35.18, and also with bromine in the ratio 31.83 to 79.36, sug- 
 gests 31.83 as the atomic weight of sulphur. 
 
 Enough has now been said to show how, aided only by the 
 laws of combination, chemists, beginning with Dalton, set about 
 
 devising tables of atomic weights and molecular consti- 
 Awgadro tutions; tables which, to be sure, were only provisional, 
 
 since it might sometimes be difficult to determine 
 whether the atomic weight of a substance should be represented 
 by a certain number, or by some simple multiple or sub-multiple 
 of that number. But as more compounds were investigated, the 
 choices became more and more restricted, and it can scarcely be 
 doubted that the study of combining powers alone would have led 
 
144 MOLECULAR PHYSICS AND HEAT 
 
 ultimately to most of the now accepted values of atomic weights 
 and chemical formulae, even if Avogadro's Law had never been 
 discovered. The discovery of this law, however, facilitated 
 greatly the work of fixing these quantities. The law was 
 announced in 1811 by the Italian chemist whose name it bears. 
 It asserts that at a given temperature and pressure all gases contain 
 the same number of molecules per cubic centimeter. The proof of 
 the law rests upon a remarkable relation which is found to exist 
 between the combining powers of substances and their gas or 
 vapor densities. The following table* brings out clearly this 
 striking relation. The column headed "Density" represents the 
 results of experiments upon the relative densities, at a given 
 temperature and pressure, of a number of the gases already men- 
 tioned. For convenience of representation, the density of hydro- 
 gen gas is taken as 2. 
 
 GAS DENSITY ATOMIC WEIGHT 
 
 Hydrogen 2 1 
 
 Nitrogen 27.82 13.93 
 
 Oxygen 31.80 15.88 
 
 Chlorine 70.72 35.18 
 
 Bromine 159.54 79.36 
 
 Iodine 254.73 125.90 
 
 Sulphur 64.06 ' 31.83 
 
 MOLECULAR WEIGHT 
 
 Hydrochloric acid (HC1) 36.30 36.18 U + 35.18) 
 
 Marsh gas (CH 4 ) 16.08 15.91 (11.91 + 4) 
 
 Carbon monoxide (CO) 27.95 27.79 (11.91 + 15.88) 
 
 Carbon dioxide (C0 2 ) 44.10 43.67 (ii.Qi + 15.88 x 2) 
 
 Nitric oxide (NO) 29.95 29.81 (13.93 + 15.88) 
 
 Nitrous oxide (N 2 0) 44.10 43.74 (13.93 x 2 + 15.88) 
 
 Water (H 3 0) 18.03 17.88 (2+15.88) 
 
 It is seen that in the lower group of substances the numbers 
 which represent vapor densities in terms of a gas one-half as dense 
 as hydrogen are throughout almost identical with the numbers 
 which represent the weights of the molecule in terms of the weight 
 of the hydrogen atom, as above defined. But if the weights of 
 the individual molecules of a number of gases bear the same ratios 
 
 *See Landolt and Bornstein, Physikalisch-chemische Tabellen, pp. 
 115, 116; and Wullner, Experimental Physik, Vol. II, p. 802. 
 
DENSITIES OF GASES AND VAPORS 145 
 
 as the weights of the gases per cc., then evidently the number of 
 molecules per cc. must be the same in all the gases. This 
 remarkably simple conclusion, which applies necessarily to all of 
 the gases of the second group, provided the conclusions above 
 reached as to their molecular constituents are correct, is seen to 
 apply also to all the gases of the first group, if only the molecules 
 of the gases, hydrogen, nitrogen, oxygen, chlorine, bromine, 
 iodine, and sulphur, be assumed to be twice as heavy as the atoms . 
 of these substances ; that is, if these molecules are composed each 
 of two atoms, thus, H 2 , X 2 , 2 , C1 2 , Br 2 , I 2 , S 2 ; for then the 
 molecular weights become 2, 27.86, 31.76, 70.36, 158.72, 251.80, 
 arid 63.66 respectively, numbers which are in remarkably close 
 agreement with those given in the column of densities. 
 
 Xow, it is found that with equally simple choices as to the 
 molecular constitutions of those gases in which thajcombining 
 powers of the constituent elements leave two or m6ra choices 
 open, the densities of all known gases become identicg]Lwith their 
 molecular weights. This constitutes overwhelmm^evidence for 
 
 the truth of Avogadro's hypothesis. It is this fa$ of agreement 
 between molecular weights and vapor densities v&y&Js the experi- 
 mental basis for the law of Avoyadro.* Tliis agreement is least 
 
 perfect in the cases of those gases which Sh the largest depar- 
 tures from Boyle's Law. For actual ga^es this law, therefore, 
 like those of Boyle,and Gay-Lussac, is only a close approximation. 
 
 -Experiment 
 
 1. T<^ ue^mine the density of C0 2 and to compare 
 the same wjth its molecular weight. 
 
 2. To determine,*,,the density ot^Vfater vapor and to compare 
 the same with its. molecular weight.^ * 
 
 A glass globe^bf known volume V is weighed, first when full of 
 air at temperature T (absolute), pressure P x , then when full 
 of the unknowi^ts at temperature T 2 , pressure P t . 
 
 In these weighings a*closed bulb of the same volume as 
 
 *The proof of this law* which Maxwell first drew from the kinetic 
 theory a proof which rests upon the Maxwell-Boltzman law of the dis- 
 tribution of energies between two sets of unlike particles in a gaseous 
 mixture, and which has since been given a place in a large number of 
 chemical and physical texts can not be recognized as adequate. (See 
 Note by Rayleigh in Maxwell's Theory of Heat, 10th ed., p. 326.) 
 
140 MOLECULAR PHYSICS AND HEAT 
 
 the density globe is used as a counterpoise so as to eliminate all 
 effects due to changes in the buoyancy of the air. If the differ- 
 ence between the first and second weighings be represented by w 
 (w being of course negative if the second weight exceeds the 
 first), the density of the gas at T Z P Z by d 02 , the density of air 
 at T^ by d ai , then evidently 
 
 Vd ai - Vd n =w* (145) 
 
 Further, since density is directly proportional to pressure when 
 the temperature remains constant (see Boyle's Law), and inversely 
 proportional to absolute temperature when the pressure remains 
 constant (see Ex. XVI, Problem 2) , it is evident that the equation 
 which expresses the relation between the densities of air at T l P l 
 and at T 2 P 2 is 
 
 d ~ p 
 
 #a 2 r< * 2 1 
 
 Now, the quantity which will be first sought in this experiment is 
 the density of the unknown gas in terms of the density of air at 
 
 the same temperature and pressure, i.e., -p-- This is obtained 
 
 dai 
 
 easily from (145) and (146). Thus, substitution in (145) of the 
 value of d ai obtained from (146) gives 
 
 TTTl ai) i Z W 
 
 whence f: 2 = p s Trv^- 
 
 All of the quantities on the right side of (147), excepting^, 
 are measured directly in the experiment. d a2 is obtained from 
 (146) and the result of Ex. XV; or, if it is found more convenient 
 to determine d ai than d a ^ (147) may, with the aid of (146), be 
 thrown into the form 
 
 *In this equation the expansion of the bulb is neglected, because in 
 neither of the following determinations will it affect the result by more 
 than a small fraction of one per cent. If it is desired to take it into 
 account it is only necessary to replace (145) by Vd ai V(l -}- yt)d aa = w 
 (in which 7 is the expansion coefficient of glass and t the number of 
 degrees between T l and T 2 ), and then to solve precisely as above. 
 
DENSITIES OF GASES AND VAPORS 147 
 
 Finally, since air is 14.44 times heavier than hydrogen, it is only 
 necessary to multiply the density of the unknown gas in terms of 
 air by 28.88 in order to obtain its density in terms of a gas one- 
 half as heavy as hydrogen. This is the quantity which, accord- 
 ing to the law of Avogadro, should agree with the molecular 
 weight. 
 
 DIRECTIONS. 1. For convenience in filling and weighing, the 
 
 density globe, the capacity of which is about 250 cc., is provided 
 
 with two taps a and b (see Fig. 80). The volume of 
 
 this globe is first to be found by filling it with water 
 
 and weighing upon the trip scales. The density of water at the 
 
 observed temperature is to be taken from the table of water 
 
 FIGURE 80 
 
 densities in the Appendix. V is, of course, the ratio of the 
 weight and density of the water. 
 
 Having found F, dry the globe by carefully rinsing it with 
 alcohol and then forcing through it a current of air from a bel- 
 lows, at the same time heating srentlv bv means of a 
 
 Filling with r J L . . 
 
 air and rapidly moving Bunsen flame. Continue this opera- 
 tion until all odor of alcohol has disappeared from the 
 bulb. Then, after carefully lubricating the stopcocks, connect 
 the bulb with the bellows through a calcium chloride drying- 
 tube, as in Fig. 80, and force through the combination a very 
 gentle current of air for about one minute. Then close tap b and 
 allow the apparatus to stand in this condition for about five min- 
 utes, shielding the bulb as much as possible from temperature 
 changes. Xext close tap , read the barometer, and take the 
 temperature by means of a thermometer hung near the bulb. 
 Then detach the bulb, carefully remove all dust and grease from 
 
148 MOLECULAR PHYSICS AND HEAT 
 
 its surface, and weigh upon an analytical balance, using a counter- 
 poise of the same volume as the globe and following the directions 
 given in Ex. XV for the first weighing. It is particularly impor- 
 tant that no mercury be allowed to touch the bulb, as it is nearly 
 impossible to remove small mercury globules from a glass sur- 
 face. 
 
 After weighing, put the bulb into connection, through the 
 drying-tube, with a reservoir. of C0 2 or with a vessel in which the 
 
 gas is being generated; and, keeping the bulb in such 
 co, and position that the exit tap is on top, allow a gentle current 
 
 of the gas to flow through the bulb for about two min- 
 utes. Find the temperature of the issuing gas by placing, at the 
 orifice of the exit tap, the bulb of the thermometer previously used. 
 Then close first the entrance, then the exit tap ; detach the bulb 
 and re-weigh, following the directions given in Ex. XV for the sec- 
 ond weighing. If neither the temperature nor the pressure differs 
 appreciably from the values taken in connection with the first 
 weighing, (147) reduces to 
 
 rfw-i " 
 
 d a ,~ ~Vd a ,' 
 
 w being itself negative in this case, since C0 2 is heavier than air.* 
 
 2. The method here used for determining the density of water 
 
 vapor in terms of air does not differ in principle from that 
 
 employed with C0 2 . Since, however, the maximum 
 Filling the pressure which water vapor is able to exert at ordinary 
 
 temperatures is less than atmospheric pressure (see Ex. 
 
 XVIII), it is evident that it must be impossible under 
 ordinary atmospheric conditions wholly to replace the air in the 
 density globe by water vapor. This can be done easily at any tem- 
 perature at which the maximum pressure of water vapor is more 
 than atmospheric, e.g., at 150C. Hence the following direc- 
 tions : After carefully drying the bulb B (see Fig. 81), by repeatedly 
 
 *Precisely the same method may be used with gases lighter than air, 
 save that in this case the exit tap should be at the bottom during the 
 filling. However, all of the standard determinations of gas densities have 
 been made with bulbs provided with but one tap rather than with two as 
 here described. The bulb has then been completely exhausted before 
 being put into connection with the gas reservoir. 
 
DENSITIES OF GASES AND VAPORS 
 
 149 
 
 warming and exhausting through a calcium chloride tube as in 
 Ex. XV, make a first weighing upon an analytical balance, using, 
 as above, a counterpoise of the 
 same volume as B. Then place 
 the capillary orifice o beneath 
 the surface of distilled water, 
 and warm the bulb slightly by 
 means of the hand. Upon 
 cooling, one or two cubic cen- 
 timeters of water will be drawn 
 into the bulb. Next, com- 
 pletely immerse the bulb in 
 melted paraffin (see Fig. 81), 
 leaving but a few centimeters 
 of the capillary tube projecting 
 above the surface of the liquid. 
 Keep the temperature of the 
 bath constant, at, say, 120 
 C., by very thorough and con- 
 tinuous stirring, and by a 
 proper regulation of the Bun- 
 sen burner. The rapid vapor- 
 ization of the water will drive FIGURE si 
 all air from B. If a flame 
 
 "be held in front of 0, it will be seen to be deflected by the rapid 
 current of issuing steam. When the cessation of this deflec- 
 tion indicates that the water in B has entirely boiled a\vay, seal 
 the globe by means of a fine blow-pipe flame. This is best don 
 by heating the tube to softness just above the paraffin and then 
 drawing off the tip. As soon as the sealing is complete, take 
 readings of the temperature of the bath and of the barometer 
 height. Then remove the bulb, clean it thoroughly with a cloth 
 while the paraffin is still hot, and test for a leak by allowing the 
 condensed steam to run down to the tip of the tube and observing 
 whether or not fine bubbles enter the bulb. Then, after cooling 
 to the temperature of the room, again weigh the bulb together 
 with the drawn-off tip. 
 
 To find the volume of the bulb, file off under water the sealed 
 tip. The bulb will at once fill with water. Weigh this bulb full 
 
150 MOLECULAR PHYSICS AND HEAT 
 
 of water upon the trip scales, and compute the volume as in the 
 experiment with C0 2 . If the bulb does not completely fill with 
 
 water, the filling may be completed by means of a 
 buib meof Pipette. It is true that (147) and (148) are not then 
 
 rigorously correct, but unless the bubble is quite large, 
 the error introduced will be negligible. 
 
 Record 
 
 1. Weight on trip scales of bulb = - of bulb -f water = 
 Temperature of water = . . density of water = . . V = 
 
 Wt. added to counterpoise to balance globe -f- air = rest'g pt. = 
 
 Wt. added to counterpoise to balance globe -f- CO 2 = rest'g pt. = 
 
 Rest'g pt. after addition of 2 mg.= .-. sensitiveness = .\w = 
 
 'fa-= -X 28.88 = - - molecular wt. = - % error = - 
 
 Wt. added to counterpoise to balance globe -f- air = rest'gpt. = 
 
 Wt. added to counterpoise to balance globe -f- H 2 O = rest'g pt. = 
 
 Rest'g pt. after addition of 2 mg. = .. sensitiveness = ..? = 
 
 Weight on trip scales of bulb = of bulb -f water = 
 Temperature of water = . . density of water = .: V= 
 
 X 28.88 = -- molecular wt. = 
 
 error = -- 
 
 Problems 
 
 1. One gram of air was introduced into an empty spherical 
 bulb of radius 10 cm. Find the pressure, in mm. of mercury, 
 which the gas exerted against the walls of the bulb when the 
 temperature was 25 C. 
 
 2. Find the pressure which the same weight of N^O gas would 
 exert in the same vessel at the same temperature. Compute 
 similarly for hydrogen gas; for CH 4 gas. 
 
 3. One gram of nitrogen and 1 gram of HJS are introduced 
 together into the bulb used in the first Problem. Find the pres- 
 sure at 25 C 
 
DENSITIES OF GASES AND VAPORS 151 
 
 4. Find the volume which 2 grams of hydrogen will occupy at 
 0C. 76 cm. The same for 32 grams of oxygen; for 30 grams 
 NO. Explain the connection between the results. 
 
 5. Find the density of air at 100 C. 76 cm. ; of water vapor. 
 If the density of water at 100C. 76 cm. is .95852, find how 
 many times water expands upon vaporizing. 
 
 6. Find the density of alcohol vapor at 72 76 cm., the formula 
 for alcohol being C 2 H 6 0. If the density of alcohol at 72 is .8, 
 find how many times alcohol expands upon vaporizing. 
 
XVIII 
 
 THE PRESSURE - TEMPERATURE CURVE OF A 
 SATURATED VAPOR 
 
 Theory 
 
 If the molecules of gases and of vapors are in rapid motion, the 
 molecules of liquids must be also, for no fundamental distinc- 
 tion exists between the liquid and the gaseous con,di- 
 theoryof tions. At high temperatures the two states become 
 absolutely identical. At temperatures below a cer- 
 tain critical value, however, the possession of a clearly marked 
 surface may be taken as the distinguishing feature of a liquid. 
 
 Figures 82 and 83 illustrate the probable differences between 
 the motions of the molecules in gases and in liquids at ordinary 
 temperatures and pressures. In the former (Fig. 82) the mole- 
 
 FlGURE 82 
 
 cules are so far apart that their mutual attractions may in general 
 be neglected. They move in straight lines through distances 
 which are large in comparison with their own dimensions. Their 
 motions change direction at collision only. The zigzag line repre- 
 sents a possible path of one single molecule in going from position 1 
 to position 1' and making impacts in so doing against molecules 
 2, 3, 4, etc., up to 12. The mean distances traversed between 
 
 152 
 
TENSION OF SATURATED VAPOR 153 
 
 successive collisions by a molecule of air at 0C., 76 cm., is only 
 about .00006 mm., but this is a distance which, small as it is, is 
 at least 100 times as large as the diameter of the molecule. In 
 liquids, on the other hand (see Fig. 83), the molecules are crowded 
 so closely together that their mo- 
 tions between impacts are extremely 
 minute of the same order of mag- 
 nitude as the molecules themselves 
 and at the surface of the liquid, 
 where there is greater freedom of 
 motion, the paths of the particles 
 are influenced not only by collisions 
 but also by the attractions of the 
 other molecules. On account of FIGURE 83 
 
 the enormous number of molecules 
 
 present in or near the surface, this downward force upon a mole- 
 cule just above the surface is doubtless very large; so large, in 
 fact, that the molecules which are moving away from the surface 
 are in general unable to leave it. They simply rise to a certain 
 distance by virtue of their velocities, after the manner of project- 
 iles shot up from the earth, and then fall back again into the 
 liquid. Their paths near the surface thus become similar to the 
 forms shown in Fig. 83. The zigzag line in the figure repre- 
 sents a possible path of a particle in the body of a liquid. 
 
 But conditions may arise which render possible the escape of a 
 
 molecule from the liquid; for it must not be assumed, either in 
 
 the case of liquids or of erases, that the molecules of 
 
 The kinetic 
 
 theory and the same substance all have the same velocity, for even 
 
 vaporization. . 
 
 if they were all given a common velocity to begin with, 
 the collisions would at once create differences. Again, although 
 constancy of temperature means that the mean velocity reniains 
 the same, the velocity of each individual molecule will in general 
 change at each impact. . The conditions of impact must often be 
 such that a molecule receives a velocity much greater than the 
 mean value. If the substance be a liquid, and if one of these 
 more rapidly moving molecules be near the surface, it maybe able 
 to break through the thin space in which the powerful attraction 
 exists, and to move off as an independent molecule into the space 
 above. It thus happens that the space above the liquid in a 
 
154 MOLECULAR PHYSICS AND HEAT 
 
 closed vessel gradually becomes filled with the gaseous form of the 
 substance of the liquid. This gas becomes more and more dense 
 as more molecules escape, but there is evidently a limit to its pos- 
 sible density, for many of the escaped molecules chance, in their 
 wanderings, to return to the surface and reenter the liquid. The 
 number of molecules thus returning per second evidently increases 
 as the number of molecules above the liquid increases, i.e., it is 
 proportional to the density of the vapor. When this density has 
 reached a certain limit, there is set up a condition of "active 
 equilibrium" in which as many molecules reenter the liquid per 
 second as escape. When this condition is reached, the vapor is 
 .said to be saturated, that is, it has the largest density which it is 
 ever able to have at the existing temperature, and it therefore 
 exerts the largest pressure which it ever can exert at this tem- 
 perature. 
 
 But, if the temperature be raised, the vapor can evidently have 
 
 a larger density, for the number of molecules escaping per second 
 
 must be greater at the higher temperature, i.e., at the 
 
 Dependence ... _ ^ ' 
 
 of the den- higher mean velocity, and hence the density ot the 
 pressure of vapor must be greater before the condition of equilib- 
 
 a saturated ... ., . ,. , i -, , i 
 
 vapor upon rium is set up. Also, since the pressure exerted by the 
 
 temperature. . 
 
 vapor is proportional both to the density and to the 
 mean kinetic energy of each impact, and since both density and 
 kinetic energy increase with temperature, it is evident that the 
 pressure must rise with two-fold rapidity as the temperature rises. 
 But, if the temperature be held constant, all attempts to 
 increase the density or the pressure of a vapor which is in con- 
 Density and ^ ac ^ w ^ n ^ s liquid in a closed vessel must be futile. 
 independent ^ see ^is clearly, conceive of a few drops of ether 
 of volume. inserted into a barometer so as to fill the space above 
 the mercury with ether and saturated ether vapor (see Fig. 84). 
 As soon as the density of this vapor is temporarily increased by 
 compression, i.e., by lowering the tube in the cistern, the equi- 
 librium at the surface is destroyed, and immediately more mole- 
 cules begin to enter the liquid per second than escape from it. 
 Hence, in a very short time, enough ether condenses to restore the 
 old condition of density and pressure. Similarly, raising the tube 
 and thus increasing the volume occupied by the vapor diminishes 
 the number of molecules which reenter the liquid per second, 
 
TENSION OF SATURATED VAPOR 155 
 
 while leaving the number which escape, unchanged ; hence 
 
 equilibrium is soon reestablished at the old density and 
 
 pressure. This can be proved easily by observing that 
 
 raising or lowering the tube does not alter the height of 
 
 the mercury in the tube above the mercury in the cistern. 
 
 This readjustment to equilibrium conditions takes place 
 
 almost instantly when the space contains only the liquid 
 
 and its vapor. The presence of air or of any 
 
 Retarding , , . a 
 
 influence other gas exerts a very marked influence upon 
 the time required for adjustment, but does not 
 affect the ultimate result. Thus, in Fig. 84, the density 
 and pressure of the ether vapor at a given temperature are 
 ultimately the same whether air is present in the tube or 
 not, i.e., just as much liquid will evaporate into a space 
 filled with air as into a vacuum. But whereas, when 
 the ether is introduced into a vacuum, the maximum 
 density of the vapor is reached in a few seconds, when it 
 is introduced into a space containing air, the condition of 
 saturation may not be reached for a number of hours. 
 Of course, if air be present, the total pressure against the 
 walls is the sum of the pressures of the air and of the 
 vapor. 
 
 If the liquid be contained in a vessel which is open to FIGUB 
 the air, so that the pressure can not rise above the atmospheric 
 condition, the vapor which forms is continually being 
 
 Effect of air ' . . , , J ? 
 
 currents upon removed by diffusion and by air currents, so that it 
 never reaches its maximum density, i.e., the rate of 
 exit of molecules from the liquid always remains greater than the 
 rate of entry. Hence the liquid gradually disappears or ''evap- 
 orates." The rate of this evaporation evidently depends upon th 
 rapidity with which the vapor above the liquid is removed. Hence 
 it is that fanning greatly facilitates evaporation. 
 
 The cooling effect of evaporation is very easily understood in 
 
 the light of the kinetic theory. For, since it is only the more 
 
 rapidly moving molecules which are able to break away 
 
 The cooling * %_ ,. ,. , , ,,. - ,, 
 
 effect of from the attraction which exists near the surface, the 
 
 mean kinetic energy of the molecules of the liquid is 
 
 continually being diminished by the loss of the most energetic 
 
 members. And since temperature is a function of the average 
 
 j 
 
156 MOLECULAR PHYSICS AND HEAT 
 
 molecular energy, this loss means, of course, a continual reduction 
 of the temperature of the liquid. This fall of temperature would 
 continue indefinitely if the liquid did not soon become so much 
 cooler than the surrounding objects that it receives heat from 
 them as rapidly as it loses it by evaporation. 
 
 This passage from the liquid to the vaporous condition by sur- 
 face evaporation takes place to some extent at all temperatures 
 The toning wnenever tne space above the liquid is not' saturated. 
 temperature. AS the temperature is increased, outside conditions 
 remaining the same, it takes place more and more rapidly, until 
 finally a temperature is reached at which it begins to. take place 
 not simply at the surface but also within the body of the liquid, 
 i.e., bubbles of vapor begin to form beneath the surface upon the 
 sides of the containing vessel, whence they rise to the top, grow- 
 ing rapidly as they ascend. It is at once evident that this condi- 
 tion can not be reached until the maximum pressure exerted by 
 the vapor which is formed from the liquid is at least equal to the 
 outside pressure; for, if the pressure exerted by the vapor in the 
 bubbles were less than that outside, these bubbles, even if formed, 
 would at once collapse., This temperature, then, at which the 
 pressure of the saturated vapor becomes equal to the outside pressure, 
 is called the boiling temperature. It does not follow, however, that 
 a liquid will always boil as soon as its temperature reaches the 
 boiling point as here defined. In fact, the temperature of a 
 boiling liquid must always be at least a trifle higher than that at 
 which the pressure of the saturated vapor is equal to the outside 
 pressure, for the pressure within the bubble must be sufficient to 
 overcome not only the outside pressure but also the weight of the 
 superimposed liquid and the surface tension of the bubble (see 
 Ex. XXI). When the bubble, however, rises to the surface and 
 breaks, the pressure exerted by the vapor which was contained 
 within it must fall exactly to the atmospheric condition, and the 
 temperature of this vapor must also fall, by virtue of expansion, 
 to that temperature at which the pressura of th^ saturated vapor 
 is equal to the existing atmospheric pressure. Hence, a ther- 
 mometer ivhich is to indicate the true boiling temperature according 
 to the above definition must be placed not in the boiling liquid itself 
 but in the vapor rising from it. 
 
 The temperature of the liquid itself is a very uncertain quan- 
 
TENSION OF SATURATED VAPOR 157 
 
 tity. Gay-Lussac found that the temperature of boiling water in 
 a glass vessel was usually 1 to 3 higher than in a metal ves- 
 sel. For the reasons above mentioned, it must always be 
 
 The temper- , . i . , ,-, ., . ... , , -, 
 
 ature of a trine higher than the boiling; point ; but under some 
 
 theliquid. . 
 
 circumstances it may rise many degrees above this 
 temperature. For it is by no means necessary that bubbles of 
 vapor begin to form as soon as the temperature is reached at which 
 they are able to exist after being formed. The presence of air in 
 the water or occluded in the walls of the containing vessel is 
 found to be essential to the genesis of bubbles. A Frenchman 
 named Donny found, in 1846, that when he very carefully removed 
 this air he could raise the temperature of water in a glass vessel to 
 138 C. before boiling began. But in all such cases, since the 
 pressure of the saturated vapor corresponding to the temperature 
 of the water is much more than the atmospheric pressure, as soon 
 as a bubble once starts it grows with explosive rapidity and 
 produces the familiar phenomenon of "boiling with bumping." 
 In 1861 another Frenchman, Dufour, succeeded in raising globules 
 of water immersed in oil to a temperature of 175 C. 
 
 Experiment 
 
 To determine the variation of the boiling point with 
 the pressure, (1) by the static method; (2) by the 
 dynamic method. 
 
 DIRECTIONS. 1. The apparatus used in the static method 
 is shown in Fig. 85. The bulb J9, originally open at c, was first 
 half filled with mercury. The long arm (about 5 mm. in diam- 
 eter), also originally open at the top, was then exhausted and 
 inclined till the mercury completely filled it up to a point at which 
 it had been drawn down to capillary dimensions. The tube was 
 then sealed off at this point, so that whe.n the instrument was 
 vertical the difference between the levels of the mercury in the bulb 
 and in the tube was equal to the barometric height. 
 the bulb Water was then inserted at c and boiled until the air was 
 all driven from the bulb, when the opening at c was 
 sealed off. Since, then, only water and water vapor exist above 
 the mercury in the bulb, the difference between the levels in the 
 
158 
 
 MOLECULAR PHYSICS AND HEAT 
 
 The 
 readings. 
 
 bulb and in the tube gives at once the pressure of 
 the saturated water vapor in the bulb. It is, there- 
 fore, only necessary to vary the temperature of 
 the bulb in order to obtain the curve expressing 
 the relation between the temperature and the pres- 
 sure of saturated water vapor. 
 
 The whole bulb is to be placed in a jar of 
 water whose temperature is first to be lowered, 
 by the insertion of ice, to nearly 0C., 
 and then slowly raised to about 50 C. 
 by pouring in hot water and siphoning 
 off the cold. At about 50 it is well to replace 
 the glass jar by a metal pail and thenceforth to 
 heat slowly to 100 by means of a Bunsen flame. 
 Between and about 70C. readings of the pres- 
 sure are to be taken at intervals of 10 to 12, 
 between 70 and 100 at intervals of about 4. The 
 water must be very vigorously stirred throughout 
 the experiment', and the temperature should be 
 held constant for at least one minute before a 
 reading is taken. 
 
 The thermometer readings must all be cor- 
 rected. (1) for the errors of the instrument itself, 
 
 and ( 2 ) f r the len g tn of the exposed 
 
 thread of mercury. If the first cor- 
 thermometer. re ction is to be made accurately, the 
 FIGURE 85 thermometer should be one which has been com- 
 
 pared with a standard, as in Ex. XVI. If, how- 
 ever, this comparison has not been made, the corrections 
 may be obtained with a moderate degree of accuracy by observ- 
 ing the corrections at the freezing and boiling points and 
 interpolating between these points for the corrections at other 
 temperatures. Thus, suppose that when wrapped in one layer of 
 flannel and packed in shaved ice over which a little water has 
 been poured, the thermometer reads -0.2, and that when com- 
 pletely immersed to the top of the thread in a steam bath upon a 
 day on which the boiling point of water should be 99.4, the reading 
 is 100.5. The corrections at and 100 are then +.2 and -1.1 
 respectively. If these two corrections be plotted as ordinates 
 
 Corrections 
 
TENSION" OF SATURATED VAPOR 
 
 159 
 
 upon a horizontal line representing temperatures, and if the ends 
 of these ordinates be connected by a straight line, as in Fig. 86, 
 the corrections for intermediate temperatures may be read off 
 
 FIGURE 86 
 
 Correction 
 for the ex- 
 posed thread. 
 
 from this curve. Thus, in this case, the correction at 10 is-f.l, 
 at 40 it is -.3, at 95 it is -1.05, etc. 
 
 The correction for the exposed thread is obtained by adding to 
 the observed temperature .OOOWl(t ^ ), in which / is the length 
 in degrees of the exposed thread of mercury, t the 
 observed temperature corrected according to (1), t Q the 
 mean temperature of the exposed column obtained from 
 a second thermometer whose bulb hangs about the middle of this 
 column, and .00016 the apparent expansion coefficient of mercury 
 in glass [.000181 (= coef. of Hg) - .000025 (= coef. of glass) 
 - .00016, approximately]. 
 
 In order to compare the results with tabulated values of vapor 
 pressures, it is necessary to express all pressures in terms of col- 
 umns of mercury at 0C. This correction is made as 
 usual by multiplying the observed heights by the ratio 
 of the densities of mercury at the mean temperature 
 of the observed column and at (see Appendix). 
 This correction may be made very roughly, for it is only at the 
 higher temperatures that it amounts to more than the observa- 
 tional errors. The mean temperature of the column may be 
 taken from a third thermometer hung near its middle point. 
 
 The observed pressure will need a still further cor- 
 
 Correctfon .,, _ . . , 
 
 for capillary rection on account of the capillary depression of the 
 
 depression. 
 
 mercury m the tube. This correction may be taken 
 from the table in the Appendix. 
 
 2. The dynamic method consists in the direct observation of 
 the temperatures of the steam which rises above a boiling liquid 
 
 Reduction 
 of the ob- 
 served pres- 
 sures to oC. 
 
160 
 
 MOLECULAR PHYSICS AND HEAT 
 
 made to boil under varying pressures. In Fig. 87, A represents 
 an air-tight metal boiler, which may be replaced if need be by a 
 simple long-necked glass flask. B is a condenser through which 
 
 \D 
 
 FIGURE 87 
 
 a slow current of water is passed from the tap TI . It is only by 
 The water virtue of the immediate condensation of the steam 
 as it forms that the pressure within the boiler can 
 be kept constant. C is an air-tight chamber of sufficient capac- 
 ity (in this case about 4 liters) to prevent irregularities in 
 the boiling from producing appreciable changes in the pressure. 
 The only other essential features of the apparatus are a manom- 
 eter D and any sort of an air pump. That shown in the figure 
 is a Bunsen water pump, such as may be very conveniently used 
 
TENSION OF SATUBATED VAPOR 161 
 
 in any laboratory which is supplied with running water. As soon 
 as the tap T z is opened, a jet of water rushes through the orifice 
 at o and draws with it the air from the chamber^, thus exhaust- 
 ing any vessel with which E is connected. H is a trap to prevent 
 water from being sucked back into the manometer when the 
 pump is stopped. F is a three-way stopcock (see Fig. 77b, 
 p. 131) by means of which the boiler may be put into connection 
 either with the pump or with the air, or cut off entirely from out- 
 side communication. G is also a three-way cock, which is inserted 
 so that it may be unnecessary to disconnect the boiler if it is 
 desired to use the water pump for exhausting purposes in other 
 experiments. 
 
 First, start the circulation in the condenser, then turn F until 
 A is in communication with the air, and start the water to boil- 
 * n &' After the conditions have become stationary, 
 rea( j the barometer and the boiling temperatures. 
 Then turn F so that the boiler communicates with the pump, and 
 allow the water to run until a difference of 5 or 10 cm. has been 
 produced in the arms of the open mercury manometer D. In this 
 regulation of pressure, strive to duplicate as nearly as possible some 
 pressure used in the static method. Next close ^entirely, stop 
 the pump, and after waiting about two minutes take several 
 observations of the new boiling point and the corresponding 
 pressure. Then again start the pump, put the boiler into con- 
 nection with it, and reduc'e the pressure to some second value used 
 in 1. Continue in this way until the boiling temperature has 
 fallen to about 75 C. In this method the observations can not 
 be conveniently carried to temperatures lower than 75, because, 
 with much further exhaustion, the difficulty of boiling with 
 bumping is encountered. By attaching a bicycle pump to F, the 
 boiling point for pressures somewhat higher than 76 cm. can be 
 investigated. Correct the thermometer readings exactly as in 1, 
 and tabulate results in the form shown in the Record. Finally, 
 it is required to plot in the note-book a full-page curve in which 
 temperatures are represented by abscissae, and pressures by ordi- 
 nates. The book values are to be indicated by dots, the values 
 obtained in 1 by crosses, and those obtained in 2 by circles. 
 The smooth curve, which comes as nearly as possible to touching 
 all these points, is the curve required. 
 
162 MOLECULAR PHYSICS AND HEAT 
 
 Record 
 
 IST METHOD 2D METHOD 
 
 ' , , ., Book 
 
 TEMPERATURES PRESSURES TEMPERATURES PRESSURES val- 
 
 cor- cor- pres- 
 
 rected rected red'c'd cor- cor- baro- mano- total sures 
 
 obs'd for (1) for (2) obs'd to rected obs'd rected meter meter red'c'd 
 
 Problems 
 
 1. Assume that the laws which hold for ideal gases hold also 
 for vapors up to the very point of saturation, i.e., assume that 
 equation (146) is applicable to saturated vapors. Then, with the 
 aid of the known density of air at 0, 76 cm., viz., .001293, the 
 density of water vapor in terms of air, viz., .624, equation (146), 
 and the above values for the pressures of saturated water vapor, 
 calculate the densities of saturated water vapor at 10 C., at 40 
 C., at 70C., at 100C., and compare with the observed densities 
 given in the table in the Appendix. The results will show how 
 closely the gas laws apply even to saturated vapors. 
 
 2. In a uniform barometer tube in which the mercury stands 
 but 40 cm. high, the space above the mercury is 40 cm. long, and 
 contains at first only dry air. A few drops of ether are then 
 introduced into the tube. If the tension of saturated ether vapor 
 at the temperature of the room is 30 cm., find to what height 
 
TENSION OF SATURATED VAPOR 163 
 
 above the mercury in the cistern the mercury in the tube will 
 ultimately fall. 
 
 3. If the bulb of the apparatus shown in Fig. 85 were gradually 
 heated above 100, would any temperature ever be reached at 
 which the water within the bulb would be observed to boil? 
 
 4. Explain why, from the standpoint of the kinetic theory, a 
 lower temperature can be reached by fanning an open vessel of 
 ether than by fanning an open vessel of water. 
 
 
XIX 
 
 HYGKOMETRY 
 
 Theory 
 
 Hygrometry is that branch of Physics which relates to the 
 study of the water vapor contained in the earth's atmosphere. 
 The urvose From the considerations presented in Ex. XVIII, it is 
 metrffobser- ev ident that, were it not for the presence of the air, the 
 vations. earth would always be covered with this vapor in a 
 saturated condition, and precipitation in the form of fog, dew, or 
 rain would accompany every fall in temperature, however slight. 
 But the presence of air so retards the process of evaporation that 
 even in the immediate neighborhood of lakes or oceans the condi- 
 tion of saturation does not usually exist. Hence it is, that precip- 
 itation often fails to occur even when the thermometer falls 
 suddenly through many degrees. Nevertheless, a knowledge of 
 the hygrometric state, i.e., the state of dryness, or wetness, of 
 the atmosphere, or, what amounts to the same thing, a knowledge 
 of the number of degrees through which the temperature must 
 fall before dew can form, is of considerable importance not only 
 for scientific but also for practical purposes, such, for example, 
 as the forecasting of the probability of frost,' or the maintenance 
 within green-houses, drying-rooms, sick-rooms, and dwellings, of 
 ^ suitable climatic conditions. 
 
 The four Tke four quantities involved in hygrometric deter- 
 
 qumSStaF minations are: 
 
 sought. x. The density of the water vapor in the air, i.e., 
 
 the weight in grams of the water vapor contained in 1 cc. of 
 space. This is usually called the absolute humidity. It is here 
 represented by the letter d. 
 
 2. The relative humidity or the degree of saturation. This is 
 represented by the letter r, and is defined as the ratio between the 
 density of the water vapor existing in the atmosphere at any given 
 time, and the largest donsity which it could possibly have at the 
 
 164 
 
HYGROMETRY 165 
 
 existing temperature; i.e., if D represent the density of saturated 
 water vapor at the existing temperature, then the relative humid- 
 ity r is given by r = - Since the Desalts of Problem 1, page 162, 
 
 have shown that at ordinary temperatures the density of saturated 
 water vapor can be calculated with sufficient accuracy from the 
 pressure which it exerts (obtained in Ex. XVIII), D may always 
 be considered a known quantity (see also Appendix, table 6). 
 
 3. The dew-point T, or the point to which the temperature 
 must fall in order that the water vapor existing in the atmosphere 
 may be in the saturated condition. Of course, as soon as the 
 temperature falls below this point, condensation must ensue. 
 
 4. The tension or pressure, p, which the wafer vapor in the air 
 exerts at the existing temperature. 
 
 As will presently appear, the experimental determination of 
 any one of these four quantities taken in connection with the 
 pressure-temperature curve of a saturated vapor (see Ex. XVIII), 
 suffices for the calculation of all the rest. 
 
 The first attempt to construct an instrument for measuring 
 hygrometric conditions was made about 1600, when an Italian 
 named Sanctorius devised what is 
 
 Absorption . 
 
 hygrome- now known as an absorption hygrom- 
 eter, an instrument usually asso- 
 ciated with the name of de Saussure, a Genevan, 
 who brought it into prominence in 1783. It 
 is well known that many organic substances 
 expand with an increase in the dampness of 
 the atmosphere. De Saussure's hygrometer 
 consisted of a human hair attached as in Fig. 
 88, so that changes in its length caused a 
 pointer to move over a scale which was con- 
 structed by marking the position of the pointer 
 in a saturated atmosphere 100, its position in 
 a perfectly dry atmosphere 0, and then divid- 
 ing the intervening space into 100 equal parts. 
 The position of the pointer at any time was FIGURE 88 
 
 assumed to indicate directly the degree of 
 saturation of the atmosphere (r). These instruments are still in 
 common use, but they are now always graduated empirically by 
 
166 MOLECULAR PHYSICS AND HEAT 
 
 comparison with a dew-point hygrometer (see below); for careful 
 experiments made by Regnault in 1845 showed that instruments 
 constructed as above agree accurately neither with one another 
 nor with the indications of dew-point hygrometers. Further, it is 
 found that a given absorption hygrometer does not remain com- 
 parable even with itself for any long interval of time. Hence its 
 indications can only be relied upon if it is frequently recalibrated. 
 Accurate measurements in hygrometry began with the intro- 
 duction by the Englishman Daniell, in 1820, of the dew-point 
 
 hygrometer, the essential principle of which had been 
 h ew rometer em ply e( l by the Frenchman Le Eoy as early as 1751. 
 
 This instrument in one or another of its numerous 
 modifications has become the standard of comparison for the 
 testing and graduation of all other hygrometers. It consists 
 essentially of a polished metal tube, the temperature of which is in 
 some way lowered until dew is observed to form upon its surface. 
 From this temperature of condensation T, it is possible to deter- 
 mine all the other hygrometric constants. 
 
 Thus the pressure^ is obtained at once from T and the pres- 
 sure-temperature curve of a saturated vapor. It is simply the 
 TO obtain pressure of saturated water vapor at the temperature T. 
 
 For, although the cooling of the layers of atmosphere 
 
 rom r the P r wo ^ cn are ^ n contact with the metal surface causes an 
 dew-point. increase in the density both of the air and of the water 
 vapor of which these layers are composed, yet, since the baro- 
 metric pressure is in no way affected by the cooling, it is evident 
 that the pressure both of the air and of the water vapor within 
 these layers must remain precisely the same as outside, where no 
 cooling takes place. The beginning of precipitation means only 
 that in the layers adjoining the surface the density and pressure 
 corresponding to saturation have been reached. If, then, the 
 pressure within these layers is the same as outside, it is clear that 
 table 6 gives the correct value of p. 
 
 Not so, however, with d. The table gives, indeed, the value 
 of this quantity within the cooled layer, but the density at 
 a distance is the density within the cooled layer multiplied 
 
 , T r r 273 + n 
 
 by = ^ ; for when pressure remains constant, density is 
 inversely proportional to absolute temperature. Instead of obtain- 
 
HYGROMETRY 
 
 167 
 
 ing d in this way from the table, it may be directly calculated 
 from p by the ordinary gas laws, viz., by (146). Thus, since 
 the densit j of air at 0, 760 mm., is .001293, and 
 g i nce under like condition of temperature and pres- 
 e density f water vapor in terms of air is .624, 
 point. (146) becomes, when d ai , 2h and TI ar e replaced by 
 
 d, p and T tt and d a . 2 , 2h a nd T 2 by .001293 x .624, 760 and 273, 
 
 ro obtain 
 of e water tv 
 l thedew rom stire 
 
 
 .001293 x. 624x273 _ . 00029 p 
 760 . T t ~ P ~ ~^T~ 1 
 
 (149) 
 
 in which p must, of course, be expressed in mm. of mercury, 
 since the barometric pressure has been so expressed. This exten- 
 sion to unsaturated vapor of the laws which hold rigorously only 
 for ideal gases must be permissible in practice, since the results of 
 Problem 1, page 162, have shown that at ordinary temperatures 
 equation (149) may be applied without appreciable error even 
 to saturated water vapor. The rela- 
 tive humidity r can be at once obtained 
 from either p or d and table 6; for 
 d p 
 
 r = 
 
 One of the most perfect forms of 
 the dew-point hygrometer is due to the 
 Frenchman All uard (1880), 
 is shown ir 
 
 SSmeter. The nickel tube A ) u P on 
 
 which the dew is formed, is 
 about 2 cm. in diameter, and has one 
 flat, highly polished side which is placed 
 in close juxtaposition to a strip B of 
 equally well polished nickel upon which 
 no dew is formed. Tube A is filled 
 with ether, the temperature of which 
 may be lowered by causing a current of 
 air to bubble through it. This is 
 accomplished by means of an aspirator 
 attached by a rubber tube to C. A 
 bellows attached to F serves the pur- 
 pose equally well. The experimenter 
 
 FIGURE 89 
 
168 
 
 MOLECULAR PHYSICS AND HEAT 
 
 The chem- 
 ical hygrom- 
 eter. 
 
 sits at a distance of 10 or 12 feet, so that the moisture from his 
 breath or body may not affect the result, and observes the tube and 
 thermometer through a telescope. At the instant at which A begins 
 to look cloudier than B, he takes the temperature indicated by the 
 thermometer E. He then stops the current of air and observes 
 again the temperature at which the cloudiness disappears from A. 
 With a little practice the temperatures of appearance and disap- 
 pearance of the dew can be made to approach to within .1G. 
 The mean of these two temperatures is taken as the dew-point. 
 This form of instrument should not be used in rapidly moving 
 air, for then the layers of air which are in contact with A are 
 removed before they can take up the temperature of the nickel, 
 and in consequence the observed dew-points are too low. 
 
 The indications of a dew-point hygrometer may be very nicely 
 checked by means of the chemical method, first used for hygro- 
 metric determinations by the Swiss chemist Brunner in 
 1844. It consists in slowly drawing a known volume 
 of air V through drying-tubes, preferably of anhydrous 
 phosphorus pentoxide, and measuring the increase w in the weight 
 of the tubes. Let v represent the volume of water which has 
 been drawn out of the aspirator R during the experiment (see 
 Fig. 90). The gas which has replaced this water consists of the 
 perfectly dry air which 
 has emerged from the 
 drying-tubes and the 
 water vapor which has 
 formed from the water 
 in R. Since this vapor 
 may be assumed to be 
 saturated, the pressure 
 which it exerts is P, 
 the pressure of satu- 
 rated vapor corresponding to the temperature of the room. Hence 
 the pressure exerted by the air alone which is within R is H P, 
 H being the barometric pressure. When this same air was 
 outside, it exerted a pressure Hp, p being, as above, the 
 pressure exerted by the water vapor in the outside air. Hence 
 the volume which the air in R occupied before it entered 
 the drying-tubes, i.e., the volume of air Fin which the weight 
 
 FIGURE 90 
 
HY GEOMETRY 
 
 169 
 
 w of water vapor was contained, is given by (see Boyle's Law) 
 
 V H-P 
 v ~ H- p 
 w 
 
 (150) 
 
 From (149), (150), and the relation = d, there results 
 
 .00029;? 
 
 H-P 
 H-p 
 
 (151) 
 
 an equation which contains only one unknown quantity, viz., 7?. 
 After p has been determined from (151), d and r may be found as 
 above, while r is taken from table 6 ; i.e., it is the temperature- of 
 saturation of water vapor corresponding to the pressure p. In 
 case P and p have nearly the same value, (150) gives V- v, in 
 
 
 which case d is obtained at once from d = -, and p is then found 
 
 v 
 
 from (149). 
 
 This chemical method leaves nothing to desire in the matter of 
 accuracy, but since an observation usually requires from 1 to 3 
 hours, the result is, of course, only a mean value of 
 the humidity during this time. It is little employed 
 in practice. 
 
 The instrument which is now most extensively 
 used in meteorological observations was first con- 
 ceived by the Scotch physicist John Leslie 
 
 
 The wet-and- . T , . . . 
 
 dry huib m 1790. It was given its present form 
 by August, in Berlin, about 1825. It is 
 called the wet-and-dry bulb hygrometer, and con- 
 sists of two sensitive thermometers mounted side by 
 side, one of which has its bulb wrapped in muslin, 
 which is kept moist by means of a cotton wick 
 immersed in a water reservoir c (see Fig. 91). The 
 temperature t' indicated by the wet-bulb thermom- 
 eter is always lower than the temperature t shown 
 by its dry-bulb neighbor, unless the water vapor in 
 the air is already in a state of saturation, for the 
 evaporation which otherwise takes place produces 
 cooling (see page 155). The amount of this cool- 
 ing evidently increases with the rapidity of evaporation, which is 
 in turn directly proportional to the dryness of the atmosphere, 
 
 FIGURE 91 
 
170 MOLECULAR PHYSICS AND HEAT 
 
 i.e., to D d, or, what amounts to the same thing, to P p. But 
 it also depends upon the density of the air into which the evapo- 
 ration takes place, i.e., upon the barometric height H, as well as 
 upon the velocity of the air currents (see page 155). A long 
 series of comparisons of this instrument with the dew-point hygrom- 
 eter has given for air in moderate motion the empirical formulae 
 
 (for f above 0) (for t' below 0) 
 
 p = P-0. 00080 H(t-t'), and p = P-0. 00009 H (t-t f ), (152) 
 in which p, P and H are expressed in mm. of mercury and (t t') 
 in degrees centigrade. When^? has been determined from this 
 equation, d is given by (149), and then r and T from table 6. 
 For air at rest, the numerical quantities in (152) are considerably 
 too small. In order to make (152) applicable to indoor observa- 
 tions, it is recommended to hang the instrument from a long cord 
 and to let it swing as a pendulum for several minutes before tak- 
 ing the readings. Hygrometer makers usually specify carefully 
 the conditions under which instruments of this type are to be 
 used, and furnish with them empirical tables. 
 
 Experiment 
 
 To compare the indications of a dew-point and a 
 wet-and-dry bulb hygrometer. 
 
 Place the Alluard hygrometer in a room which is free from 
 evaporating water, pour ether into G (Fig. 89) until the liquid 
 surface is above the window in A, turn the polished 
 metal face into as favorable a light as possible, set the 
 observing-stand, telescope, and aspirator at a convenient distance, 
 and take, first, a rough observation of the dew-point. In subse- 
 quent observations regulate the evaporation of the ether so that 
 the temperature falls very slowly in the neighborhood of the point 
 sought, and take the reading of the thermometer E when the first 
 cloudiness begins to show upon A. In taking observations with 
 a rising temperature, an occasional bubble may be allowed to pass 
 through the ether in order to keep it stirred. Tabulate results 
 as in the Record. Then bring into the room a wet-and-dry bulb 
 hygrometer, and take a set of observations in the manner indicated 
 in the Theory. 
 
No. of 
 obs'n 
 
 HYGROMETRY 
 
 Record 
 
 DEW-POINT HYGROMETER, 
 
 Room 
 tern. (E') 
 
 Dew 
 app'd (E) 
 
 Dew 
 disapp'd (E) 
 
 Mean 
 
 P = 
 
 .-. d = 
 
 171 
 
 Wet-and- 
 dry bulb 
 
 H= 
 
 Problems 
 
 1. When the relative humidity is .47 at 21 C., what will be 
 the dew-point? 
 
 2. If the temperature of the air at sundown on a clear day be 
 10, and if the wet-bulb thermometer read 8C., at what tem- 
 perature will dew form? Need there be fear of frost during the 
 night? (Bar. ht. 750 mm.) 
 
 3. If the wet-bulb thermometer of Problem 2 had read 4.5C., 
 what would have been the dew-point? In this case frost would 
 have been almost certain. Why? 
 
 4. Dry air at 18, 755 mm., weighs .001205 gm. per cc. Find 
 the density of the atmosphere at this temperature and pressure 
 when the dew-point is 10 C. 
 
XX 
 
 ARCHIMEDES' PRINCIPLE 
 
 Theory 
 
 The law which asserts that the loss in loeight experienced by 
 
 any body ivhen immersed in a fluid is equal to the iveight of the 
 
 displaced fluid, was discovered by the immortal Greek 
 
 Proof of law * J ' . J . . 
 
 ofArcM- philosopher Archimedes, who perished in the siege of 
 Syracuse in 212 B.C. Unlike many of the laws which 
 have preceded, it is not an approximation, nor is it primarily 
 empirical, experiment having only served to confirm results which 
 follow with certainty from theory. The work of Archimedes was 
 not known in the middle ages, and the law was rediscovered in 
 1586 by the Flemish scientist Stevin, who advanced for it the 
 following proof : Within a body of fluid, isolate in thought some 
 mass by means of any imaginary bounding sur- 
 face 8 (see Fig. 92) 4 . Since the mass of liquid 
 within this boundary is in equilibrium, its 
 weight must be neutralized by forces whose 
 existence is due to the surrounding liquid. 
 But these forces which are exerted by the 
 
 surrounding liquid upon the surface S depend 
 only upon the conditions which exist outside of S, and are wholly 
 independent of the nature of the substance within S. Hence 
 any immersed body whatever which has the surface S must be 
 buoyed up by forces the resultant of which is equal to the weight 
 of the displaced liquid. 
 
 It follows from this law that when a body is balanced upon 
 scales in air, the balancing weights do not accurately represent 
 Application ^ e ^ nie we ig n ^ ^ ^ ne body, i.e., its weight in vacuo. 
 of. ArS -^ or kth ^ e body and the weights are buoyed up by 
 etermina- e ^ e a ^ r ' an( ^ s i nce i n general the volume of air dis- 
 ^rlct placed by the body is not the same as that displaced 
 weight. by the weights, the buoyant effects upon the two sides 
 must be different. However, with the aid of the "principle of 
 moments" the true weight X can be easily obtained from the 
 apparent weight JF(i.e., the weight of the weights), the volume 
 
 172 
 
r 
 
 ARCHIMEDES 7 PRINCIPLE 173 
 
 Fof the body, the volume V of the weights and the density <r of 
 air. For, since the resultant downward force on the side of the 
 body is (X- F<r) grams, and that on the other side (W- V'a) 
 grams, the equation of balance for equal balance arms becomes 
 
 X- V<r= W- V'a- or X= PF+(F- F>. (153) 
 
 Hence the so-called air correction, i.e., the correction which must 
 be applied to the apparent weight W in order to obtain the true 
 weight X, is simply the difference between the weight of air dis- 
 placed by the body and that -displaced by the weights. This 
 correction is evidently positive if V > V , negative if F < V. 
 Since the density of the weights is always known (for brass it is 
 
 W 
 
 8.4), it is usually convenient to replace F' by :, and to write 
 
 O.4: 
 
 (153) in the form 
 
 (154) 
 
 Or again, if the density d of the body happens to be roughly 
 known, (153) takes' the approximately correct form (since, on 
 account of the smallness of the correction term, Xis usually very 
 nearly equal to JF), 
 
 In deducing the above expressions for a correct weight X, the 
 balance arms were assumed to be equal. Although this is, of 
 course, seldom rigorously the case, it was shown on p. 39 that 
 the error arising from any inequality can be completely eliminated 
 by taking a mean of the weighings made on both pans. Hence, in 
 order to obtain a very accurate weight, it is necessary first to find 
 W by means of a double weighing, and then to apply to W the air 
 correction as shown in (153). 
 
 In case the ^quantity sought is a small increase or decrease in 
 weight, as in Exs. XV and XVII, the rigorous process described 
 in this section is not to be recommended; for neither the error in 
 the balance arms nor the air correction due to the weights is in 
 such cases likely to be an appreciable quantity. For most pur- 
 poses single uncorrected weighings, in which, however, the body 
 weighed always hangs from the same balance arm (see p. 118), are 
 sufficiently accurate. 
 
174 MOLECULAR PHYSICS AND HEAT 
 
 Archimedes' principle also furnishes the most convenient and 
 accurate method of determining densities. For, if any body, 
 regular or irregular, whose absolute weight is X grams, 
 ke f un( i to lose L grams when weighed in water of 
 densitv P ^ is evident at once from the statement of 
 Archimedes' principle that the volume of the body is 
 
 denxity of 
 
 a solid. ? an( j hence that its density d, which is bv definition 
 
 P 
 mass , . 
 
 = , is given by 
 
 volume' 
 
 *-- (6) 
 
 P 
 
 But, in order to find L accurately, an air correction of the ordi- 
 nary form must be applied to the apparent loss of weight. For 
 let the body whose true weight is X and whose apparent weight 
 is W be immersed in water and weighed, first when hung from one 
 balance arm, then from the other, and let this mean apparent weight 
 in water be W^. The equation of balance for this case, in which 
 the body hangs in water while the weights hang in air, is evidently 
 
 X- V P = Wi-^r. (157) 
 
 Substituting the value of X found in (154), there results at once 
 
 (W 
 V- Ztj 
 
 which is an equation of the same form as (154), the apparent 
 weight W having been simply replaced by the apparent loss of 
 weight (W-WJ. 
 
 The very statement of Archimedes' principle also suggests its 
 
 use for determining the densities of liquids. For, if L represent 
 
 the loss of weight of a body in water of density /o, and 
 
 af P the C iaw n ^ ^he loss of weight of the same body in another 
 
 medestothe ^ u ^ ^ unknown density d, then, since the volume of 
 
 determina- J^ 
 
 denrf? */ ^ e bcly is , the density of the unknown liquid is 
 
 a liquid. P 
 
 given by 
 
ARCHIMEDES' PRINCIPLE 
 
 175 
 
 The weighings in the two liquids may, of course, be made with 
 an ordinary balance, precisely as above, but for the sake of 
 rapidity a modified form of balance due to Mohr (see Fig. 93) is 
 commonly used. 
 
 The principal features of this balance are (1) the division of 
 one arm into ten equal parts, and (2) the use of weights of con- 
 venient shape to hang 
 from any of the ten 
 notches, the tenth of 
 which is the hook c. 
 Since each weight can 
 be given ten differ- 
 ent values, a very 
 small number of 
 weights is required. 
 There are never more 
 than five. Further, 
 as will presently ap- 
 pear, the absolute 
 value of these weights 
 is of no importance 
 
 if only their ratios are FIGURE 93 
 
 accurately represented 
 
 by the numbers 1, .1, .01, and .001 (the fifth weight is a 
 duplicate of 1, used for the sake of extending the range of 
 the instrument). For, suppose that the body B is of such 
 weight that, when hung in air from the. hook c, it is possible 
 by means of a little adjustment (see Experiment) to balance the 
 beam so that the two points at a are exactly together. Next 
 suppose that when B is immersed in water as in the figure, it 
 is necessary, in order to bring the points together again, i.e., 
 in order just to counterbalance the buoyancy of the water, to 
 hang weight 1 and weight .1 from the hook c, and weights .01 
 and .001 from notch 4. The loss of weight L of the bulb, 
 expressed, not in grams, but in terms of weight 1, is then evi- 
 dently 1.1044. Suppose now that when the water is replaced by 
 another liquid of the same temperature, it is found necessary, in 
 order to bring the points again together, to hang 1 from notch 9, 
 .1 from notch 2, .01 from notch 4, and .001 from notch 9. The 
 
176 MOLECULAR PHYSICS AND HEAT 
 
 loss of weight of the bulb in this liquid, expressed as above in 
 terms of 1, is then .9249. Hence (159) gives 
 
 j_ .9249 
 ~ri044 p ' 
 
 If p is known d is at once obtained, no matter what happens to be 
 the weight of 1. In practice this weight is usually arranged to be 
 as nearly as possible equal to the weight of the water displaced by 
 the bulb at 15C. In this case the reading for water at this 
 temperature is evidently 1.0000, and if no high degree of accuracy 
 is required, the reading of the instrument when the bulb is 
 immersed in the unknown liquid gives at once the density of that 
 liquid. Eigorously the apparent losses in weight L and ', 
 obtained by means of a Mohr's balance, are subject to air correc- 
 tions, but since in practice the balance is only used to compare 
 densities which differ comparatively little from one another, the 
 influence of these corrections upon the result is negligible. They 
 could be made, if necessary, by reducing L and L' to grams and 
 then applying (158), F being in this case the volume of the bulb. 
 
 Experiment 
 
 1. To compare a density determination made upon an accurately 
 turned cylinder of aluminium by means of weight and dimension 
 measurements, with a determination made upon the 
 same cylinder by means of observations of weight and 
 loss of weight in water. 2. To compare the results of measure- 
 ments upon the density of a liquid made by a Mohr's balance, 
 with those made by an ordinary constant-weight hydrometer. 
 
 DIRECTIONS. 1. First find the volume of the cylinder from 
 very careful measurements of its dimensions, made by 
 
 Volume. 
 
 means of calipers. 
 
 Then suspend the cylinder from a very fine platinum wire and 
 make an absolute weighing in air of the cylinder and wire as follows : 
 Absolute CO Choose any convenient zero to which to refer your 
 weighing. weighings, e.g., the 10 mark.* 
 
 *The true zero of the unloaded balance may be determined if desired, 
 but this is never necessary with a double weighing; for evidently, if the 
 arbitrarily chosen zero differs from the true zero, the weighing upon one 
 
ARCHIMEDES' PRINCIPLE 177 
 
 (2) Hang the body from the left arm and, using all the precau- 
 tions mentioned on page 116, find the resting point JKj, which 
 corresponds to such a weight in the right pan as will keep the 
 pointer swinging within three or four divisions of the chosen zero. 
 
 (3) Add a 2 mg. weight to the lighter side and take the new 
 resting point J? 2 ; then at once raise the arrest and compute the 
 sensitiveness. 
 
 (4) Exchange the positions of the cylinder and the weights, at 
 the same time making and recording a careful count of the latter, 
 exclusive of the two added milligrams. 
 
 (5) Find the new resting point 7? 3 ; then raise the arrest and 
 count again the weights as they are returned to their respective 
 compartments in the box of weights. 
 
 (6) Calculate from the sensitiveness the corrections necessary to 
 apply to the counted weight upon each side, in order to make 
 both of the resting points coincide with tke- chosen zero. Thus, 
 if the zero is 10, the sensitiveness 2.1, and the resting point 
 when the cylinder is on the right pan 10.9, the correction, in this 
 
 case to be subtracted from the counted weight, is - = .4. 
 
 /. 1 
 
 Represent the mean corrected weight by W and the result obtained 
 by applying to W the air correction (154) by X. 
 
 Next immerse the aluminium cylinder, in the manner shown 
 in Fig. 94, in a beaker of distilled water which has been recently 
 
 freed from air by boiling, but which 
 in water. has again regained the temperature of 
 
 the room. Carefully remove from the 
 immersed body all bubbles, even the smallest, by 
 means of a camel's hair brush, then weigh on both 
 sides, following exactly the directions 1 to 6 above. 
 Represent the mean weight in water by Tf l5 and 
 let the result obtained by applying the air cor- 
 rection (156) to the apparent loss in weight, 
 W- WH be represented by L. FIGUUE 
 
 side will be just as much too large as that upon the other is too small, 
 and the mean will therefore be the same. If, however, a weighing is 
 made upon balances which are known to have arms so nearly alike that a 
 double weighing is unnecessary, this single weighing must be referred to 
 the true zero obtained by the method of oscillations (see p. 116). 
 
 
178 MOLECULAR PHYSICS A^D HEAT 
 
 In order to obtain Jf, the weight of the cylinder alone, the 
 weight w of the suspending wire must, of course, be obtained and 
 
 subtracted from th.e joint weight X obtained above. 
 the suspend- Since this is a very small quantity, a single weighing 
 
 made upon one pan, and uncorrected for displaced air, 
 is sufficient. To make this weighing, find: 
 
 (a) The true zero of the balance (see note, p. 170); 
 
 (b) The resting point /*, when the wire alone is on one pan, 
 and some nearly equal weight on the other ; 
 
 (c) The resting point r a (after 2 mg. have been added to 
 determine the sensitiveness for this load) ; 
 
 (d) The weight w of the wire (obtained from the weight in the 
 pan and the sensitiveness). 
 
 The loss of weight L represents, of course, the weight of water 
 displaced both by the immersed cylinder and the immersed por- 
 tion of the suspending wire. Estimate roughly the 
 
 Correcting ^ - i i -,. 
 
 Lforim- volume v oi this immersed wire by measuring its diam- 
 
 mersed wire. *,>,,-, . x 
 
 eter 8 (with the micrometer cahper) and its approxi- 
 mate length L This volume v is approximately the weight of 
 the displaced water. Hence the loss of weight L of the cylinder 
 is given by L = L v. 
 
 2. Loosen set screw s' of the Mohr's balance (Fig. 
 93), and turn the base until the leveling screw s lies in 
 the vertical plane which includes the beam. Adjust ver- 
 tically at s f until a convenient height for immersion is 
 reached. Then from the hook c hang the float B, in air, 
 and level by means of s until the two points at a are 
 very accurately together. Then, by means of the weights 
 bring the points again together, first when the float is 
 immersed in distilled water, then when immersed in the 
 liquid of unknown density, and take the corresponding 
 readings (see Theory). The density of the water at the 
 observed temperature is taken from the table of water 
 densities. The temperatures of the two liquids compared 
 must be the same, otherwise a correction must be applied 
 because of the expansion of the float. 
 
 Compare the density given by the Mohr's balance with 
 the indication of a direct-reading, constant-weight hydrom- 
 FIG. 95 eter (see Fig. 95). The theory of this instrument is 
 
ARCHIMEDES' PRINCIPLE 179 
 
 too simple to require explanation. The reading is made through 
 the liquid, the eye being placed as little as possible beneath the 
 level of the surface. If the instrument does not read the correct 
 density of distilled water at the observed temperature, a correc- 
 tion amounting to the difference must be applied to its indication 
 of the density of the other liquid. For very accurate density 
 determinations with hydrometers of this sort it is important that 
 the stem be wet above the point of contact with the liquid, since 
 otherwise the capillary forces between the liquid and the stem may 
 give rise to very considerable errors. Hence, before taking a read- 
 ing, push the instrument down below its natural position of 
 equilibrium and then let it rise. If in this operation the liquid 
 be observed to be depressed about the stem, instead of elevated, 
 the stem should be carefully cleaned with an alkali, e.g., potas- t 
 sium hydrate. 
 
 Record 
 
 1. 'Bar. ht. = Temp, of room = ..% Density of air = 
 Diam. of cylinder 1st obs. 2d 3d 4th mean = 
 Height of cylinder " - mean = -- . . V= 
 
 Weighing of cyl. -f wire cyl. + wire wire 
 
 in air in water alone 
 
 Chosen zero = Zero t * = 
 
 R! (cyl. left) = - Rest. pt. r 1 = - - P = 
 
 E 2 (cyl. left) = - Rest. pt. r 2 = - -I 
 
 jR 3 (cyl. right) = . *. Sensitiv's = -5 = 
 
 Counted wts. = Counted wt. v = 
 
 . : Sensitiv's (8) = - Cor'd wt. (w) = - 
 
 Cor'dwt. left = - - .:X(=Xw) = - 
 
 Cor'd wt. right = - ~L(= L v) = - 
 
 Mean(W) = - W, = - d = ^ = - d = ^ = 
 
 True wt. (X) = L = % difference in d's = 
 
 2. Reading of Mohr's balance in water = tern. = p = 
 Read'g of Mohr's balance in unknown liq.= tern. = -- . . d = 
 Hydrom'r in water = . *. corr'n = in liquid = -- .'. d (cor'd) = 
 
 * t tern, and p = den. of water. I, 5, and v are length, diameter, and 
 volume of immersed portion of wire. 
 
180 MOLECULAR PHYSICS AND HEAT 
 
 Problems 
 
 1. If the length measurements made upon the cylinder were 
 3.021, 3.023, and 3.024 cm., and if the diameter measurements 
 were 2.567, 2.563, 2.564, and 2.562 cm., find what is the first 
 uncertain figure in the number which represents the volume. 
 (Results should never be recorded farther than to one place beyond 
 the first uncertain figure.) 
 
 2. If the weighings can all be made with such accuracy that 
 the tenths ing. place is the first place of uncertainty, find to how 
 many more places of certainty the density is given by the loss of 
 weight method than by the direct measurement method (weight 
 of cylinder about 12 gm.). 
 
 3. Archimedes discovered his principle when seeking to detect 
 a suspected fraud in the construction of a crown made for the 
 tyrant of Syracuse. It was thought to have been made from an 
 alloy of gold and silver instead of from pure gold. If the crown 
 weighed 1000 gm. in air and 940 gm. in water, find how many 
 gm. of gold and how many of silver were used in its construction. 
 
 Assume that the volume of an alloy is the combined volumes of the 
 components, and take the density of gold as 19.3 and that of silver as 10.5. 
 
 4. A 10-gm. weight placed upon a block of wood weighing 30 
 gm. sinks it to a certain point in water. In a salt solution it 
 requires 15 gm. to sink the wood to the same point. Find the 
 density of the salt solution. 
 
 5. If the density of ice is .918 and that of sea water is 1.03, 
 find what fraction of the total volume of an iceberg is above 
 water. 
 
 6. A cylinder of cork 10 cm. high and of density .2 floats upon 
 water. If the air above the water be removed, will the cork sink 
 or rise in the liquid? How much? 
 
 Assume incornpressibility in both cork and water. 
 
 7. Suppose that a constant-weight hydrometer which it is 
 desired to calibrate is immersed in two liquids whose densities are 
 known to be 1. and 1.1, that the two points of immersion are 
 accurately marked, and that the intervening stem is then divided 
 into 10 equal parts. Assuming that the stem is accurately cylin- 
 drical, will this hydrometer give correct readings in liquids of 
 intermediate densities? Why? 
 

 XXI 
 
 CAPILLARITY 
 
 Theory 
 
 One of the fundamental assumptions made in elementary 
 
 hydrostatics is that a liquid, like so much sand, exerts pressure 
 
 merely by virtue of its weight, and by virtue of the 
 
 Ordinary J J J 
 
 law of liquid property, which it possesses in common with all fluids, 
 
 of transmitting pressure in all directions.* Thus if 
 
 p be the pressure (force in grams per unit area) exerted upon the 
 
 surface of a liquid of density d, then the number of grams of 
 
 *This property follows at once from the fact of fluidity and the fun- 
 damental laws of mechanics. Thus let A (Fig. 96) be a substance concern- 
 ing which the one assumption is made that it is capable of 
 adjusting itself with perfect ease to any change in the shape 
 of the containing vessel. Let /and/' be two forces acting 
 upon frictionless pistons 1 and 2 of areas a and a' respectively. It is 
 required to find the ratio which must exist 
 between the forces /and/', in the condition 
 of equilibrium. Let piston 1 move uniformly 
 down a distance s thus crowding out of cyl- 
 inder 1 a volume of fluid as. /' must then 
 move uniformly up a distance s' such that 
 as a ' s ' . But from the ' 'principle of work" 
 (scholium to Third Law) in the condition of 
 equilibrium (rest or uniform motion) fs = 
 
 f's'. Hence -p-= r. i- e-. the forces which 
 must act on the pistons in the condition FIGURE 96 
 
 of equilibrium are directly proportional to 
 
 their areas. Since the directions of the forces /and/' are wholly arbi- 
 trary, there results the law first announced in 1653 by the French 
 philosopher, mathematician, and man of letters, Pascal, "The forces 
 transmitted by fluids act equally in all directions and are proportional to 
 the areas of the surfaces upon which they act," a law which finds its 
 most beautiful experimental demonstration in the hyrdaulic press. 
 
 181 
 
182 
 
 MOLECULAR PHYSICS AND HEAT 
 
 Capillary 
 phenomena. 
 
 pressure P, which exists at any depth z beneath the surface, is 
 given by (see Fig. 97) 
 
 P=p Q + zd. (160) 
 
 There follows at once then the result, in general confirmed by 
 experiment, that a liquid contained in a series of 
 communicating vessels must take the same level in 
 all of them, no matter how different they may be in 
 size or shape. 
 
 But it was observed as early as 1500 by that most 
 universal of all geniuses, Leonardo da Vinci, that 
 when a tube approaches capillary dimen- 
 sions water rises in it far above the level 
 in the outside vessel (see Fig. 101). Later 
 and more careful investigation has shown the existence 
 FIGURE 97 of a large number of different phenomena to which 
 the ordinary laws of hydrostatics do not apply. These 
 are usually all called "capillary phenomena," because they were 
 first observed in connection with capillary tubes. They are all 
 manifestations of those intermolecular forces which were assumed 
 in Ex. XVIII in order to reconcile the existence of liquid surfaces 
 with the theory of molecular motion. This section is devoted 
 to a study of the effects of these forces ; and since these effects 
 would of necessity be just the same whether the molecules are 
 at rest or in motion, the fact of motion will for the present be 
 disregarded. 
 
 The simplest experiments place the existence of these inter- 
 molecular forces beyond the possibility of. doubt, and show at the 
 same time that, while they have enormous values at 
 short range, they diminish so rapidly with the distance 
 as to become wholly inappreciable at distances which 
 still amount to extremely minute fractions of a milli- 
 meter. Thus a drop of mercury, instead of spreading out into an 
 infinitely thin layer, as it would do if gravity alone acted upon its 
 molecules, is held together in globular form. A sheet of glass 
 may be brought extremely close to a surface of water without 
 appearing to be attracted toward it in the slightest degree, but as 
 soon as contact is made the glass clings to the water with remark- 
 able tenacity. The surface of two metal blocks may be brought 
 to within a thousandth of an inch without showing any appre- 
 
 Evidence as 
 to the exist- 
 ence and 
 nature of 
 molecular 
 forces. 
 
CAPILLARITY 
 
 183 
 
 pressure. 
 
 ciable attraction, but as soon as they are brought somewhat 
 nearer, as by pressure or welding, it requires tons of weight 
 to pull them apart again. The operation of the aspirator 
 pump described in Ex. XVIII is due to the attraction between 
 the air about the orifice o and the outpouring current of 
 water. 
 
 Starting,' then, with these two facts, (1) the existence of 
 intermolecular forces, and (2) the rapid diminution of these 
 Laplace's f orces with the distance, * the great French geometrician, 
 Laplace, first developed, about 1807, a theory of capil- 
 lary action. His reasoning was somewhat as follows: 
 Let r represent the distance within which one molecule attracts 
 another with a force which is large enough to deserve consid- 
 eration. Laplace called it the radius of influence of molecular 
 force. It is, of course, not a quantity the magnitude of which is 
 definitely fixed, but it probably never exceeds .00005 mm. 
 
 Now a molecule m' in the interior of a liquid is indeed acted 
 upon by all the multitude of molecules lying within a sphere 
 of radius r (see Fig. 98); but, by virtue of symmetry, the 
 resultant of all these forces is evidently zero ; so that m' may 
 be treated as 
 though it were 
 under the in- 
 fluence of no 
 molecular force 
 whatever. Not 
 so, however, with 
 any molecule m 
 which is nearer 
 
 to the surface than the distance r. For while the molecules within 
 the space cefd exactly neutralize the effects of the molecules within 
 the space acdl, the downward resultant of the forces of all the 
 
 FIGURE 98 
 
 *In order to account for this rapid diminution with the distance it is 
 not necessary to assume that these intermolecular forces are any other 
 than those concerned in ordinary gravitation. For the law of inverse 
 squares can be considered to hold for the attractions of masses of finite 
 volume only so long as the distance between the nearest points of the 
 attracting bodies is infinitely large in comparison with the distances 
 between the molecules of the bodies. 
 
184 MOLECULAR PHYSICS AND HEAT 
 
 molecules in efg is wholly unbalanced. This force continually 
 urges m into the interior of the liquid. But all the other mole- 
 cules in the same horizontal layer with m are urged inward with 
 the same force and all the molecules in other layers which are 
 within a distance r of the surface are urged in with other forces. 
 The result of all these unbalanced forces acting upon all the mole- 
 cules contained in the surface layer of thickness r (called the 
 active layer) must be, then, an interior pressure of uncertain, 
 perhaps enormous, magnitude. It has been estimated for water 
 at something like 10,000 atmospheres, but it has never been 
 measured directly and never can be. For since a liquid is always 
 bounded on all sides by a surface, this molecular pressure usually 
 balances itself and therefore cancels out in hydrostatic measure- 
 ments. Hence it is that equation (160), which leaves the existence 
 of molecular pressure altogether out of account, and treats the 
 liquid molecules as though they were so many independent grains 
 of sand, nevertheless gives, in general, correct results. But it 
 is exactly such apparent violations of the ordinary hydrostatic 
 laws as are shown in capillary phenomena, which furnish a beau- 
 tiful proof of the existence of Laplace's molecular pressure. 
 
 For it is easy to show that this pressure must be greater 
 
 underneath a convex, and less underneath a concave surface, than 
 
 it is beneath a flat one. Thus, let M (Fig. 99) represent 
 
 Variation ' . r 
 
 of molecular a molecule which lies in the active layer at a given dis- 
 tance beneath a surface, and let the circle drawn about 
 
 M represent the sphere of influence of molecular 
 forces. The surface will first be assumed to be flat (acb), then 
 convex (ecf), and then concave (gcli). In the first case, since 
 pidjq neutralizes pacbq, the resultant downward force acting 
 upon Mis due to the attraction of the molecules lying within 
 the segment iojd of the sphere. In the 
 second case plcdlq neutralizes pecfq, and 
 the resultant downward force is due to 
 kold, a volume which is greater than 
 iojd. In the third case the resultant force 
 is due to mond, a volume which is less 
 than iojd. Hence the resultant downward 
 force upon the molecules in the active layer 
 FIGURE 99 is greatest beneath the convex, and least 
 
CAPILLARITY 
 
 185 
 
 beneath the concave, surface. It is evident also from the same 
 kind of reasoning that the greater the convexity, the greater this 
 pressure. Thus, if the pressure beneath a plane surface be repre- 
 sented by P Q (Laplace named this the normal pressure), that 
 beneath a curved surface is P p, in which the magnitude of 
 p depends upon the nature of the liquid and the magnitude of 
 the curvature, while its sign is + or according as the surface is 
 convex or concave. Laplace proved by a mathematical analysis 
 of the forces exerted by segments of the kind shown in Fig. 99, 
 that p, expressed in terms of a characteristic constant A of the 
 liquid (called its capillary constant), and the two principal radii 
 of curvature R and R' of the surface, is 
 
 (161) 
 
 This makes the ascension or depression of liquids in capillary 
 tubes perfectly intelligible. For, starting with the fact of obser- 
 va ^ 011 (accounted for below) that a liquid in a small 
 * u ^ e assnmes a curved instead of a flat surface, its rise 
 depression. or f a ;Q j n the tube, according as the surface is concave 
 or convex, follows as a matter of course from a very simple con- 
 sideration of the pressures involved. 
 
 Thus, the correct value of the pressure at a distance z below a 
 plane surface is not p Q + zd, as assumed in (160), but rather 
 p + P + zd, and the pres- 
 sure at the same distance 
 z beneath a concave surface 
 in a capillary tube (see Fig. 
 100) is 
 
 
 V^ 
 
 >! 
 
 
 : 
 
 )o 
 **% 
 
 }m 
 
 ~ 
 
 I 
 
 FI 
 
 ( 
 
 ;i 
 
 r* 
 
 HI 
 
 3 100 
 
 z 
 > a 
 
 FlGTJ 
 
 RE 101 
 
 Hence from Pascal's Law of 
 
 the equal transmission of 
 
 pressure (see note, p. 181) 
 
 there can be no equilibrium 
 
 until the stronger molecular 
 
 pressure beneath the flat surface has pushed up the liquid in 
 
 the tube to such a height h (see Fig. 101) that the total pressures 
 
186 MOLECULAR PHYSICS AND HEAT 
 
 at any two points a and # in the same horizontal plane are the 
 same, i.e., until 
 
 p Q + PQ + Zd = PQ + \P Q - A (-jj- + jj; j + Z(l + lid \ 
 
 or 
 
 R and R' are the curvatures at the point considered, e.g., 1, 2, 
 or 3 (Fig. 101), and h is the elevation of this point above the out- 
 side plane surface. 
 
 It thus appears that, correctly speaking, a liquid does not 
 rise in a capillary tube because of a capillary attraction, any more 
 than it rises in a suction pump because of the attraction of the 
 vacuum created by the lifting of the piston. In both cases the 
 liquid is pushed up by a pressure existing outside. In the case of 
 the pump this is the atmospheric pressure acting on top of the 
 water in the cistern; in the case of the capillary tube it is the 
 normal pressure P acting in the surface layer of the outside 
 liquid. 
 
 If it were possible to remove entirely the molecular pressure 
 
 within the tube, the height of rise would be a measure of P , just 
 
 as the height of rise of the water in a long tube from 
 
 Measurement ,.,,,..,., , . 
 
 ofthecapti- which the air is entirely removed, is a measure of the 
 
 lar-y constant. 1 . . 
 
 atmospheric pressure. Since, however, nothing more 
 can be done than to obtain a curved surface within the capillary 
 tube, it is only the capillary constant A which can be found from 
 observations of the height of ascension li, the density d, and the 
 radii of curvature R and R' [see (162)]. 
 
 In the general case it would be difficult to measure R and R ', 
 but if the tube is cylindrical, then it follows from symmetry that 
 at the middle of the meniscus R = R ', and (162) reduces to 
 
 9 A 
 
 hd=^- (163) 
 
 If, farther, the tube is so small that the height of the meniscus m 
 (see Fig. 101) is negligible in comparison with 'h, i.e., if h is 
 practically constant for all points of the surface, then it follows 
 
 from (162) that the curvature l-^- + - ) is also practically con- 
 stant. But the only surface of constant curvature which can ful- 
 
CAPILLARITY 187 
 
 fill the condition imposed by (163) is a section of a sphere. If 
 finally, then, the liquid can be made to wet completely the interior 
 of the tube, so that its angle of contact a with the walls is 180, 
 then the meniscus must be a hemisphere, and the radius R is 
 simply the radius of the tube. 
 
 Equation (163) is then applicable to all cases for which these 
 conditions hold.* It shows that the height of rise h is inversely 
 proportional to the radius of the capillary a law discovered 
 experimentally by an Englishman in 1718 and called after him the 
 law of Jurin. Equation (163) thus makes the measurement of the 
 capillary constant a very simple matter in the case of liquids 
 which wet solids of which capillary tubes can be made. 
 
 Another interesting conclusion which 
 can be drawn from the above reasoning 
 is that, since in equilibrium the height 
 h depends only upon the curvature and 
 the density, the dimensions of the capil- 
 lary above or below the point of contact __ 
 have no effect whatever upon the phe- 
 nomena. Thus, if water be drawn up FIGURE 102 
 into tubes of such different shape as a 
 
 and # (Fig. 102), it should come to rest in the descent at pre- 
 cisely the same level in both. This conclusion is wholly con- 
 firmed by experiment. 
 
 It only remains to show why a liquid in a capillary tube 
 assumes a curved surface a task of no difficulty when it is 
 remembered that a liquid surface can be in equilibrium 
 'contact U f on ^y wnen it is perpendicular to the resultant force 
 acting upon its molecules. This property follows sim- 
 ply from the fact of mobility of the particles. For, if the force 
 acting upon the surface molecules had any component parallel to 
 
 *If the height of the meniscus is not wholly negligible in comparison 
 with h, the mean value of h can be obtained by adding ^R to the height 
 of the lowest point of the meniscus. For the volume of the liquid above 
 this lowest point is the volume of a cylinder of radius R and height R, 
 minus the volume of a hemisphere of radius R; or, irR 3 %irR 3 = %irR 3 . 
 This volume divided by the area of the base, viz., -n-R 2 , gives the mean 
 
 Tf 
 
 height, viz., _-. Formula (163) thus modified holds for tubes of as much 
 o 
 
 as 2 mm. diameter. 
 
188 
 
 MOLECULAR PHYSICS AND HEAT 
 
 the surface, the molecules would move over the surface in obedi- 
 ence to this component, i.e., equilibrium would not exist. If, 
 then, on (Fig. 103) represent the line of junction of a liquid with a 
 
 solid wall, /! the resultant of all the 
 forces exerted upon the molecules at 
 o by such portion of the liquid as 
 lies within the molecular range when 
 the liquid surface is assumed hori- 
 zontal, and / 2 the resultant of the 
 forces exerted upon the same mole- 
 cules by the molecules of the wall 
 which lie either above or below the 
 horizontal line passing through o, 
 FIGURE 103 then three cases may be distinguished : 
 
 (1) That in which / t = 2/ 2 . In 
 
 this case, as appears from Fig. 103, the cohesion of the liquid is 
 exactly equal to twice the adhesion of the solid and liquid, 'and the 
 final resultant R is parallel with the wall. Hence equilibrium 
 exists in the condition assumed, i.e., the angle of contact a is 90. 
 
 (2) That in which /j > 2/ 2 . The resultant R then falls to 
 the right of on. Hence equilibrium can not exist until the 
 surface near o has become convex, i.e., until the angle of con- 
 tact a has become acute. This is the 
 
 case of liquids which do not wet the 
 wall. If the substances be mercury and 
 glass (Fig. 104), equilibrium is reached 
 when a is about 43. It is to be ob- 
 served, in general, that this angle a must 
 always be the same for the same two 
 substances. For, on account of the 
 extreme minuteness of the sphere of 
 influence, the weight of the particles 
 contained within it is wholly negligible 
 in comparison with the molecular forces, 
 i.e., it is simply the relation between these molecular forces 
 which determines the angle of contact. 
 
 (3) That in which /i < 2/ 2 . The resultant R then falls to 
 the left of on (see Fig. 105). Hence equilibrium can not exist 
 until the surface near o has become concave and the angle of con- 
 
 FlGUBB 104 
 
CAPILLARITY 
 
 189 
 
 tact obtuse. In all cases in which the liquid 
 
 completely wets the solid, the angle of contact 
 
 is necessarily 180, i.e., a thin film of the liquid 
 
 lies flat up against the face of the solid. This 
 
 is evident from the consideration that when a 
 
 partially immersed body is raised from a liquid, 
 
 the angle of contact can not remain constant 
 
 at any value less than 180 unless the liquid 
 
 retreats down the side of the body as rapidly 
 
 as the body rises, i.e., unless the liquid be one 
 
 which does not wet the solid. 
 
 The law of transmission of pressure by liquids easily accounts 
 
 for this, at first view, somewhat surprising result. For, in 
 accordance with this law, the molecular pressure P' 
 existing because of adhesion at a point in the liquid 
 
 close to the limit of contact ( see , Fig. 106 )> is trans- 
 mitted undiminished in a direction parallel to the sur- 
 face of the solid, and therefore constitutes a force pushing out the 
 
 molecules at c. The only oppos- 
 ing force acting to prevent the 
 limiting molecules from moving 
 up along the surface is the verti- 
 cal component of the attraction 
 / exerted upon these molecules 
 by such portion of the liquid as 
 lies within the sphere of in- 
 fluence drawn about c. Hence, 
 unless the ratio of the cohesion 
 to the adhesion exceeds a cer- 
 tain limit, &> thin film of the 
 liquid must spread out indefi- 
 nitely over the surface of the 
 
 solid. This conclusion is not surprising, since it means simply 
 that a body which attracts a liquid strongly enough will draw 
 every particle of it as near as possible to itself. 
 
 Thus it is that a drop of water spreads out indefinitely over a 
 perfectly clean glass or mercury surface, that a drop of olive oil 
 spreads over water, or, in general, that any liquid spreads out over 
 any perfectly clean surface which it wets. But such perfectly 
 
 FIGURE 106 
 
190 
 
 MOLECULAR PHYSICS AND HEAT 
 
 clean surfaces are very difficult to obtain, and that on account of 
 the prevalence of this very phenomenon. Thus, the least drop of 
 oil touching a mercury or a glass surface spreads over it very 
 quickly and completely changes the effect of adding a drop of 
 water. However, such familiar facts as the creeping of salt solu- 
 tions over battery jars, of kerosene over lamps, or the rapid 
 spreading of oil over water, attest the correctness of the above 
 conclusions. Of course, when but a drop of the liquid is present 
 a limit to the spreading must be reached when the liquid attains a 
 thickness of the. order of magnitude of the diameter of the mole- 
 cule. Rayleigh measured oil films on water which had a thickness 
 of but .000002 mm. The diameter of an oil molecule can not, 
 therefore, be more than this. 
 
 Another interesting result which may be deduced from 
 Laplace's theory of molecular pressure is that, in general, a liquid 
 must behave as though its surface were a stretched 
 of surface elastic membrane. For, since every molecule in -the 
 active layer is always being urged into the interior, it 
 follows that as many molecules as can possibly do so will leave 
 this layer and pass within., i.e., a liquid, like a distended rubber 
 balloon, will always tend to draw together into the form which 
 has the smallest possible surface for a given volume. Thus it is 
 that all bodies of liquid which are not distorted by gravity or 
 other outside forces always assume the spherical form, e.g., a 
 raindrop, a soap bubble, a globule of oil floating beneath the sur- 
 face of a liquid of the same density. 
 
 It follows again, from the tendency to assume the form of 
 smallest surface, that a liquid film, a form of enormous surface, 
 
 must exhibit a sensible contractil- 
 ity. Experiment amply supports 
 the conclusion. Thus, a soap bub- 
 ble may be observed to begin to 
 draw back into the 
 bowl of the pipe as 
 soon as the blower 
 removes his mouth. 
 
 A wet loop of cotton thread laid upon a soap 
 film, as in Fig. 107a, is snapped out at once into 
 circular form, as in Fig. 107b, as soon as the FIGURE ios 
 
 FIGURE 107 
 
CAPILLARITY 191 
 
 film within the loop is pricked with a pin. A film formed in 
 the frame abdc (Fig. 108) snaps the piece ab back to cd as soon 
 as the stretching force F is removed. 
 
 Further, Laplace's theory leads to the remarkable conclusion 
 that the contractility of liquid films is wholly independent of 
 Tension of their thickness. For the work which is performed by 
 Sendenfo/ an ^ a S erL ^ which is increasing the surface of a liquid 
 thickness. \ consists solely in bringing new molecules from the 
 interior to the surface, against the force of the molecular pressure. 
 Similarly the contractility of the film when the stretching force is 
 removed is nothing but a manifestation of the force of molecular 
 pressure drawing back molecules into the interior. Hence the 
 work done by the outside agent when the surface is increasing, 
 or by the molecular pressure when it is decreasing, is simply pro- 
 portional to the increase or decrease in surface; i.e., the work 
 required to pull down ab (Fig. 108) a given distance, e.g., 1 mm., 
 must always be the same, whether the film has been stretched 
 little or much, i.e., whether it is thick or thin. It is because 
 this conclusion is at variance with the law which governs the 
 stretching of solids (stretching force proportional to cross-section) 
 that it appears strange. It is, however, completely confirmed by 
 experiment. Thus the fact that the loop of Fig. 107b takes 
 the accurately circular form shows that it is subjected to pre- 
 cisely the same force at all points on its circumference; yet 
 the varying colors of the film show that it has a widely varying 
 thickness. 
 
 It is to be observed, however, that this conclusion as to 
 the constancy of F should hold only so long as the film is 
 more than twice as thick as the active layer; for after this 
 thickness has been reached the molecular pressure, and hence 
 also the work required to bring a new molecule from the middle to 
 the surface, must begin to diminish. It is probable, however, that 
 the film must break at this point. Hence it is that the smallest 
 thickness which a soap film can have is usually taken as a measure 
 of the diameter of the sphere of molecular influence. This 
 quantity, as measured by Johonnott at the Ryerson Laboratory in 
 1897, is .000012 mm. 
 
 It follows from the constancy of F that if ab (see Fig. 108) be 
 pulled down a distanced, the work done by ^is equal, to Fd. 
 
192 MOLECULAK PHYSICS AND HEAT 
 
 But this work is proportional to the increase in surface. If, then, 
 Relation ^ represent the amount of work which must be done 
 moiSar against the molecular pressure in order to bring enough 
 mrface 6 and m lecules into the active layer to form one new sq. cm. 
 tension. o f sur f ace? then, since the total increase in surface (con- 
 sidering both sides of the film) is %ab'd, it follows that 
 
 Fd = 2ab-d'T, or T=-> (164) 
 
 But %ab is simply the length of the line of surface to which the 
 stretching force F is applied. And since the value of .F depends 
 not at all upon the thickness of the liquid, but only upon the 
 length of the attached surface line 2ab, and upon a quantity T 
 which is proportional to the normal molecular pressure, it is 
 obvious that it is merely the surface of the liquid down to a 
 depth r, i.e., the active layer, which is to be regarded as the seat 
 of the contractile force F of the film. Finally then, since when 
 %ab = 1, 
 
 T= F, (165) 
 
 it follows that Laplace^s molecular pressure manifests itself in any 
 liquid surface as a tangential contractile 
 force (see Fig. 109) which is numerically 
 equal, in grams per cm. of length, to the 
 quantity of work, expressed in gm. cm., 
 FIGURE 109 which is required to bring up into the active 
 
 layer, against the molecular pressure, 
 enough molecules to form one new sq. cm. of surface. 
 
 A rather interesting experiment has been devised to illustrate 
 this fact of contractility when but 
 one surface of a thick film is al- ' 
 lowed to contract. A shallow ves- 
 sel with one side cd movable about c 
 c is filled with water (see Fig. 110). FIGURE no 
 As soon as the thread t is burned, 
 
 the side cd is pulled over into the vessel, in spite of the fact that 
 the weight of the liquid would tend to press it more firmly against 
 the support e. 
 
 Now the pressure existing within a rubber balloon may be 
 
CAPILLARITY 193 
 
 easily calculated from a knowledge of the tension in its elastic 
 envelope. Since, then, the molecular pressure in liquids man- 
 
 ifests itself as a surface contractility, it ought to be 
 of Laplace's possible to obtain, from the value of this contractility, 
 
 the quantity p of (161), i.e., the increase in 
 internal pressure which is due to a curvature of the surface. 
 Thus let o-o-' (Fig. Ill) represent an infinitely small rectangular 
 element of a convex surface. Let the arcs 
 <r and a-' correspond to the two principal 
 radii of curvature R and R '. If the ten- 
 sion in the surface has a force of T grams 
 per unit length, then the number of grams 
 of force F, which act on each of the arcs <r, is 
 To-. These forces are, of course, tangential 
 to the surface and perpendicular to the 
 arcs. Similarly, Ta grams of force act 
 upon each of the arcs </. Since the sur- FIGURE 111 
 
 face is curved, all four of these forces have 
 slight components in the direction of the normal on. The 
 pressure beneath the element is evidently the sum of these 
 normal components divided by the area o-o-' of the element, for 
 pressure is, by definition, force per unit area. The compo- 
 nent "of F parallel to on is F cosa = Fsin ft. But in the limit 
 
 sin ft = ^57- Hence the sum of the normal components of the two 
 
 fT* f 
 
 forces F is Similarly, the sum of the normal components of 
 
 rj*i ' 
 
 the two forces F' is ^ Hence the pressure^ due to curvature 
 is given by 
 
 7W 7W 
 
 <"> 
 
 That this formula is identical with that denned by Laplace by a 
 more direct but much more difficult method attests the correctness 
 of the reasoning, and also shows that Laplace's capillary constant 
 A is simply the tension in the surface per unit length or the 
 amount of work required to add one sq. cm. to the surface. 
 
194 
 
 MOLECULAR PHYSICS AND HEAT 
 
 Object. 
 
 Experiment 
 
 To compare the values of the surface tensions of 
 water and alcohol, as given by the capillary tube 
 method, with the results obtained by measuring directly the con- 
 tractility of films. 
 
 DIRECTIONS. 1. Fill two small beakers, one with pure dis- 
 tilled water, the other with absolute alcohol. Then prepare a 
 number of fresh capillary tubes by heating to softness 
 kits of clean glass tubing in a Bunsen flame, and draw- 
 ing them down to diameters of from .1 to .5 mm. 
 Select several tubes which seem to be most nearly circular in form, 
 and attach them by means of a rubber band to a mirror millimeter 
 scale, as shown in Fig. 112. Take the reading 
 r Q of the fixed point o upon the scale, by placing 
 the eye so that the image of o comes into coin- 
 cidence with o itself.* Then immerse the lower 
 ends of the tubes in the liquid and raise and 
 lower the clamp c several times (a rack and pinion 
 adjustment is to be preferred) so as to wet thor- 
 oughly the capillaries above the points reached 
 by the liquid. Next bring o exactly into contact, 
 from below, with the liquid surface, slip up the 
 capillary alone a trifle, and take the reading r of 
 the bottom of the meniscus as soon as the level 
 has settled back to its position of equilibrium. 
 It is clear that the height of rise li is given by 
 h = r r . Mark by means of a bit of wax the 
 point to which the liquid rises in the tube, then remove it from 
 the scale, scratch it very carefully with a sharp file at this 
 point, and break it off as squarely as possible. Stick 
 of the tube ^ e ^ ro ^en tube upright against the side of a block of 
 wood by means of soft wax. Then focus upon the 
 broken end a microscope which is provided with a micrometer 
 eyepiece. Count the number of turnsf and fractions of a turn 
 
 *If a mirror-scale is not available the eye may be set in the correct 
 position for reading upon an ordinary scale, by pressing a small piece of 
 mirror glass against the scale, behind the point o. 
 
 fThe counting is greatly facilitated by means of the toothed edge 
 which is found in the field of view of the eyepiece. Each tooth cor- 
 responds to one revolution. 
 
 FIGURE 112 
 
CAPILLARITY 
 
 195 
 
 Calculation. 
 
 Sources 
 of error 
 
 increases 
 a rise in 
 
 which must be given to the micrometer screw in order to cause 
 the movable cross-hairs to traverse exactly the internal diameter 
 of the tube. Repeat several times, using in each case a different 
 diameter. Then replace the capillary tube by a standard mil- 
 limeter scale, and find in the same way the number of turns and 
 fractions of a turn corresponding to 1 mm. From the two 
 observations find the diameter D of the tube in mm. 
 
 In (163) lid represents a pressure expressed in grams per square 
 centimeter. Hence, in order to obtain A in absolute units, the 
 lid of (163) must be multiplied by 980, and both R and 
 h must be expressed in centimeters. The best deter- 
 minations have given for water at 15, A = 75 ; for alcohol, A = 25.5. 
 If the results obtained by this method are not uniform, it will 
 be because, on account of the presence of impurities, the wetting 
 of the tube is not perfect, or because the tube has not 
 a circular section. It is to be observed also that A is 
 a function of the temperature, diminishing as the latter 
 This was to have been expected, since 
 temperature corresponds to a pushing 
 apart of the molecules. 
 
 2. In order to make a direct measurement of 
 
 the surface tension, attach a very light wire frame 
 
 a (Fig. 113) to a delicate helical spring 
 
 The direct 
 
 measurement s, and by means of an elevating table #, 
 raise a vessel of liquid till the frame 
 is immersed. Next lower the table carefully by 
 means of a rack and pinion r, until a film forms be- 
 tween the prongs of the frame. Then quickly take 
 the reading of the index i upon the mirror- scale 
 m. Before repeating, stir the liquid vigorously by 
 means of a glass rod which has been carefully 
 cleaned in a Bunsen flame. Continue this operation 
 until a number of consistent readings can be 
 obtained. The difference between this reading 
 and that taken when the spring and frame hang 
 freely, is, of course, a measure of the force of ten- 
 sion F possessed by the two surfaces of the film. 
 In order to reduce this force to dynes observe the elongation pro- 
 duced by a known weight of the same order of magnitude as 
 
 FIGURE 113 
 
196 MOLECULAR PHYSICS AND HEAT 
 
 F. Then apply Hook's Law to determine F accurately in grams. 
 
 Finally measure the distance I between the vertical wires of the 
 
 frame a with an ordinary metric scale and calculate Tfrom (164). 
 
 Since the presence of the least particle of oil upon the surface 
 
 changes completely the value of T, it is of great importance that 
 
 the frame and vessel be thoroughly cleaned with caustic 
 
 Precautions. ,, . . . . , ,, 
 
 potash beiore the experiment is begun, and that care 
 be taken not to touch the liquid at any time with the fingers. 
 The purpose of the stirring after each observation is to break up 
 any film of impurity which may be present in spite of all precau- 
 tions. It will usually be found to increase the reading somewhat. 
 The readings are to be taken only when a distinct film is visible 
 between the prongs. If the frame continually snaps up without 
 forming a film, clean it again with caustic potash and lower the 
 table more slowly. 
 
 Record 
 
 1. r = No. turns of microscope micrometer to one mm. = 
 
 Water Alcohol Density = 
 
 r l (tube 1) r z (tube 2) = r 3 (tube 3) = - - r 4 (tube 4) = - 
 
 .:h l = - h 2 = - .". h a = h 4 = - 
 
 D l (in screw turns)= D 2 = 
 
 Mean A[=T]= Mean A[=T] = - 
 2. Rd'g with film = zero = Rd'g with film = zero = 
 
 Problems 
 
 1. The gifted American physicist, Joseph Henry, first suggested 
 in 1848 the determination of the capillary constant by attaching a 
 manometer to a soap bubble, and thus measuring the pressure 
 existing within the bubble. Assuming the surface tension of a 
 soap solution to be 70 dynes, find what would be the difference in 
 the levels in the arms of a water manometer when attached to a 
 bubble of 5 cm. diameter. 
 
CAPILLARITY 197 
 
 2. Find how many ergs of work must be expended to blow a 
 soap bubble of 15 cm. diameter. 
 
 3. A drop of water placed in a conical 
 tube (see Fig. 114) is observed to travel 
 rapidly toward the small end; but a drop 
 
 of mercury travels toward the large end. FIGURE 114 
 
 Explain 
 
 4. How high will water rise in pores which are .001 mm. in 
 diameter? 
 
 5. Deduce formula (163) from the consideration that, in a 
 capillary tube in which the angle of contact is 180, the total 
 upward force is the surface tension acting upon a line whose 
 length is the circumference of the tube, while the balancing 
 downward force is the weight of the liquid raised. 
 
 6. Explain from considerations of molecular pressure, how a 
 needle or any small body which is much more dense than water, 
 may yet float upon it provided a < 90. Could it ever float 
 if o> 90? 
 
XXII 
 CALORIMETRY 
 
 Theory 
 
 Calorimetry is that branch of Physics which deals with the meas- 
 urement of heat quantity. It had its beginning about the year 
 1760 with the work of the Scotch chemist and physicist, 
 J se P n Black, the originator of the caloric theory, and 
 the ^ rs ^ investigator to draw a sharp distinction between 
 heat and temperature; in fact the first to give any care- 
 ful definition of the term heat. 
 
 That bodies change in temperature is a fact of direct observa- 
 tion, but the notion that a something called heat passes between 
 bodies of changing temperature is of the nature of an hypothesis. 
 This hypothesis has taken two forms. With Black and his fol- 
 lowers, the so-called calorists, heat was an imponderable fluid, the 
 caloric, the passing of which into or out of a body was the cause 
 of temperature change. The unit of heat, the calorie, was then 
 defined as the amount of heat which must enter or leave 1 gram of 
 water in order to produce 1 degree of change in its temperature. 
 
 With Joule, Clausius, and practically all physicists of the latter 
 half of the nineteenth century, a rise in temperature represents an 
 increase, not in the quantity of a contained heat fluid, but simply 
 in the mean kinetic energy of the molecules. The calorists' defi- 
 nition of the heat unit has, however, been retained in its original 
 form, their concept of the transfer of a heat fluid being simply 
 replaced by the concept of a transfer of molecular energy, kinetic 
 or potential, or both. A knowledge of the caloric theory is now 
 important only because of the light which it throws upon the 
 terminology of heat. The theory was altogether abandoned after 
 Joule's demonstration of the equivalence of heat and work. 
 
 Up to Black's time it was generally supposed that the rise in 
 temperature of a substance in contact with a hot body was con- 
 tinuous; but Black pointed out that while ice or snow is changing 
 
 198 
 
CALORIMETRY 199 
 
 into water it maintains, if well stirred, a perfectly constant tem- 
 perature, no matter how hot may be the stove with which it is 
 in contact. In order to explain this phenomenon, 
 
 Origin of the 
 
 term patent together with the inverse one that the condensation 
 of steam or the freezing of water is accompanied by a 
 large evolution of heat, Black assumed that a certain amount of 
 the caloric always became hidden or latent at the time of a change 
 from the solid to the liquid, or from the liquid to the gaseous 
 condition. For example, since it was found that the mixing of 1 
 gram of ice at and 1 gram of water at 80 C. would yield 2 
 grams of water at 0, or. that 2 grams of water at 40 was 
 required to just melt 1 gram of ice at 0, 80 calories was taken to 
 be the latent lieat of fusion of ice. 
 
 According to the modern mechanical theory, the temperature 
 of a substance remains constant while the change of state is going 
 on simply because the energy of motion communicated 
 V latent to the molecules in contact with the hot body is at 
 once transformed into energy of position ; that is, the 
 heated molecules immediately break away from the forces which 
 have been holding them in the given state (solid or liquid, as the 
 case may be), and thereby lose their increased velocities (i.e., 
 their increased temperature) as rapidly as they receive them. The 
 operation is wholly analogous to that in which a body shot up 
 from the earth loses its velocity in raising itself against gravity. 
 Thus, although the old terms of the calorists, latent heat of fusion 
 and latent heat of vaporization, are still retained, these latent heats 
 represent to-day only given changes in the potential energy of the 
 molecules, just as a given rise in temperature represents a given 
 change in their mean kinetic energy. 
 
 But it must not be supposed that changes in the kinetic and 
 potential energies of the molecules may not take place simultane- 
 ously. In fact, there is but a limited number of substances, those 
 in general which are of a crystalline structure, which show at any 
 points a change in potential energy unaccompanied by a change in 
 kinetic, i.e., by a rise in temperature. Thus wax, resin, gutta 
 percha, glass, alcohol, carbon, and a great number of other sub- 
 stances pass gradually through all stages of viscosity in melting or 
 solidifying. In such cases the temperature changes continually; 
 i.e., there is no definite point at which the substance may be said 
 
200 MOLECULAR PHYSICS AND HEAT 
 
 to begin to melt. On the other hand, every increase in the tem- 
 perature of a solid is accompanied by a certain amount of expan- 
 sion, and hence by^ome increase in the potential as well as the 
 kinetic energy of the molecules. 
 
 The following table shows the values of the latent heats of 
 some of the commoner substances : 
 
 Water 
 
 Melting Latent heat 
 point, of fusion 
 C. (calories) 
 
 0. 79.9 
 
 Boiling Latent heat 
 point, of vaporization 
 C. (calories) 
 
 100. 536. 
 80.2 94. 
 118. 97. 
 357. 62. 
 447. 362. 
 
 Bsnzol 
 
 5.3 
 
 30. 
 46. 
 
 2.8 
 9. 
 21. 
 6 
 
 27. 
 
 Acetic acid. 
 
 ...'. 16.5 
 
 Mercury 
 
 - 39.5 
 
 Sulphur 
 
 114. 
 
 Silver 
 
 .. . 970. 
 
 Lead 
 
 328. 
 
 Platinum. . 
 
 ..1780. 
 
 It was discovered very early that the quantity of heat given up 
 by 1 gram of water in falling through 1 degree would raise very 
 
 different weights of other substances through one 
 Specific heat, degree, e.g., 33 gm. of mercury, 10.5 gm. of copper, 
 
 8.9 gm. of iron, 2.3 gm. of turpentine, etc. The 
 calorists explained these facts by the assumption that different 
 substances possess per unit weight different capacities for the heat 
 fluid. The heat capacity of a body was then defined as the number of 
 calories required to raise the body through 1 degree, and the specific 
 heat of a substance, as the number of calories required to raise 1 
 gram of that substance through 1 degree. These definitions are 
 still retained now that heat is regarded as molecular energy; but 
 the fact that different amounts of this energy must be communi- 
 cated to gram weights of different substances in order to produce 
 the same increase in temperature, i.e., the same increase in the 
 average kinetic energy of translation of the molecules, is attrib- 
 uted to 
 
 (1) The differences in the number of molecules contained in 
 gram weights of different substances ; and 
 
 (2) The differences in the internal work which are incidental 
 to an increase in temperature. By internal work is meant the 
 work done in increasing the distances between the molecules, in 
 augmenting the energy, kinetic or potential, of the atoms within 
 
CALORIMETRY 201 
 
 the molecules, or, in general, any work other than that repre- 
 sented in the increase in the kinetic energy of translation of the 
 molecules themselves, or in the expansion against the force 
 of atmospheric pressure. 
 
 The first element can be easily investigated, for if this were the 
 only cause of difference in the specific heats of different sub- 
 stances, these differences would disappear in a com- 
 ^eafs ular parison of quantities which represent, not equal 
 weights, but equal numbers of molecules. Such quan- 
 tities can evidently be obtained by taking, in each case, a number 
 of grams which is equal to the molecular weight of the substance. 
 This quantity has been given the name of a gram-molecule, and 
 the number of calories of heat required to raise 1 gram-molecule of 
 a substance through 1 C. is called its molecular specific heat, or 
 simply its molecular heat. The molecular heat is evidently, there- 
 fore, simply the product of the specific heat per gram and the 
 molecular weight. The following table contains a comparison of 
 a very few specific heats per given weight, and specific heats per 
 given number of molecules : 
 
 SPECIFIC MOLECULAR MOLECULAR 
 GASES HEAT WEIGHT HEAT 
 
 I Oxygen (O 2 ) 2175 32. 6.95 
 
 Nitrogen (N 2 ) 2435 28. 6.81 
 
 I Hydrogen (II 2 ) 3.4090 2. 6.82 
 
 Hydrochloric acid (HC1) 1845 36.4 6.72 
 
 Carbon monoxide (CO) 2450 28. 6.86 
 
 I Nitric oxide (NO) 2317 30. 6.95 
 
 Nitrous oxide (N 2 O) 2262 44. 9.95 
 
 Carbon dioxide (CO 2 ) 2163 44. 9.52 
 
 Water vapor (H 2 O) 4805 18. 8.64 
 
 SOLIDS 
 Potassium (K 2 ) 1655 78.2 12. 94 
 
 Sodium (Na 2 ) 2934 46. 13.50 
 
 3<! Silver (Ag 2 ) 0570 216. 12.32 
 
 Copper (Cu 2 ) 0952 126.8 12.08 
 
 Mercury (Hg 2 ) 0332 400. 13.28 
 
 f Oxide of copper (CuO) 1420 79.4 11.27 
 
 4J Oxide of nickel (NiO) 1588 74.8 11.87 
 
 I Oxide of mercury (HgO) 0518 216. 11.19 
 
202 MOLECULAR PHYSICS AND HEAT 
 
 SPECIFIC MOLECULAR MOLECULAR 
 SOLIDS HEAT WEIGHT HEAT 
 
 f Chloride of calcium (CaCl 2 ) 1642 111. 18.2 
 
 5J Chloride of zinc (ZnCl 2 ) 1362 136.2 18.6 
 
 I Chloride of barium (BaCl 2 ) 0896 208. 18.6 
 
 f Sulphate of lead (PbSO 4 ) 0872 303. 26.4 
 
 6 \ Sulphate of barium (BaSO 4 ) 1128 233. 26.3 
 
 I Sulphate of calcium (CaSO 4 ) 1966 130. 26.7 
 
 This table shows that many of the differences in specific heats 
 do in fact disappear upon comparison of equal numbers of mole- 
 cules. The differences which are still left must be attributed 
 wholly to the second cause, viz., differences in internal* work 
 because of differences in molecular structure or molecular attrac- 
 tion. Since an extensive series of observations, of which the above 
 table is a small fraction, has shown that, in general, the molecular 
 heats of chemically similar substances, the molecules of which 
 possess the same number of atoms (see table) , are nearly the same 
 in a given state of aggregation (law of Neumann), it must of course 
 be inferred that the internal works are also the same for such 
 similar substances. 
 
 Since the molecules of gases are not subjected to appreciable 
 mutual attractions it might be expected that with molecules of 
 equal complexity the molecular work would be less in the gaseous 
 than in the solid or liquid condition (see groups 1 and 3). Again, 
 it would be natural to conclude that, for substances in the same 
 state of aggregation, the internal work would in general increase 
 with the complexity of the molecule. Both of these inferences 
 are seen to be in accordance with the facts presented in the table. 
 
 The law of Xeumannf is not exact, nor could it be expected to 
 
 * The external work done in expanding against atmospheric pressure 
 may be neglected for solids and liquids. For gases it is a constant quan- 
 tity. 
 
 f The law of Neumann is an extension of a law discovered in 1818 by 
 
 Dulong and Petit in accordance with which the atomic heats (products of 
 
 specific heat and atomic weight) of all the solid elements are 
 
 heats nearly the same, amounting to about 6.3 (see group 3 of 
 
 table). This law was extended in another direction in 1848 
 
 by Woestyn who found that the molecular heat of a compound is often 
 
 equal to the sum of the atomic heats of the elements contained in the 
 
 compound. It is difficult to recognize in this discovery anything more 
 
 than an interesting empirical law. 
 
CALORIMETRY 203 
 
 be iii view of the differences in the attractions which exist between 
 
 different sorts of molecules. In fact, experiment has shown that 
 
 the specific heat of a given substance is not constant, 
 
 Variation 
 
 of specific but that it in general increases steadily with the tem- 
 
 heate with J 
 
 thetemper- perature. Hence, save in the case of the permanent 
 gases, the quantities given in the table are only to be 
 regarded as mean specific heats between certain temperatures, e.g., 
 15 and 100. The rate of increase is fortunately slight for water 
 and for most solids, so that in ordinary work at moderate tem- 
 peratures, it may be disregarded. But for most liquids it is far from 
 negligible; for example, the specific heat of alcohol is .54 at 
 and .64 at 40C. The fact of the dependence of the specific heat 
 of water upon the temperature was first established by Regnault 
 by the method of mixture. Thus, for example, it was found that 
 equal quantities of water at different temperatures do not, when 
 mixed, yield exactly the mean temperature; or, again, that the 
 fall of 10 gm. of water from 60 to 30 does not heat a given 
 quantity of cold water quite as much as the fall of 5 gm. from 90 
 to 30. The first exact work upon the nature of this variation 
 was, however, done by Professor Rowland of Johns Hopkins in 
 1879, in accordance with whose results the specific heat of water 
 diminishes by about 1 per cent from to 29, and then increases 
 again slowly to 100. 
 
 In view of these facts it has been found necessary to define the 
 calorie as the amount of heat required to raise 1 gm. of water, not 
 through 1 degree, but from 15 to 16 C. The defini- 
 tions of lieat ca P acit J an( i of specific heat remain 
 unchanged, but the quantity usually obtained by exper- 
 iment is not the specific heat at a given temperature, but rather 
 the mean specific heat between two specified temperatures. Thus, 
 if Q represent the amount of heat passing into or out of a mass m 
 while it changes in temperature from ^ to t^ then 
 
 r^r = mean heat capacity between ti and 2 ; (167) 
 
 
 
 and -,-^- 7-r = mean specific heat between t v and t 2 . (168) 
 m (1% ?i) 
 
 Measurement There are three methods which have been used for 
 the measurement of specific and latent heats. These are : 
 
204 MOLECULAR PHYSICS AND HEAT 
 
 (1) The method of mixture, 
 
 (2) The method of cooling, 
 
 (3) The method of fusion of ice or condensation of steam. 
 TJie method of mixture is most common, most simple, and, in 
 
 many cases, most accurate. It consists in mixing known weights 
 of bodies of different temperatures, observing the 
 qTrnSSe^ resulting temperature and then writing out an equation 
 which contains on one side all of the heat quantities 
 lost by the cooling bodies, and on the other, all of the heat quantities 
 gained by the warming bodies. The specific heat, or the latent 
 heat, sought is the unknown quantity of this equation. For 
 example, suppose that it be required to find the latent heat of 
 steam x from an experiment in which p gm. of steam at 100 are 
 condensed in M gm. of water. Let t { be the initial temperature 
 of the water and t f the final temperature attained by the mixture. 
 Then the number of calories lost by the steam in condensing is px ; 
 that lost by the condensed steam, in passing from its temperature 
 of condensation, 100, down to t f , is p (100 - t f ). The heat gained 
 by the water is M(t f t^). Hence, if no heat went into the vessel, 
 the thermometer, the stirrer, or the atmosphere, the equating of 
 the heat losses and heat gains would give 
 
 px +p (100 -t f ) = M(t f - t t ). (169) 
 
 But, as a matter of fact, the calorimeter gains heat as well as 
 the water. If its heat capacity be represented by (7, the number 
 Calorimeter ^ ca l r i es so gained is G (t f t t ). Similarly, if C' and 
 'mometer cor- ^" are ^ ne heat capacities of the thermometer and 
 rections. stirrer respectively, (0' + C")(t f t t ) calories go into 
 them. These terms must therefore be added to the right side of 
 (169). The quantities (7, C", and C" are determined either from 
 direct observation (see Experiment) or from the product of the 
 known weights and specific heats. 
 
 To determine the number of calories gained by the atmosphere 
 through radiation from the calorimeter it is necessary to know (1) 
 
 the mean temperature t m of the water during the exper- 
 ^orrection inient, (2) the mean temperature tj of the surrounding 
 
 atmosphere, (3) the number of minutes of duration JV 
 of the experiment, and (4) a quantity Tc called the radiation con- 
 stant of the calorimeter. This constant represents the number of 
 
CALORIMETRT 205 
 
 calories which will pass per minute into the atmosphere from the 
 calorimeter when the temperature of the latter is 1 above that of 
 the surrounding air. Then in the above experiment the number 
 of calories L lost by radiation is 
 
 L = k(t m -tj)N. (170) 
 
 This also must be added to the right side of (169). Its sign will 
 of course be negative if t m <.tj. 
 
 t m is determined by averaging the temperature readings made at 
 intervals of 15 seconds throughout the experiment, k is obtained 
 from an observation of the number of minutes n required for the 
 temperature of the water to fall, e.g., .5, after the conclusion of 
 the experiment. If, then, d be the mean difference in temper- 
 ature between the water and the room during this time of fall, 
 since the total number of calories lost during the n minutes is 
 MX .5, the number of calories which would be lost per minute for 
 a difference of temperature of 1 is 
 
 It will be observed that in this method of calculation it is assumed 
 that a body radiates heat, i.e., falls in temperature, at a rate 
 which is proportional to the difference between its temperature 
 and that of the surrounding atmosphere. This is a law of cooling 
 which was announced by Newton, a law which is not even 
 approximately correct. However, where the radiation correction 
 is small and the quantity (t m tj) not more than 5, the incorrect- 
 ness of Newton's law will not generally introduce an appreciable 
 error into the result. 
 
 The radiation correction, however, can never be determined 
 with great certainty. Hence the effort is always made to reduce 
 it as much as possible. This is done (1) by using such 
 l ar g e quantities of water that the temperature change 
 * s sma ^ (^) ^J making the initial temperature about 
 as much below the room temperature as the final is 
 above it, (3) by highly polishing the outside surface of the calorim- 
 eter so that it radiates very slowly, and (4) by inclosing it in a 
 still air space surrounded by constant temperature walls. With 
 the form of calorimeter shown in Fig. 115, the radiation correc- 
 tion is ordinarily negligible unless the experiment lasts more than 
 
206 
 
 MOLECULAR PHYSICS AND HEAT 
 
 FIGURE 115 
 
 2 minutes or unless the difference of 
 temperature between the calorimeter 
 H and the water-jacket J exceeds 2C. 
 A very sensitive thermometer is of 
 course necessary for the accurate meas- 
 urement of such small temperature 
 changes. 
 
 |) The above discussion applies in gen- 
 eral to the method of mixture, whether 
 it be a latent or a specific 
 
 The tempera- . , . , . , . ^, . 
 
 ture of the heat which is sought. But 
 
 hot, body. 
 
 m the latter case some 
 special device must often be employed 
 
 for obtaining accurately the temperature of the hot body, and 
 for transferring it quickly to the calorimeter. Solids which are 
 not acted upon by water are placed in finely divided form, in a 
 wire net, and heated to a tempera- 
 ture indicated by a thermometer 5 
 in an inclined air tube F which is 
 surrounded by a water or steam bath 
 P (see Fig. 116). In cases in which 
 there might be any heat-producing 
 action between the water and the 
 solid, the latter is first inclosed in 
 some thin-walled vessel of known 
 heat capacity, then heated in F and 
 dropped into H as before. In the 
 case of liquids, no special heating 
 
 device is necessary, since thermometers can be plunged directly 
 into them ; but such liquids as can not be brought into contact 
 with water must be inclosed, like similar solids, in thin-walled 
 vessels of known heat capacity. 
 
 The second method of determining specific heats is the 
 method of cooling. It consists in comparing the times required 
 
 for a given closed vessel to cool, in air, through 
 of l cooiirw )d a gi ven number of degrees, first when filled with 
 
 water and then when filled with the substance whose 
 specific heat is sought. If Z and Z z are the two times of 
 cooling, and C l and C 9 the two corresponding heat capacities 
 
 FIGURE lie 
 
CALOBIMETRY 207 
 
 of the vessel and contents in the two cases, then it may be shown 
 that 
 
 For, since radiation takes place from the surface layers only, it is 
 clear that, with given outside conditions, a given surface, at a 
 given temperature, must always lose heat at the same rate, no 
 matter what substance may be inclosed by the walls. Let, then, 
 q denote the number of calories which passes out of a given sur- 
 face at temperature t in an infinitely short element of time. This 
 loss in heat ' will be associated with a small fall in temperature, 
 viz. 8 1? which will be determined by [see (167)] 
 
 q = CA. (173) 
 
 If now the heat capacity be changed from C to C 2 by a change in 
 the contents of the vessel, the new fall in temperature 8 2 in the 
 same infinitely short interval of time and at the same temperature 
 t will be determined by 
 
 q = C&. (174) 
 
 Hence, from (173) and (174), 
 
 i.e., at any given temperature of the surface the two changes in 
 temperature during the same small, interval of time are inversely 
 proportional to the heat capacities. This is exactly equivalent to 
 the statement that, at a given temperature, the two intervals of 
 time required for the same small change in temperature are directly 
 proportional to the heat capacities. Since, then, the times required 
 to pass through each small element of the scale, e.g., from 60 to 
 59, are proportional to the heat capacities, the total times required 
 to pass through any interval of temperature made up of these 
 small elements must also be proportional to the heat capacities. 
 It is to be observed that this conclusion involves no assumption 
 whatever regarding the nature of the law of cooling, or regarding 
 the relation between the roles played by convection cur- 
 
 Cmditions ,1 ,. 
 
 in which rents and by true radiation in the cooling process. It 
 
 method of 
 
 cooling is rests solely upon the assumption 01 similarity in the 
 
 outside temperature conditions and uniformity of tem- 
 
 perature in all parts of the cooling body. This last condition is 
 
208 
 
 MOLECULAR PHYSICS AND HEAT 
 
 difficult to fulfil when the cooling vessel contains solids. Hence 
 the method has not proved satisfactory for the determination of 
 
 the specific heats of such substances ; 
 but for liquids it has been found 
 both accurate and convenient. 
 
 The third method of measuring 
 specific or latent heats consists in 
 either determining the 
 
 The ice and 
 
 steam cai- amount oi ice m which a 
 
 orimeters. 
 
 heated body will melt 
 while its temperature is falling to 
 0C., or finding the amount of steam 
 m' which a cold body will condense 
 while its temperature is rising to 
 100C. The heat given up by the 
 body in the first case is then evi- 
 FIGURE 117 dently 80m. That taken up in the 
 
 second case is 5'36m'. 
 
 The ice calorimeter is as old as Black, but the modern form is 
 due to Bunsen (see Fig. 117). The ice cap b is formed in the 
 water W by inserting a freezing mixture into the tube D. The 
 point to which the mercury M rises in the graduated capillary 
 tube Tis then noted. The hot substance is next dropped into Z>, 
 where it melts a certain amount of ice. The movement of the 
 mercury in T to the right because of the contraction due to change 
 of state is proportional to the amount of ice melted. The value 
 in calories of one division of T is determined 
 by inserting into D a substance of known 
 heat capacity. This calorimeter has proved 
 very valuable in determining the specific 
 heat of very small bodies. 
 
 The steam calorimeter (Fig. 118) has 
 proved of especial value only in the determi- 
 nation of the specific heats of gases. A 
 light metal globe , full of the gas, is sus- 
 pended from the arm of a balance within 
 a chamber of known temperature t. When 
 steam is suddenly admitted into this chamber through E it con- 
 denses upon the globe and the walls until their temperature reaches 
 
 FIGURE 118 
 
CALORIMETRY 
 
 209 
 
 100. With proper precautions to prevent the loss of drops of con- 
 densed steam (see pan s), the increase tri in the weight of the globe 
 represents the amount of steam which must be condensed in order 
 to raise the temperature of globe and contained gas from t to 100. 
 If the heat capacity (7 of the globe is known, that of the con- 
 tained gas C' can evidently be found from ((7 + (?')(100 t) = 
 536w'. Accurate results can not be obtained with either of these 
 latent heat calorimeters without the use of greater precautions 
 than can be taken ordinarily in intermediate laboratory courses. 
 
 Object. 
 
 Experiment 
 
 To compare the method of cooling and the method 
 of mixture in the determination of the specific heat 
 of turpentine. 
 
 DIRECTIONS. 1. By means of the trip scales* find the .weight 
 w c of the nickel-plated brass vessel A of about 30 cc. capacity (see 
 Fig. 119). Then fill it with 
 
 Observations ... ' . 
 
 upon rates DOi ling water and weigh 
 
 of cooling. . 8 . 
 
 again (w h ). Subtract and 
 obtain the weight of the water w w . 
 Treat the blackened vessel B in the 
 same way, filling it with the same num- 
 ber of grams of water. Suspend both 
 vessels from the wooden cover D by 
 means of corks and thermometers, as 
 shown in the figure. Attach the cyl- 
 indrical brass vessels E and F by 
 means of the catches at e t and immerse 
 in a large pail or battery jar full of 
 water at about the room temperature. 
 
 Take a very exact reading of each thermometer about once a 
 minute while the temperatures are falling from 71 to 39 C. 
 The hour, minute, and second of each reading must be accurately 
 taken. Since the eye can not be upon both the watch and the 
 thermometer at the same time, it is advisable to have in the room 
 some audible second marker with which the watch can be com- 
 pared just before each reading. This may be dispensed with if 
 there are' two observers, one to record temperatures and the other 
 
 FIGURE '119 
 
210 MOLECULAR PHYSICS AND HEAT 
 
 to record times. In this case the first observer should tap sharply 
 upon the table at the instant of each temperature reading, at the 
 same time calling out either A or B, while the second observer 
 records in the A or B column the hour, minute, and second of each 
 tap. The mean temperature tj of the water-jacket may be found 
 from observations taken at the beginning and end of the cooling 
 process. . v 
 
 Next pour out the water from A and B and dry them thor- 
 oughly; fill to the same level as before, this time with hot tur- 
 pentine; obtain the weights (w' b ), and by subtraction the 
 corresponding weights of the turpentine w t ; replace in the bath 
 and take a new set of observations upon the rate of cooling 
 between 71 and 39. 
 
 With the observed values of the times and temperatures, plot 
 upon a large sheet of coordinate paper four smooth full-page 
 curves, using times as abscissae and temperatures as 
 curves of ordinates. In so doing, choose the scale of temper- 
 ature so that the lowest observed temperature is repre- 
 sented by a line near the bottom of the page, the highest by a line 
 near the top. Choose the scale of times so that the time of begin- 
 ning of observations upon the cooling of the polished vessel when 
 filled with water is represented by a line (the zero of time for this 
 case) which is near the left side of the page, while the time of 
 conclusion of observations upon this case is represented by a line 
 near the right side of the page. Plot the other curves upon the 
 same sheet to the same scale, the zero of times being in each case 
 ttie time of beginning of observations. 
 
 Now read oif very carefully upon the four smooth curves the 
 four times included between any two temperature lines, e.g., 70 
 and 40, and thus obtain from both A and B the 
 Computa- 
 
 uon. quantity -^- If, then, c and c' represent the respective 
 
 *i 
 heat capacities of the empty vessel A and of the thermometer, and s, 
 
 Z C 
 
 the specific heat of turpentine, the equation i = -^ becomes 
 
 Z 2 2 
 
 (176) 
 
 From (176) s can easily be obtained as soon as c and c' have been 
 found. 
 
CALOKIMETKY , 211 
 
 To obtain c, multiply iv c by the specific heat of brass (.095). To 
 obtain c', fill vessel A with water at the room temperature, immerse 
 and read carefully the thermometer; then plunge it 
 ^ wa ^ er ? withdraw, wipe off the adhering water ; 
 
 m meters. read quickly and again instantly immerse in A. Stir 
 and note the rise in the temperature of the water. If 
 then w' is the weight of the water, t t and t f its initial and final 
 temperatures, and t the initial temperature of the thermometer, 
 then [see (167)] 
 
 From any two of the four cooling curves test the incorrectness 
 
 of Newton's law of cooling as follows: If z l and z z are the times 
 
 required to cool from 71 to 69 and from 41 to 39 
 
 Newton's 
 
 law of respectively, and if t s is the temperature of the jacket, 
 
 then if Newton's law were correct it would follow that 
 
 - m &b& / 178 x 
 
 ^~w~t; 
 
 2. To find the mean specific heat of turpentine between 70 
 and 40 C. by the method of mixture, first heat the water in the 
 jacket J (Fig. 115) to about 37 and hold it constant 
 at that temperature by the gentle application of heat 
 and by such stirring as is found necessary. Find the 
 weight W c of the nickel-plated calorimeter H, including the 
 stirrer, then fill half -full of water and weigh again ( W b ). Repre- 
 sent the weight of the water alone by W w . Give the water a 
 temperature which is about 3 below that of the jacket and set it 
 in place within the jacket as in Fig. 115. From the specific heat 
 found by the method of cooling, calculate about how many grams 
 of turpentine will need to fall from 70 to 40 in order to raise 
 the water in the calorimeter from 34 to 40. From this and the 
 density of turpentine (.87) estimate very roughly to what height 
 the mixture of water and turpentine will raise the level of the 
 liquid in the calorimeter. Then heat to about 73 a half -liter or 
 more* of turpentine in a dipper provided with a lip ; stir it thor- 
 oughly with a thermometer (No. 1) until the temperature falls to 
 about 70, then take an accurate reading t' t and very quickly pour 
 
 * A large quantity is used so that the cooling may not be too rapid. 
 
212 MOLECULAR PHYSICS AND HEAT 
 
 about the estimated volume into the water. The temperature t t 
 of this water should have been taken but an instant before with 
 the aid of a more sensitive thermometer (Xo. 2). 
 
 Cover the calorimeter as quickly as possible after the mixing 
 and stir very thoroughly with the wire-net stirrer R (see Fig. 
 115), keeping it, however, always below the surface. Take a read- 
 ing every 15 seconds from the time of mixing until the temper- 
 ature has passed its highest point and begun to fall. The first 
 few of these readings will of course be uncertain, but they have a 
 very small influence upon the result. Record the highest temper- 
 ature reached t f , then withdraw the thermometer, shake back into 
 the calorimeter the drops which adhere, and take the weight W m 
 of the calorimeter and contents, including stirrer. By subtrac- 
 tion obtain the weight W t of the turpentine. 
 
 To find the radiation constant pour out the mixture and fill the 
 calorimeter with a volume of water about equal to the volume of 
 the mixture. Find the weight M of the water alone, 
 for radio,- then raise it to about the final temperature ^/, replace 
 it in the jacket and note the time n required, with con- 
 tinual stirring, to fall .5C. From this and the temperature of 
 the jacket, compute first the radiation constant (equation 171), 
 then the number of calories lost by radiation (equation 170). 
 
 Calculate the heat capacities of the calorimeter and stirrer by 
 multiplying their joint weight by .095, obtain the heat capacity of 
 Correction.-? ^ ne thermometer either by the method employed in 1 
 *eteran im ~ or ky the following process : Estimate the volume in 
 thermometer. C ubic centimeters of that portion of the thermometer 
 which is immersed, by noting the rise of water in a narrow gradu- 
 ate when the thermometer is sunk in it up to the point to which 
 it was wet in the experiment. Multiply this by the specific heat 
 of mercury per cubic centimeter, viz. 13.6 x .033 = .45. It is 
 because this is about the same as the specific heat of glass per 
 cubic centimeter, viz. 2.5 x .19 that the thermometer may be 
 treated as though it were made entirely of mercury. 
 
CALORIMETRY 213 
 
 Record 
 A B 
 
 W c = W c = 
 
 Wb= W'b W = W'b = 
 
 W w = . '. Wt= . '. Wiv= . '. Wt= 
 
 ti= 
 
 c = c' = t = tf = 
 
 . '. sp. h't of turpentine = . '. sp. h't of turpentine = 
 
 Zmn- M-tj_ *,_ 4Q-t,_ 
 
 z 2 70- tj~ z 2 ~ 70 */~ 
 
 2. 1st trial 2d trial 
 
 W c = W c = - 
 
 .;W W = - t f =- W w =- t f =- 
 
 W m =- t'i = - W m =- t'i=- 
 
 .:W t =- t' f = * W t = t' f = * 
 
 Radiation cons' t (171) M === ?i = d = .-.k = 
 
 Radiation correct'n (170) t m = tj= N= . . L = 
 
 Heat capacity of cal'r and stirrer = of thermoin'r = 
 
 . . sp. h't of turpentine = . . sp. h't of turpentine = 
 
 Problems 
 
 1. 50 gm. of ice are dropped into a brass calorimeter contain- 
 ing 800 gm. of water at 27 C. The calorimeter weighs 150 gm. 
 Find the final temperature. 
 
 2. A 50 kgm. block of ice fell 30 meters. How many grams 
 of ice were melted by the heat generated by the fall? (4.2 x 10 7 
 ergs = 1 calorie.) 
 
 3. According to very careful determinations made by Violle 
 the quantity of heat required to raise 1 gm. of platinum from 
 to t is given for all temperatures by the formula Q = .0317^ + 
 .OOOOOG^ 2 . Find the temperature of a Bunsen flame if a 20 gm. 
 platinum ball dropped from the flame into 377 gm. of water at 
 raised the temperature of the water 2. 
 
 * t'f is tf reducecf to terms of thermometer No. 1. It is found by com- 
 paring No. 1 and No. 2 at about the temperature tf. It is evident that 
 this proceeding diminishes errors due to imperfect thermometers. 
 
214 MOLECULAR PHYSICS AND HEAT 
 
 4. What should be the result of mixing 10 gm. of snow at 
 with 10 gm. of water at 35C.? 
 
 5. The globe of a steam calorimeter is made of brass, weighs 
 50 gm. and has a volume of 1 liter. It contains nitrogen at a 
 pressure of 2 atmospheres. The temperature of the chamber is 
 10. What will be the increase in weight upon the admission of 
 steam? 
 
XXIII 
 EXPANSION 
 
 Theory 
 
 The true coefficient of expansion c of any body at any temper- 
 ature is defined as the ratio between the volume increase, pro- 
 duced by an infinitely small rise in temperature, and 
 the volume of the body at zero. Thus if F , F <p and 
 F, 2 represent the volumes at 0, ti and t z , then, if ^ and t z are 
 infinitely close together, 
 
 c= J^~^- (179) 
 
 For hydrogen this quantity is the same for all temperatures 
 simply by virtue of the definition of temperature; for it is the 
 expansion of hydrogen which has been made the meas- 
 es constant ure of temperature change (see p. 126). For the other 
 
 for gases. ... J . . . . A J . ' . _ T 
 
 gases it is constant by virtue of the law of G-ay-Lussac 
 (see p. 121). Hence, for gases which follow closely this law, equa- 
 tion (179) gives the correct definition of the expansion coefficient, 
 even when ti and t 2 represent widely different temperatures. For 
 such cases the volume at tf, viz. F t , expressed in terms of the 
 volume at 0, viz. F , is [cf. (179) when ^ = 0] 
 
 F,= V,(l + ct}. (180) 
 
 The density at t, viz. D t , in terms of the density at 0, viz. D , 
 
 "ivr*i ^1^1 
 
 is found by substituting in (180) the relation =- r = Volume. 
 
 This gives 
 
 n _ /'1ft1'\ 
 
 jL/ 1 ~ : I J.OJ.I 
 
 l + ct 
 
 That c can not be constant for most liquids and solids is evi- 
 dent from the fact, already mentioned on p . 123, that ther- 
 mometers made from liquids or solids by dividing the increase in 
 volume between and 100 into 100 equal parts, do not in gen- 
 
 215 
 
216 
 
 MOLECULAR PHYSICS AND HEAT 
 
 
 stant for 
 
 solids and 
 than .2 
 
 eral agree at intermediate temperatures with the hydrogen ther- 
 mometer. For most solids, however, and for mercury, the depart- 
 ures are slight for temperatures below 100. For 
 example, a mercury in glass thermometer graduated in 
 
 , . ,. , , ,., , , , 
 
 the manner just indicated diners from a hydrogen 
 thermometer at no point between and 100 by more 
 Hence, in ordinary work with these substances, at ordi- 
 nary temperatures, c is usually considered constant, and equations 
 (180) and (181) are applied precisely as in the case of gases. For 
 high temperatures, however, this approximation can not be used. 
 The expansion coefficients of most liquids other than mercury 
 increase rapidly with the temperature. For example, in passing 
 from to 40 the coefficient of turpentine increases 
 t ffl c<Sant 3.9% ; of alcohol, 4.4% ; of bisulphide of carbon, 4.9% ; 
 liquids^ of ether, 6.6%. Between and 4 water possesses 
 the peculiar property, which is also shown by certain 
 alloys, of contracting as the temperature rises, i.e., it has a nega- 
 tive coefficient. It is evident, then, that in general if t^ and t% 
 represent widely different temperatures, (179) gives 
 the mean coefficient between ti and t z , rather than 
 the true coefficient at any particular temperature. 
 It is to be observed also that in the case of both 
 solids and liquids (not, however, in the case of gases), 
 % the increase in volume ( V t2 V tl ) [see 
 
 (179)] is so small, even when t 1 and t z 
 differ from each other by as much as 100, 
 that the error introduced into (179) by replacing 
 V by the volume at any ordinary temperature is 
 less than the necessary observational error in obtain- 
 ing the increase ( V h V t} ). 
 
 The most accurate and convenient method of 
 studying the coefficients of liquids at various tem- 
 Method of peratures is to fill a glass vessel having a 
 pamionof x ' lar g e bulb and a capillary neck (see Fig. 
 liquid*. 120), to raise the temperature from t l 
 
 to t z and to observe the corresponding rise I of the liquid in the 
 neck. If a represent the area of a cross-section of the tube, then 
 the apparent increase in the volume of the liquid is al. This 
 apparent increase is, however, in reality the differences between 
 
 V replace- 
 able by V. 
 
 FIGURE 120 
 
EXPANSION 217 
 
 the expansions of the glass and of the liquid; for the increase in 
 the capacity of the bulb is precisely the same as would be the 
 increase in the volume of a solid glass vessel of the same volume 
 as the interior of the bulb. This is evident from the consideration 
 that a solid bulb may be conceived as made up of a series of con- 
 centric hollow bulbs, each of which expands independently of all 
 the rest. Hence the increase in the interior capacity of each hol- 
 low shell is the same as the increase in volume of the solid bulb 
 which fills it. Hence, if F represent the interior volume of the 
 bulb at and y, the expansion coefficient of glass, the increase 
 in capacity of the bulb, viz. (F, 2 V tl ) is, by (179), equal to 
 "Po(4 t\)y- Similarly, if c be the coefficient of the liquid, the 
 real increase in its volume is V Q (t z t^c. Since, then, the 
 apparent increase al is the difference between these quantities, 
 there results 
 
 (182) 
 
 an equation which shows that it is impossible to obtain c from the 
 measurement of $, Z, F , ti and t%, unless y is already known. 
 
 A similar condition exists with respect to the volume coeffi- 
 cients of solids. For one of the most precise methods of measur- 
 Measurement i n g this quantity consists in finding the densities D tl 
 ( /oefflcient ime and A 2 of the solid at t, and by the loss of 
 of solids. W eight method (see Ex. XX), and then substituting in 
 the equation which results from replacing the volumes in (179) by 
 
 . ,. Mass m , . 
 
 the relation .=- - This gives 
 
 Density 
 
 But the density of a solid can not be found by the loss of weight 
 method unless the density of the liquid in which it is immersed is 
 known. If the expansion coefficient c of one single liquid could 
 be accurately investigated by a method which was independent of 
 the expansion of glass or of any other substance, this liquid could 
 be used for determining y for any glass bulb [see (182)], and this 
 bulb could thereafter be used for investigating the coefficients of 
 all other liquids. This would in turn render the method of (183) 
 serviceable for determining the coefficients of solids. 
 
218 
 
 MOLECULAR PHYSICS AND HEAT 
 
 B 
 
 Absolute 
 expansion 
 of mercury. 
 
 E 
 
 FIGURE 121 
 
 The liquid which has been chosen for this 
 special investigation is mercury. The method 
 used was devised early in the last 
 century by Dulong and Petit. Reg- 
 nault afterward repeated the deter- 
 minations and obtained results in close agree- 
 ment with those of the earlier investigators. 
 The principle of the method is very simple, 
 though its application is less so. The appa- 
 ratus consists essentially of two tubes A B and 
 CE (see Fig. 121) which are connected at the 
 bottom by a capillary tube BE. Tube AB is 
 surrounded with melting ice and CE with a 
 water-jacket of known temperature t. The 
 levels at A and C must adjust themselves so 
 that the pressures at B and J? are equal; but the pressure at B 
 is DJi Q (see Fig. 121), and that at E is D t h t , D and D t being the 
 densities of mercury at and t respectively. Hence 
 
 h 9 D 9 =h t D t . (184) 
 
 It follows from (184) and (181) that the value of c between 
 and t is given by 
 
 c = ^-- (185) 
 
 Regnault obtained for the mean coefficient of mercury between 
 and 100, .0001815. 
 
 The volume (or cubical) coefficient which has been thus far 
 discussed is the only expansion coefficient which liquids possess ; 
 unear but for solids there exists also a linear coefficient which 
 
 expansion . _ , 
 
 of solids. is defined by 
 
 (186) 
 
 in which ? , l tv and ?, 2 are the lengths of any given line of the 
 body at 0, ^, and / 2 respectively. It is this coefficient which 
 is usually made the subject of measurement in the case of solids; 
 for the cubical coefficient c can be deduced from it by means of 
 the simple relation 
 
 c = 3X. (187) 
 
EXPANSION 219 
 
 To prove that this relation exists, let /, T, and I" be the three 
 edges of a rectangular block at a temperature of t ' . These three 
 Proof that lengths may be expressed in terms of X and the cor- 
 responding lengths at 0, thus [see (186)], 
 
 Multiplication gives 
 
 m" J J' J" f-l i O\/ _i_ Q\ 2 / 2 _l_ \ 8 / 3 \ (1 ft\ 
 ^Qt-Qt n \ -^- ~T" d'T'V I O/V 6 I /V t' I 4 1 XOOI 
 
 or 
 
 F t = Fo (1 + 3X + 3X 2 f + X 3 / 3 ). (189) 
 
 But since X is always very small, scarcely ever greater than 
 .00003, the terms in X 2 and X 3 are wholly negligible in comparison 
 with the term in X. Hence 
 
 V t = Fo (l + 3X2f). (190) 
 
 Comparison with (180) shows that c = 3X. Q. E. D. 
 
 Linear expansion coefficients have for the most part [been 
 determined by one of two very simple methods. The first was 
 introduced by \ i 
 
 Lavoisier and 
 coefficients. Laplace to- 
 ward the end 
 
 of the eighteenth cen- f| 
 tury. It consists in plac- Wjj^*^^ 
 ing against a rigid wall FIGUKE m 
 
 (see Fig. 122) one end of 
 
 the bar which is to be heated, attaching an optical lever to the 
 other end, and measuring the expansion precisely as the elonga- 
 tion of the wire was measured in Ex. VIII. 
 
 The other method, which is the one now in use at the Inter- 
 national Bureau of Weights and Measures, consists in focusing 
 two fixed microscopes upon fine scratches near the ends of the bar 
 to be investigated, and measuring directly with the aid of a microm- 
 eter eyepiece and a comparison scale, the elongation produced by 
 a given rise in temperature (see Fig. 123). Different bars of the 
 same material show very considerable differences in expansion 
 coefficients. Hence, in general, the numbers given in tables as 
 
220 MOLECULAR PHYSICS AND HEAT 
 
 the coefficients of solids must be looked upon merely as mean 
 values. 
 
 Experiment 
 To determine the linear coefficient of expansion of a 
 
 Object. ._ _. . 
 
 hollow brass tube. 
 
 Place the tube a (see Fig. 123) in the non-conducting hair- 
 felt covering #, mount it upon supports as in the figure, and focus 
 
 the two micrometer microscopes A and B upon fine 
 preliminary scratches near the two ends of a. In this setting see to 
 
 adjustments. , . . . , , , . , '-, 
 
 it that the microscope is placed so that tne scratch 
 appears on that side of the field of view which is farthest from the 
 
 middle of the rod. This is done so that the expansion may not 
 cause the scratches to move out of the field. It will be observed 
 also that it takes into account the inversion of image produced by 
 the microscope. Connect the rubber tube m with the water tap 
 and take the temperature of the water by holding a thermometer 
 in the stream which emerges from o. Then set the movable cross- 
 hairs of the microscopes accurately upon the scratches and take 
 the readings upon the scales in the micrometer eyepieces. 
 
 This is done by first rotating the micrometer screws in such a 
 direction that the reading upon the circular head continually 
 increases, i.e., passes from toward 100, at the same 
 microscope time observing in what direction the movable cross- 
 hairs pass over the field of view. If it be found that 
 they move from right to left, then the reading of the scratch is 
 the number of turns (i.e., teeth) and fractions of a turn (see 
 micrometer head) through which the micrometer screw has 
 turned in bringing the cross-hairs from the extreme right-hand 
 tooth up to its present position. This is read off directly upon 
 
EXPANSION 221 
 
 the toothed scale and the micrometer head. It will be observed 
 that in order to facilitate the counting of the teeth every fifth 
 notch is more deeply indented than the rest. If the cross-hairs 
 had been observed to move from left to rignt, the extreme left- 
 hand tooth would of course have been chosen as the point of 
 reference. Let the recorded reading be a mean of a number of 
 settings, and in each setting, in order to avoid the error due to 
 the play, or back-lash, between the nut and the screw, bring the 
 cross-hairs up to the scratch from the same side. 
 
 As soon as the two readings have been taken, turn off the cur- 
 rent of cold water, using extreme precautions against disturb- 
 ing the tube #, transfer the rubber connecting tube from the 
 
 water tap to a steam boiler, and allow a rapid current 
 Swtraerjfe ^ steaDi to pass through a until expansion ceases. 
 
 Take again a set of readings at both en Is. Then find 
 the reducing factors of each microscope by focusing upon a 
 standard scale and observing the number of turns to the. mil- 
 limeter. Reduce the observed expansion to millimeters. Measure 
 the distance between the scratches by means of an ordinary meter 
 stick. Obtain the temperature of the steam from the barometer 
 height and the table in the Appendix. Then compute A. [see 
 (186)]. 
 
 Record 
 No. of screw turns to mm. in microscope A = in B = - 
 
 1st trial 3d trial 
 
 Temper- Reading Temper- Reading 
 
 ature in A in B ature in A in B 
 
 Water Water 
 
 Steam Steam 
 
 Differences Differences 
 
 Diff's in mm. Diff's in mm. 
 
 .-. Expansion coef. c= .*. Expansion coef. c = 
 
 Problems 
 
 1. In compensated pendulums the expansion of one set of rods 
 lowers the bob, while that of another raises it. Suppose the first 
 set to be made of iron and to have a total length of 90 cm. If the 
 material of the second set is zinc, what must be their total length? 
 
 2. If a clock which has an uncompensated pendulum made of 
 
222 MOLECULAR PHYSICS AND HEAT 
 
 brass keeps correct time at 15, how many seconds will it lose per 
 day at 25? 
 
 3. A surveyor's steel tape which is 10 meters long is correct 
 at 15. What is its error at 0? at 100? 
 
 4. A glass bulb of 10 cm. capacity is exactly full of mercury at 
 20. How many grams of mercury will run out if it is heated 
 to 100? 
 
 5. A barometer provided with a brass scale correct at 15 
 read 735.65 mm. on a day on which the temperature was 25. By 
 means of the expansion coefficients of brass and mercury, find the 
 correct barometer height at upon the day in question. 
 
 6. A cubical block of steel 30 cm. on an edge floats on mer- 
 cury. How far will it sink when the temperature rises from 15 
 to 75? 
 
 o; ';,;_' 
 
APPENDIX 
 
 THE MICROMETER CALIPEB 
 
 In the micrometer caliper (see Fig. 124) the divisions upon the 
 scale c correspond to the pitch of the screw s. When the jaws ab 
 are in contact the edge of the sleeve d coincides with the zero line 
 of the scale c, and the zero division upon the graduated edge d 
 coincides with the line ec. Hence, when the jaws are separated 
 by a rotation of the milled head h, the whole number of screw 
 turns of separation is read off directly upon the scale c, while the 
 
 FIGURE 124 
 
 fraction of a screw turn is given by the graduation upon the 
 edge d. For example, if the pitch of the screw is .5 mm., and if 
 there are fifty divisions upon the circumference of d, then each of 
 these divisions corresponds to a motion of .01 mm. at ~b. By esti- 
 mating to tenths of a division upon d, the separation of the jaws 
 can be read to .001 mm. 
 
 To make a measurement, place the object between the jaws ab 
 and turn up the milled head li until, with light pressure between 
 thumb and finger, the head slips through the fingers instead of 
 rotating farther. In the best instruments the head is arranged to 
 slip upon the screw as soon as a certain pressure has been reached. 
 If this arrangement is absent, take great care not to crowd the 
 screw. Without removing the object from ab read upon the 
 scales c and d the separation of the jaws to .001 mm. Then 
 
 223 
 
224 
 
 APPENDIX 
 
 remove the object, close the jaws, using the same pressure as 
 before, and if the zero of d does not coincide exactly with the line 
 ec, observe the difference in thousandths millimeters, and correct 
 the first reading for this zero error of the caliper. 
 
 THE VERNIER CALIPER 
 
 A vernier is an auxiliary sliding scale which enables an observer 
 to increase somewhat his accuracy in estimating the fractional 
 portion of the last small division of the main scale. For example, 
 it is seen at once from the main scale DE (Fig. 125), that the 
 
 length of the rod 
 AB is about 2.7 
 units. The use of 
 the auxiliary scale 
 A C removes all 
 
 uncertainty as to this tenths place, and even makes possible 
 an estimate in hundredths place. For AC is so divided that 
 ten of its divisions are exactly equal to nine divisions on DE. 
 Hence, if the zero of AC were back in coincidence with the 
 mark 2 of DE, the mark 1 of A C would be just one- tenth of 
 one of the DE divisions behind the mark 3 on DE. Similarly, 
 2 would be two tenths behind 4, 3 three tenths behind 5, and so 
 on. Hence, if the vernier scale were moved up so as to bring its 1 
 mark into coincidence with the mark nearest to it on DE, the 
 distance which its zero would move up beyond 2 would be one- 
 
 E 
 
 SCALE 
 
 D 
 
 113 
 
 [IS 
 
 ill 
 
 no is 
 
 
 s IT 
 
 6 
 
 5, |4- 
 
 I3J2I1 JO 
 
 . 
 
 ilo 
 
 i 
 
 8l 7l 
 
 e; 
 
 Sl 4| 
 
 si 
 
 2| 1 
 
 Q^'~'-" -"^- " "'"j 
 
 
 VERNIER 
 FlGUKB 
 
 125 
 
 
 A 
 
 B 
 
 FIGURE 
 
 tenth of a unit. If it were moved up so as to bring the 5 into 
 coincidence, the zero of A C would have moved up five tenths 
 
APPENDIX 225 
 
 units beyond 2. In general, then, it is only necessary to observe 
 which mark on the vernier A is in coincidence with a mark on 
 DE in order to know how many tenths the zero of AC has moved 
 up beyond the last division passed on DE. A glance at Fig. 125 
 shows that 6 has already passed coincidence, while 7 has not quite 
 reached the coincidence. The length of AB is then 2.6+. The 
 next figure beyond the 6 is nearer to 7 than to 6, i.e., it is between 
 2.65 and 2.70. If then the length be recorded as 2.67, the only 
 estimate will be in the figure 7, i.e., in hundredths instead of in 
 tenths place as at first. Thus a vernier which has 10 divisions to 
 9 divisions on the main scale reads directly to tenths of the main 
 scale divisions. In the same way a vernier which has 25 divisions 
 equal to 24 main scale divisions reads directly to twenty-fifths of 
 the main scale divisions. If the ratio is 50 to 49 the vernier reads 
 to fiftieths, etc. Fig. 126 shows a vernier caliper in which the 
 vernier has 25 divisions to 24 half -millimeter divisions on the main 
 scale. It therefore reads to .02 mm. The body is placed between 
 the jaws a and #, and the reading taken directly upon the scale 
 and vernier. 
 
226 
 
 APPENDIX 
 
 TABLES 
 
 Table 1 
 
 COEFFICIENTS OF ELASTICITY 
 
 SUBSTANCES 
 
 VOLUME ELAS- 
 TICITY = K 
 
 SIMPLE RIGIDITY 
 = n 
 
 YOUNG'S MODULUS 
 
 Water 
 
 2.2 X 1 
 2.6 X 1 
 4.1 X 
 10.8 X 
 18.4 X 
 14.6 X 
 9.6 X 
 16.8 X 
 5.5 X 
 
 Oio 
 U 11 
 
 
 
 Mercury 
 
 
 6.6 X 1 
 10.8 X 
 21.4 X 
 19.6 X 
 13.5 X 
 12.3 X 
 6.5 X 
 
 6 n 
 
 Glass 
 
 2.4 X 1 
 3.7 X 
 8.2 X 
 7.7 X 
 5.3 X 
 4.5 X 
 2.5 X 
 
 W 
 
 Brass drawn 
 
 Steel 
 
 Wrought iron 
 
 Cast iron 
 
 Copper . . 
 
 Aluminium 
 
 
 Table 2 
 
 DENSITIES 
 
 Solids 
 
 Aluminium 2. 58 
 
 Brass (about) 85 
 
 Brick 2.1 
 
 Copper 8.92 
 
 Cork 0.24 
 
 Diamond 3.52 
 
 Glass (common crown) 2.6 
 
 " (flint) 3.0-6.3 
 
 Gold 19.3 
 
 Ice at C. 0.91 
 
 Iron, c.ast 7.4 
 
 Iron, wrought. 
 
 Lead 
 
 Nickel 
 
 Oak 
 
 Pine 
 
 Platinum 
 
 Quartz 
 
 Silver 
 
 Sugar 
 
 Tin 
 
 Zinc . . 
 
 7.86 
 
 11.3 
 
 8.9 
 
 0.8 
 
 0.5 
 
 21 50 
 
 2.65 
 
 10.53 
 
 1.6 
 
 7.29 
 
 7.15 
 
 Mean density of earth is 5.5270. 
 
 Liquids 
 
 Alcohol at 20 C 0.789 
 
 Carbon bisulphide 1.29 
 
 Ethyl ether at C 0. 735 
 
 Glycerine 1 26 
 
 Turpentine 0.87 
 
 Mercury 
 
 Sulphuric acid 
 
 Hydrochloric acid. 
 
 Nitric acid 
 
 Olive oil 
 
 13.596 
 1.85 
 1.27 
 1.56 
 0.91 
 
 Gases at C. and 76 Centimeters of Mercury Pressure 
 
 Air, dry 0.001293 
 
 Ammonia 0.000770 
 
 Carbon dioxide 0.001974 
 
 Chlorine 0.003133 
 
 Hydrogen 0.0000895 
 
 Marsh gas 0.000715 
 
 Nitrogen 0.001257 
 
 Oxygen 0.001430 
 
APPENDIX 
 
 22? 
 
 Table 3 
 
 DENSITY OF DRY Am AT TEMPERATURE t AND PRESSURE Hmm. OF MERCURY 
 
 t 
 
 #=720 
 
 730 
 
 740 
 
 750 
 
 763 
 
 770 
 
 DIFFERENCE: 
 PER mm. 
 
 10 
 
 .001181 
 
 .001198 
 
 .001214 
 
 .001231 
 
 .001247 
 
 .001263 
 
 16 
 
 11 
 
 1177 
 
 1194 
 
 1210 
 
 1226 
 
 1243 
 
 1259 
 
 1 2 
 
 12 
 
 1173 
 
 1189 
 
 1206 
 
 1222 
 
 1238 
 
 1255 
 
 2 3 
 
 13 
 
 1169 
 
 1185 
 
 1202 
 
 1218 
 
 1234 
 
 1250 
 
 3 5 
 
 14 
 
 1165 
 
 1181 
 
 1197 
 
 1214 
 
 1230 
 
 1246 
 
 4 6 
 
 15 
 
 .001161 
 
 .001177 
 
 .001193 
 
 .001209 
 
 .001225 
 
 .001242 
 
 5 8 
 
 16 
 
 1157 
 
 1173 
 
 1189 
 
 1205 
 
 1221 
 
 1237 
 
 6 10 
 
 17 
 
 1153 
 
 1169 
 
 1185 
 
 1201 
 
 1217 
 
 1233 
 
 7 11 
 
 18 
 
 1149 
 
 1165 
 
 1181 
 
 1197 
 
 1213 
 
 1229 
 
 8 13 
 
 19 
 
 1145 
 
 1161 
 
 1177 
 
 1193 
 
 1209 
 
 1224 
 
 9 14 
 
 20 
 
 .001141 
 
 .001157 
 
 .001173 
 
 .001189 
 
 .001204 
 
 .001220 
 
 15 
 
 21 
 
 1137 
 
 1153 
 
 1169 
 
 1185 
 
 1200 
 
 1216 
 
 1 2 
 
 22 
 
 1133 
 
 1149 
 
 1165 
 
 1181 
 
 1196 
 
 1212 
 
 2 3 
 
 23 
 
 1130 
 
 1145 
 
 1161 
 
 1177- 
 
 1132 
 
 1208 
 
 3 4 
 
 24 
 
 1126 
 
 1141 
 
 1157 
 
 1173 
 
 1188- 
 
 -- 1204 
 
 4 6- 
 
 ' 25 
 
 .001122 
 
 .001138 
 
 .001153 
 
 .001169 
 
 .001184 
 
 .001200 
 
 5 7 
 
 26 
 
 1118 
 
 1134 
 
 1149 
 
 1165 
 
 1180 
 
 1196 
 
 6 9 
 
 27 
 
 1114 
 
 1130 
 
 1145 
 
 1161 
 
 1176 
 
 1192 
 
 7 10 
 
 28 
 
 1110 
 
 1126 
 
 1142 
 
 1157 
 
 1172 
 
 1188 
 
 8 12 
 
 29 
 
 1107 
 
 1122 
 
 1138 
 
 1153 
 
 1169 
 
 1184 
 
 9 13 
 
 30 
 
 .001103 
 
 .001119 
 
 .001134 
 
 .001149 
 
 .001165 
 
 .001180 
 
 
 Correction for Moisture in Above Table 
 
 Dew-point 
 
 Subtract 
 
 Dew-point 
 
 Subtract 
 
 Dew-point 
 
 Subtract 
 
 Dew-point 
 
 Subtract 
 
 10 
 8 
 6 
 
 4 
 
 2 
 
 .000001 
 .000002 
 .000003 
 .000002 
 .000003 
 
 
 
 fc 
 
 +6 
 +8 
 
 .000003 
 .000003 
 .000004 
 .000004 
 .000005 
 
 +10 
 
 + 13 
 +14 
 + 16 
 +18 
 
 .000006 
 .000006 
 .000007 
 .000008 
 .000009 
 
 +20 
 
 +22 
 +24 
 +26 
 +28 
 
 .000010 
 .000012 
 .000013 
 .000015 
 .000016 
 
 Table 4 
 DENSITY OF WATER 
 
 Table 5 
 
 DENSITY OF MERCURY 
 
 TEMP. C 
 
 DENSITY 
 
 TEMP. C 
 
 DENSITY 
 
 TEMP. C 
 
 DENSITY 
 
 
 
 0.999884 
 
 13 
 
 0.999443 
 
 
 
 13.596 
 
 1 
 
 0.999941 
 
 14 
 
 0.999312 
 
 10 
 
 13.572 
 
 2 
 
 0.999982 
 
 is 
 
 0.999173 
 
 12 
 
 13.567 
 
 3 
 
 1.000004 
 
 16 
 
 0.999015 
 
 14 
 
 13.562 
 
 3.95 
 
 1.000000 
 
 17 
 
 0.998854 
 
 16 
 
 13.557 
 
 4 
 
 1.000013 
 
 18 
 
 0.998667 
 
 18 
 
 13.552 
 
 5 
 
 1.000003 
 
 19 
 
 0.998473 
 
 20 
 
 13.547 
 
 6 
 
 0.999983 
 
 20 
 
 0.998272 
 
 22 
 
 13.542 
 
 7 
 
 0.999946 
 
 22 
 
 0.997839 
 
 24 
 
 13.537 
 
 8 
 
 0.999899 
 
 24 
 
 0.997380 
 
 26 
 
 13.532 
 
 9 
 
 0.999837 
 
 26 
 
 0.996879 
 
 28 
 
 13.528 
 
 10 
 
 0.999760 
 
 28 
 
 0.996344 
 
 30 
 
 13.523 
 
 11 
 
 0999668 
 
 30 
 
 0.995778 
 
 32 
 
 13.518 
 
 12 
 
 0.999562 
 
 100 
 
 0.958860 
 
 34 
 
 13.513 
 
228 
 
 APPENDIX 
 
 Table 6 
 SATURATED WATER VAPOR 
 
 Showing pressure P (in mm. of mercury) and density I) of aqueous vapor saturated 
 at temperature t; or showing boiling point t of -water and density D of steam cor- 
 responding to an outside pressure P. 
 
 t 
 
 P 
 
 D 
 
 t 
 
 P 
 
 D 
 
 t 
 
 P 
 
 I) 
 
 10 
 
 2.2 
 
 2.3X10- 6 
 
 30 
 
 31.5 
 
 30.1X10- 6 
 
 88.5 
 
 496. 
 
 
 9 
 
 2.3 
 
 2.5 " 
 
 35 
 
 41.8 
 
 39.3 " 
 
 89 
 
 505. 
 
 
 8 
 
 2.5 
 
 2.7 " 
 
 40 
 
 54.9 
 
 50.9 " 
 
 89.5 
 
 515. 
 
 
 7 
 
 2.7 
 
 2.9 " 
 
 45 
 
 71.4 
 
 65.3 " 
 
 90 
 
 525. 
 
 428.4X10- 6 
 
 - 6 
 
 2.9 
 
 3.2 " 
 
 50 
 
 92.0 
 
 83.0 " 
 
 90.5 
 
 535. 
 
 
 5 
 
 3.2 
 
 3.4 " 
 
 55 
 
 117.5 
 
 104.6 " 
 
 91 
 
 545. 
 
 
 4 
 
 3.4 
 
 3.7 " 
 
 60 
 
 148.8 
 
 130.7 " 
 
 91.5 
 
 556. 
 
 
 3 
 
 3.7 
 
 4.0 " 
 
 65 
 
 187.0 
 
 162.1 " 
 
 92 
 
 566. 
 
 
 2 
 
 3.9 
 
 4.2 " 
 
 70 
 
 233.1 
 
 199.5 " 
 
 92.5 
 
 577. 
 
 
 1 
 
 4.2 
 
 4.5 " 
 
 71 
 
 243.6 
 
 
 93 
 
 588. 
 
 
 
 
 4.6 
 
 4.9 " 
 
 72 
 
 254.3 
 
 
 93.5 
 
 599.6 
 
 
 1 
 
 4.9 
 
 5.2 " 
 
 73 
 
 265.4 
 
 
 94 
 
 610.6 
 
 
 2 
 
 5.3 
 
 5.6 " 
 
 74 
 
 276.9 
 
 
 94.5 
 
 622.0 
 
 
 3 
 
 5.7 
 
 6.0 " 
 
 75 
 
 288.8 
 
 243.7 " 
 
 95 
 
 633.6 
 
 511.1 " 
 
 4 
 
 Si 
 
 6/4 " 
 
 75.5 
 
 294.9 
 
 
 95.5 
 
 645.4 
 
 
 5 
 
 6.5 
 
 6.8 " 
 
 76 
 
 301.1 
 
 
 96 
 
 657.4 
 
 
 6 
 
 7.0 
 
 7.3 
 
 76.5 
 
 307.4 
 
 
 96.5 
 
 669.5 
 
 
 7 
 
 7.5 
 
 7.7 " 
 
 77 
 
 313.8 
 
 
 97 
 
 681.8 
 
 
 8 
 
 8.0 
 
 8.2 " 
 
 77.5 
 
 320.4 
 
 
 97.5 
 
 694.2 
 
 
 9 
 
 8.5 
 
 8.7 " 
 
 78 
 
 327.1 
 
 
 98 
 
 707.1 
 
 
 10 
 
 9.1 
 
 9.3 " 
 
 78.5 
 
 333.8 
 
 
 98.2 
 
 712.3 
 
 
 11 
 
 9.8 
 
 10.0 " 
 
 79 
 
 340.7 
 
 
 98.4 
 
 717.4 
 
 
 12 
 
 10.4 
 
 10.6 " 
 
 79.5 
 
 347.7 
 
 
 98.6 
 
 722.6 
 
 
 13 
 
 11.1 
 
 11,2 " 
 
 80 
 
 354.9 
 
 295.9 " 
 
 '98.8 
 
 727.9 
 
 
 14 
 
 11.9 
 
 12.0 " 
 
 80.5 
 
 362.1 
 
 
 99 
 
 733.2 
 
 
 15 
 
 12.7 
 
 12.8 " 
 
 81 
 
 369.5 
 
 
 99.2 
 
 738.5 
 
 
 16 
 
 13.5 
 
 13.o " 
 
 81.5 
 
 377.0 
 
 
 99.4 
 
 743.8 
 
 
 17 
 
 14.4 
 
 14.4 " 
 
 82 
 
 384.6 
 
 
 99.6 
 
 749.2 
 
 
 18 
 
 15.3 
 
 15.2 " 
 
 82.5 
 
 392.4 
 
 
 99.8 
 
 754.7 
 
 
 19 
 
 16.3 
 
 16.2 " 
 
 83 
 
 400.3 
 
 
 100 
 
 760.0 
 
 606.2 " 
 
 20 
 
 174 
 
 17.2 " 
 
 83.5 
 
 408.3 
 
 
 100.2 
 
 765.5 
 
 
 21 
 
 18.5 
 
 18.2 " 
 
 84 
 
 416.5 
 
 
 100.4 
 
 771.0 
 
 
 22 
 
 19.6 
 
 19.3 " 
 
 84.5 
 
 424.7 
 
 
 100.6 
 
 776.5 
 
 
 23 
 
 20.9 
 
 20.4 " 
 
 85 
 
 433.2 
 
 357.1 " 
 
 100.8 
 
 782.1 
 
 
 24 
 
 22.2 
 
 21.6 " 
 
 85.5 
 
 441.7 
 
 
 101 
 
 787.7 
 
 
 25 
 
 23.5 
 
 22.9. " 
 
 86 
 
 450.5 
 
 
 102 
 
 816.0 
 
 
 26 
 
 25.0 
 
 24.2 " 
 
 86.5 
 
 459.3 
 
 
 103 
 
 845.3 
 
 
 27 
 
 26.5 
 
 25.6 " 
 
 87 
 
 468.3 
 
 
 105 
 
 906.4 
 
 715.4 " 
 
 28 
 
 28.1 
 
 27.0 " 
 
 87.5 
 
 477.4 
 
 
 107 
 
 971.1 
 
 
 29 
 
 29.7 
 
 28.5 " 
 
 88 
 
 486.8 
 
 
 110 
 
 1075.4 
 
 840.1 " 
 
APPENDIX 
 
 229 
 
 Table 7 
 REDUCTION OF BAROMETRIC HEIGHT TO C. 
 
 (The table corrections represent the number of millimeters to be subtracted from 
 the observed height h. They are obtained from the formula (.000181 .000019)^, the 
 first number being the cubical expansion coefficient of mercury, the second the linear 
 coefficient of brass.) 
 
 t 
 
 OBSERVED HEIGHT IN mm. 
 
 680 
 
 690 
 
 700 
 
 710 
 
 720 
 
 730 
 
 740 
 
 750 
 
 760 
 
 770 
 
 
 mm. 
 
 mm. 
 
 s mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 10 
 
 1.10 
 
 1.12 
 
 1.13 
 
 1.15 
 
 1.17 
 
 1.18 
 
 1.20 
 
 1.22 
 
 1.23 
 
 1.25 
 
 11 
 
 1.21 
 
 1.23 
 
 1.25 
 
 1.27 
 
 1.28 
 
 1.30 
 
 1.32 
 
 1.34 
 
 1.35 
 
 1.37 
 
 12 
 
 .32 
 
 1.34 
 
 1.36 
 
 1.38 
 
 1.40 
 
 1.42 
 
 1.44 
 
 1.46 
 
 1.48 
 
 1.50 
 
 13 
 
 .43 
 
 1.45 
 
 1.47 
 
 1.50 
 
 1.52 
 
 1.54 
 
 1.56 
 
 1.58 
 
 1.60 
 
 1.62 
 
 14 
 
 .54 
 
 1.56 
 
 1.59 
 
 1.61 
 
 1.63 
 
 1.66 
 
 1.68 
 
 1.70 
 
 1.72 
 
 1.75 
 
 15 
 
 .65 
 
 1.68 
 
 1.70 
 
 1.73 
 
 1.75 
 
 1.77 
 
 1.80 
 
 1.82 
 
 1.85 
 
 1.87 
 
 16 
 
 .76 
 
 1.79 
 
 1.81 
 
 1.84 
 
 1.87 
 
 1.89 
 
 1.92 
 
 1.94 
 
 1.97 
 
 2.00 
 
 17 
 
 1.87 
 
 1.90 
 
 1.93 
 
 1.96 
 
 1.98 
 
 2.01 
 
 2.04 
 
 2.07 
 
 2.09 
 
 2.12 
 
 18 
 
 1.98 
 
 2.01 
 
 2.04 
 
 2.07 
 
 2.10 
 
 2.13 
 
 2.16 
 
 2.19 
 
 2.22 
 
 2.25 
 
 19 
 
 2.09 
 
 2.12 
 
 2.15 
 
 2.19 
 
 2.22 
 
 2.25 
 
 2.28 
 
 2.31 
 
 2.34 
 
 2.37 
 
 20 
 
 2.20 
 
 2.24 
 
 2.27 
 
 2.30 
 
 2.33 
 
 2.37 
 
 2.40 
 
 2.43 
 
 2.46 
 
 2.49 
 
 21 
 
 2.31 
 
 2.35 
 
 2.38 
 
 2.42 
 
 2.45 
 
 2.48 
 
 2.52 
 
 2.55 
 
 2.59 
 
 2.62 
 
 22 
 
 2.42 
 
 2.46 
 
 2.49 
 
 2.53 
 
 2.57 
 
 2.60 
 
 2.64 
 
 2.67 
 
 2.71 
 
 2.74 
 
 23 
 
 2.53 
 
 2.57 
 
 2.61 
 
 2.65 
 
 2.68 
 
 2 72 
 
 2.76 
 
 2.79 
 
 2.83 
 
 2.87 
 
 24 
 
 2.64 
 
 2.68 
 
 2.72 
 
 2.76 
 
 2.80 
 
 2.84 
 
 2.88 
 
 2.92 
 
 2.95 
 
 2.99 
 
 25 
 
 2.75 
 
 2.79 
 
 2.84 
 
 2.88 
 
 2.92 
 
 2.96 
 
 3.00 
 
 3.04 
 
 3.08 
 
 3.12 
 
 Table 8 
 
 CAPILLARY DEPRESSION OF MERCURY IN GLASS 
 
 DIAM- 
 
 HEIGHT OF THE MENISCUS IN mm. 
 
 ETEK 
 
 &4 
 
 0.6 
 
 0.8 
 
 1.0 
 
 1.2 
 
 1.4 
 
 1.6 
 
 1.8 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 mm. 
 
 4 
 
 0.83 
 
 1.22 
 
 1.54 
 
 ' 1.98 
 
 2.37 
 
 
 
 
 5 
 
 0.47 
 
 0.65 
 
 0.86 
 
 1.19 
 
 1.45 
 
 1.80 
 
 
 
 6 
 
 0.27 
 
 0.41 
 
 0.56 
 
 0.78 
 
 0.98 
 
 1.21 
 
 1.43 
 
 
 7 
 
 0.18 
 
 0.28 
 
 0.40 
 
 0.53 
 
 0.67 
 
 0.82 
 
 0.97 
 
 1.13 
 
 8 
 
 
 0.20 
 
 0.29 
 
 0.38 
 
 0.46 
 
 0.56 
 
 0.65 
 
 0.77 
 
 9 
 
 
 0.15 
 
 0.21 
 
 0.28 
 
 0.33 
 
 0.40 
 
 0.46 
 
 0.52 
 
 10 
 
 
 
 0.15 
 
 0.20 
 
 0.25 
 
 0.29 
 
 0.33 
 
 0.37 
 
 11 
 
 
 
 0.10 
 
 0.14 
 
 0.18 
 
 0.21 
 
 0.24 
 
 0.27 
 
 12 
 
 
 
 0.07 
 
 0.10 
 
 0.13 
 
 0.15 
 
 0.18 
 
 0.19 
 
 13 
 
 
 
 0.04 
 
 0.07 
 
 0.10 
 
 0.12 
 
 0.13 
 
 0.14 
 
230 
 
 APPENDIX 
 
 Table 9 
 
 SURFA CE ' TENSIONS 
 in dynes per cm. 
 
 Alcohol 
 
 at 20 
 
 25 5 
 
 Olive Oil . 
 
 at 20 
 
 31 7 
 
 Benzine . . 
 Glycerine 
 
 at 15 
 
 at 17 
 
 28.8 
 63 1 
 
 Petroleum 
 Water 
 
 at 20 
 at 
 
 23.9 
 76 6 
 
 Mercury 
 
 at 20 
 
 4500 
 
 Water 
 
 at 20 
 
 740 
 
 
 
 
 
 
 
 Table 10 
 
 AVERAGE SPECIFIC HEATS 
 
 Alcohol 
 
 at 40 
 
 648 
 
 Lead 
 
 0-100 
 
 0315 
 
 Aluminium . . 
 
 0-100 
 
 2185 
 
 Mercury 
 
 20- 50 
 
 0333 
 
 Brass 
 
 
 0.094 
 
 Paraffine 
 
 
 683 
 
 Copper 
 
 0-100 
 
 0095 
 
 Platinum 
 
 0-100 
 
 0323 
 
 Germa n si 1 ver 
 
 
 00946 
 
 Silver 
 
 0-100 
 
 0568 
 
 Glass 
 
 
 20 
 
 Steel 
 
 
 118 
 
 Gold 
 
 
 0316 
 
 Tin . . 
 
 0-100 
 
 0559 
 
 Ice 
 
 
 504 
 
 Turpentine .... 
 
 
 467 
 
 Iron 
 
 0-100 
 
 0.1130 
 
 Zinc 
 
 
 0935 
 
 
 
 
 
 
 
 Table 11 
 
 AVERAGE COEFFICIENTS OF EXPANSION BETWEEN AND 100 C. 
 
 Glass 
 
 Cubical 
 0.000025 | Mercury.... 0.0001815 | Turpentine... 0.00105 
 
 Aluminium.. 0.000023 
 
 Brass 0.000018 
 
 Copper 0.000017 
 
 Gold . . . . 0.000014 
 
 Linear 
 
 Iron (soft) . . 0.000012 
 
 Iron (cast).. 0.0000105 
 
 Lead 0.000029 
 
 Platinum.. . 0.000009 
 
 Silver 0.000019 
 
 Steel 0.000011 
 
 Tin 0.000022 
 
 Zinc 0.000029 
 
 Table 12 
 IMPORTANT NUMBERS 
 
 ic = 3.1416. 7T 2 = 9.8696. = 0.31831. Logarithm IT = .49715. 
 
 Base of the natural system of logarithms e = 2.7183. 
 
 1 inch = 25.4 millimeters. 1 meter = 39.37 inches. 1 mile == 1.609 kilo- 
 meters. 
 
 1 kilogram = 2.2 pounds. 1 ounce = 28.35 grams. 1 grain = 64.8 milli- 
 grams. 
 
 Mechanical equivalent of 1 calorie (15) = 4.19 X 10 7 ergs. 
 
APPENDIX 
 
 231 
 
 NATURAL SINES 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Complement 
 Difference 
 
 
 
 0.0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 0175 
 
 89 
 
 1 
 
 0175 
 
 0192 
 
 0209 
 
 0227 
 
 0244 
 
 0262 
 
 0279 
 
 0297 
 
 0314 
 
 0332 
 
 0349 
 
 88 
 
 2 
 
 0349 
 
 0366 
 
 0384 
 
 0401 
 
 0419 
 
 0436 
 
 0454 
 
 0471 
 
 0488 
 
 0506 
 
 0523 
 
 87 
 
 3 
 
 0523 
 
 0541 
 
 0558 
 
 0576 
 
 0593 
 
 0610 
 
 0628 
 
 0645 
 
 0663 
 
 0680 
 
 0698 
 
 86 
 
 4 
 
 0698 
 
 0715 
 
 0732 
 
 0750 
 
 0767 
 
 0785 
 
 0802 
 
 0819 
 
 0837 
 
 0854 
 
 0872 
 
 85 
 
 5 
 
 0.0872 
 
 0889 
 
 0906 
 
 0924 
 
 0941 
 
 0958 
 
 0976 
 
 0993 
 
 1011 
 
 1028 
 
 1045 
 
 84 
 
 6 
 
 1045 
 
 1063 
 
 1080 
 
 1097 
 
 1115 
 
 1132 
 
 1149 
 
 1167 
 
 1184 
 
 1201 
 
 1219 
 
 83 
 
 7 
 
 1219 
 
 1236 
 
 1253 
 
 1271 
 
 1288 
 
 1305 
 
 1323 
 
 1340 
 
 1357 
 
 1374 
 
 1392 
 
 82 
 
 8 
 
 1392 
 
 1409 
 
 1426 
 
 1444 
 
 1461 
 
 1478 
 
 1495 
 
 1513 
 
 1530 
 
 1547 
 
 1564 
 
 81 
 
 9 
 
 1564 
 
 1582 
 
 1599 
 
 1616 
 
 1633 
 
 1650 
 
 1668 
 
 1685 
 
 1702 
 
 1719 
 
 1736 
 
 80 
 
 10 
 
 0.1736 
 
 1754 
 
 1771 
 
 1788 
 
 1805 
 
 1822 
 
 1840 
 
 1857 
 
 1874 
 
 1891 
 
 1908 
 
 79 
 
 11 
 
 1908 
 
 1925 
 
 1942 
 
 1959 
 
 1977 
 
 199-1 
 
 2011 
 
 2028 
 
 2045 
 
 2062 
 
 2079 
 
 78 
 
 12 
 
 2079 
 
 2096 
 
 2113 
 
 2130 
 
 2147 
 
 2164 
 
 2181 
 
 2198 
 
 2215 
 
 2233 
 
 2250 
 
 77 17 
 
 13 
 
 2250 
 
 2267 
 
 2284 
 
 2300 
 
 2317 
 
 2334 
 
 2351 
 
 2368 
 
 2385 
 
 2402 
 
 2419 
 
 76 
 
 14 
 
 2419 
 
 2436 
 
 2453 
 
 2470 
 
 2487 
 
 2504 
 
 2521 
 
 2538 
 
 2554 
 
 2571 
 
 2588 
 
 75 
 
 15 
 
 0.2588 
 
 2605 
 
 2622 
 
 2639 
 
 2656 
 
 2672 
 
 2689 
 
 2706 
 
 2723 
 
 2740 
 
 2756 
 
 74 
 
 16 
 
 2756 
 
 2773 
 
 2790 
 
 2807 
 
 2823 
 
 2840 
 
 2857 
 
 2874 
 
 2890 
 
 2907 
 
 2924 
 
 73 
 
 17 
 
 2924 
 
 2940 
 
 2957 
 
 2974 
 
 2990 
 
 3007 
 
 3024 
 
 3040 
 
 3057 
 
 3074 
 
 3090 
 
 72 
 
 18 
 
 3090 
 
 3107 
 
 3123 
 
 3140 
 
 3156 
 
 3173 
 
 3190 
 
 3206 
 
 3223 
 
 3239 
 
 3256 
 
 71 
 
 19 
 
 3256 
 
 3272 
 
 3289 
 
 3305 
 
 3322 
 
 3338 
 
 3355 
 
 3371 
 
 338? 
 
 3404 
 
 3420 
 
 70 
 
 20 
 
 0.3420 
 
 3437 
 
 3453 
 
 3469 
 
 3486 
 
 3502 
 
 3518 
 
 3535 
 
 3551 
 
 3567 
 
 3584 
 
 69 
 
 21 
 
 3584 
 
 3600 
 
 3616 
 
 3633 
 
 3649 
 
 3665 
 
 3681 
 
 3697 
 
 3714 
 
 3730 
 
 3746 
 
 68 
 
 22 
 
 3746 
 
 3762 
 
 3778 
 
 3795 
 
 3811 
 
 3827 
 
 3843 
 
 3859 
 
 3875 
 
 3891 
 
 3907 
 
 67 
 
 23 
 
 3907 
 
 3923 
 
 3939 
 
 3955 
 
 3971 
 
 3987 
 
 4003 
 
 4019 
 
 4035 
 
 4051 
 
 4067 
 
 66 16 
 
 24 
 
 4067 
 
 4083 
 
 4099 
 
 4115 
 
 4131 
 
 4147 
 
 4163 
 
 4179 
 
 4195 
 
 4210 
 
 4226 
 
 65 
 
 25 
 
 0.4226 
 
 4242 
 
 4258 
 
 4274 
 
 4289 
 
 4305 
 
 4321 
 
 4337 
 
 4352 
 
 4368 
 
 4384 
 
 64 
 
 26 
 
 4384 
 
 4399 
 
 4415 
 
 4431 
 
 4446 
 
 4462 
 
 4478 
 
 4493 
 
 4509 
 
 4524 
 
 4540 
 
 63 
 
 27 
 
 4540 
 
 4555 
 
 4571 
 
 4586 
 
 4602 
 
 4617 
 
 4633 
 
 4648 
 
 4664 
 
 4679 
 
 4695 
 
 62 
 
 28 
 
 4695 
 
 4710 
 
 4726 
 
 4741 
 
 4756 
 
 4772 
 
 4787 
 
 4802 
 
 4818 
 
 4833 
 
 4848 
 
 61 
 
 29 
 
 4848 
 
 4863 
 
 4879 
 
 4894 
 
 4909 
 
 4924 
 
 4939 
 
 4955 
 
 4970 
 
 4985 
 
 5000 
 
 60 
 
 30 
 
 0.5000 
 
 5015 
 
 5030 
 
 5045 
 
 5060 
 
 5075 
 
 5090 
 
 5105 
 
 5120 
 
 5135 
 
 5150 
 
 59 15 
 
 31 
 
 5150 
 
 5165 
 
 5180 
 
 5195 
 
 5210 
 
 5225 
 
 5240 
 
 5255 
 
 5270 
 
 5284 
 
 5299 
 
 58 
 
 32 
 
 5299 
 
 5314 
 
 5329 
 
 5344 
 
 5358 
 
 5373 
 
 5388 
 
 5402 
 
 5417 
 
 5432 
 
 5446 
 
 57 
 
 33 
 
 5446 
 
 5461 
 
 5476 
 
 5490 
 
 5505 
 
 5519 
 
 5534 
 
 5548 
 
 5563 
 
 5577 
 
 5592 
 
 56 
 
 34 
 
 5592 
 
 5606 
 
 5621 
 
 5635 
 
 5650 
 
 5664 
 
 5678 
 
 5693 
 
 5707 
 
 5721 
 
 5736 
 
 55 
 
 35 
 
 0.5736 
 
 5750 
 
 5764 
 
 5779 
 
 5793 
 
 5807 
 
 5821 
 
 5835 
 
 5850 
 
 5864 
 
 5878 
 
 54 
 
 36 
 
 5878 
 
 5892 
 
 5906 
 
 5920 
 
 59a4 
 
 5948 
 
 5962 
 
 5976 
 
 5990 
 
 6004 
 
 6018 
 
 53 14 
 
 37 
 
 6018 
 
 6032 
 
 6046 
 
 6060 
 
 6074 
 
 6088 
 
 6101 
 
 6115 
 
 6129 
 
 6143 
 
 6157 
 
 52 
 
 38 
 
 6157 
 
 6170 
 
 6184 
 
 6198 
 
 6211 
 
 6225 
 
 6239 
 
 6252 
 
 6266 
 
 6280 
 
 6293 
 
 51 
 
 39 
 
 6293 
 
 6307 
 
 6320 
 
 6334 
 
 6347 
 
 6361 
 
 6374 
 
 6388 
 
 6401 
 
 6414 
 
 6428 
 
 50 
 
 40 
 
 0.6428 
 
 6441 
 
 6455 
 
 6468 
 
 6481 
 
 6494 
 
 6508 
 
 6521 
 
 6534 
 
 6547 
 
 6561 
 
 49 
 
 41 
 
 6561 
 
 6574 
 
 6587 
 
 6600 
 
 6613 
 
 6626 
 
 6639 
 
 6652 
 
 6665 
 
 6678 
 
 6691 
 
 48 13 
 
 42 
 
 6691 
 
 6704 
 
 6717 
 
 6730 
 
 6743 
 
 6756 
 
 6769 
 
 6782 
 
 6794 
 
 6807 
 
 6820 
 
 47 
 
 43 
 
 6820 
 
 6833 
 
 6845 
 
 6858 
 
 6871 
 
 6884 
 
 6896 
 
 6909 
 
 6921 
 
 6934 
 
 6947 
 
 46 
 
 44 
 
 6947 
 
 6959 
 
 6972 
 
 6984 
 
 6997 
 
 7009 
 
 7022 
 
 7034 
 
 7046 
 
 7059 
 
 7071 
 
 45 
 
 Complement 
 
 .9 
 
 .8 
 
 .7 
 
 .6 
 
 .5 
 
 .4 
 
 .3 
 
 .2 
 
 .1 
 
 .0 
 
 Angle 
 
 NATURAL COSINES 
 
232 
 
 APPENDIX 
 
 NATURAL SINES 
 
 
 
 
 
 
 
 
 
 
 
 n Complement 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Difference 
 
 45 
 
 0.7071 
 
 7083 
 
 7096 
 
 7108 
 
 7120 
 
 7133 
 
 7145 
 
 7157 
 
 7169 
 
 7181 
 
 7193 
 
 44 
 
 46 
 
 7193 
 
 T206 
 
 7218 
 
 7230 
 
 7242 
 
 7254 
 
 7266 
 
 7278 
 
 7290 
 
 7302 
 
 7314 
 
 43 12 
 
 47 
 
 7314 
 
 7325 
 
 7337 
 
 7349 
 
 7361 
 
 7373 
 
 7385 
 
 7396 
 
 7408 
 
 7420 
 
 7431 
 
 42 
 
 48 
 
 7431 
 
 7443 
 
 7455 
 
 7466 
 
 7478 
 
 7490 
 
 7501 
 
 7513 
 
 7524 
 
 7536 
 
 7547 
 
 41 
 
 49 
 
 7547 
 
 7559 
 
 7570 
 
 7581 
 
 7593 
 
 7604 
 
 7615 
 
 7627 
 
 7638 
 
 7649 
 
 7660 
 
 40 
 
 50 
 
 0.7660 
 
 7672 
 
 7683 
 
 7694 
 
 7705 
 
 7716 
 
 7727 
 
 7738 
 
 7749 
 
 7760 
 
 7771 
 
 39 
 
 51 
 
 7771 
 
 7782 
 
 7793 
 
 7804 
 
 7815 
 
 7826 
 
 7837 
 
 7848 
 
 7859 
 
 7869 
 
 7880 
 
 38 " 
 
 52 
 
 7880 
 
 7891 
 
 7902 
 
 7912 
 
 7923 
 
 7934 
 
 7944 
 
 7955 
 
 7965 
 
 7976 
 
 7986 
 
 37 
 
 63 
 
 7986 
 
 7997 
 
 8007 
 
 8018 
 
 8028 
 
 8039 
 
 8049 
 
 8059 
 
 8070 
 
 8080 
 
 8090 
 
 36 
 
 54 
 
 8090 
 
 8100 
 
 8111 
 
 8121 
 
 8131 
 
 8141 
 
 8151 
 
 8161 
 
 8171 
 
 8181 
 
 8192 
 
 35 
 
 55 
 
 0.8192 
 
 8202 
 
 8211 
 
 8221 
 
 8231 
 
 8241 
 
 8251 
 
 8261 
 
 8271 
 
 8281 
 
 8290 
 
 34 10 
 
 56 
 
 8290 
 
 8300 
 
 8310 
 
 8320 
 
 8329 
 
 8339 
 
 8348 
 
 8358 
 
 8368 
 
 8377 
 
 838^3 
 
 33 
 
 57 
 
 8387 
 
 8396 
 
 8406 
 
 8415 
 
 8425 
 
 8434 
 
 8443 
 
 8453 
 
 8462 
 
 8471 
 
 8480 
 
 32 
 
 58 
 
 8480 
 
 8490 
 
 8499 
 
 8508 
 
 8517 
 
 8526 
 
 8536 
 
 8545 
 
 8554 
 
 8563 
 
 8572 
 
 31 
 
 59 
 
 8572 
 
 8581 
 
 8590 
 
 8599 
 
 8607 
 
 8616 
 
 8625 
 
 8634 
 
 8643 
 
 8652 
 
 8660 
 
 30 9 
 
 60 
 
 0.8660 
 
 8669 
 
 8678 
 
 8686 
 
 8695 
 
 8704 
 
 8712 
 
 8721 
 
 8729 
 
 8738 
 
 8746 
 
 29 
 
 61 
 
 8746 
 
 8755 
 
 8763 
 
 8771 
 
 8780 
 
 8788 
 
 8796 
 
 8805 
 
 8813 
 
 8821 
 
 8829 
 
 28 
 
 62 
 
 8829 
 
 8838 
 
 8846 
 
 8854 
 
 8862 
 
 8870 
 
 8878 
 
 8886 
 
 8894 
 
 8902 
 
 8910 
 
 27 8 
 
 68 
 
 8910 
 
 8918 
 
 8926 
 
 8934 
 
 8942 
 
 8949 
 
 8957 
 
 8965 
 
 8973 
 
 8980 
 
 8988 
 
 26 
 
 64 
 
 8988 
 
 8996 
 
 9003 
 
 9011 
 
 9018 
 
 9026 
 
 9033 
 
 9041 
 
 9048 
 
 '9056 
 
 9063 
 
 25 
 
 65 
 
 0.9063 
 
 9070 
 
 9078 
 
 9085 
 
 9092 
 
 9100 
 
 9107 
 
 9114 
 
 9121 
 
 9128 
 
 9135 
 
 24 
 
 66 
 
 9135 
 
 9143 
 
 9150 
 
 9157 
 
 9164 
 
 9171 
 
 9178 
 
 9184 
 
 9191 
 
 9198 
 
 9205 
 
 23 7 
 
 67 
 
 9205 
 
 9212 
 
 9219 
 
 9225 
 
 9232 
 
 9239 
 
 9245 
 
 9252 
 
 9259 
 
 9265 
 
 9272 
 
 22 
 
 68 
 
 9272 
 
 9278 
 
 9285 
 
 9291 
 
 9298 
 
 9304 
 
 9311 
 
 9317 
 
 9323 
 
 9330 
 
 9336 
 
 21 
 
 69 
 
 9336 
 
 9342 
 
 9348 
 
 9354 
 
 9361 
 
 9367 
 
 9373 
 
 9379 
 
 9385 
 
 9391 
 
 9397 
 
 20 6 
 
 70 
 
 0.9397 
 
 9403 
 
 9409 
 
 9415 
 
 9421 
 
 9426 
 
 9432 
 
 9438 
 
 9444 
 
 9449 
 
 9455 
 
 19 
 
 71 
 
 9455 
 
 9461 
 
 9466 
 
 9472 
 
 9478 
 
 9483 
 
 9489 
 
 9494 
 
 9500 
 
 9505 
 
 9511 
 
 18 
 
 72 
 
 9511 
 
 9516 
 
 9521 
 
 9527 
 
 9532 
 
 9537 
 
 9542 
 
 9548 
 
 9553 
 
 9558 
 
 9563 
 
 17 
 
 73 
 
 9563 
 
 9568 
 
 9573 
 
 9578 
 
 9583 
 
 9588 
 
 9593 
 
 9598 
 
 9603 
 
 - 9608 
 
 9613 
 
 16 5 
 
 74 
 
 9613 
 
 9617 
 
 9622 
 
 9627 
 
 9632 
 
 9636 
 
 9641 
 
 9646 
 
 9650 
 
 9655 
 
 9659 
 
 15 
 
 75 
 
 0.9659 
 
 9664 
 
 9668 
 
 9673 
 
 9677 
 
 9681 
 
 9686 
 
 9690 
 
 9694 
 
 9699 
 
 9703 
 
 14 
 
 76 
 
 9703 
 
 9707 
 
 9711 
 
 9715 
 
 9720 
 
 9724 
 
 9728 
 
 9732 
 
 9736 
 
 9740 
 
 9744 
 
 13 * 
 
 77 
 
 9744 
 
 9748 
 
 9751 
 
 9755 
 
 9759 
 
 9763 
 
 9767 
 
 9770 
 
 9774 
 
 9778 
 
 9781 
 
 12 
 
 78 
 
 9781 
 
 9785 
 
 9789 
 
 9792 
 
 9796 
 
 9799 
 
 9803 
 
 9806 
 
 9810 
 
 9813 
 
 1 9816 
 
 11 
 
 79 
 
 9816 
 
 9820 
 
 9823 
 
 9826 
 
 9829 
 
 9833 
 
 9836 
 
 9839 
 
 9842 
 
 9845 
 
 9848 
 
 10 
 
 80 
 
 0.9848 
 
 9851 
 
 9854 
 
 9857 
 
 9860 
 
 9863 
 
 9866 
 
 9869 
 
 9871 
 
 9874 
 
 9877 
 
 9 3 
 
 81 
 
 9877 
 
 9880 
 
 9882 
 
 9885 
 
 9888 
 
 9890 
 
 9893 
 
 9895 
 
 9898 
 
 9900 
 
 9903 
 
 8 
 
 82 
 
 9903 
 
 9905 
 
 9907 
 
 9910 
 
 9912 
 
 9914 
 
 9917 
 
 9919 
 
 9921 
 
 9923 
 
 9925 
 
 7 
 
 83 
 
 9925 
 
 9928 
 
 9930 
 
 9932 
 
 9934 
 
 9936 
 
 9938 
 
 9940 
 
 9942 
 
 9943 
 
 9945 
 
 6 2 
 
 84 
 
 9945 
 
 9947 
 
 9949 
 
 9951 
 
 9952 
 
 9954 
 
 9956 
 
 9957 
 
 9959 
 
 9960 
 
 9962 
 
 5 
 
 85 
 
 0.9962 
 
 9963 
 
 9965 
 
 9966 
 
 9968 
 
 9969 
 
 9971 
 
 9972 
 
 9973 
 
 9974 
 
 9976 
 
 4 
 
 86 
 
 9976 
 
 9977 
 
 9978 
 
 9979 
 
 9980 
 
 9981 
 
 9982 
 
 9983 
 
 9984 
 
 9985 
 
 9986 
 
 3 l 
 
 87 
 
 9986 
 
 9987 
 
 9988 
 
 9989 
 
 9990 
 
 9990 
 
 9991 
 
 9992 
 
 9993 
 
 9993 
 
 9994 
 
 2 
 
 88 
 
 9994 
 
 9995 
 
 9995 
 
 9996 
 
 9996 
 
 9997 
 
 0997 
 
 9997 
 
 9998 
 
 9998 
 
 9998 
 
 1 
 
 89 
 
 9998 
 
 9999 
 
 9999 
 
 9999 
 
 9999 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 1.0000 
 
 
 
 Complement 
 
 .9 
 
 .8 
 
 .7 
 
 .6 
 
 .5 
 
 .4 
 
 .3 
 
 .2 
 
 .1 
 
 .0 
 
 Angle 
 
 NATURAL COSINES 
 
 
APPENDIX 
 
 233 
 
 NATURAL TANGENTS 
 
 
 
 
 
 
 
 
 
 
 
 
 Complement 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Difference 
 
 
 
 0.0000 
 
 0017 
 
 0035 
 
 0052 
 
 0070 
 
 0087 
 
 0105 
 
 0122 
 
 0140 
 
 0157 
 
 0175 
 
 89 
 
 I 
 
 0175 
 
 0192 
 
 0209 
 
 0227 
 
 0244 
 
 0262 
 
 0279 
 
 0297 
 
 0314 
 
 0332 
 
 0349 
 
 88 
 
 2 
 
 0349 
 
 0367 
 
 0384 
 
 0402 
 
 0419 
 
 0437 
 
 0454 
 
 0472 
 
 0489 
 
 0507 
 
 0524 
 
 87 
 
 3 
 
 0524 
 
 0542 
 
 0559 
 
 0577 
 
 0594 
 
 0612 
 
 0629 
 
 0647 
 
 0664 
 
 0682 
 
 0699 
 
 86 
 
 4 
 
 0699 
 
 0717 
 
 0734 
 
 0752 
 
 0769 
 
 0787 
 
 0805 
 
 0822 
 
 0840 
 
 0857 
 
 0875 
 
 85 
 
 5 
 
 00875 
 
 0892 
 
 0910 
 
 0928 
 
 0945 
 
 0963 
 
 0981 
 
 0998 
 
 1016 
 
 1033 
 
 1051 
 
 84 
 
 6 
 
 1051 
 
 1069 
 
 1086 
 
 1104 
 
 1122 
 
 1139 
 
 1157 
 
 1175 
 
 1192 
 
 1210 
 
 1228 
 
 83 
 
 7 
 
 1228 
 
 1246 
 
 1263 
 
 1281 
 
 1299 
 
 1317 
 
 1334 
 
 1352 
 
 1370 
 
 1388 
 
 1405 
 
 82 
 
 8 
 
 1405 
 
 1423 
 
 1441 
 
 1459 
 
 1477 
 
 1495 
 
 1512 
 
 1530 
 
 1548 
 
 1566 
 
 1584 
 
 81 
 
 9 
 
 1584 
 
 1602 
 
 1620 
 
 1638 
 
 1655 
 
 1673 
 
 1691 
 
 1709 
 
 1727 
 
 1745 
 
 1763 
 
 80 
 
 10 
 
 0.1763 
 
 1781 
 
 1799 
 
 1817 
 
 1835 
 
 1853 
 
 1871 
 
 1890 
 
 1908 
 
 1926 
 
 1944 
 
 79 w 
 
 11 
 
 1944 
 
 1962 
 
 1980 
 
 1998 
 
 2016 
 
 2035 
 
 2053 
 
 2071 
 
 2089 
 
 2107 
 
 2126 
 
 78 
 
 12 
 
 2126 
 
 2144 
 
 2162 
 
 2180 
 
 2199 
 
 2217 
 
 2235 
 
 2254 
 
 2272 
 
 2290 
 
 2309 
 
 77 
 
 13 
 
 2309 
 
 2327 
 
 2345 
 
 2364 
 
 2382 
 
 2401 
 
 2419 
 
 2438 
 
 2456 
 
 2475 
 
 2493 
 
 76 
 
 14 
 
 2493 
 
 2512 
 
 2530 
 
 2549 
 
 2568 
 
 2586 
 
 2605 
 
 2623 
 
 2642 
 
 2661 
 
 2679 
 
 75 
 
 15 
 
 0.2679 
 
 2698 
 
 2717 
 
 2736 
 
 2754 
 
 2774 
 
 2792 
 
 2811 
 
 2830 
 
 2849 
 
 2867 
 
 74 
 
 16 
 
 . 2867 
 
 2886 
 
 2905 
 
 2924 
 
 2943 
 
 2962 
 
 2981 
 
 3000 
 
 3019 
 
 3038 
 
 3057 
 
 73 19 
 
 17 
 
 3057 
 
 3076 
 
 3096 
 
 3115 
 
 3134 
 
 3153 
 
 3172 
 
 3191 
 
 3211 
 
 3230 
 
 3249 
 
 72 
 
 18 
 
 3249 
 
 3269 
 
 3288 
 
 3307 
 
 3327 
 
 3346 
 
 3365 
 
 3385 
 
 3404 
 
 3424 
 
 3443 
 
 71 
 
 19 
 
 3443 
 
 3463 
 
 3482 
 
 3502 
 
 3522 
 
 3541 
 
 3561 
 
 3581 
 
 3600 
 
 3620 
 
 3640 
 
 70 
 
 20 
 
 0.3640 
 
 3659 
 
 3679 
 
 3699 
 
 3719 
 
 3739 
 
 3759 
 
 3779 
 
 3799 
 
 3819 
 
 3839 
 
 69 
 
 21 
 
 3839 
 
 3859 
 
 3879 
 
 3899 
 
 3919 
 
 3939 
 
 3959 
 
 3979 
 
 4000 
 
 4020 
 
 4040 
 
 68 20 
 
 22 
 
 4040 
 
 4061 
 
 4081 
 
 4101 
 
 4122 
 
 4142 
 
 4163 
 
 4183 
 
 4204 
 
 4224 
 
 4245 
 
 67 
 
 23 
 
 4245 
 
 4265 
 
 4286 
 
 4307 
 
 4327 
 
 4348 
 
 4369 
 
 4390 
 
 4411 
 
 4431 
 
 4452 
 
 66 
 
 24 
 
 4452 
 
 4473 
 
 4494 
 
 4515 
 
 4536 
 
 4557 
 
 4578 
 
 4599 
 
 4621 
 
 4642 
 
 4663 
 
 65 21 
 
 25 
 
 0.4663 
 
 4684 
 
 4706 
 
 4727 
 
 4748 
 
 4770 
 
 4791 
 
 4813 
 
 4834 
 
 4856 
 
 4877 
 
 64 
 
 26 
 
 4877 
 
 4899 
 
 4921 
 
 4942 
 
 4964 
 
 4986 
 
 5008 
 
 5029 
 
 5051 
 
 5073 
 
 5095 
 
 63 
 
 27 
 
 5095 
 
 5117 
 
 5139 
 
 5161 
 
 5184 
 
 5206 
 
 5228 
 
 5250 
 
 5272 
 
 5295 
 
 5317 
 
 62 22 
 
 28 
 
 5317 
 
 5340 
 
 5362 
 
 5384 
 
 5407 
 
 5430 
 
 5452 
 
 5475 
 
 5498 
 
 5520 
 
 5543 
 
 61 
 
 29 
 
 5543 
 
 5566 
 
 5589 
 
 5612 
 
 5635 
 
 5658 
 
 5681 
 
 5704 
 
 5727 
 
 5750 
 
 5774 
 
 60 23 
 
 30 
 
 0.5774 
 
 5797 
 
 5820 
 
 5844 
 
 5867 
 
 5890 
 
 5914 
 
 5938 
 
 5961 
 
 5985 
 
 6099 
 
 59 
 
 31 
 
 6009 
 
 6032 
 
 6056 
 
 6080 
 
 6104 
 
 6128 
 
 6152 
 
 6176 
 
 6200 
 
 6224 
 
 6249 
 
 58 2 * 
 
 32 
 
 6249 
 
 6273 
 
 6297 
 
 6322 
 
 6346 
 
 6371 
 
 6395 
 
 6420 
 
 6445 
 
 6469 
 
 6494 
 
 57 
 
 33 
 
 6494 
 
 6519 
 
 6544 
 
 6569 
 
 6594 
 
 6619 
 
 6644 
 
 6669 
 
 6694 
 
 6720 
 
 6745 
 
 56 25 
 
 34 
 
 6745 
 
 6771 
 
 6796 
 
 6822 
 
 6847 
 
 6873 
 
 6899 
 
 6924 
 
 6950 
 
 6976 
 
 7002 
 
 55 
 
 35 
 
 0.7002 
 
 7028 
 
 7054 
 
 7080 
 
 7107 
 
 7133 
 
 7159 
 
 7186 
 
 7212 
 
 7239 
 
 7265 
 
 54 26 
 
 36 
 
 7265 
 
 7292 
 
 7319 
 
 7346 
 
 7373 
 
 7400 
 
 7427 
 
 7454 
 
 7481 
 
 7508 
 
 7536 
 
 53 27 
 
 37 
 
 7536 
 
 7563 
 
 7590 
 
 7618 
 
 7646 
 
 7673 
 
 7701 
 
 7729 
 
 7757 
 
 7785 
 
 7813 
 
 52 28 
 
 38 
 
 7813 
 
 7841 
 
 7869 
 
 7898 
 
 7926 
 
 7954 
 
 7983 
 
 8012 
 
 8040 
 
 8069 
 
 8098 
 
 51 28 
 
 39 
 
 8098 
 
 8127 
 
 8156 
 
 8185 
 
 8214 
 
 8243 
 
 8273 
 
 8302 
 
 8332 
 
 8361 
 
 8391 
 
 50 29 
 
 40 
 
 0.8391 
 
 8421 
 
 8451 
 
 8481 
 
 8511 
 
 8541 
 
 8571 
 
 8601 
 
 8632 
 
 8662 
 
 8693 
 
 49 30 
 
 41 
 
 8693 
 
 8724 
 
 8754 
 
 8785 
 
 8816 
 
 8847 
 
 8878 
 
 8910 
 
 8941 
 
 8972 
 
 9004 
 
 48 31 
 
 42 
 
 9004 
 
 9036 
 
 9067 
 
 9099 
 
 9131 
 
 9163 
 
 9195 
 
 9228 
 
 9260 
 
 9293 
 
 9325 
 
 47 32 
 
 43 
 
 9325 
 
 9358 
 
 9391 
 
 9424 
 
 9557 
 
 9490 
 
 9523 
 
 9556 
 
 9590 
 
 9623 
 
 9657 
 
 46 33 
 
 44 
 
 9657 
 
 9691 
 
 9725 
 
 9759 
 
 9793 
 
 9827 
 
 9861 
 
 9896 
 
 9930 
 
 9965 
 
 1.0000 
 
 45 3 * 
 
 Complement 
 
 .9 
 
 .8 
 
 .7 
 
 .6 
 
 .5 
 
 .4 
 
 .3 
 
 .2 
 
 .1 
 
 .0 
 
 Angle 
 
 NATURAL COTANGENTS 
 
234 
 
 APPENDIX 
 
 NATURAL TANGENTS 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 Dif. 
 
 45 
 
 1.0000 
 
 1.0035 
 
 1.0070 
 
 1.0105 
 
 1.0141 
 
 1.0176 
 
 1.0212 
 
 1.0247 
 
 1.0283 
 
 1.0319 
 
 36 
 
 46 
 
 1.0355 
 
 1.0392 
 
 1.0428 
 
 1.0464 
 
 1.0501 
 
 1.0538 
 
 1.0575 
 
 1.0612 
 
 1.0649 
 
 1.0686 
 
 37 
 
 47 
 
 1.0724 
 
 1.0761 
 
 1.0799 
 
 1.0837 
 
 1.0875 
 
 1.0913 
 
 1.0951 
 
 1.0990 
 
 1.1028 
 
 1.1067 
 
 88 
 
 48 
 
 1.1106 
 
 1.1145 
 
 1.1184 
 
 1.1224 
 
 1.1263 
 
 1.1308 
 
 1.1343 
 
 1.1383 
 
 1. 1423 
 
 1.1463 
 
 40 
 
 49 
 
 1. 1504 
 
 1.1544 
 
 1.1585 
 
 1.1626 
 
 1.1667 
 
 1.1708 
 
 1.1750 
 
 1.1792 
 
 1.1833 
 
 1.1875 
 
 41 
 
 50 
 
 1.1918 
 
 1.1960 
 
 1.2002 
 
 1.2045 
 
 1.2088 
 
 1.2131 
 
 1.2174 
 
 1.2218 
 
 1.2261 
 
 1.2305 
 
 43 
 
 51 
 
 1.2849 
 
 1.2393 
 
 1.243" 
 
 1.2482 
 
 1.2527 
 
 1.2572 
 
 1.2617 
 
 1.2662 
 
 1.2708 
 
 1.2753 
 
 45 
 
 52 
 
 1.2799 
 
 1.2846 
 
 1.2892 
 
 1.2938 
 
 1.2985 
 
 1.3032 
 
 1.3079 
 
 1.3127 
 
 1.3175 
 
 1.3222 
 
 47 
 
 53 
 
 1.3270 
 
 1.3319 
 
 1.3367 
 
 1.3416 
 
 1.3465 
 
 1.3514 
 
 1.3564 
 
 1.3613 
 
 1.3663 
 
 1.3713 
 
 49 
 
 54 
 
 1.3764 
 
 1.3814 
 
 1.3865 
 
 1.3916 
 
 1.3968 
 
 1.4019 
 
 1.4071 
 
 1.4124 
 
 1.4176 
 
 1.4229 
 
 52 
 
 55 
 
 1.4281 
 
 1.4335 
 
 1.4388 
 
 1.4442 
 
 1.4496 
 
 1.4550 
 
 1.4605 
 
 1.4659 
 
 1.4715 
 
 1.4770 
 
 54 
 
 56 
 
 1.4826 
 
 1.4882 
 
 1.4938 
 
 1.4994 
 
 1.5051 
 
 1.5108 
 
 1.5166 
 
 1.52M 
 
 1.5282 
 
 1.5340 
 
 57 
 
 57 
 
 1.5399 
 
 1.5458 
 
 1.5517 
 
 1.5577 
 
 1.5637 
 
 1.5697 
 
 1.5757 
 
 1.5818 
 
 1.5880 
 
 1.5941 
 
 60 
 
 58 
 
 1.6003 
 
 1.6066 
 
 1.6128 
 
 1.6191 
 
 1.6255 
 
 1.6319 
 
 1.6383 
 
 1.6447 
 
 1.6512 
 
 1.6577 
 
 64 
 
 59 
 
 1.6643 
 
 1.6709 
 
 1.6775 
 
 1.6842 
 
 1.6909 
 
 1.6977 
 
 1.7045 
 
 1.7113 
 
 1.7182 
 
 1.7251 
 
 68 
 
 60 
 
 1.7321 
 
 1.7391 
 
 1.7461 
 
 1.7532 
 
 1.7603 
 
 1.7675 
 
 1.7747 
 
 1.7820 
 
 1.7893 
 
 1.7966 
 
 72 
 
 61 
 
 1.8040 
 
 1.8115 
 
 1.8190 
 
 1.8265 
 
 1.8341 
 
 1.8418 
 
 1.8495 
 
 1.8572 
 
 1.8650 
 
 1.8728 
 
 77 
 
 62 
 
 1.880 r / 
 
 1.8887 
 
 1.8967 
 
 1.9047 
 
 1.9128 
 
 1.9210 
 
 1.9292 
 
 1.93751.9458 
 
 1.9542 
 
 82 
 
 63 
 
 1.9626 
 
 1.9711 
 
 1.9797 
 
 1.9883 
 
 1.9970 
 
 2.0057 
 
 2.0145 
 
 2.0233 
 
 2.0323 
 
 2.0413 
 
 88 
 
 64 
 
 2.0503 
 
 2.0594 
 
 2.0686 
 
 2.0778 
 
 2.0872 
 
 2.0965 
 
 2.1060 
 
 2.1155 
 
 2.1251 
 
 2.1848 
 
 94 
 
 65 
 
 2.145 
 
 2.154 
 
 2.164 
 
 2.174 
 
 2.184 
 
 2.194 
 
 2.204 
 
 2.215 
 
 2.225 
 
 2.236 
 
 10 
 
 66 
 
 2.246 
 
 2.257 
 
 2.267 
 
 2.278 
 
 2.289 
 
 2.300 
 
 2.311 
 
 2.322 
 
 2.333 
 
 2,344 
 
 11 
 
 67 
 
 2.356 
 
 2.367 
 
 2.379 
 
 2.391 
 
 2.402 
 
 2.414 
 
 2.426 
 
 2.438 
 
 2.450 
 
 2.463 
 
 12 
 
 68 
 
 2.475 
 
 2.488 
 
 2.500 
 
 2.513 
 
 2.526 
 
 2.539 
 
 2.552 
 
 2.565 
 
 2.578 
 
 2.592 
 
 13 
 
 69 
 
 2.605 
 
 2.619 
 
 2.633 
 
 2.646 
 
 2.660 
 
 2.675 
 
 2.689 
 
 2.703 
 
 '2.718 
 
 2.733 
 
 14 
 
 70 
 
 2.747 
 
 2.762 
 
 2.778 
 
 2.793 
 
 2.808 
 
 2.824 
 
 2.840 
 
 2.856 
 
 2.872 
 
 2.888 
 
 16 
 
 71 
 
 2.904 
 
 2.921 
 
 2.937 
 
 2.954 
 
 2.971 
 
 2,989 
 
 1006 
 
 3.024 
 
 3.042 
 
 3.060 
 
 17 
 
 72 
 
 3.078 
 
 3.096 
 
 3.115 
 
 3.133 
 
 3.152 
 
 3.172 
 
 3.191 
 
 3.211 
 
 3.230 
 
 3.250 
 
 19 
 
 73 
 
 3.271 
 
 3.291 
 
 3.312 
 
 8.333 
 
 3.354 
 
 3.376 
 
 3.398 
 
 3.420 
 
 3.442 
 
 3.465 
 
 22 
 
 74 
 
 3.487 
 
 3.511 
 
 3.534 
 
 3.558 
 
 3.582 
 
 3.606 
 
 3.630 
 
 3.655 
 
 3.681 
 
 3.700 
 
 25 
 
 75 
 
 3.732 
 
 3.758 
 
 3.785 
 
 3.812 
 
 3.839 
 
 3.867 
 
 3.895 
 
 3.923 
 
 3.952 
 
 3.981 
 
 28 
 
 76 
 
 4.011 
 
 4.041 
 
 4.071 
 
 4.102 
 
 4.134 
 
 4.165 
 
 4.198 
 
 4.230 
 
 4.264 
 
 4.297 
 
 32 
 
 77 
 
 4.331 
 
 4.366 
 
 4.402 
 
 4.437 
 
 4.474 
 
 4.511 
 
 4.548 
 
 4.586 
 
 4.625 
 
 4.665 
 
 37 
 
 78 
 
 4.705 
 
 4.745 
 
 4.787 
 
 4.829 
 
 4.872 
 
 4.915 
 
 4.959 
 
 5.005 
 
 5.050 
 
 5.097 
 
 44 
 
 79 
 
 5.145 
 
 5.193 
 
 5.242 
 
 5.292 
 
 5.343 
 
 5.396 
 
 5.449 
 
 5.503 
 
 5.558 
 
 5.614 
 
 52 
 
 80 
 
 5.67 
 
 5.73 
 
 5.79 
 
 5.85 
 
 5.91 
 
 5.98 
 
 6.04 
 
 6.11 
 
 6.17 
 
 6.24 
 
 7 
 
 81 
 
 6.31 
 
 6.39 
 
 6.46 
 
 6.54 
 
 6.61 
 
 6.69 
 
 6.77 
 
 6.85 
 
 6.94 
 
 7.03 
 
 8 
 
 82 
 
 7.12 
 
 7.21 
 
 7.30 
 
 7.40 
 
 7.49 
 
 7.60 
 
 7.70 
 
 7.81 
 
 7.92 
 
 8.03 
 
 10 
 
 83 
 
 8.14 
 
 8.26 
 
 8.39 
 
 8.51 
 
 8.64 
 
 8.78 
 
 8.92 
 
 9.06 
 
 9.21 
 
 9.36 
 
 14 
 
 84 
 
 9.51 
 
 9.68 
 
 9.84 
 
 10.0 
 
 10.2 
 
 10.4 
 
 10.6 
 
 10.8 
 
 11.0 
 
 11.2 
 
 
 85 
 
 11.4 
 
 11.7 
 
 11.9 
 
 12.2 
 
 12.4 
 
 12.7 
 
 13.0 
 
 13.3 
 
 13.6 
 
 14.0 
 
 3 
 
 86 
 
 14.3 
 
 14.7 
 
 15.1 
 
 15.5 
 
 15.9 
 
 16.3 
 
 16.8 
 
 17.3 
 
 17.9 
 
 18.5 
 
 6 
 
 87 
 
 19.1 
 
 19.7 
 
 20.4 
 
 21.2 
 
 22.0 
 
 22.9 
 
 23.9 
 
 24.9 
 
 26.0 
 
 27.3 
 
 
 88 
 
 28.6 
 
 30.1 
 
 31.8 
 
 33.7 
 
 35.8 
 
 38.2 
 
 40.9 
 
 44.1 
 
 47.7 
 
 52.1 
 
 
 89 
 
 57. 
 
 64. 
 
 72. 
 
 82. , 
 
 95. 
 
 115. 
 
 43. 
 
 191. 
 
 286. 
 
 573. 
 
 
 Angle 
 
 .0 
 
 .1 
 
 .2 
 
 .3 
 
 .4 
 
 .5 
 
 .6 
 
 .7 
 
 .8 
 
 .9 
 
 
 NATURAL TANGENTS 
 
.*. 
 
 APPENDIX 
 
 LOGARITHMS 
 
 235 
 
 
 
 
 1 1 2 
 
 3 
 
 4 
 
 5 
 
 0212 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 2 3 
 
 456 
 
 789 
 
 10 
 
 0000 
 
 0043|0086 
 
 012 
 
 50170 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 4 8 12 
 
 17 21 25 
 
 29 33 37 
 
 11 
 
 12 
 13 
 
 14 
 15 
 16 
 
 0414 
 0792 
 
 1461 
 1761 
 
 0453 
 
 082S 
 1173 
 
 I 
 
 0492 
 
 0864 
 1206 
 
 Ii523 
 
 1 
 
 053l|0569 
 
 08990934 
 12391271 
 
 15531584 
 
 18471875 
 21222148 
 
 0607 
 0969 
 1303 
 
 1614 
 1903 
 2175 
 
 0645 
 1004 
 1335 
 
 1644 
 1931 
 2201 
 
 0682 
 1038 
 1367 
 
 1673 
 1959 
 
 2227 
 
 0719 
 1072 
 
 I3y 
 
 1703 
 19"87 
 2253 
 
 0755 
 1*06 
 1430 
 
 1732 
 2014 
 2279 
 
 4- 8 11 
 3 7 10 
 3 6 10 
 
 369 
 368 
 358 
 
 15 19 23 
 14 17 21 
 13 16 19 
 
 12 15 18 
 11 14 17 
 11 13 16 
 
 26 30 34 
 24 28 31 
 23 26 29 
 
 21 24 27 
 20 22 25 
 18 21 24 
 
 17 
 
 18 
 19 
 
 304 
 ,553 
 
 a?88 
 
 gD^Kj 
 
 2810J3833 
 
 2380Mtf 2430 
 2625^02672 
 2856^82900 
 
 2455 
 269o 
 2923 
 
 2480 
 2718 
 2945 
 
 2504 
 2742 
 2967 
 
 2529 
 2765 
 
 2989 
 
 257 
 257 
 247 
 
 10 12 15 
 9 12 14 
 9 11 13 
 
 17 20 22 
 16 19 21 
 16 18 20 
 
 20 
 
 3010 
 
 3032 
 
 3054 
 
 307530963118 
 
 3139 
 
 3160 
 
 3181 
 
 3201 
 
 246 
 
 8 11 13 
 
 15 17 19 
 
 21 
 22 
 23 
 
 3222 
 3424 
 3617 
 
 3243 
 3444 
 3636 
 
 3263 
 3464 
 3655 
 
 35^433043324 
 348335023522 
 3674|3692|3711 
 
 3345 
 3541 
 3729 
 
 3365 
 3560 
 3747 
 
 3385 
 3579 
 3766 
 
 3404 
 3598 
 
 3784 
 
 246 
 246 
 246 
 
 8 10 12 
 8 10 12 
 7 9 11 
 
 14 16 18 
 14 15 17 
 13 15 17 
 
 24 
 25 
 26 
 
 27 
 
 28 
 29 
 
 3802 
 31)79 
 
 "*' 
 
 43U 
 4472 
 4624 
 
 3820 
 3997 
 416 
 
 433 
 
 448 
 46S 
 
 3838 
 4014 
 4183 
 
 4346 
 4502 
 4654 
 
 385C 
 4031 
 420C 
 
 4362 
 466 
 
 3874 
 4048 
 4216 
 
 4378 
 4533 
 4683 
 
 3892 
 4065 
 4232 
 
 4393 
 
 4548 
 4698 
 
 3909 
 4082 
 4249 
 
 4409 
 4564 
 4713 
 
 3927 
 4099 
 4265 
 
 4425 
 4579 
 
 4728 
 
 3945 
 4116 
 
 4281 
 
 4440 
 4594 
 
 4742 
 
 3962 
 4133 
 4298 
 
 4456 
 4609 
 
 4757 
 
 245 
 235 
 235 
 
 235 
 235 
 
 1 3 4 
 
 7 9 11 
 7 9 10 
 
 7 8 10 
 
 689 
 689 
 679 
 
 12 14 16 
 12 14 15 
 11 13 15 
 
 11 13 14 
 11 12 14 
 10 12 13 
 
 30 
 
 4771 
 
 478 
 
 4800 
 
 481 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 488$ 
 
 4900 
 
 1 3 4 
 
 679 
 
 10 11 13 
 
 31 
 32 
 33 
 
 4914 
 5051 
 
 5185 
 
 492 
 506 
 519 
 
 4942 
 5079 
 5211 
 
 495 
 509 
 522 
 
 4969 
 5105 
 5237 
 
 4983 
 5119 
 5250 
 
 4997 
 5132 
 5263 
 
 5011 
 5145 
 5276 
 
 5024 
 5159 
 5289 
 
 5038 
 5172 
 5302 
 
 1 8 4 
 1 3 4 
 1 3 4 
 
 678 
 578 
 568 
 
 10 11 12 
 9 11 12 
 
 9 10 12 
 
 34 
 35 
 36 
 
 5315 
 5441 
 5563 
 
 532 
 545 
 557 
 
 5340 
 5465 
 
 5587 
 
 535 
 547 
 559 
 
 5366 
 5490 
 5611 
 
 5378 
 5502 
 5623 
 
 5391 
 5514 
 5635 
 
 5403 
 5527 
 5647 
 
 5416 
 5539 
 
 56-58 
 
 5428 
 5551 
 5670 
 
 1 3 4 
 1 2 4 
 
 1 2 4 
 
 568 
 567 
 567 
 
 9 10 11 
 9 10 11 
 8 10 11 
 
 37 
 
 38 
 39 
 
 5682 
 
 5798 
 5911 
 
 569 
 580 
 
 592 
 
 5705 
 5821 
 5933 
 
 571 
 583 
 5944 
 
 5729 
 5843 
 5955 
 
 5740 
 5855 
 5966 
 
 5752 
 5866 
 5977 
 
 5763 
 
 5877 
 5988 
 
 57f5 
 
 5888 
 5999 
 
 5786 
 5899 
 6010 
 
 1 2 3 
 1 2 3 
 1 2 3 
 
 567 
 567 
 457 
 
 8 9 10 
 8 9 10 
 
 8 9 10 
 
 40 
 
 6021 
 
 603 
 
 6042 
 
 605 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 1 2 3 
 
 456 
 
 8 9 10 
 
 41 
 42 
 43 
 
 6128 
 6232 
 6335 
 
 613 
 624 
 634 
 
 6149 
 6253 
 6355 
 
 616 
 626 
 636 
 
 6170 
 6274 
 6375 
 
 6180 
 6284 
 6385 
 
 6191 
 6294 
 6395 
 
 6201 
 6304 
 6405 
 
 6212 
 6314 
 6415 
 
 6222 
 6325 
 6425 
 
 123 
 123 
 1 2 3 
 
 456 
 456 
 456 
 
 789 
 789 
 789 
 
 44 
 45 
 46 
 
 6435 
 6532 
 
 6628 
 
 644^ 
 654 
 663 
 
 6454 
 6551 
 6646 
 
 646 
 656 
 665 
 
 6474 
 6571 
 6665 
 
 6484 
 6580 
 6675 
 
 6493 
 6590 
 6684 
 
 6503 
 6599 
 6693 
 
 6513 
 6009 
 6702 
 
 6522 
 6618 
 6712 
 
 123 
 1 2 3 
 L 2 3 
 
 456 
 456 
 456 
 
 789 
 789 
 
 778 
 
 47 
 
 48 
 49 
 
 6721 
 6812 
 6902 
 
 673 
 
 682 
 691 
 
 6739 
 6830 
 6920 
 
 674 
 
 683 
 692 
 
 6758 
 6848 
 6937 
 
 6767 
 
 6857 
 6946 
 
 6776 
 6866 
 6955 
 
 6785 
 6875 
 6964 
 
 6794 
 
 6884 
 6972 
 
 6803 
 6893 
 6981 
 
 1 2 3 
 123 
 1 2 3 
 
 455 
 
 445 
 445 
 
 678 
 678 
 678 
 
 50 
 
 6990 
 
 699 
 
 7007 
 
 701 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 1 2 3 
 
 345 
 
 678 
 
 51 
 52 
 53 
 
 7076 
 7160 
 7243 
 
 7084 
 7168 
 7251 
 
 7093 
 
 7177 
 7259 
 
 710 
 718 
 726 
 
 7110 
 7193 
 
 7275 
 
 7118 
 7202 
 
 7284 
 
 7126 
 7210 
 722 
 
 7135 
 
 7218 
 7300 
 
 7143 
 7226 
 
 7308 
 
 7152 
 7235 
 7316 
 
 1 2 3 
 1 2 2, 
 122 
 
 345 
 3 4 5 
 345 
 
 678 
 677 
 667 
 
 54 
 
 7324 
 
 7332 
 
 7340 
 
 734 
 
 7356 
 
 7364 
 
 7372 
 
 7380 
 
 7388 
 
 7396 
 
 1 2 2 
 
 345 
 
 667 
 
 fei 
 
236 
 
 APPENDIX 
 
 LOGARITHMS 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 2 3 
 
 456789 
 
 55 
 
 7404 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 122 
 
 345567 
 
 56 
 
 7482 
 
 7490 
 
 7497 
 
 7505 
 
 7513 
 
 7520 
 
 7528 
 
 7536 
 
 7543 
 
 7551 
 
 122 
 
 345567 
 
 57 
 
 7559 
 
 7566 
 
 7574 
 
 7582 
 
 7589 
 
 7597 
 
 7604 
 
 7612 
 
 7619 
 
 7627 
 
 122 
 
 3 4 5L5 6 7 
 
 58 
 
 7634 
 
 7642 
 
 7649 
 
 7657 
 
 7664 
 
 7672 
 
 7679 
 
 7686 
 
 7694 
 
 7701 
 
 1 1 2 
 
 3 4 4m 6 7 
 
 59 
 
 7709 
 
 7716 
 
 7723 
 
 7731 
 
 7738 
 
 7745 
 
 7752 
 
 7760 
 
 7767 
 
 7774 
 
 1 l^jW -1 4r~5 6 7 
 
 60 
 
 7782 
 
 7789 
 
 7796 
 
 7803 
 
 7810 
 
 7818 
 
 7825 
 
 7832 
 
 7839 
 
 7846 
 
 1 iBt. ! 4566 
 
 61 
 
 7853 
 
 7860 
 
 7868 
 
 7875 
 
 7882 
 
 7889 
 
 7896 
 
 7963 
 
 7910 
 
 7917 
 
 1 W 4 5 6 6 
 
 62 
 
 7924 
 
 7931 
 
 7938 
 
 7945 
 
 7952 
 
 7959 
 
 7966 
 
 7973 
 
 79A&87 
 
 1 R 4 5 6 6 
 
 63 
 
 7993 
 
 8000 
 
 8007 
 
 8014 
 
 8021 
 
 8028 
 
 8035 
 
 8041 
 
 <S( ).").-, 
 
 1 1WG3 4| 5 5 6 
 
 64 
 
 8062 
 
 8069 
 
 8075 
 
 8082 
 
 8089 
 
 8096 
 
 8102 
 
 8109 
 
 81160*122 
 
 112 
 
 334556 
 
 65 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 112 
 
 334556 
 
 66 
 
 8195 
 
 8202 
 
 8209 
 
 8215 
 
 8222 
 
 8228 
 
 8235 
 
 8241 
 
 8248 
 
 8254 
 
 112 
 
 334556 
 
 67 
 
 8261 
 
 8267 
 
 8274 
 
 8280 
 
 8287 
 
 8293 
 
 8299 
 
 8306 
 
 8312 
 
 8319 
 
 1 1 2 
 
 3 3 4l 5 5 6 
 
 68 
 
 8325 
 
 8331 
 
 8838 
 
 8344 
 
 8351 
 
 8357 
 
 8363 
 
 8370 
 
 8376 
 
 8382 
 
 112 
 
 334 
 
 456 
 
 69 
 
 8388 
 
 8395 
 
 8401 
 
 8407 
 
 8414 
 
 8420 
 
 8426 
 
 8432 
 
 8439 
 
 8445 
 
 112 
 
 234 
 
 456 
 
 70 
 
 8451 
 
 8457 
 
 8463 
 
 8470 
 
 8476 
 
 8482 
 
 8488 
 
 8494 
 
 8500 
 
 8506 
 
 112 
 
 234 
 
 456 
 
 71 
 
 8513 
 
 8519 
 
 8525 
 
 8531 
 
 8537 
 
 8543 
 
 8549 
 
 8555 
 
 8561 
 
 8567 
 
 112 
 
 23| 
 
 455 
 
 72 
 
 8573 
 
 8579 
 
 8585 
 
 8591 
 
 8597 
 
 8603 
 
 8609 
 
 8615 
 
 8621 
 
 8627 
 
 1 2 
 
 234 
 
 455 
 
 
 8633 
 
 8639 
 
 8645 
 
 8651 
 
 8657 
 
 86C& 
 
 8669 
 
 8675 
 
 8681 
 
 686 
 
 1 2 
 
 234 
 
 455 
 
 ?i 
 
 8692 
 
 8698 
 
 8704 
 
 8710 
 
 8716 
 
 8722 
 
 8727 
 
 8733 
 
 8739 
 
 8745 
 
 1 2 
 
 3 4 
 
 455 
 
 75 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8802 
 
 1 2 
 
 233 
 
 455 
 
 76 
 
 8808 
 
 8814 
 
 8820 
 
 8825 
 
 8831 
 
 8837 
 
 8842 
 
 8848 
 
 8854 
 
 8859 
 
 1 2 
 
 233 
 
 4 5 5 
 
 77 
 
 8865 
 
 8871 
 
 8876 
 
 8882 
 
 8887 
 
 8893 
 
 88*99 
 
 8904 
 
 8910 
 
 8915 
 
 1 2 
 
 233 
 
 4 4 5 
 
 78 
 
 8921 
 
 8927 
 
 8932 
 
 8938 
 
 8943 
 
 8949 
 
 8954 
 
 8960 
 
 8965 
 
 8971 
 
 1 2 
 
 233 
 
 445 
 
 79 
 
 8976 
 
 8982 
 
 8987 
 
 8993 
 
 8998 
 
 9004 
 
 9009 
 
 9015 
 
 9020 
 
 9025 
 
 1 1 2 
 
 233 
 
 4 4 5 
 
 80 
 
 9031 
 
 9036 
 
 9042 
 
 9047 
 
 9053 
 
 9058 
 
 9063 
 
 9069 
 
 9074 
 
 9079 
 
 112 
 
 233 
 
 445 
 
 81 
 
 9085 
 
 9090 
 
 9096 
 
 9101 
 
 9106 
 
 9112 
 
 9117 
 
 9122 
 
 9128 
 
 9133 
 
 112 
 
 233 
 
 445 
 
 82 
 
 9138 
 
 9143 
 
 9149 
 
 9154 
 
 9159 
 
 9165 
 
 9170 
 
 9175 
 
 9180 
 
 9186 
 
 112 
 
 233 
 
 445 
 
 83 
 
 9191 
 
 9196 
 
 9201 
 
 9206 
 
 9212 
 
 9217 
 
 9222 
 
 9227 
 
 9232 
 
 9238 
 
 112 
 
 233 
 
 445 
 
 84 
 
 9243 
 
 9248 
 
 9253 
 
 9258 
 
 9263 
 
 9269 
 
 9274 
 
 9279 
 
 9284 
 
 9289 
 
 1 1 2 
 
 233 
 
 445 
 
 85 
 
 .9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 1 1 2 
 
 233 
 
 445 
 
 86 
 
 9345 
 
 9350 
 
 9355 
 
 9360 
 
 9365 
 
 9370 
 
 9375 
 
 9380 
 
 9385 
 
 9390 
 
 112 
 
 233 
 
 445 
 
 87 
 
 9395 
 
 9400 
 
 9405 
 
 9410 
 
 94 1 5 
 
 9420 
 
 9425 
 
 9430 
 
 9435 
 
 9440 
 
 1 1 
 
 223 
 
 344 
 
 88 
 
 9445 
 
 9450 
 
 9455 
 
 94-60 
 
 9465 
 
 9469 
 
 9474 
 
 9479 
 
 9484 
 
 9489 
 
 1 1 
 
 223 
 
 344 
 
 89 
 
 9494 
 
 9499 
 
 9504 
 
 9509 
 
 9513 
 
 9518 
 
 9523 
 
 9528 
 
 9533 
 
 9538 
 
 1 1 
 
 223 
 
 344 
 
 90 
 
 9542 
 
 9547 
 
 9552 
 
 9557 
 
 9562 
 
 9566 
 
 9571 
 
 9576 
 
 9581 
 
 9586 
 
 1 1 
 
 223 
 
 344 
 
 91 
 
 9590 
 
 9595 
 
 9600 
 
 9605 
 
 9609 
 
 9614 
 
 9619 
 
 9624 
 
 9628 
 
 9633 
 
 1 1 
 
 223 
 
 344 
 
 92 
 
 9638 
 
 9643 
 
 9647 
 
 9652 
 
 9657 
 
 9661 
 
 9666 
 
 9671 
 
 9675 
 
 9680 
 
 1 1 
 
 223 
 
 344 
 
 93 
 
 9685 
 
 9689 
 
 9694 
 
 9699 
 
 9703 
 
 9708 
 
 9713 
 
 9717 
 
 9722 
 
 9727 
 
 1 1 
 
 223 
 
 344 
 
 94 
 
 9731 
 
 9736 
 
 9741 
 
 9745 
 
 9750 
 
 9754 
 
 9759 
 
 9763 
 
 97681 9773 
 
 Oil 
 
 223 
 
 344 
 
 95 
 
 9777 
 
 9782 
 
 9786 
 
 ,9791 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 9818 
 
 Oil 
 
 223 
 
 344 
 
 96 
 
 9823 
 
 9827 
 
 9832 
 
 9836 
 
 9841 
 
 9845 
 
 9850 
 
 9854 
 
 9859 9863 
 
 1 1 
 
 223 
 
 344 
 
 97 
 
 9868 
 
 9872 
 
 9877 
 
 9881 
 
 9886 
 
 9890 
 
 9894 
 
 9899 
 
 9903 9908 
 
 1 1 
 
 223 
 
 344 
 
 98 
 
 9912 
 
 9917 
 
 9921 
 
 9926 
 
 9930 
 
 9934 
 
 9939 
 
 9943 
 
 9948| 9952 
 
 1 1 
 
 223 
 
 344 
 
 99 
 
 9956 
 
 9961 
 
 9965 
 
 9969 
 
 9974 
 
 9978 
 
 9983 
 
 9987 
 
 9991 
 
 9996 
 
 Oil 
 
 223 
 
 334 
 
INDEX 
 
 Absolute expansion of mercury, 218. 
 
 humidity, 164. 
 
 temperature, 136, 145. 
 
 zero, 128. 
 Acceleration, angular, 78, 85. 
 
 linear, 9. 
 
 uniform, 9. 
 
 Acceleration machine, 11. 
 Adjustment of cathetometer, 109. 
 Air, correction in weighing, 173. 
 
 table of density of dry, 227. 
 Alluard, form of hygrometer, 167, 
 
 170. 
 
 Apparent expansion, 216. 
 Archimedes' principle, 172. 
 Aspirator, 160. 
 Atomic theory, 138. 
 
 weights, 144. 
 
 August, wet-and-dry bulb hygrom- 
 eter, 169. 
 Avogadro, law of, 131, 138, 143. 
 
 Balance, analytical, 116. 
 
 Mohr's, 175. 
 
 simple, 36, 37, 38. 
 
 spring, 26. 
 Barometer, 108. 
 
 corrections, table of, 229. 
 Bernouilli, work on the kinetic the- 
 ory, 106. 
 
 Black, theory of heat, 198. 
 Blow-pipe, 149. 
 
 Boiling temperature. 133, 156. 
 Boyle's Law, 105. 
 
 departures from, 107. 
 Brake, Prony, 48. 
 Brunner, the chemical hygrometer, 
 
 168. 
 
 Bunsen ice calorimeter, 208. 
 Buoyancy, correction for, 173 
 
 Cailletet, liquefaction of gases 
 
 128 
 Caliper, micrometer, 223. 
 
 vernier, 224: 
 Caloric, 198. 
 
 Calorie, definition of, 198, 203. 
 Calorimeter, ice, 208 
 
 steam, 208 
 
 Calorimetric methods, 204. 
 Calorimetry, 198. 
 
 Capillary constant, measurement of, 
 186, 194. 
 
 correction, 159. 
 
 depressions, table of, 229. 
 
 phenomena, 182. 
 
 tubes, change of level in, 185. 
 Carbon dioxide, solid, 128. 
 Cathetometer, 109. 
 Center of gravity, 34, 35, 37. 
 
 of inertia, 33, 35. 
 
 of mass, definition of, 33, 34. 
 
 of oscillation, .96. 
 Centripetal force, 100. 
 Chappuis, determinations of gas 
 
 coefficients, 126. 
 Charles's Law, 122. 
 Chemical hygrometer, 168. 
 Circular motion, uniform, 101. 
 Clausius, development of the ki- 
 netic theory, 106. 
 
 Coefficient of expansion, definition 
 of, 215. 
 
 of solids and liquids, 216. 
 
 of gases, 124. 
 
 of glass, 133. 
 
 table of, 230. 
 Coefficient of restitution, 58, 59. 
 
 of rigidity, definition of, 72. 
 
 of elasticity, table of, 226. 
 Coincidences, method of, 96. 
 
 237 
 
238 
 
 IXDEX 
 
 Component of a force, definition of, 
 
 23. 
 Composition and resolution of 
 
 forces, 21. 
 
 Compound pendulum, 95. 
 Conservation of energy, 45. 
 Constant proportions, law of, 138. 
 Constant of radiation, 204. 
 Constant-volume thermometer, 125. 
 Contact angle, 187. 
 Contractility of a surface film, 192. 
 Cooling, curves of, 210., 
 
 by evaporation, 155. 
 
 law of, 205. 
 
 method of, 206. 
 
 Correction of a mercury thermome- 
 ter, 135. 
 Curve of saturated vapor, 152. 
 
 Dalton, development of the atomic 
 theory, 138. 
 
 law of multiple proportions, 
 
 140. 
 
 Daniell, dew-point hygrometer, 166. 
 Density, of air, 114, 
 
 of dry air, table of, 227. 
 
 of carbon dioxide, 145. 
 
 of gases and vapors, 138. 
 
 of gases, table of, 144. 
 
 of a liquid, 174. 
 
 of mercury, table of, 227. 
 
 of a solid, 174. 
 
 of solids, liquids, and gases, table 
 of, 226. 
 
 of water, table of, 227. 
 Dewar, liquefaction of gases, 128. 
 Dew-point, 165. 
 
 Dew-point hygrometer, 166, 167. 
 Dimensional formulae, 84, 85. 
 Disk, inertia, 82. 
 
 Donny, superheating of water, 157. 
 Double weighing, 39, 173. 
 Dufour, superheating of water, 157. 
 Dulong and Petit, law of, 202. 
 
 method for expansion of mer- 
 cury, 218. 
 Dyne, definition of, 17. 
 
 Efficiency, definition of, 46. 
 Elasticity, 65. 
 
 coefficients of, 65. 
 
 table of, 226. 
 Energy, definition of, 4 
 
 conservation of, 45 
 
 and efficiency, 42 
 
 kinetic, 42, 43. 
 
 molecular, 129 
 
 potential, 44. 
 
 of rotation, 84 
 
 unit of, 42. 
 
 Equation of moments, 32, 37. 
 Equili brant, 33 
 
 Equivalence, principle of, 139. 
 Evaporation, 155. 
 
 Expansion, apparent and absolute, 
 216 
 
 of mercury, absolute, 218. 
 
 coefficient of, 215 
 
 coefficient of gases, 123, 126, 215. 
 
 coefficients of solids and liquids, 
 215, 218. 
 
 coefficients, table of, 230. 
 
 Faraday, liquefaction of gases, 
 
 128. 
 
 Field of force, uniform, 34. 
 Films, theory of thin liquid, 189. 
 Force, 13, 15 
 
 absolute unit of, 17. 
 
 centripetal, 100. 
 
 composition and resolution of, 21, 
 22, 23. 
 
 constant of a spring, 90. 
 
 definition of, 17. 
 
 gravitational unit of, 17. 
 
 inter molecular, 182. 
 
 table, 26. 
 Freezing mixture, 135. 
 
 "g, determination of, 95. 
 
 "gr," value of, 13 
 
 Galileo's definition of force, 15. 
 
 Gases, theory of, 106. 
 
 Gas thermometers, 123. 
 
INDEX 
 
 239 
 
 Gay-Lussac, law of, 122. 
 
 experiments on temperature of 
 
 water and its vapor, 157. 
 Glass, coefficient of expansion of, 
 
 133. 
 
 Gram, definition of, 17. 
 Gram -molecule. 201. 
 Gravitation, law of, 20. 
 Gravity, center of, 34, 35, 37. 
 Gyration, radius of, 96. 
 
 Heat capacity, definition of, 200. 
 
 latent, 199. 
 
 table of latent, 200. 
 
 mechanical equivalent of, 45. 
 
 molecular, 201. 
 
 table of molecular, 201. 
 
 definition of specific, 200. 
 
 table of specific, 201, 230. 
 
 early theory of, 198. 
 Heats, specific, variation with tem- 
 perature, 203. 
 Henry, measurement of capillary 
 
 constant, 196. 
 Hooke's Law, 66, 91. 
 Horse -power, 51. 
 Humidity, absolute, 164. 
 
 relative, 164. 
 Hydrogen, liquid, 128. 
 
 solid, 128. 
 
 thermometer, 127. 
 Hydrometer, constant-weight, 178. 
 Hygrometer, absorption, 165. 
 
 chemical, 168. 
 
 dew-point, 166, 167. 
 
 wet-and-dry bulb, 169. 
 Hygrometry, 164. 
 
 Ice calorimeter, 208. 
 Ice, latent heat of, 199. 
 Impact, elastic, 61. 
 
 inelastic, 54. 
 
 laws of, 52, 53. 
 Inertia, 78. 
 
 center of, 33, 35. 
 
 moment of, 78, 79. 
 Inertia disk, 82. 
 
 International Bureau of Weights and 
 Measures, method of meas- 
 uring linear expansion coeffi- 
 cients, 219. 
 
 Johonnott, thickness of the black 
 
 spot, 191. 
 Joule, equivalent of heat, 45, 46. 
 
 development of the kinetic the- 
 ory, 106. 
 
 relation between pressure and 
 molecular energy of a gas, 
 129. 
 definition of, 51. 
 
 Kinetic energy, 42, 43. 
 
 energy in impact, loss of, 53, 59. 
 energy of rotation, 84. 
 energy of translation, 84. 
 theory of gases, 106. 
 theory of liquids, 152. 
 theory and temperature, 127. 
 
 Laplace, formula and its deduction, 
 
 193. 
 theory of molecular pressure, 
 
 183. 
 
 normal pressure, 185. 
 Latent heat, 199. 
 table of, 200. 
 
 Lavoisier and Laplace, linear coeffi- 
 cient, 219. 
 
 LeRoy, dew-point hygrometer, 166. 
 Leslie, wet-and-dry bulb hygrom- 
 eter, 169. 
 
 Lever arm, definition of, 32. 
 Lever, law of, 31, 32. 
 
 optical, 68. 
 
 Linear expansion coefficient, 218. 
 Liquid, density of, 174. 
 Logarithms, table of, 235. 
 
 Manometer, 48, 49. 
 Mariotte's Law, 105. 
 Mass, unit of, 17. 
 
 gravitational unit of, 17. 
 
 definition of center of, 33, 34. 
 Maxwell, proof of Avogadro's Law, 
 145. 
 
240 
 
 INDEX 
 
 Mayer, conservation of energy, 45. 
 Mean free path, 153. 
 Mechanical equivalent of heat, 45. 
 Mercury, table of densities of, 227. 
 Method of coincidences, 96 
 
 of cooling, 206. 
 
 of mixture, 204, 211. 
 
 of oscillation, 117. 
 Micrometer caliper, 223. 
 Modulus of torsion, 74. 
 Modulus, volume, 66. 
 
 Young's, 67. 
 Mohr's balance, 175. 
 Molecular energy and pressure of a 
 gas, 129. 
 
 force, 182. 
 
 heat, 201. 
 
 heats, table of, 201. 
 
 pressure, 183. 
 
 pressure, relation to surface ten- 
 sion, 192. 
 
 pressure, variation with curva- 
 ture, 184. 
 
 velocity, 100, 130. 
 
 weights, 144, 201. 
 Molecule, 139, 152. 
 Moment of force, definition of, 32. 
 
 of inertia, definition of, 78, 79 
 
 of torsion, definition of, 75. 
 Moments, equation of, 32, 37. 
 
 principle of, 30, 31. 
 Momentum, 15, 43. 
 
 conservation of, 52, 53. 
 Motion, rotary, 81. 
 
 simple harmonic, 87. 
 
 uniformly accelerated, 9. 
 
 uniform circular, 101. 
 Motor, water, 48. 
 Multiple proportions, law of, 140. 
 
 Neumann, law of, 202. 
 Newton, interpretation of Third Law, 
 42. 
 
 coefficient of restitution, 58. 
 
 conservation of energy, 45. 
 
 law of cooling, 205, 211. 
 
 law of gravitation, 20. 
 
 Newton, laws of motion, 15, 17. 
 Nitrous oxide, liquid, 128. 
 Normal pressure, 185. 
 
 Olszewski, liquefaction of gases, 128. 
 Optical lever, 68. 
 Oscillation, center of, 96. 
 
 method of, 117. 
 Oxygen, liquefaction of, 128. 
 
 Parallax, error of, 12. 
 Parallelogram of forces, 22. 
 Pascal's Law, 181 
 Pendulum, compound, 95. 
 
 simple, 96. 
 
 torsion, 92. 
 
 Pictet, liquefaction of gases, 128. 
 Potential energy, 44. 
 Pressure coefficient of gases, 125, 126. 
 
 exerted by a gas, 129, 130. 
 
 law of liquid, 181. 
 Prony brake, 48. 
 Pulleys, system of, 47. 
 Pump, water, 160. 
 
 Radian, 68, 75, 79. 
 Radiation constant, 204. 
 Radius of gyration, 96. 
 
 of influence of molecular force, 
 
 183. 
 Rayleigh, measurement of thickness 
 
 of oil films, 190. 
 
 Reduction of vapor pressures, 159. 
 Regnault, experiments on Boyle's 
 
 Law, 107. 
 
 device of counterpoise in weigh- 
 ing, 115. 
 experiments on hygrometers, 
 
 166. 
 
 coefficient of expansion of mer- 
 cury, 218. 
 
 Relative humidity, 164. 
 Repulsion theory of gases, 105. 
 Resolution of forces, 21. 
 Resultant, definition of, 22. 
 of forces, 23, 24. 
 
INDEX 
 
 241 
 
 Restitution, coefficient of, 58, 59, 61. 
 Rigidity, coefficient of, 65, 71, 72. 
 Rotary motion, 81. 
 
 Rowland, experiments on specific 
 heat of water, 203. 
 
 Sanctorius, absorption hygrometer, 
 
 165. 
 
 Saturated vapor, 154. 
 Saussure, absorption hygrometer, 165. 
 Scholium to third law of motion, 29. 
 Sensitiveness of a balance, 118, 177. 
 Simple harmonic motion, 87, 90. 
 
 pendulum, 96. 
 
 Specific heat, definition of, 200. 
 table of, 201, 230. 
 variation with temperature, 203 
 Spring balance, 26. 
 Steam calorimeter, 208. 
 latent heat of, 199. 
 Stevin, proof of Archimedes' princi 
 
 pie, 172. 
 
 Substitution, principle of, 141. 
 Superheating of water, 157. 
 Surface tension, 156, 190. 
 measurement of, 195. 
 relation to molecular pressure 
 
 192. 
 
 table of, 230. 
 theory of, 190. 
 
 Table of capillary depressions o 
 
 mercury, 229. 
 
 of coefficients of expansion, 23 
 of densities of dry air, 227. 
 of densities of mercury, 227. 
 of densities of solids, liquids, an 
 
 gases, 226. 
 of densities and tensions 
 
 water vapor, 228. 
 of densities of water, 227. 
 of elastic coefficients, 226. 
 of logarithms, 235. 
 of reductions of baromet 
 
 heights, 229. 
 of specific heats, 230. 
 
 able of surface tensions, 230. 
 
 of trigonometric functions, 231. 
 emperature, absolute, 136, 145. 
 
 boiling, 156. 
 
 measurement of, 122. 
 "ension in a cord, 19. 
 
 surface, 156, 190. 
 
 measurement of surface, 19."). 
 
 theory of surface, '190. 
 
 table of surface, 230. 
 
 of saturated vapor, 158, 165. 
 hermometer, gas, 123. 
 
 corrections, 135, 158. 
 
 corrections, curve of, 159. 
 
 hydrogen, 127. 
 orsion, modulus of, 74. 
 
 moment of, 75. 
 
 pendulum, 92. 
 Trigonometric functions, table of> 
 231. 
 
 Uniform accelerated motion, 9, 10. 
 circular motion, 101. 
 field of force, 34. 
 
 Vapor pressures, reduction of, 159. 
 saturated, 154. 
 tension, 158. 
 tension, table of, 228. 
 Vaporization, 153. 
 
 Variation of specific heats with tem- 
 perature, 203. 
 Vector, 100. 
 Velocity, 21 
 angular, 84. 
 mean, 10. 
 
 molecular, 106, 130. 
 Vernier, 224. 
 
 caliper, 225. 
 
 Vinci, capillary phenomena, 182. 
 Violle, specific heat of platinum, 213. 
 Volume coefficients of solids, 217. 
 modulus, 66. 
 
 Water density, table of, 227. 
 motor, 48. 
 pump, 160. 
 
242 
 
 INDEX 
 
 Watt, definition of, 51. 
 
 Weighing, method of double, 39, 173. 
 
 with false balances, 38. 
 Weight of body in vacuo, 172. 
 Weights, atomic and molecular, 144. 
 Wet-and-dry bulb hygrometer, 169. 
 Woestyn, law of, 202. 
 Work, definition of, 29. 
 
 Work, principle of, 29. 
 
 unit of, 42. 
 Wroblewski, liquefaction of 
 
 128. 
 
 Young's modulus, 67. 
 Zero, absolute, 128. 
 
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