: 
 
THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 PRESENTED BY 
 
 PROF. CHARLES A. KOFOID AND 
 MRS. PRUDENCE W. KOFOID 
 
THE 
 
 MECHANICS 
 
 OF 
 
 LAPLACE 
 
 // 
 
 TRANSLATED WITH NOTES AND ADDITIONS 
 . BY 
 
 REV. J. TOPLIS, B.D., 
 
 LONDON: 
 LONGMANS BROWN & CO. 
 
 1814. 
 
/ "* m 
 
 THE 
 
 MECHANICS 
 
 OP 
 
 LAPLACE. 
 
 CHAP. I. 
 
 Of the equilibrium and of the composition of the forces 
 which act upon a material point. 
 
 1. A BODY appears to us to be in motion when it 
 changes its situation relative to a system of bodies 
 which we suppose to be at rest : but as all bodies, even 
 those which seem to be in a state of the most absolute 
 rest, may be in motion ; we conceive a space, bound- 
 less, immoveablc, and penetrable to raatter : it is to the 
 parts of this real or ideal space that we by imagination 
 refer the situation of bodies ; and we conceive them to 
 be in motion when they answer successively to different 
 parts of space. 
 
 The nature of that singular modification in conse- 
 quence of which bodies are transported from one place 
 to another, is, and always will be unknown : we have 
 designated it by the name of force ; and we are not able 
 
2 LAPLACE'S MECHANICS. 
 
 to determine any thing more than its effects, and tlie 
 laws of its action. The effect of a force acting upon a 
 material point is, if no obstacle opposes, to put it into 
 motion ; the direction of the force is the right line which 
 it tends to make the point describe. It is evident that 
 if two forces act in the same direction, their effect is the 
 sum of the two forces, and that if they act in a contrary 
 direction, the point is moved by a force represented by 
 their difference. If their directions form an angle with 
 each other, a force results the direction of which is a 
 mean between the directions of the composing forces. 
 Let us see what is this resultant and its direction. 
 
 For this purpose, let us consider two forces x and y 
 acting at the same time upon a material point M, and 
 forming a right angle with each other. Let % repre- 
 sent their resultant, and the angle which it makes with 
 the direction of the force x ; the two forces x and y 
 being given, the angle will be determined, as well as 
 the resultant % ; in short there exists amongst the three 
 quantities #, *,. and a relation which it is required 
 to know. 
 
 Let us then suppose the forces x, and y infinitely 
 small, and equal to the differentials dx and dy\ let us 
 suppose again that x becoming successively dx, QdX) 
 3dx> &c. y becomes dy> Qdy^ Sflfy, &c. ; it is evident 
 that the angle will be always the same, and that the 
 resultant % will become successively dz, 2dz, 3dz, &c. ; 
 therefore in the successive increments of the three forces 
 #,^, and s, the ratio of x to 2 will be constant, and 
 can be expressed by a function * of which we will re- 
 
 * Every expression in which any number of indeterminate 
 quantities enter in any manner, is called a function of the 
 indeterminate quantities. Thus # J , a*, a-\-bx, sin. x and 
 
MECHANICS. 
 
 present by $ (6) ; we shall therefore have x=z $ (S) ; 
 an equation in which x may be changed into^, pro- 
 vided that at the same time we change the angle 9 into 
 
 -- Q 9 TT being the semi-circumference of a circle whose 
 
 radius is unity. 
 
 Moreover the force x may be considered as the re- 
 sultant of two forces x 1 and #", of which the first x 1 is 
 directed along the resultant z, and the second x 11 per- 
 pendicular to it. The force x which results from these 
 two new forces forming the angle 9 with the force x' 
 
 and the angle with the force x", we shall have: 
 
 these two forces may be substituted for the force x. 
 In like manner two new forces y' and y" may be sub- 
 
 stituted for y> the first being equal to and directed 
 
 3~?/ 
 
 along s, and the second equal to -- and perpendicular 
 
 logarithm of (a-\-bx) are called functions of x; and fltf-j-^j 
 (x-\~y)*-> s i n - (<M"\-y) an( l log- (fuc+y*) are called func- 
 tions of x and y. One quantity is called an explicit function 
 of another quantity or quantities, when we directly perceive 
 how it is formed from the other quantity or quantities. 
 Thus in the expressions yax z -\-bx-{-c and yaxz-}-bx 2 -\~ 
 cz z ; we have first y an explicit function of x, and next^/ an 
 explicit function of x and z. When we do not directly see 
 how one quantity is formed from others, tut must find it by 
 an algebraical process, we call that quantity an implicit 
 function of the others. Thns in the first of the foregoing 
 equations, x is an implicit functien of y. and in the second, 
 an implicit function of y and x. 
 
& LAPLACE'S MECHANICS. 
 
 to 2 ; we shall therefore have, instead of the two forces 
 x and^y, the four following ; 
 
 a?* y z xy xy 
 
 T ' V ' T ' 
 
 the two last acting in contrary directions destroy each 
 other ; the two first acting in the same direction, when 
 added together form the resultant 2 ; we have therefore 
 
 *?-&=** ; 
 
 from which it follows that the resultant of the two forces 
 x and^ is represented in quantity by the diagonal of a 
 rectangle whose sides represent these forces. 
 
 Let us proceed to determine the angle 9. * If we in- 
 crease the force x by its differential dx y without altering 
 thf* force y, this angle will be diminished by the inde- 
 finitely small quantity d9, but it is possible to suppose 
 the force dx resolved into two, one dx 1 in the direction 
 of 2, and the other dx" perpendicular to it ; the point 
 M will then be acted upon by the two forces z-\-dx' 
 and dx" which are perpendicular to each other, and the 
 resultant of these two forces which we represent by s/ 
 
 will make with dx' 1 the angle - dQ ; we shall have 
 therefore by what precedes 
 
 the function 01 - dQ J is consequently indefinitely 
 
 * That the reasoning in the proof of the direction of the 
 resulting force may be more readily comprehended, I have 
 given a diagram, (Jig* i.J in which we may suppose Mr 
 or bz=X) Mb or xz or c==y, Ms=z 9 xa or zcdx 9 zz'=z 
 dx 1 , z'cdx", Me resultant of z+dd and dd', angle zMx 
 pr0 ; and angle zMc= dO. 
 
LAPLACE'S MECHANICS. 5 
 
 small, and of the form HO, k being a constant quantity 
 independent of the angle ; we have therefore 
 
 s' differing only by an indefinitely small quantity from. 
 %\ moreover as dx" forms with dx the angle- 9, 
 we Lave 
 
 therefore 
 
 .z- 
 
 Jf we increase the force y by o^, supposing x constant ; 
 we shall have the corresponding variation of the angle 
 
 6 by changing x intoy,^ into x, and into - 9, in 
 
 the preceding equation, which then gives dQ= '- ; by 
 
 k,z 
 
 making x and y vary at the same time, the whole vari- 
 ation of the angle will be *~ ? and we shall have 
 
 K % 
 
 By substituting for s 2 us value x^-^y* and* integrating 
 
 * The integral or fluent of the quantity 
 
 may easily be found by substituting ux for y, and udx-\~xdu 
 for dy, which gives 
 
 du uxdr _ du _ 
 u z x*~~ ~~l+7 2 ~~ 
 therefore an arc whose tang. is w,is equal to kQ+( , consequent. 
 
 ly MZZ- = tang. (kQ+ p). In the above it is hardly necessary 
 
 to observe, that dx, dy, du, and dQ represent #, y 9 u, and d, 
 the fluxions of .r, ?/, u, and 9. 
 
6 LAfctACE'S MECHANICS. 
 
 we shall have -= tang. (kQ-\-p) y p being a constant 
 
 quantity. This equation being combined with that of 
 x*-\-y*=z* ', gives jr=n~. cos. (k6-}-p)* 
 
 It is now only required to find the two constant quan- 
 tities k and p ; but if we suppose y to vanish, then evi- 
 dently s=jr, and 9=0 ; therefore cos. f=l,and x=z. 
 cos. k9. If we suppose x to vanish, then s=y and 
 Q=:7r' cos. kQ being then equal to nothing. A: ought 
 to be equal to 2n-{-l 9 n being a whole number, and in 
 this case x will vanish whenever 6 shall be equal to 
 
 it 
 - ; but x being nothing, we have evidently 6- f ?r - 
 
 therefore 2w-j-l=l or n-=0, consequently 
 
 x=.z. cos. 0. 
 
 From which it follows, that the diagonal of the rect- 
 angle constructed upon the right lines which represent 
 (he two forces .rand ^, represents, not only the quantity, 
 but likewise the direction of their resultant. In like man- 
 ner we are able for any force whatever to substitute two 
 other forces, which form the sides of a rectangle having 
 ihat force for the diagonal ; and from thence it is easy 
 to conclude, that it is possible to resolve a force into 
 three others which form the sides of a rectangular pa* 
 rallelepiped of which it is the diagonal *. 
 
 * For if MA (fig. %.) represent any force, it may be 
 resolved into two others, MB and JMF, by means of the 
 rectangular parallelogram MBAF, also MF may in like 
 manner be resolved into the forces MG and ME 9 by form, 
 ing the rectangular parallelogram MGFE ; then if a paral- 
 lelepiped be constructed having MEFG and MB for its 
 base and altitude, the force represented by its diagonal MA 
 will have been resolved into three other forces represented 
 in quantity and directiom by its three edges MG 9 ME, and 
 MB. These three lines arc called the co-ordinates of the 
 line 
 
LAPLACE'S MECHANICS. 7 
 
 Let therefore , #, c be the three rectangular co- 
 ordinates of the extremity of the right line, which re- 
 presents any force whatever, of which the origin is that 
 of the co-ordinates; this force will be represented by 
 
 the function 1/a 2 4"^ 2 4~ c % and by resolving it parallel 
 to the axes of 0, &, and c, the partial forces will be re- 
 spectively represented by these co-ordinates. 
 
 Let a', &', c f be the co-ordinates of a second force ; 
 -}-' j b-\-b' 9 c-j-c' will be the co-ordinates of the re- 
 sultant of the two forces, and will represent the partial 
 ones into which it can be resolved parallel to three 
 axes ; from which it is easy to conclude, that this re- 
 sultant is the diagonal of the parallelogram constructed 
 upon the two forces. 
 
 In general , b, c ; a', b f , c 1 '; &c. being the co-or- 
 dinates of any number whatever of forces ; a-\-a!-\-a ff 
 &c. ; b-{-b'-{-b /l &c ; c+c'+c" &c. will be the co-or- 
 dinates of their resultant ; the square of which will be the 
 sum of the squares of these last co-ordinates ; we shall 
 therefore, by this means, have both the magnitude and 
 'the position of the resultant. 
 
 2. From any point whatever of the direction of a 
 force S, which point we shall take for the origin of this 
 force, let us draw to the material point M a right line 
 which we will call s ; let JT, y, and % be the three rect- 
 angular co-ordinates, which determine the position of 
 the point M 9 and , by and c the co-ordinates of the 
 origin of the force ; we shall then have * 
 
 * In (f l > 3 J let Ax, Ay, and Az represent the three 
 rectangular co-ordinates of #, #, and 2, and MS the line s- 9 
 fron? the points S and M let fall the perpendiculars Sm and 
 MN upon tye plane ,yAx 9 join m and JV, draw SB. pcrpendi. 
 
If we resolve the force S parallel to the axes of r, of 
 4 y, and of z ; the corresponding partial forces will be by 
 
 the preceding n. S. , S. 1 ^, and S. , * 
 
 cular to MN, from m and JV, in the plane yAx, draw the 
 perpendiculars mP and AT to ^, from m draw wzQ per. 
 pendicular to NT ; then because Sm and .MJV are perpen- 
 dicular to the same plane, they are parallel to each other ; 
 also as mN meets M.N in the plane yAx 9 it is perpendicular 
 to it, and parallel and equal to SR, as is Sm to RN ; in 
 the rectangular figure PTQm we have PT=.mQ, and Pm=. 
 TQ. In *he figure SM=s 9 Sm=RN=c, MN=z, MR 
 r=MNNR=zc, AT==y, mPTQ=b, NQ=NT 
 TQyb, ATx, APa,PTmQ:=ATAP=xa, 
 and as MRS is a right angled triangle 
 
 but SR z =mN z =, as the triangle mQN is right angled, 
 *= PT Z +QN*, therefore 
 
 or by substitution, * < (x a)*+(y b} 2 +(z c) 2 . If 
 S coincides with A, then a, 6, and c vanish, and 
 
 Let a, /3, and 7 respectively represent the angles which 
 5 in this case makes with the axes of #, y, and s, then it is 
 evident from fig. 2. in which we may suppose MA~s 9 
 MG=x, MEy, and MB=z, that we have the following 
 proportion 5 : x : : rad. (1) : cos. <z, consequently 
 
 cos. -; in like manner cos. -, and cos. yzr-; if 
 
 5 S S 
 
 these cosines be substituted for their values in the equation 
 
 it will be changed into the following, 
 
 * By the preceding number s : x a : : S : ; 
 
 or the force in the direction of the axis x. 
 
LAPLACE'S MECHANICS 
 
 or a. I r 
 
 and ( - - I denoting, according to the received nofa- 
 
 \d Zj 
 
 tion, the co-efficients of the variations $x 9 y, and Jg, 
 in the variation of the preceding expression of s. 
 
 If, in like manner, we name s 1 the distance of M from 
 
 any point in the direction of another force S', that point 
 
 ^^ s i^ 
 being taken for the origin of the force ; S' I r I \vili 
 
 be this force resolved parallel to the axis of #, and so 
 on of the rest ; the sum of the forces S, S', S /7 , &c. re- 
 
 solved parallel to this axis will therefore be 2. S.( : J 
 the characteristic 2 of finite integrals denoting here, 
 the sum of the terras S. (^), S'. Q, &c. 
 
 Let Fbe the resultant of all the forces 5, S ; , &c., 
 and w the distance of the point J/from a point in the 
 
 S$ s\ f% s\ f^s\ 
 * The expressions t J, ^ J, ^J enclosed be. 
 
 tween parentheses, represent the co-efficients of the partial 
 ilifferentiations of the equation 
 
 s= (x a 
 
 taken by making Xy y^ an d z rary separately ; thus differ. 
 entiating the equation, by supposing y and s constant, we 
 
 obtain fc = ^ therefore^ ^= , in like man. 
 
 t \oxJ s 
 
 1 ^~ 1^3-^ - and f | ^ C : these expressions 
 
 \oyJ s \$zj s 
 
 are evidently equiralent to the co-sines of the angles which 
 the line s makes with the co-ordinates x, y^ and z respec- 
 tively. 
 
10 LAPLACE'S MECHANICS. 
 
 direction of this resultant, which is taken for its origin ; 
 F". ( , ] will be the expression of this resultant re- 
 go'ved parallel to the axis of x ; we shall therefore havt 
 by <he preceding number F. ( ^J=2 S. ( r~ ) 
 
 We shall have in like manner, 
 
 from which we may obtain, by multiplying these three 
 equations respectively by S.r, Sy, and &s, and then add- 
 ing them together, 
 
 F. $u 2. S. Ss ; . . . r>) 
 As this last equation has place, whatever may be the 
 variations &r, 5y, and 5s, it is equivalent to the three 
 preceding ones. 
 
 If its second member is an exact variation of a func- 
 tion p, we shall have F. &?/=&<, and consequently, 
 
 that is to say, the sum of all the forces S, S 7 , &c. re- 
 solved parallel to the axis oi x is equal to the partial 
 
 differential ( r - V This case generally takes placa 
 
 when these forces are respectively functions of the dist- 
 ance of their origin from the force M. In order to have 
 the resultant of all these forces resolved parallel to any 
 right line whatever, we shall take the integral 2.f.S$$, 
 and naming it p, we will consider it as a function of x, 
 and of two other right lines perpendicular to x and to 
 
 ech other; the partial differential ( r-J will then be 
 
LAPLACES MECHANICS. It 
 
 the resultant of the forces S, S', &c. resolved parallel to 
 the right line x.* 
 
 * The following expressions of the equilibrium of a point 
 follow from the above equations. Suppose that the powers 
 are represented both in magnitude and direction by &, S' 9 
 S", &c. whose directions form the following angles, 
 with the axis of x ... a, a', a", . . 
 with the axis of,y . . . /3, /3'. /3", . . . 
 with the axis of z ... y+ y', y", 
 
 By resolving each of these forces into three others whose di. 
 rections are parallel to the axes, we shall have for the com. 
 posing forces parallel 
 
 to 0? . . S. COS. #, 5'. COS. a', S". COS. a", &C. . 
 
 toy . . S. cos. /3, S'. cos. /3', S". cos. /3", &c. . 
 
 to Z . . S. COS. y, 5'. COS. y', S". COS. y", &C. . 
 
 Each of these three collections of forces is equivalent to a 
 single force, equal to their sum^ because these components 
 are directed in 'the same right line. Naming P. (2, and R 
 the three forces respectively parallel to x^ y^ and z 9 we 
 have 
 
 P S.cos 
 
 lil=6\COS.y-fS >/ .COS.y / -f.^".COS.y"-|-&C. 
 
 Let , 6, and c represent the unknown angles which the 
 direction of the resultant V forms with the three axes ; 
 F. cos. a, V. cos. 6, V. cos c will be its components in the 
 directions of the axes ; we shall have therefore V. cos, u P 3 
 F. cos. 6m Q, and F. cos. cR. If we add the squares of 
 these equations together, remembering that cos.*a-\-cos. 2 b-{- 
 cos. 2 c 1, we shall obtain F 2 = P*+Q*+R^ which gives 
 
 V\/( P 2 -f Q 2 +# Z J ; the direction of the resultant may 
 be obtained from the equations 
 
 cos. a , cos. b t=jr, cos. c nr-p. 
 These equations determine both the magnitude and the di- 
 
19 LAPLACfi'S MECHANICS. 
 
 3. When the point M is in eqnilibrio by the action 
 of all the forces which solicit it, their resultant is no- 
 thing, and the equation (a) becomes 
 
 S. S. 3s ; (b) 
 which shews, that in the case of the equilibrium of a 
 
 point acted upon by any number whatever of forces, 
 the sum of the products of each force by the element of 
 its direction is nothing. 
 
 If the point M is forced to be upon a curved surface, 
 it will experience a re-action which we shall denote by 
 JR. This re-action is equal and directly contrary to the 
 pressure with which the point presses upon the surface ; 
 for by supposing it acted upon by two forces, R and 
 /L, it is possible to conceive, that the force R is 
 destroyed by the re-action of the surface, and that the 
 point M presses upon the surface with the force R / 
 but the force of pressure of a point upon a surface is 
 perpendicular to it, otherwise it would be possible to 
 resolve <the force into two, one perpendicular to the sur- 
 face, which would be destroyed by it, the other pa- 
 rallel to the surface, in consequence of which the point 
 would have no action upon it, which is contrary to the 
 
 rection of the resultant F, which is evidently the diagonal 
 of the parallelepiped constructed upon P, Q, and R. If 
 the system be in equilibrio, it is manifest that each collection 
 of forces parallel^ the axes should likewise be in equilibrio, 
 which gives 
 
 Pr=0, Q=0, R0. 
 
 With respect to the signs of the components S. cos. , 
 S', cos. a/, c., it may be observed that those which tend 
 to increase the co-ordinates should be reckoned positive, 
 and those which act in a contrary direction negative. 
 
LAPLACE'S MECHANICS. 13 
 
 supposition ; naming * r therefore the perpendicular 
 drawn from the point 3/at the surface, and terminated 
 in any point whatever of its direction, the force R will 
 be directed along this perpendicular; it will be neces- 
 sary therefore to add R.$r to the second member of (lie 
 equation (b) which will become 
 
 0=2. S. $s+R. Jry (c) 
 
 7? being then the resultant of all the forces S, S 7 , &c 
 it is perpendicular to the surface. 
 
 If we suppose that the arbitrary variations &r, y ? 
 and 2 appertain to the curve surface upon which the 
 point is forced to remain, we shall have, by the nature 
 of the perpendicular to this surface, SrzzzO, which makes 
 A 1 , or vanish from the preceding equation : the equa- 
 tion (b) has place therefore in this case, provided that 
 we extract one of the three variations r, 5y, and o^,by 
 means of the equation to the surface ; but then the 
 equation (b) which in the general case is equivalent to 
 three, is not equivalent to more than two distinct equa- 
 tions, which we may obtain by equalling separately t 
 nothing the co-efficients of the two remaining differ- 
 entials, t Let u-=D be the equation of the surface- 
 
 * In (Jig. 4.) let a point be in equilibrio at 37, on the 
 curve AMB 9 by means of the forces MP 9 J\lQ y and the re. 
 action of the curve ; then if AJR 9 supposed perpendicular to 
 the curve at Af, be the resultant of the forces MP, and 3/Q, 
 it will represent the pressure of the point upon the curve, 
 and if RM be produced tor, and 3/r Af/v, then Mr will 
 represent the re-action of the curve upon the point, which 
 may be supposed in equilibrio in consequence of the force? 
 MP, MQ, and Mr. 
 
 t The nature of a surface may be determined by three rect- 
 angular co-ordinates, as that of a line may by two ; thus fet 
 
J4 LAPLACE'S MECHANICS. 
 
 the two equations SrzziO and $u=.Q will have place at 
 the same time, this requires that r should be equal to 
 f N being a function of.r, y, and z. Naming a, 
 
 u~0 be an equation to a surface, and let its co-ordinates or, 
 ^, and z be respectively measured upon the lines AX^ AY^ 
 and AZ (jig. 5.) ; if the values of x and y are given and 
 represented by a and #, by taking on the axes of a: and,y 
 APa, 4Q=b, and drawing the parallels QAP, PM' to 
 these axes, the point M' 9 which is the projection of the point 
 M of 1 he surface upon the plane of#, ?/, will be determined ; 
 the equation by substitution will then give the corresponding 
 value of z, which determines the length of the co-ordinate 
 MM, and, consequently the point M of the surface. 
 
 Curves of double curvature are formed by the intersection 
 of two surfaces. Thus, let the equations to two surfaces be 
 represented by F(oc, ?/, zJO, and/fa?, #, zJrzO, then the 
 curve of double curvature formed by their intersection will 
 have the same co-ordinates \ if therefore the variable x be 
 extracted by means of these equations, the resulting one will 
 represent the projection of the curve of double curvature 
 upon the plane of 3/2; in like manner, if y had been ex. 
 tracted, the resulting one would have been that of the pro- 
 jection upon the plane of are; and if z had, that of the pro. 
 jection upon the plane of xy. The resulting equations 
 likewise represent cylindrical surfaces elevated upon these 
 projections respectively perpendicular to the planes of the 
 co-ordinates, and the curve of double curvature will be the 
 intersection of any two of these surfaces. 
 
 Those who are desirous of being acquainted with the pro. 
 perties of surfaces, or of curves of double curvature, or of 
 lines supposed in space, and considered with reference to 
 their projections upon three planes at right angles to each 
 other, may consult the Traite' du Calcul Differentiel et 
 Integral par S. F. Lacroix, the Application de 1'Algebre a la 
 Geometric par MM. Monge et Hachette, &c. 
 
LAPLACE'S MECHANICS. 15 
 
 ft, and c the co-ordinates of the origin of r, we shall 
 have to determine it 
 
 from which we may obtain 
 
 1 5 and consequently 
 
 by making therefore 
 
 the term R. Sr of the equation ( c) will be changed info 
 X&w, and this equation will become 
 
 0=2. S. Ss+x. *w / 
 
 in which we ought to equal separately to nothing the 
 co-efficients of the variations ojc, y, and 2, which gives 
 three equations ; but they are only equivalent to two 
 between x^y, and s, on account of the indeterminate 
 quantity X which they contain. We may therefore 
 instead of extracting from the equation (b) one of the 
 variations &x, ^y s or Ss, by means of the differential 
 equation to the surface, add to it this equation multi- 
 plied by an indeterminate quantity X, and then con- 
 sider the variations for, 2>y, and $z as independant quanti- 
 ties. This method, which likewise results from the theory 
 of elimination, unites to the advantage of simplifying 
 the calculation, that of making known the pressure 
 R with which the point JJ/acts against the surface*. 
 
 * Let the point be supposed in equilibrio upon a surface 
 whose equation is M 0, and let the forces ,9, ', &", &c. be 
 reduced to three P, Q 9 and R acting in the directions of the 
 
16 LAPLACF/S MECHANICS. 
 
 Supposing this point to be contained in a canal. of 
 simple or double curvature, it will prove on the part 
 
 three rectangular co-ordinates ; then the sum of the moment* 
 P.&c-l-Q.fy+jR.S* will be equivalent to S$s+S'W+S"Ss" 
 -f-&c. and by adding %u, multiplied by the indeterminate 
 quantity A, the equation of equilibrium becomes 
 
 P.S*-f Q.fy+U.fcf-f-xSMiz: ; 
 
 but u being a known function of x 9 y t and z 9 we shall have 
 l>y differentiation 
 
 %u\ f$u\ . f$u\ 
 
 ^ I, I 1, ana I * I representing the co-efficients of 
 
 &e, 5y, and ?. By substituting this value of $u in the pre. 
 ceding eqnation, it becomes 
 
 which gives, by equalling separately to nothing each sum of 
 the terms multiplied respectively by ^JF, fy ? and oz 9 the three 
 following equations 
 
 e+ ,(L ;)=0 
 
 from which, by extracting A, we shall obtain the two follow 
 ing equations 
 
 that contain the conditions of the equilibrium of a ppint 
 upon a surface. 
 
17 
 
 of the canal, a re-action which we shall denote by /;, 
 that will be equal and directly contrary to the pressure 
 with which the point acts against the canal, the direc- 
 tion of which will be perpendicular to its side; but the 
 curve formed by this canal is the intersection of two 
 
 In the case of a point acted upon by certain forces, the 
 conditions of its equilibrium upon a surface may be found 
 with more ease, by directly substituting in the equation 
 
 the value of oz obtained from the differential equation 
 
 of the surface, and then equalling separately to nothing the 
 co-efficients of the differentials <$x and S^. By this method 
 we shall immediately get th* 1 equations 
 
 P R. 
 
 which are equivalent to the equations found by the other 
 method. 
 
 In like manner, if a body is forced to be upon a line of a 
 given description, determined by the two differential equa- 
 tions Jyzrp&e, SzqSx of the projections of the line upon the 
 planes of xy and a:*-, we have only to substitute these values 
 of ly and $2, in the equation P.&e-f-Q.fy-f/OzzzO, which, 
 on being divided by &r, gives the equation, 
 
 for the condition of equilibrium 
 
18 LAPLACE'S MECHANICS. 
 
 surfaces, of which the equations express its nature ; we 
 may therefore consider the force k as the resultant 
 of the two forces 7? and R', which re-act upon the 
 point jl/from the two surfaces; moreover as the di- 
 rections of the three forces /?, R l , and k are perpendi- 
 cular to the side of the curve, they are in the same 
 plane. By naming therefore or and r' the elements of 
 the directions of the forces R and R', which directions 
 are respectively perpendicular to each surface, it will 
 be necessary to add to the equation (b) the two terms 
 R.$r and R'3r' which will change it into the following 
 Q==S,.S3s+R3r+R'. V. (d) 
 
 If we determine the variations .r, Sy, and $z so that 
 they may appertain at the same time to two surfaces, 
 and, consequently, to the curve formed by the canal ; 
 r and r' will vanish, and the preceding equation will 
 be reduced to the equation (b) y which therefore has 
 place again in the case where the point M is forced to 
 move in a canal ; provided, that by means of the two 
 equations which express the nature of the canal, we 
 make two of the variations S v r, Sy, and $z to disappear. 
 
 Let us suppose that w=0 and u'=D are the equations 
 of the two surfaces, whose intersection forms the canal, 
 If we make 
 
 R 
 
 and 
 
 R 1 
 
 X' -- 
 
 /()'<)'+(' 
 
 the equation (d) will become 
 0=2. S. 
 
LAPLACE'S MECHANICS. 
 
 an equation in which the co-efficients of each of the 
 variations &r, ?/, and Ss will be separately equal to no- 
 thing; in this manner three equations will "be obtained, 
 by means of which the values of X and X 1 may be deter- 
 mined, which will give the re-actions R and R' of the 
 two surfaces ; and by composing them we shall have 
 the re-action 7; of the canal upon the poi: t JIT, and, 
 consequently, the pressure with which this point acts 
 against the canal. This re-action, resolved parallel to 
 the axis of jc is equal to 
 
 the equations of condition z/zzrJ), and w'zzzO, to which 
 the motion of the point M is subjected, express, there- 
 fore, by means of the partial differentials of functions, 
 which are equal to nothing because of these equations, 
 the resistances which act upon this point in consequence 
 of the conditions of its motion. 
 
 It appears from what precedes, that the equation (b) 
 of equilibrium has generally place, provided, that the 
 variations o.r, Sy, and Iz are subjected to the conditions 
 of equilibrium,, This equation may therefore be made 
 the foundation of the following principle. 
 
 If an indefinitely small variation is made in the posi- 
 tion of the point My so that it still remains upon the 
 surface, or the curye along which it would move if it 
 were not entirely free ; the sum of the forces which t so- 
 licit it, each multiplied by the space which the point 
 moved in its direction, is equal to nothing in the case 
 of equilibrium *. 
 
 * Let the forces S, S", S" 9 &c. be supposed to act npon 
 the point M in the directions of the Hues s, s', s", &c. r. 
 
20 LAPLACE'S MECHANICS. 
 
 The variations &r, y, and $z being supposed arbi- 
 trary and independent, it is possible in the equation 
 
 spectively drawn from that point to the origins of these 
 forces; let V iHp.-esen; rheir resuKant, aod u a line drawn 
 from the point M in us direction ; also, let an) line AiN 9 
 be supposed to be drawn trom the point A/, and each of *he 
 forces V* Si &', &c. to ue resolved into two others, one in 
 the direction of this line, and the other perpendicular to it. 
 Because Fis the resultant of a.11 the other forces, its compo- 
 nent along the line MN will hf equal to the sum of the com- 
 ponents of the other forces along the same Jine ; let m, , 
 ', ", &c. denote the angles which the directions of the 
 forces V) , S', &", &c. respectively make with the line 
 ) we shall then have the following equation, 
 
 Let any point N be taken upon the line MJV, then if this 
 line be represented by 6, and its respective projections upon 
 the lines wz, $, *', s", &c. or their continuations by AM, 
 A*, A*', A*", &c. we shall have 
 
 A// 6. cos. z, A^zz&.cos.fl, A-y'm&.cos.a', &c. 
 If both sides of the preceding equation are multiplied by &, 
 it will, by substitution, become 
 
 V AwzuS.As-f S.'A*'-f ".A5-"-f &c. 
 
 If the point M be supposed to move to*the point JV, and the 
 line MN be regarded as representing the virtual velocity of 
 this point, the quantities AM, As, As', &c. will denote the 
 virtual velocitie^ of 'his point in the directions of the forces 
 V. S, S 1 , S", &c. ; the last equation therefore shews, that the 
 product of the resultant of any number of forces applied to 
 the same pbint. by the virtual velocity of this point estimated 
 in its direction, is equal to the sum of the products of these 
 forces by their re-pective virtual velocities, estimated in the 
 directions of the forces. It is not absolutely necessary that 
 
LAPLACE'S MECHANICS. 51 
 
 (a) to substitute for the co-ordinates r, y and * three 
 other quantities which arc functions of them, and to 
 equal the variations of these quantities to nothing. 
 
 Thus naming p the radius drawn from the origin of 
 the co-ordinates to the projection of the point M upon 
 the plane of x and y^ and CT the angle formed by p and 
 the axis of x, we shall have 
 
 By considering therefore in the equation (a), U 9 s, s', 
 &c. as functions of/?, ro, and ~, and comparing the co- 
 efficients of Jw, we shall have 
 
 pfL?V-2 sf V 
 
 ( j is the expression of the force F resolved ia 
 
 the direction of the element p.tizs. Let V be this force 
 resolved parallel to the plane of .r. and y, and p the per- 
 pendicular let fall from the axis of s, upon the direction 
 
 the virtual velocities should be supposed indefinitely small, 
 if they are, the last equation becomes 
 
 In the case of equilibrium, we have J^mO, and consequent]? 
 
 o = s.fc+s / .a*'+,& i 'V, &c. 
 
 If the point is forced to remain upon a given curve orsur^ 
 face, this Aquation will be proper, the virtual yelocities $s 9 
 $.?', s", &c. being supposed indefinitely small, and S^nrO in- 
 stead of F. 
 
 The products S.$S) S'.$s f , S".W, &c. are called by some 
 authors the moments of the powers S, S 1 , S", &c. ; and their 
 sum being equal to nothing, which has place not only for one 
 point, but, as will be proved hereafter, for any system what. 
 ever in equilibrio, is called the principle of virtual ve 
 loci ties. 
 
LAPLACE'S MECHANICS. 
 
 of V 1 parallel to the same plane ; - will be a second 
 
 p 
 
 expression of the force V resolved in the direction of 
 the element pfa *; we shall therefore have 
 
 If we suppose the force V 1 to be applied to the extremity 
 of the perpendicular p, it will tend to make it turn about 
 the axis of 3 ; the product of this force by the perpen- 
 dicular is what is called the moment of the force V with 
 respect to the axis of s \ this moment is therefore equal 
 
 to V.( ~- 1 ; and it appears from the equation (e) 9 
 
 \diyj 
 
 that the moment of the resultant of any number what- 
 ever of forces is equal to the sum of the moments of 
 these forces 
 
 * Let MB (fig* 6.J represent F', or the force V resolved 
 along a plane parallel to that of xy 9 let be the point where 
 this plane cuts the axis of s, join OM 9 then OM will be pa- 
 rallel and equal to p , draw OA or p perpendicular to V or 
 MB produced, also BD perpendicular to OM; then, as the 
 right angled triangles MBD, MO A have a common angle at 
 M, they are similar, consequently 
 
 M0( f ) : OA(p) : : MB(V') 
 
 P 
 
 but the line DB represents the force V resolved in the di- 
 rection fift perpendicular to p ? therefore that force is repre- 
 
 pV 
 seated by ~ , 
 
 f 
 
LAPLACE'S MECHANICS. 
 
 CHAP. II. 
 
 Of the motion of a material point. 
 
 4. A POINT at rest is not able to give itself any mo- 
 tion, because it does not contain within itself any cause 
 why it should move in one direction rather than another. 
 When it is solicited by any force whatever, and after- 
 wards left to itself, it will move constantly in an uniform 
 manner in the direction of that force, if not opposed by 
 any resistance. This tendency of matter to persevere 
 in its state of motion or rest, is called its inertia. This 
 is the first law of the motion of bodies. 
 
 That the direction of the motion is a right line fol- 
 lows evidently from this, that there is not any reason 
 why the point should change its course more to one 
 side than the other of its first direction : but the cause 
 of the uniformity of its motion is not so evident. The 
 nature of the moving force being unknown, it is impos- 
 sible to discover a priori it it ought to continue with- 
 out ceasing. In fact, as a body is incapable of giving 
 itself any motion, it appears equally incapable of alter* 
 in<j that which it has received : so that the law of 
 
2k LAPLACE'S MECHANICS. 
 
 inertia is the most simple and natural which it is pos- 
 sible to imagine ; it is also confirmed by experience, 
 for we observe upon the earth that the more the ob- 
 stacles are diminished, which oppose the motions of 
 bodies, the longer these motions are continued ; which 
 leads us to believe, that if the obstacles were removed 
 they would never cease. But the inertia of matter is 
 most remarkable in the motions of the celestial bodies, 
 which have not during a great number of ages experi- 
 enced any perceptible alteration. Thus we may regard 
 the inertia of bodies as a law of nature, and when we 
 shall observe any alteration in the motion of a body, we 
 will suppose that it is owing to the action of a different 
 cause. 
 
 In uniform motion the spaces gone over are in pro- 
 portion to the times, but the time employed in describ- 
 ing a given space, is longer or shorter according to the 
 magnitude of the moving force. These differences have 
 given rise to the notion of velocity, which, in uniform 
 motion, is the ratio of the space to the time passed in 
 going over it: thus, s representing the space, t the 
 
 time, and v the velocity, we have v=.-. Time and 
 
 space being heterogeneal, and consequently, not com- 
 parable quantities, a determinate interval of time is 
 chosen, such as a second for an unit of time; in like 
 manner, some unit of space is chosen, as a metre; and 
 then space and time become abstract numbers, which 
 express how often they contain the units of their species, 
 that are tli us rendered comparable to each other. By 
 this means the velocity becomes the ratio of two ab- 
 stract numbers, and its unity is the velocity of a body 
 which passes over the space of a metre in one second. 
 5. The force being only known by the space which 
 
LAPLACE'S MECHANICS. 25 
 
 it causes a body to describe in a given time, it is natural 
 to fake this space for its measure ; but this supposes that 
 many forces acting in the same direction, should cause 
 the body to pass over a space equal to the sum of the 
 spaces which each of them would have made it go 
 over separately; or what comes to the same, that the 
 force is proportional to the velocity. This is what we 
 are not able to know a priori, owing to our ignorance 
 of the nature of the moving force ; it is therefore ne- 
 cessary to have again recourse to experience on this 
 occasion, for all that which is not a necessary conse- 
 quence of the little which we know respecting the na- 
 ture of things, must be to us but a result of observation*. 
 
 * Mr. Knight, in the ninth No. of the Mathematical Re- 
 pository, has attempted to prove the law of the proportion, 
 ality of the force to the velocity, by supposing two straight 
 lines at right angles to each other, to represent the magni- 
 tudes and the directions of two forces, and taking parts from 
 these lines, measured from their junction, to represent the 
 velocities which these forces would respectively cause ; and 
 bv completing two parallelograms, one about the lines de- 
 noting the forces, and the other about those denoting the 
 velocities, and drawing diagonals in each of them represent- 
 ing the respective resultants of the forces and the velocities ; 
 he has shewn that if the diagonals are in the same right line 
 the parallelograms will be similar, and consequently the 
 forces and the respective velocities proportional ; if they 
 are not, it must be supposed that the resultant of the forces 
 has caused a motion in a different direction to its own, which 
 is absurd. 
 
 Mr, Knight has, T think, proved that the force varies as 
 the velocity, if it be taken for granted, that the proofs re. 
 
26* LAPLACE'S MECHANICS. 
 
 Naming t> the velocity of the earth, which is common 
 to all the bodies upon its surface, let /"be the force by 
 which one of these bodies M is actuated in consequence 
 of this velocity, and let us suppose that 7izif<p (f) is the 
 relation which exists bet ween the velocity and the force; 
 (p (f) being a function of/ which it is necessary to de- 
 termine by experiment. Let a, h, and c be the three 
 partial forces into which the force / may be resolved 
 parallel to three axes which are perpendicular to each 
 other. Let us then suppose the moving b >dy M to be 
 solicited by a new force/ 7 , which may be resolved into 
 three others ', b', and c' parallel to the same axes. 
 The forces by which this body will be actuated in the 
 directions of these axes, are a-{-a' 9 &-|-//, and c-f-c'/ 
 naming JFthe sole resulting force, it will become, from 
 what precedes, 
 
 If the velocity corresponding to F be named U; 
 ~rr - will represent this velocity resolved pa- 
 
 specting the composition arid the resolution of forces, and 
 those respecting the composition and the resolution of velo. 
 cities, are satisfactorily demonstrated independent of each 
 other. 
 
 * If U represents the velocity of the body corresponding 
 to F, we shall find that part of it relative to the axis of a-\-a r 
 by the proportion 
 
 tc+c 1 ) 2 : +' : : U : 
 (a+a')U 
 
 V a-l-tt y--t-( wfw y--f-( G-J-CV 
 but JF f =1/faH-a / J a +r*+* / J a 4-('cH-c / JS therefore, hj 
 
 Substitution the expression becomes- .-'^ > -> 
 
 r< 
 
LAPLACE'S MECHANICS. 27 
 
 rallel to the axis of a; also the relative velocity of the 
 body upon the earth parallel to this axis will be 
 
 ~ 9 or (a+a').<?(F) a. $ (f). The 
 
 * 
 
 greatest forces which we are able to impress upon bodies 
 
 at the surface of the earth, being much smaller than 
 those by which they are actuated in consequence of the 
 motion of the earth, we may consider a', V, and c' as 
 indefinitely small quantities relative to /: we shall 
 
 therefore have F=f+- ^ ' I and 
 
 * If ', &', and c' are supposed indefinitely small relative 
 fo/ 3 th^ir squares and products may be neglected: the PX- 
 
 pression (a+a'y+lb+b'y+^c+cf)* will then become 
 '^c l ; let a 2 -f 6 2 -f-c 2 :rr 2 and 
 
 ^cc / = < y, then if the expression \/f z -\-y is 
 
 ex- 
 
 panded by the binomial theorem, it will become /-}- 
 
 / 
 
 & c . or ^1 & c . but as all the terms after the 
 
 / 
 
 two first of the series contain the squares, products, or 
 higher powers of a', 6', and c, they may be neglect d. 
 
 If in the expression 9 (a) we substitute x-^-k for x 9 it be. 
 
 . 
 
 7c 4 -f &c. \\Iuch gives, if (pf.rj be represented 
 
 by M, 9fjc+ ArJ = n + ~ .Ar+_^ .A.- 2 ^ - f- -- ^+ &C 
 a'.c l/J.a'.t 2 1.2.3f/^ 
 
 this is called the theorem of Taylor, and is prored a variety 
 of ways in different mathematical works. If /be substituted 
 
LAPLACE'S MECHANICS. 
 
 ; tf(f) ; ?Y/) being the 
 
 differential of <p(f) divided by df. The relative velo- 
 city of M in the direction of the axis of a, will, in like 
 manner, become 
 
 a'.<t(f)-{j.{ a a>+bb>+cc'}. Wf). 
 
 Its relative velocities in the directions of the axes b and 
 c will be 
 
 The position of the axes a,b, and c being arbitrary, we 
 may take the direction of the impressed force for the 
 axis of 0, and then b' and c' will vanish ; and the pre- 
 ceding relative velocities will be changed into the fol- 
 lowing, 
 
 does not vanish, the moving body in conse- 
 quence of the impressed force ', wilt have a relative 
 velocity perpendicular to the direction of this force, 
 provided that b and c do not vanish ; that is to say, 
 provided that the direction of this force does not coin- 
 
 for a?, and -- i_ fork in the above expression, it becomes 
 
 but as the quantities a', &', and c' are indefinitely small, the 
 products and higher powers than the irst of ^ a - . C -. 
 way be neglected. 
 
LAPLACE'S MECHANICS. 29 
 
 cide with that of the motion of the earth. Thus, sup- 
 pose that a globe at rest upon a very smooth horizontal 
 plane is struck by the base of a right angled cylinder, 
 moving in the direction of its axis, -which is supposed 
 horizontal, the apparent relative motion of the globe 
 will not be parallel to this axis in all its positions with 
 respect to the horizon : thus we have an easy means of 
 discovering by experiment if $'(f) has a perceptible 
 value upon the earth ; but the most exact experiments 
 have not shewn in the apparent motion of the globe any 
 deviation from the direction of the impressed force ; 
 from which it follows that upon the earth, <p'(f) is very 
 nearly nothing. If its small value were perceptible, it 
 would particularly be shewn in the duration of the oscil- 
 lations of the pendulum, which would alter as the posi- 
 tion of the plane of its motion differed from the direction 
 of the motion of the earth. As the most exact observations 
 have not discovered any such difference, we ought to 
 conclude that Q'(f) is insensible, and may be supposed 
 equal to nothing upon the surface of the earth. 
 
 If the equation q>'(f)=0 has place, whatever the 
 force f may be, q(f) will be constant, and the velocity 
 will be proportional to the force ; it will also be pro- 
 portional to it if the function Q(f) is composed of only 
 one term, as otherwise $(f} would not vanish except/" 
 did : it is necessary, therefore, if the velocity is not 
 proportional to the force, to suppose that in nature the 
 function of the velocity which expresses the force is 
 formed of many terms, which is hardly probable ; it is 
 also necessary to suppose that the velocity of the earth 
 is exactly that which belongs to the equation <p f (f}=Q, 
 which is contrary to ail probability. Moreover the ve- 
 locity of the earth varies during the different seasons of 
 the year ; it is about one thirtieth part greater in winter 
 
SO LAPLACE'S MECHANICS. 
 
 than in summer. This variation is still more consider- 
 able, if, as every thing appears to indicate, the solar 
 system be in motion in space ; for according as this pro- 
 gressive motion agrees with that of the earth, or is ron- 
 trary to it, (here should result during the course of the 
 year very great variations in the absolute motion of 
 the earth, which would alter the equation that we nrc 
 considering, nnd the relation of the impressed force to 
 the absolute velocity \vhich results, if thisequation and 
 this ratio were not independent of the motion of the 
 earth ; nevertheless, the smallest difference has not been 
 discovered by observation. 
 
 Thus we have two laws of motion, the law of inertia, 
 and that of the force being proportional to the velocity, 
 which arc .given from experience. They are the most 
 natural and the most simple which it is possible to 
 imagine, and are without doubt derived from the na- 
 ture itself, of ,maitGr ; bat this nature being unknown, 
 they are with .respect to us solely the consequences of 
 observation, ancLthconly ones which the science of me- 
 chanics requires ;from experience. 
 
 6. As the velocity is proportional to the force, these 
 two quantities, may be represented by each other, and 
 all that! has been previously established respecting the 
 composition of forces, , may be applied to the composi- 
 tion of velocities^*. It therefore results that the rela- 
 
 * Let I/, u", and v'" represent the uniform velocities im- 
 pressed upon^a body in the directions of three rectangular 
 coordinates, #, ^, and s, the, spaces respectively passed over 
 in the time t in consequence of them ; we shall then have the 
 three following equations 
 
 y~v"t, z^tf'f, 
 
LAPLACE'S MECHANICS. 31 
 
 live motions of a system of bodies actuated by any 
 forces whatever, are the same whatever may be their 
 common motion ; for this last motion resolved into three 
 others parallel to three fixed axes, causes the partial 
 
 and the resulting motion will be uniform and rectilinear, and 
 determined hy the equation 
 
 sV^f +**)=t\/(v' 2 + f H *'" z )y 
 
 in which * represents the space gone over. If v represents 
 f.he velocity of the body, it will be equal to 
 
 The co. sines of the angles which the direction s of the mo- 
 tion forms with the co-ordinates #, #, and s respectively, 
 are 
 
 Let 5, ar, #, and z have the same significations as above, 
 and g', ", and #'" denote the constant accelerating forces 
 impressed parallel to the axes of .r, #, and z ; then the equa- 
 tions of the motion of the point being 
 
 _ \affZ. .. - 1 0JI/1 . 9 - ifl."//2. 
 x - 2'S l > y - TS l ) * - TS t 9 
 
 the equations of the projections of the line passed over upon 
 the planes of xy and yz t will be, as appears by extracting t* 
 from the two first and two last equations 
 
 the line passed over will therefore be a right line, and we 
 shall have 
 
LAPLACES MECHANICS. 
 
 velocities of each body parallel to these axes to increase 
 by the same quantity, and as their relative velocity 
 only depends upon the difference of these partial velo- 
 cities, it is the same whatever may be the motion com- 
 
 If a, /3, and y represent the co-sines of the angles which 
 makes with j?, y^ and z respectively, then 
 
 cos . g= . _/,.., cos./3= 
 
 cos.v 
 
 The accelerating force in the direction of s is constant and 
 equal to y'tf 2 -f-g" z -f g 1 " 2 ), and composed of the three given 
 accelerating forces, as is the case with uniform motions. 
 
 Retaining the foregoing notation, and supposing that v 1 , 
 ", and v 1 " represent the initial velocities of a point acted 
 upon by constant accelerating forces parallel to the three 
 axes ; the equations of the impressed motion will be 
 
 [x=*ft + tet*; y=v"t+g"*; *=*"'<+&"/. 
 The projection of the curve passed over by the point upon 
 the plane of xy^ found by extracting t from the two first 
 equations, is 
 
 A, JB, and C being constant quantities, a comparison of the 
 co-efficients of x z and xy will shew that this projection is a 
 parabola. The projection upon one of the other planes will 
 also give a parabola, consequently the line passed over is a 
 parabola, 
 
 It may be proved that the curve passed over is of single 
 curvature or upon a plane, without obtaining the equations 
 
LAPLACE'S MECHANICS. 33 
 
 mon to all the bodies ; it is therefore impossible to judge 
 concerning the absolute motion o! the system of which 
 we make a part, by the appearances we observe, and it 
 is this which characterises the law of the proportion- 
 ality of the force to the velocity. 
 
 Again, it results from No. 3, that if we project each 
 force and their resultant upon a fixed plane, ihe sum of 
 the moments of the composing forces thus projected, 
 with resp ct to a fixed point taken upon the plane, is 
 equal to the moment of the projection of the resultant; 
 but if we draw a radius, which we shall call a radius 
 vector, from this point to the moving body, this radius 
 projected upon the fixed plane will trace, in conse- 
 quence of each force acting separately, an area equal to 
 the product of the projection of the line which the 
 moving body is made to describe, into one half of the 
 perpendicular drawn from the fixed point to this pro- 
 jection : this area is therefore proportional to the time. 
 
 of projection, by extracting t from the three equations of 
 motion, which gives one of the form 
 
 ox -f by -\- cz- Q 
 that belongs to a plane surface. 
 
 If the velocities A, and parallel to the three axes 
 at dt at 
 
 at any instant whatever, are composed into one, it will be 
 
 The accelerating force in the direction of the motion, or 
 
 dv 
 
 -, is, as may easily be proved, 
 
34i LAPLACE'S MECHANICS. 
 
 If is also in a given time proportional to the moment of 
 IK projection of the force ; thus, the sum of the areas 
 which the projection of the radius vector would describe 
 in consequence of each composing force, if it ai I 
 a one, is equal to the area which the resultant \v i 
 make the same projection describe. It therefore fol- 
 lows, that if a body is projected in a right line, and 
 afterwards solicited by any forces whatever directed fo 
 wards a fixed point, its radius vector will always de- 
 srribf abou this point areas proportional to the times ; 
 because the areas which the new composing quantifies 
 cause this radius to describe will be nothing. Inversely, 
 ^vve may see that if the moving body describes areas 
 proportional to the times about the fixed point, the re- 
 sultant of the new forces which solicit it is always di- 
 rected towards this point. 
 
 7. Let us next consider the motion of a point solicited 
 by forces, such as gravity, which seem to act continu- 
 ally 
 
 The causes of this, and similar forces which have 
 
 place in nature, being unknown, it is impossible to dis- 
 cover whether they act without interruption, or, after 
 successive imperceptible intervals of time ; but it is easy 
 to be assured that the phenomena ought to be very 
 nearly the same in the two hypotheses ; for if we repre- 
 sent the velocity of a body upon which a force acts in- 
 cessantly by the ordinate of a curve whose abscissa re- 
 presents the time, this curve in the second hypothesis 
 will be changed into a polygon of a very great number 
 of sides, which for this reason may be confounded with 
 the curve. We shall, with geometers, adopt the first 
 hypothesis, and suppose that the interval of time which 
 separates two consecutive act ions of any force -whatever 
 is equal to the element dtot the time, which we will de- 
 
LAPLACE'S MECHANICS. 35 
 
 note by t. It is evidently necessary to suppose that the 
 action of the force is more considr rable as the interval 
 is greater which separates its successive actions ; in 
 order that after the same time ;, the velocity may be the 
 same ; the instantaneous action of a force ought there- 
 fore to be supposed in the ratio of its intensity, and of 
 the element of the time during which it is supposed to 
 act. Thus, representing this intensity by P, we ought 
 to suppose at the commencement of each instant dt, the 
 moving body to be solicited by a force P.dt, and moved 
 uniformly during this instant. This agreed upon : 
 
 It is possible to reduce all the forces which solicit a 
 point M to three, P, (?, and /?, acting parallel to three 
 rectangular co-ordinates x 9 y, and z, which determine 
 the position of this point ; we shall suppose these forces 
 to act in a contrary direction to the origin of the co- 
 ordinates, or to tend to increase them. At the com- 
 mencement of a new instant cfr, the moving body re- 
 ceives in the direction of each of its co-ordinates, the 
 increments of force or of velocity, P.dt, Q.dt, R.dt. 
 The velocities of the point M parallel to these co-orJi- 
 
 nates are ^ ^anci^ ; for during an indefinitely 
 
 small time, they may be supposed to be uniform, and, 
 therefore, equal to the elementary spaces divided by the 
 element of the time. The velocities by which the 
 moving body is actuated at tiie commencement of a 
 new instant, are consequently, 
 
 dt dt dt 
 
 or 
 
36 LAPLACE'S MECHANICS. 
 
 but at this new instant, the velocities by which the 
 moving body is actuated parallel to the co-ordinates 
 
 ,, fix . , dx dii , du 
 x,y, and s, are evident!/ f- f +d. ; -f f +d. -,- nnd 
 
 7t+* ' Tt : the forces -*+ p '*>-*;n + Q ' dt > 
 and d.-^-\-R.dt, ought therefore to be destroyed, so 
 
 that the moving body may in consequence of these 
 sole forces be in equilibrio. Thus denoting by &r, Sy, 
 and s any variations whatever of the three co-ordinates 
 x, y, and 2, variations which it is not necessary to con- 
 found with the differentials cte, dy, and cfz, that ex- 
 press the spaces which the moving body describes pa- 
 rallel to the co-ordinates during the instant dt ; the 
 equation ( b) of No. 3 will become 
 
 -P.dl \ 
 
 -2Q.dt 
 
 (f) 
 
 If the point 71/be free, we shall equal the co-efficients 
 o* .r, y, and <$z separately to nothing, and, supposing 
 the element dt of the time constant^ the differential 
 equations \vill become * 
 
 d* x n d 2 - y ~ , d z z n 
 
 * The equations P 5 ^3=Q, and zz: .ft, are 
 
 dt 2 ' dt* dt 1 
 
 sufficient to enable us t* discover the velocity, the trajectory 
 
LAPLACE'S MECHANICS. 37 
 
 If the point M be not free but subjected to move 
 upon a curve line or a surface, (here must be extracted 
 
 and the place at any given time, of a point not constrained 
 to move along a line or a surface, but continually acted upon 
 by forces which are given every instant both in magnitude 
 and direction. 
 
 Thus supposing for greater simplicity, that thp point 
 moves in the plane xy, P and Q being constant or variable 
 but given ; by extracting the time from the two equations 
 
 zrP, - Q> and integrating them twice, we shall 
 
 find a relation between x and y which will give the trajectory 
 of the point. In a like manner, the rela'ion between x aud 
 , or y and t may be found, which will give the position of 
 the point for any given value of the time t. The values of 
 
 and -- will likewise give the velocities of the point in 
 at at 
 
 the directions of x and y^ from which we may obtain the 
 real velocity v of the point ; for 
 
 =:-?= i/ IGDXST 
 
 The first of the two constant quantities which the above 
 double integration requires, will be detf rmined by the value 
 of the velocity at a given instant, such as the commence, 
 ment of the time t. The second will depend upon the situa- 
 tion of the point with respect to the two axes at this instant. 
 
 If the moving body be atf raoted towards a fixed point by a 
 single force, the integrals of the equations 
 
 may be readily obtained in the following manner, 
 
 Let the origin A of the co-ordinates be placed at this 
 fixed point, and suppose the ir oving body m in any position, 
 having #, j/, and z for its rectangular co-ordinates ; then its 
 
38 LAPLACE'S MECHANICS. 
 
 from the equation (f), by means of the equations to 
 the surface or the curve*, as many of the variations 
 
 distance from the point A will be represented by y^z 1^2 \ z * 
 and the force acting upon it by , which when resolved in 
 the directions of -x } y^ and s gives 
 
 The three first mentioned equations, by propermuitiplication 
 and subtraction, evidently give the three following, 
 
 If the above values of P, (2, and H are substituted in these 
 equations, their second members Avill vanish, and their first 
 will give by integration, the following xdy ydxcdt^ 
 xdy ydx(Jdt y a.n&ydz zctyzzc"^, c, c', and c" being 
 constant quantities ; these equations shew, as will be here. 
 after demonstrated, that equal -areas are described in equal 
 times, by the projections of the line Am upon the planes of 
 the co-ordinates. If these integrals be added together, after 
 having multiplied the first by a, the second by y, and the third 
 by #, the equation cc + c^-f- 6 '"'* 0? which belongs to a plane, 
 will be obtained. 
 
 * If a point moves upon a curve line or surface, it may 
 be supposed free, and acted upon by a force equal and op. 
 posite to the perpendicular pressure upon the curre or sur- 
 face. Let us, for example, .suppose that ~~f(x,y) is rji 
 equation to a curve surface, by differentiating it, we shall have 
 
 dzpdx+qdy. Let Jfe:y'( r l+p 2 +? a - ), then Jt ma 7 be 
 easily proved that the normal of the curve surface forms with 
 the axes x 9 y, and z 9 angles, the co.sines of which are 
 
39 
 
 &r, cty, and $z as it will have equations, and the co-effi- 
 cients of the remaining variations must be equalled to 
 nothing *. 
 
 P Q ^ 
 
 , -TJ, an( ^17 respectively. If N represent a force in the 
 
 direction of the normal, equal and opposite to the pressure 
 upon the curve, its components in the directions of the axes 
 
 p N qN N 
 
 x, y^ and z will be ;~TT~? ~~~ aad ~\J respectively ; the 
 
 two first forces are negative, because they tend to diminish 
 the co-ordinates x and^y, if the curve surface have its con- 
 vexity towards the planes of xz and jys, as can easily be 
 proved. The point may therefore be regarded as free, and 
 
 acted upon by the forces P -, Q - ~ y and 
 which will give the following equations, 
 
 P_ A P 
 
 M > 
 
 N 
 
 * When the motion takes place in a resisting medium, the 
 resistance of the medium may be regarded as a force which 
 acts in a direction contrary to the motion of the body. Let 
 /represent this resistance, then its moment will be /.Si, if 
 
 i is supposed to be equal to V (& O 2 + (y m) 2 + (z n) z i 
 /, m, and n being the co-ordinates of the origin of the force 
 /. By differentiation 
 
 x / y m z n K 
 
 fe:-^. Ss-f^-r-. % y + r-.^. 
 
 If the origin of the force /is supposed to be in the tangent 
 of the curve described by the body, and indefinitely; near to 
 
40 LAPLACE'S MECHANICS. 
 
 8. It is possible in the equation (f) to suppose the 
 variations <r, Sy, and $z equal to the differentials dx 9 
 dyi and efe, because these differentials are necessarily 
 subjected to the conditions of the motion of the moving 
 
 it ; it may be conceived that x #zrrf#, y wzzJy, z n 
 zzfife, which gives, by representing the element ot the curve 
 by ds, the following equations, 
 
 x / dx y-m dy z n dz 
 
 ' ~j s : zz "r~ i and : zz ~r~ z 
 i d *' z ds' i as ' 
 
 consequently, 
 
 dx dy dz 
 
 ds*. d s' y*ds' 
 
 If the resisting medium be in motion, it will be necessary 
 to compose this motion with that of the body, in order to 
 obtain the direction of the resisting force. Let dot, dj3, and 
 dy denote the small spaces through which the medium passes 
 parallel to the axes of the co-ordinates x 9 y , and 2, during- 
 the time that the body describes the space ds, it will be 
 proper to substract these quantities from dx, dy, and dz, in 
 
 order to have the relative motions. As ds\djc* + dy* -f dz 
 if it be supposed that 
 
 d *~ 
 
 the following equation may be obtained 
 
 . iy+ 
 
 (la- do- do- 
 
 It should be observed with respect to the resistance /, that 
 
 ds 
 
 it is generally a function of the velocity ; but in this case 
 
 in which the medium is in motion, it is a function of the re. 
 
 dff 
 
 lative velocit' -r:, 
 * at 
 
LAPLACE'S MECHANICS. 41 
 
 body M. By making this supposition, and afterwards 
 integrating the equation (f), we shall have : 
 
 dx z -* dy z -\-dz z . 
 c being a constant quantify. -7- istrie square 
 
 of the velocity of M 9 which velocity we will denote by 
 v; supposing therefore that P.dx-\-Q.dy-\-R.dz is 
 the exact differential of a function (p, we shall have * 
 
 (g) 
 
 * In every case in which the formula P,dx-\-Q,,dy~\-R.dz 
 is an exact differential of the variables x t y^ and 2, the equa- 
 tion (g) will give the velocity of the point M at any part 
 of its trajectory, if we know it at any one determinate place. 
 For as <p is a function of x 9 y^ and a, let it be represented 
 by'/f*jjfy0j also suppose ^4 to denote the known velocity at 
 the point where the co-ordinates are , a', and a", then we 
 shall have the equations 
 
 and consequently 
 
 This equation shews the value of , when Jf and the co-or- 
 dinates x, y, z, 5 a' y and #" corresponding to the velocities 
 v and ^4 are given. 
 
 It appears from the above that it is possible to determine 
 the difference of the squares of the velocities at two points of 
 the trajectory, by means of the co-ordinates of thse points, 
 without knowing the curve along which the moving body 
 passes in going from one point to the other, 
 
 The above does not hold good if P.dx+Q.dy+R.dz be 
 not an exact differential, as for instance, when the forces 
 
 G 
 
42 I/APLACE'S MECHANICS. 
 
 This case has place when the forces which solicit, the 
 point M 9 are functions of the respective distances of 
 their origins from this point, which comprehends nearly 
 all the forces of nature. 
 
 In fact S, S', &c. representing these forces, s 9 s', 
 &c. being the distances of the point M from their; 
 origins, the resultant of all these forces multiplied by 
 the variation of its direction, will be equal, by No. 2. 
 toS.S.fo/ it is also equal to P.&+Q.*y+/2.&3/ we 
 Lave therefore 
 
 and as the second member of this equation is an exact 
 differential, the first is likewise. It results from the 
 equation (g), 1st. That if the point Mis not solicited 
 by any forces, its velocity is constant, because in this 
 case (p~0 *. It is easy to be assured of this otherwise 
 
 P 9 Q 9 and R arise from friction or the resistance of a fluid, 
 
 and contain in their rallies the velocities . , and ; in 
 
 dt d dt' 
 
 which case the expression is not an exact differential of a 
 function of a?, #, and 2, 'regarded as independent variables. 
 FOP to integrate p under such circumstances, it would be ne- 
 cessary to substitute the values of these variables and their 
 differentials in functions of the time, which could not be done 
 except the problem had been previously solved. 
 
 * If the point is not acted upon by any accelerating force, 
 tyut moves from an initial impulse, the forces P, Q, and R 
 are nothing and <p vanishes, therefore a a zz:r, or the velocity 
 is constant, consequently the velocities in the directions of 
 the axes # } y 9 and z- are constant, and may be supposed re. 
 
by observing that a body moving on a surface or curve 
 linef loses at each rencounter with the indefinitely small 
 plane of the surface, or of the indefinitely small side of 
 the curve, but an indefinitely small portion of its velo- 
 city of the second order *. Sndly, That the point M in 
 passing from one given point with a given velocity, to 
 arrive at another, will have at this last point the same 
 velocity, whatever may be the curve which it shall 
 have described. 
 
 But if the body is not forced to move upon a deter- 
 minate curve, the curve described by it possesses a si- 
 
 spectively equal to the inrariable quantities e', c", and c'". 
 We therefore have the equations 
 
 <*** fyji dz c '" . 
 
 Tr~ ' 7T* 1 Tt~ 
 
 from which by extracting dt, we shall obtain d'dxz^ddy, and 
 e' a dy^nd'dz ; which give by integration c^rza-ft/y, and 
 c"'y~b-\-d'z for the equations of the projections of the tra- 
 jectory upon the planes of ocy and yz ; but these equations 
 belong to straight lines, consequently the trajectory is a 
 straight line. 
 
 * It is proved in most treatises upon mechanics, that if 
 a body moves along a system of inclined planes, the velocity 
 lost in passing from one plane to another, is, as the versed 
 sine of the angle which the planes make with each other. If 
 the number of planes in a given curve are indefinitely increas- 
 ed, the supplements of their angles of inclination, and con. 
 ^frequently the chords become indefinitely small ; by the rules 
 
 f trigonometry we have versed sine zr 1 ?- f : there. 
 
 Radius 
 
 fore if the chord is an indefinitely small quantity of the first 
 order, the versed sine is an indefinitely small one of the se. 
 cond, consequently th velocity lost may be regarded as an 
 indefinitely small quantity of the second refer. 
 
44 LAPLACE'S MECHANICS. 
 
 milar property to which we have been conducted by 
 metaphysical considerations, and which is, in fact, Tmt 
 a remarkable result of the preceding differential equa- 
 tions. It consists in this, that the integral fvds com- 
 prised between the two extreme points of the curve 
 described, is less than any other curve, if the body is 
 free, or less than any other curve subjected to the same 
 surface upon which it moves, if it is not entirely free. 
 
 To make this appear, we shall observe that P.efcr-J- 
 Q'dy-\-R.dz being supposed an exact differential, the 
 equation (g) gives 
 
 the equation (f) of the preceding number becomes alsr 
 
 dx dii d z 
 
 . d. tt. > v . 
 
 Naming the element of the curve described by the 
 moving body ds, we shall have 
 
 vdt=ds ; tf=|/^+//+rf s y * 
 and by equating, 
 
 dx du d z 
 
 0=*xJ.j f \*y.d.j f t.d. Tt - <b.h> ; (h) 
 
 by di%rentiating, with respect to ^, the expression of 
 ds, we shall have 
 
 * That ds\/(d&-\-dy* + d'z t ') is evident from consider 
 ing that the co-ordinates of s and s^-ds are #, #, z and x -\- dx 
 f z-\-dz -, consequently, 
 
 which gives ds~ 
 
LAPLACE'S MECHANICS. . 4a 
 
 The characteristics d and being independant*, we 
 may place them one before the other at will ; the pre- 
 ceding equation can therefore be made to take the fol- 
 lowing form, 
 
 dz 
 -fc.rf.y-, 
 
 * It 'may here be necessary to observe, that when equa- 
 tions contain the differentials dx, dy, and dz, and the varia- 
 tions o#, y, and oz at the same time, the differentials and 
 variations are to be supposed constant wi(h respect to each 
 other, in all the various processes of differentiation or inte- 
 gration. The order in which these processes are performed 
 is also indifferent as to the result. Thus o.r/.r f/.^r, o.f/ 2 jrm 
 d.S.d-cd^x, S.J n jr f/ m . $.dn-xd r 'ox, also \fu-ftu, u 
 being here supposed a function of x 9 y, z 9 dx, dy, dz y c^jr, 
 &c., the sign /denoting the integration of the function witfe 
 respect to the characteristics dx, &c. If u be a function- 
 of jT y I/, and ~, the equation u-mf(x^y^z) gives 
 
 atso 
 
 in which we evidently have fromjjie process of differentiation 
 
 (du\__r^n\ (du\__/-!>u\ Sdu\_S$u\ 
 dxj-^xr \dyj~\lyj' \&J\*z)' 
 As the nature of these notes will not permit me to enter fully 
 upon the subject of variations, I shall refer the reader, who 
 is desirous of information respecting them, to the Traite r da 
 Calcul Differential et Integral pour S. F. Lacroix. The 
 Traite Elementaire de Calcul Diffprentiel et de Calcul In- 
 tegral, by the same author, contains an abridged account of 
 f.hem frm the largo work. 
 
45 LAPLACE'S MECHANICS. 
 
 by subtracting from the first member of this equation, 
 the second member of the equation (h), we shall have 
 
 This last equation being integrated by relation to tbo 
 characteristic d, will give 
 
 . 
 
 If we extend the integral to the entire curve describee! 
 by the moving body, and if we suppose the extreme" 
 points of this curve to be invariable*, we shall have 
 5,/ i orfsi=3); that is to say, of all the curves along 
 which a moving body, subjected to the forces P, Q 9 
 and 7?, can pass from one given point to another given 
 point, it will describe that in which the variation of 
 the integral feds is nothing, and in which, consequently 
 this integral is a minimum. If the point moves along 
 a curve surface without being acted upon by any force, 
 its velocity is constant, and the integral feds becomes 
 
 * If the point from which the body begins to move be 
 fixed, the quantities tix, oy, and Ss are there respectively 
 equal to nothing, therefore the constant quantity of the 
 equation 
 
 , dx.$x+dg.ty-{- dz.Sz 
 
 ofv ds const. f ~ J J - 
 
 dt 
 
 is equal to nothing, as its other terms vanish at that point. 
 If the quantities $,r, y, and $z are also respectively equal to 
 nothing at the end of the motion, from the point where it 
 
 dx.$x -f dy. ty -f- dz . $z 
 <?nds being fixed, we snail have - equal to 
 
 nothing, therefore Ifvds is equal to nothing, that is, tho va- 
 riation of the quantity fvds is a minimum. 
 
AAPLACE'S MECHANICS. 
 
 vfds; thus the curve described by the moving bodj is, 
 in this case, the shortest -which it is possible to trace 
 upon the surface, from the point of departure to that of 
 arrival *. 
 
 "* Maupertuis, in two memoirs, one sent to the Academy 
 f Sciences at Paris, in the year 1744, and the other to that 
 of Berlin, in the year 1746, asserted, that in all the changes 
 which take place in the situation of a body, the product of 
 the mass of the body by its Telocity and the space which ife 
 Jias passed over is a minimum. This he called tl*e principle 
 of the least action, and it was applied by him to the discovery 
 of the laws of the refraction and the reflection of light, the 
 laws of the collision of bodies, the laws of equilibrium, &c. 
 Euler afterwards shewed that in the trajectories of bodies 
 acted upon by central forces, the integral of the velocity 
 multiplied by the element of the curve is always a minimum, 
 which is an excellent application of the principle of the least 
 action to the motions of the planets. This general principle, 
 which was assumed as a metaphysical truth, appears evidently 
 to be derived from the laws of mechanics. 
 
 The following will be sufficient to shew the reader the way 
 in which the principle may be used to discover the laws of the 
 refraction and of the reflection of light. 
 
 Suppose a ray of light to pass from one point to another, 
 if the points are in the same medium, the velocity of the ray 
 is constant, the. path is a straight line and the principle ob- 
 vious ; if they are in two different mediums, let v repress;-; 
 the velocity of the ray, and s the space it passts through, in 
 the first medium, and v' its velocity and s' the space passed 
 through by it in the second medium ; we shall then have the 
 quantity v$-\-v's' , which is a minimum for the value offvds. 
 Th solution of this question is very easy and leads us to the 
 following equation, v. sin. # r. sin. &, in which a repre- 
 sents the angle f incidence and b the angle of refraction at 
 
43 
 
 9. Let us determine the pressure which a point 
 moving upon a surface exerts against it. Instead of 
 extracting from the equation (f) of No. 7, one of the 
 variations &r, Sy, and $z 9 by means of the equation of 
 the surface, we may by No. 3, add to this equation the 
 differential equation of the surface multiplied by an in- 
 determinate quantity Xefc, and afterwards consider 
 the three variations &, 2>y, and $s as independant quan- 
 tities. Let therefore u= be the equation of the sur- 
 face ; we shall add to the equation (f) the term 
 h.^u.dt, and the pressure with which t[ie point acts 
 against it will be, by No. 3, equal to 
 
 Let us now suppose that the point is not solicited by 
 any force, its velocity v will be constant ; if we observe 
 lastly, that i odt=.ds 9 the element dt of the time being 
 supposed constant, the element ds of the curve de- 
 scribed will be so likewise, and the equation (f) aug- 
 
 the surface of the second medium. The above equation 
 shews that the ratio of the two sines depends upon that of 
 the velocities of the ray in passing through the different 
 mediums. 
 
 If the ray in passing from one point to another is reflected 
 at the surface of the second medium, the velocity will be 
 constant, and the path a minimum ; in which case, it may 
 be readily proved, that the angle of incidence of the ray in 
 passing from one point to the surface of the second medium, 
 1$ equal to the angle of reflection from it to the other point. 
 
LAPLACE,'* MECHANICS. 49 
 
 mented by the terra X.Su.dt, will give the three fol- 
 
 lowing, 
 
 df- \dxj itt* 
 
 iVom which we may obtain 
 
 * 
 
 ds* 
 
 but asefo is constant, the radius of curvature of the curve 
 described by the moving body is equal to 
 
 ds* 
 
 by naming this radius r, we shall have 
 
 that is to say, the pressure exercised by the point 
 against the surface, is equal to the square of its velocity 
 divided by the radius of curvature of the curve which 
 it describes. 
 
 lithe point move upon a spherical surface, it will 
 describe (he circumference of a great circle ofthtfsphere 
 which passes by the primitive direction of its motion : 
 for there is not any reason why it should move more to 
 the right than to the left of the plane of this circle ; its 
 pressure against the surface, or, what comes to the 
 same, against the circumference which it describes, is 
 therefore equal to the square of its velocity divided by 
 the radius of this circumference. 
 
 If we imagine the point to be attached to the end of 
 a thread supposed without thickness, having the other 
 
 ii 
 
50 IMPLATF,** MErHANICS. 
 
 extremity fastened to the mitre of (he surface ; it is 
 evident that the prfs.su re exercised by this point against 
 the circumference will be eaual to the tension which 
 the thread would *xp<'rienoe if the point were retained 
 by it alone. The effort which this point makes to 
 stretch the thread, and to go farther from the centre of 
 the circumference, is, what is called the centrifugal 
 force; therefore the centrifugal force is equal 10 the 
 square of the velocity divided by th<- radius. 
 
 In the motion ot a point upon any curve whatever, 
 the centrifugal force is equal to the square of the velo- 
 city, divided by the radius of curvature of the curve, 
 because the indefinitely small arc of this curve is con- 
 founded with the circumference of the circle of curva- 
 ture ; we shall therefore have the pressure which the 
 point exerts against the curve that it describes, by 
 adding to the square of the velocity divided by the ra- 
 dius of curvature, the pressure due to the forces which 
 solicit this point. In the motion of a point upon a 
 surface, the pressure due to the centrifugal force, is 
 equal to the square of the velocity divided by the ra- 
 dius of curvature of the curve described by this point, 
 and multiplied by the sine of the inclination ot the 
 plane of the circle of curvature to the tangential plane 
 of the surface * : by adding to this pressure that which 
 
 * Suppose the radius of curvature RP or r, (Jig. 7.), of 
 the poiut P of the curve described by the body upon the sur- 
 face, to be produced to A ; let PA represent the centrifugal 
 
 V* 
 force of tke body moving in the curve ; from P draw the 
 
 line PB perpendicular to the plane tangent of the surface at 
 
LAPLACE'S MECHANICS. 51 
 
 arises from the action of 'he forces that solicit the 
 point, we shall have the whole pressure which ii exerts 
 against the surface. 
 
 We have seen that if the point is not acted upon by 
 any forces, its pressure a^ain*t the surface is equal 10 
 the square of its velocity divided by the radius o. cur- 
 
 P; draw AB perpendicular to PB let the line DPC bo tht 
 section of the plane tangent caused by a plane passing thro -^h 
 the points ABP perpendicular t the plane tangent ; then 
 as AB. CPD, are respectively perpendicular to the line P#, 
 they are parallel to each other, therefore the angle BAP is 
 equal to the angle DP ft, but this last angle is that which 
 the plane of the circle of curvature makes with the plane 
 tangent, for as the intersection of the plane of the curv< j and 
 of the plane tangent of the surface is the tangent to the curve 
 at P,the linePf? is perpendicular to it ; likewise the radius of 
 curvature of the curve is perpendicular to its tangent at the 
 same point, consequently the plane passing through the lines 
 J5P, PR, and the line PD in it drawn from the point P, 
 are perpendicular to the tangent of the curve ; therefore the 
 angle DPR is the angle which the plane tangent makes with 
 the plane of the curve. By trigonometry in the right angled 
 triangle PAB, we have 
 
 rad. (1) : sine BAP or DPR : : PA (-*} : PB, 
 
 F 2 
 
 therefore P # sine DPR, but PB represents the cen. 
 r 
 
 trifu gal force of the body moving on the surface, consequently 
 the centrifugal force of a body moving ^pon a sujjjjace, is 
 equal to the square of the velocity divided by the radius of 
 curvature of the curve described by this point an.? multiplied 
 by the sine of the inclin-ri >n >f th< utan >t the circieof cr~ 
 Tatnre to the tangential plane of the surtace. 
 
S LAPLACE'S MECHANICS. 
 
 vature of the curve described, the plane of the circle 
 of curvature, that is to say, the plane which passes 
 by two consecutive sides of the curve described by the 
 point, is, in this case, perpendicular to the surface. 
 This curve relative to the surface of the earth, is called 
 a perpendicular to the meridian, and we have proved 
 (No. S) that it is the shortest which it is possible to 
 draw from one point to another upon the surface. 
 
 10. Of all the forces which we observe upon the 
 earth, the most remarkable is gravity; it penetrates 
 into the most inward parts of bodies, and without the 
 resistance of the air, would make them fall with an 
 equal velocity. Gravity is very nearly the same at the 
 greatest heights to which we are able to ascend, and at 
 the lowest depths to which we are able to descend ; 
 its direction is perpendicular to the horizon ; but in 
 the motions of projectiles, we may suppose, without 
 sensible error, that it it constant, and that it acts along 
 parallel lines ; on account of the small extent of the 
 curves which they describe relative to the surface of 
 the earth. These bodies moving in a resisting fluid, 
 \ye shall call /3 the resistance that they experience, it 
 is directed along the side of the curve described by 
 them, which side we will denote by cfc, we shall more- 
 over call g the force of gravity. This agreed upon : 
 
 Let us resume the equation (f) of No. 7, and sup- 
 pose the plane of a: and y horizontal, and the origin of 
 % at the most elevated point; the force /3 will produce 
 in. the directions of x, y y ad 2, the three forces 
 
 _.<|fe #. , and /3.~ ; we shall therefore have, 
 s d s us 
 
 by No. 7, 
 
 /--*; <^-*.*; 
 
and the equation (f) will become 
 
 If the body be entirely free, we shall have the three 
 equations 
 
 * *, =*-'* 
 
 The two first will give 
 
 dy dx ilx dy 
 
 Tt~dt~TtTt- 
 
 from which, by inti'graf ion, we shall obtain rfcr=r 
 /being a constant quantity. This is the equation to an 
 horizontal right line; therefore ihe body moves in a 
 vertical plane. By taking for this plane that of x and 
 s, we shall have^rzziO; the two equations 
 
 will give, by making dx constant, 
 
 fls.d*t d*z dz d't . dz 
 
 from which we may obtain gdi*=.(l*z, and by differen- 
 tiating 2gdt.d*t=:d 3 z ; by substituting for d*4 its value 
 
 ' 7 5 and for dt* its value we shall have 
 <** ' - 
 
 $ _ ds d*z 
 
 g a.C^aJ*" 
 
 This equation gives the law of the resistance /3 necr*- 
 
 sary to make a projectile describe a determinate curve. 
 
 If the resistance be proportional to the square of the 
 
 ds 1 
 Velocity, & is equal to /*. , h being constant in the 
 
 Ctt" 
 
54 LAPLACE'S MECHANICS. 
 
 case where the density of the medium is uniform. We 
 shall then have 
 
 /3 hds* h.ds* 
 
 g gdi- d*z ' 
 
 and from substituting the value of-, hds-=z ; which 
 gi?'s by integration 
 
 &**.* 
 
 d x 2 
 a being a constant quantity*, and c the number whose 
 
 * The value of a may he found as follows : Let g-<fr z be 
 substituted for d*z in the equation -~^ 2ac S ? then it is 
 
 dx* g 2hs dx 
 
 evident that ^--n^.c , but is the velocity of 
 
 the projectile parallel to the axis of #, let v represent 
 the velocity of projection, and 6 the angle which its di- 
 rection makes with that axis, then ( --V t) 2 cos. 2 6 
 consequently as srrO at the commencement of the motion, we 
 have u a cos. 2 -^- and a , 
 
 d 1 z * 2hs 
 
 therefore - -^ ^-. c 
 
 By supposing dz equal to pdx, this last equation may be 
 
 dn ff 2hs 
 
 changed into the following -L~ * , c ; but as ds 
 
 dx v*. cos. 2 Q 
 
 -\-dz*~dx\/ 1-f-p 2 . hy multiplyirig the first member 
 of this equation by dx^~\+i and the second by ds it will 
 
 O' 2HS 
 
 become rfpy / T+p= P a CQS a 0- c - * Tbe inte S ral of 
 djpl/T+p* is 4:{pV / T+]p*+lo.( r jp+V / l-f-p^jjandthat of 
 
LAPLACE'S MECHANICS. 55 
 
 hyperbolical logarithm is unity. If we suppose (he re- 
 sistance of the medium to be nothing, or 7? ? we shall 
 have, by integration, thr equation to a parabola 
 
 b and e being constant quantities* 
 
 =C+--' C being a constant 
 
 quantity, which may be determined by observing, that at the 
 commencement of the projection s0 and />:z:fan.0, conse- 
 quently, fct 
 
 or 
 
 -- -r, By substituting in the above integral the 
 
 D. a C08. 2 d.A 
 
 dp g- 2hs 
 
 ralue of c 2h * obtained from the equation -7- : " :.c 
 
 a^ i>. a cos. 2 9 
 
 we shall have the following 
 
 
 
 ~ /ilpV/l-j-^ 
 
 which gives, as dz~ 
 
 By the integration of these equations the values of x and s 
 would be given in functions of p. A third equation may 
 be obtained which would give the time in a function of the 
 same quantity, by substituting the value of dx derived from 
 
 dp.dx 
 the equation <# 2 -- in one of the preceding equations, 
 
 & 
 
 and extracting the square root, which will give the following 
 
 dp 
 
 ? 2 >- 6 A 
 If these three equations could be integrated so as to have a 
 
56 LAPLACE'S MECHANICS. 
 
 The differential equation d'z=gdt* will pivf? <//* 
 .</#% from which we may obtain / r*/ ^ -\- f . 
 
 o 
 
 If x, *, and t are supposed fo begin together, we shall 
 
 finite form, the complete solution of (he problem would be 
 obtained. In this case, the two first, by the elimination of 
 j?. would give the trajectory of the curve. This problem 
 ha^ exercised the skill of many eminent mathematicians from 
 thf time of Newton to that of L"gendre, but all their solu- 
 tions are very complicated. The trajectory may be described 
 from points by means of the two first equations, and tables 
 made for forming it at any inclination. Vide the memoir 
 of Moreau, in the eleventh cahier of the Journal de 1' Ecole 
 Polytechnique 1 . The descending branch of the curve has an 
 asyintote, as appears from making p indefinitely great, which 
 
 d p dp 
 
 gives cto T ; and dz~j , or by integration, xc l ^ 
 
 and zzzrc" -{ log. p, d and c" being constant quantities. 
 li 
 
 From these two equations it appears, that if p be indefi. 
 nitely increased, the valueof z will become indefinitely great, 
 although is does not increase so fast as />, and that of a? will 
 approximate to c' as its limit. If, therefore, on the hori- 
 zontal axis of x, at a distance equal to c', from the origin of 
 the co-ordinates, perpendicular be let fall, that line will be 
 an asymtote to the descending branch of the curve. 
 
 When the angle of projection of the body is very smaW 
 with respect to the horizon, and the initial velocity not con. 
 siderable, that part of the curve above the horizontal line of 
 projection may be readily found by approximation,, and is 
 applicable to th case of ricochet firing. Vide a jmemoir of 
 Borda am augst those of the Academic des Sciences, 
 
LAPLACE'S MECHANICS. 57 
 
 have * and/'=0 ; consequently /=# y' ^ and 
 
 fc 
 
 which give 
 
 These three equations contain the whole theory of 
 projectiles in a vacua P ; it results from the above, 
 that the velocity is uniform in an horizontal direction, 
 and in a vertical one, it is the same as that \vhich would 
 be acquired by the body falling down the vertical. 
 
 If the body falls from a state of rest, b will vanish, 
 
 and we shall have ~r=g* >' *=$gf >' lae velocity 
 
 therefore increase? as the time, and the space increases 
 as the square of the time. 
 
 It is easy, by means of these formula?, to compare 
 the centrifugal force to th-it of gravity. It has been 
 shewn by what precedes, that v being the velocity of a 
 body moving in the circumference of a circle, whose 
 
 v z 
 radius is r, the centrifugal force is . J>t h be the 
 
 height from which it ought to fall to acquire the velo- 
 city v; we shall have by what precedes, v* 
 
 v 2 <2h 
 from which we may obtain =.. If ^rzr:f r, the 
 
 centrifugal force becomes equal to the gravity g ; thus 
 a heavy body attached to the extremity of a thread 
 fastened by its other extremity to an horizontal plan", 
 will stretch this thread with the same force as if it were 
 suspended vertically, provided, that it moves upon this 
 plane with the velocity which it would have acquired 
 by falling from a height equal to halt' the length of the 
 thread. 
 
MECHANICS. 
 
 11. Let us consider the motion of a heavy body en a 
 spherical surface. 
 
 By naming its radius r, and fixing the origin of the 
 co-ordinates x, y, and z at its centre; we shall have 
 r* jt? y % z*=0 ; Ihis cqnntion compared with that 
 ofw=0, gives u=s* x* ^ s 2 : by adding there- 
 fore to the equation (f) of No. 7, the function lu mul- 
 tiplyed by the indeterminate \dr, we shall have 
 
 an equation in which we may equal separately to no 
 thing, the co-efficients of each of the variations 8#, ty 
 and J, which will give the three following equations. 
 
 ! 
 
 I 
 
 dz 
 
 dt J 
 
 The indeterminate X makes known the pressure which 
 the moving body exerts against the surface. This pres- 
 
 sure is b 7 No. Oequalto 
 
 it is consequently equal to 2xr ; but by No. 8, we have 
 
 dt 2 
 
 c being a constant quantity ; by adding this equatiom 
 to the equations (A) divided by dt 9 and multiplied re- 
 spectively by j?, y, and z ;N and observing, lastly, that 
 the differential equation of the surface is Q= 
 -f-scfs, which, by differentiation, gives 
 
LAPLACE'S MECHANICS. 50 
 
 we shall find 
 
 r 
 
 If we multiply the first of the equations (A) by y, 
 and add it to the second multiplied by #, we shall have 
 from integrating their sum 
 
 ' yd* , 
 
 dt 
 c? being a new constant quantity. 
 
 The motion of a point is thus reduced to three differ* 
 ential equations of the first order 
 
 xdx-\-ydyn=. zdz y 
 xdy ydjc-=x'dt y 
 
 By raising each member of the two first equations to 
 the square, and then adding them together, we shall 
 have 
 
 if we substitute in the place ot a?~\-y* its value r 2 x% 
 
 and in the place of its value c+2gz we 
 
 shall have, by supposing that the body departs from 
 the vertical, 
 
 rdz 
 fit - ---- . 
 
 The function under the root may be changed to (he 
 following form, (a z).(z b).(2gz-\-f} y a y b> and 
 f being determined by the equations 
 
 f _ 
 
60 LAPLACE'S MECHANICS. 
 
 It is possible thus to substitute for the constant quan- 
 tiUe^ c and c' the new ones a and b ; the first of whu'h 
 is the greatest value of z, and the second the least. By 
 making afterwards 
 
 sn. 
 
 a b 
 
 the preceding differential equation will become 
 _ r.iAr a+b) d& 
 
 7 2 being equal to ~ 
 
 The angle Q gives the co-ordinate z by means of the 
 equation 
 
 siz=a . cos .* 6-|-& . sin , 2 0, 
 
 and the co-ordinate z divided by r, gives the co-sine 
 of the angle which the radius r makes with the vertical. 
 Let w be the angle which the vertical plane passing 
 by the radius r, makes with the vertical plane passing 
 by the axis of x ; we shall then have * 
 
 which give xdyydx=.(r 2 - z 2 J.d&; the equation 
 2rdyydx=.c l dt will also give 
 
 c'dt 
 
 * For \/r 2 z 2 is the projection of the line r upon the 
 plane oi xy. and if from the extremity of y r z z* a perpen- 
 dicular be drawn to the axis of #, we shall hare V 'r 2 z* 
 : x : : rad. (1) : cos, r, therefore x~\/r 2 s* . cos. w. 
 In a similar manner we shall have 5/zr-y/ r z ^2 . sin. w. 
 
LAPLACE'S MECHANICS. 61 
 
 by substituting fors and dt their preceding values in 9, 
 we shall have the angle r in a function of 6; thus we 
 may know at any time whatever the two angles andsr; 
 which is sufficient to determine the position of the mov- 
 ing body. 
 
 Naming the time which is employed in passing from, 
 the highest to the lowest value of s 9 the semi-oscillation. 
 of the body ; let \ Trepresent this time. To determine 
 it, it is necessary to integrate the preceding value of dt 
 from Q=Q to 0=f x ; <n being the semi-circumference 
 of a circle whose radius is unity : we shall thus find 
 
 Supposing the point to be suspended from the extremity 
 of a thread without mass, which is fixed at its other ex- 
 tremity, it the length of the thread is r the point will move 
 exactly as in the interior of a spherical surface, and it will 
 form with the thread a pendulum, the co-sine of whose 
 
 greatest distance from the vertical will be-. Jf we sup- 
 
 pose that in this state, the velocity of the moving body 
 is nothing, it will oscillate ^in a vertical plane, and in 
 
 this case we shall have a ir, 7 2 z^r - . The fraction 
 
 ^- is the square of the sine of half the greatest angle 
 
 which the thread forms with the vertical; the entire 
 duration Tof the oscillation of the pendulum will there- 
 fore be 
 
LAPLACE'S MECHANICS. 
 
 Jf the oscillation is very small, ~ is a very small 
 fraction, which may be neglected, and we shall have 
 
 the very small oscillations are therefore isochronous or 
 of the same duration, whatever may be thi j ir extent ; 
 and we can easily, by moans of this duration, and of 
 the corresponding length of the pendulum, determine 
 the variations of the intensity of gravity at different 
 parts of the earth's surface. 
 
 L< j t z be the height from which gravity makes a 
 body fall during the time 7 1 , we shall have, by No. 10, 
 JSzzzrg 1 ?'*, and consequently z=J a .r/ we shall there- 
 fore have with very great precision by means of the 
 length of a pendulum that beats seconds, the space 
 through which gravity will cause bodies to fall during 
 the first second of their descent. From experiments 
 Yery exactly made, it appears, that the length of a pen- 
 dulum vibrating seconds is the same, whatever may be 
 the substances which are made to oscillate : from 
 which it results that gravity acts equally upon all 
 bodies, and that it tends in the same place, to impress 
 upon them the same velocity in the same time. 
 
 12. The isoehronism of the oscillations of a pendulum 
 being only an approximation, it is interesting to know 
 the curve upon which an heavy body ought to move, 
 to arrive at the same time upon the point where its mo- 
 tion ceases, whatever may be the arc which it shall do 
 
scribe from the lowest point. But to solve this problem 
 in the most general manner, we will suppose, conform- 
 ably to what has place in nature, that the point moves 
 in a resisting medium. Let s represent the arc de- 
 scrib'd from the lowest point to the cuivr, z the verti- 
 cal abscissa reckoned from this point ; dt the element 
 of the time, and g the gravity. The retarding force 
 along the arc of the curve will be, first, the gravity re- 
 
 solved along the arc ds, which becomes equal to g. ^ ; 
 secondly, the resistance of the medium, which we shall 
 express by <p.f j~ t \t ( J^) be ing the velocity of the 
 
 moving body, and t \ being any function what- 
 ever of this velocity. The differential of this velocity 
 will be, by No. 7, equal to g. <p C^J ; we shall 
 therefore have by making dt constant 
 
 ds\ ds . ds* 
 
 Let us suppose <p. - =^. +. , and s= 
 
 . . , 
 
 if we denote by v'C^'O the differential of \}/ (s 1 ) divided 
 by ds', and by V (s') that of 4' (s 1 ) divided by ds', we 
 
 shall have 
 
 ds ds' 
 
 d-t=J-f V(s ^ 
 
 d i s d 1 s 1 d s'* 
 
 and the equation (i) will become > 
 
 o- *.<+ 
 
 - 
 
G4 LAPLACE'S MECHANICS. 
 
 ds' 1 
 tve shall cause the term multiplied by to disappear 
 
 by means of the equal ion 
 
 0=V l (s')+ 
 this equation gives by integration 
 
 h and q being constant quantity's. If we make s 1 to 
 
 commence with s, we shall have hq~=1 9 and if, for 
 greater simplicity, we mak^ //=!, we shall have 
 
 c being the number, th;- hyperbolical logarithm of 
 which is unity : the differential equation (I) then be- 
 comes 
 
 dV . ds' 
 
 * The integral of 0=y'(s')+n.[4<'(s f )Ym*y he readily 
 found, by substituting for 4"(s')) and \|/' (s 1 ) their values ; 
 
 f/Z g (Jig' 
 
 the equation then becomes Ozz -f w^, that by in 
 
 ds d s' 
 
 tegration gives h.log.Js h log. ds' -\-ns~e, or h. log. , 
 
 c ns 
 zze nsy from which we may obtain f/s'm e . efo ? c being 
 
 the number whose hyperbolical logarithm is unity; the inte. 
 
 ,ns e 
 
 gral of this equation is s 1 -{-<jcz , which gives h. log.(. 
 
 n 
 
 L r e 
 
 (s 1 4-g)) ws e, or h. log.{w,(*' -f^) j n ^- ~* ; if zrh. 
 
 lo S'^I then h >P' lo 
 
LAPLACE'S MECHANICS. 
 
 By supposing s 1 very small, we may develope Ihe last 
 term of this equation in an ascending series, with re- 
 spect to f he powers of s', which will be of this form, 
 ks'-\-ls n -\-8cc., i being greater than unity ; the last 
 equation then becomes 
 
 This equation being multiplied by 
 
 t 
 c ~T.{cos. 
 
 and afterwards integrated, supposing 7 equal to 
 ifar k > will be changed into 
 
 m ' C dd 
 
 c ~*~.{cos.y^-4-.r/~~fsin,.7 j. < U 
 
 L d t 
 
 f m \ f ^ _t 
 
 ^c. 
 
 By comparing separately, the real and the imaginary 
 
 parts, we shall have two equations, by means of which 
 
 ds 1 
 
 may be extracted ; but here it will be sufficient for 
 
 us to consider the following 
 ds' mt 
 
 m 
 
 the integrals of <he second member being supposed t 
 commence with t. Naming T the value of t at the 
 
 end of the motion, when-* is nothing ; we shall have 
 at that instant 
 
 m T C m > 
 
 C ^~' s '' \ ^sM'yTy-cos.yT^^Lfs^dt.c.-s- 
 
 s'w.yt c. 
 
 K 
 
96 LAPLACE'S MECHANICS. 
 
 Jn the case of s' beinij indefinitely small, (lie second 
 member of the equation \vill be reduced to nothing 
 "when compared with the first, an:! we shall have 
 
 -.sJn.yT y.cos.yT/ 
 
 from which we may obtain 
 
 iang.yT ^/ 
 
 and as the time T is, by the supposition, independent 
 of the arc passed over, this value of the (ang.y Triad 
 place for any arc whatever, which will give for any 
 value of s' 
 
 m t 
 
 0=l.fs n .dt. cT.sin.yf-f &c. 
 
 the integral being taken from /zzzO, to t-=.T. 
 
 By supposing s' very small, this equation will be re- 
 duced to its first term, and it can only be satisfied by 
 
 making /.nnO ; for the factor cir.sin. y t being alwayg 
 
 positive from fc=0 to t=:T, the preceding integral is 
 necessarily positive in this interval. It is not there- 
 fore possible to have tautochronism but on the suppo- 
 sition of 
 
 which gives for the equation of the tautochronous 
 
 curve 
 
 In a yacnum, and when the resistance is proportional 
 te the simple Telocity, n is nothing, and this equation 
 
LAPLACE'S MECHANICS. 67 
 
 becomes gdz-=ksds ; which is the equation to the 
 cycloid *. 
 
 It is remarkable that the co-rfficient n of the part of 
 the resistance proportional to the square of (h* 1 ve'ocify, 
 does n >f enter into the expression of the time T ; and 
 ft is evident by the preceding analysis, that this ex- 
 pression will he t!i3 same, if \ve add to the preceding 
 law of the resistance, the terms 
 ds* . 
 
 If in general, 7? represents the retarding force along the 
 curve, we shall have 
 
 =+* 
 
 i is a function of the time t, and of the whole arc passed 
 
 * The cycloid is the only curve in a plane that is fautoch* 
 ronoiisin a vacuum, but this property belongs to an indefi- 
 nite number of curves of double curvature, which may be 
 formed by applying a cycloid to a vertical cylinder of any 
 base, without changing the altitudes of the points of the 
 curve above the horizontal plane. This is evident from 
 considering the equation *:z:c-f-2p of No. 8, *vhich by 
 
 ds* i 
 
 proper substitution becomes zzc %gz and gives d/zn 
 
 lit* 
 
 ds 
 
 T^ ' ; the upper sign being taken if t and s increase 
 
 yc^gz 
 
 together, and the lower, if one increases whilst the other 
 decreases. From this last equation it appears, that the 
 value of t depends upon the initial velocity and the relation 
 between the vertical ordinates and the arcs of the ?urve If 
 therefore thii velocity and this relation be the same, vrheu 
 the curve is changed, the above equation will not be altered 
 any more than the law of motion which it denotes. 
 
LAPLACE'S MECHANICS. 
 
 orer, which is consequently a function of t and of s. 
 By differentiating this last function, we shall have a 
 differential equation of the form 
 
 V being a function of t and $, which by the condition 
 of the problem ought to be nothing, when t has a value 
 which is indeterminate and independent of the arc 
 passed over. Suppose for example F=S. T 1 , S being 
 a function of s alone, and Ta function of t alone ; w 
 shall have 
 
 <*^_ T <M di s 1_1L <*S ds* , ^dT 
 d t 2 'ds' JT"**"' dt ~~ 
 
 but the equation y=S T, gives , and consequently 
 
 --equal to a function of ; which function we will 
 
 dt ^ ,il t 
 
 denote by -^r^T* ^( "SJf ) ' 
 
 we 
 
 Such is the expression of the resistance which answer* 
 to the differential equation =ST/ and it is easy t 
 
 perceive that it comprises the case of the resistance 
 proportional to the two first powers of the velocity 
 multiplied respectively by constant co-efficientf. 
 Other differential equations would give different lawi 
 ef resistance. 
 
LAPLACE'S MECHANICS. 
 
 CHAP. III. 
 
 Of the equilibrium of a system of bodies* 
 
 33. 1 HE most simple case of the equilibrium of many 
 bodies, is that of two material points which strike each 
 other with equal and directly contrary velocities ; their 
 mutual impenetrability evidently destroys their velo- 
 cities, and reduces them to a state of rest. Let ui 
 now consider a number m of contiguous material points 
 disposed in a right line, and actuated by the velocity 
 #, in the direction of this line. Let us suppose, in like 
 manner, a number m 1 of contiguous points, disposed 
 upon the same right line, and actuated by a velocity 
 ' directly contrary to z/, so that the two systems shall 
 strike each other. In order that they may be in cqui- 
 Hbrio at the moment of the impact, there ought to be a 
 relation between u and ' which it is necessary to de- 
 termine. 
 
 For this purpose we shall observe that the system m, 
 actuated by the velocity z/, will reduce a single material 
 point to a state of equilibrium, if it be actuated by a 
 Ttlocity mu in a contrary direction ; for each point ii 
 
70 
 
 the system will destroy a velocity equal to u in this last 
 point, and consequently its m points will destroy the 
 entire velocity mn : we may therefore substitute for 
 this system a single point actuated by the velocity mu. 
 
 We can, in like manner, substitute for the system in 1 
 a single point actuated by the velocity m'u' ; but the 
 two systems being supposed to cause equilibrium, the 
 two points which take their places on^ht in like man- 
 ner to do it, this requires that their velocities should be 
 equal; we have therefore for the condition of the equi- 
 librium of the two systems wii/zizm'i/'. 
 
 The mass of a body is the number of *its material 
 points, and the product of the mass by its velocity is 
 called its quantity of motion ; this is what is understood 
 by the force of a body in motion. 
 
 For the equilibrium of two bodies or of two systems 
 of points which strike each other in contrary directions, 
 the quantities of motion, or the opposite forces, ought 
 to be equal, and consequently the velocities should be 
 inversely as the masses. 
 
 The density of bodies depends upon the number of 
 material points which are contained in a given volume. 
 In order to have thrir absolute density, it would be ne- 
 cessary to compare their masses with that of a body 
 without pores ; but as we know no bodies of that de- 
 scription, we can only speak of the relative densities of 
 bodies ; that is to say, the ratio of their density to that 
 of a given substance. It is evident that the mass is in 
 the ratio of the magnitude and the density, by naming 
 jfcfthe mass of the body, U its magnitude, and D its 
 density, we shall have generally M=DU: an equa- 
 tion in which it should be observed that the quantitiei 
 M 9 D, and U express a certain relation to the unities 
 of their species. 
 
LAPLACE'S MECHANICS. 71 
 
 What we have said, is on the supposition that bodies 
 are composed of similar material points, and that they 
 differ only by the respective positions of these points. 
 But as the nature of bodies is unknown, this hypo- 
 thesis is at least precarious ; and it is possible that there 
 may be essential differences between their ultimate par- 
 ticles. Happily the trnlh of this hypothesis is of no 
 consequence to the science of mechanics, and we may 
 make use of it without fearing any error, provided 
 that by similar material points, we understand points 
 which by striking each other with equal and opposite 
 velocities, mutually produce equilibrium whatever may 
 be their nature. 
 
 14. Two material points of which the masses are m 
 and m', cannot act upon each other but along the line 
 that joins them. In fact, if the two points are con- 
 nected by a thread which passes over a fixed pulley, 
 their reciprocal action cannot be directed along this line. 
 But the fixed pulley may be considered as having at 
 its centre a mass of infinite density, which re-acts upon 
 the two bodies mand m 1 , whose action upon each other 
 may be considered as indirect. 
 
 Let p denote the action which m exercises upon m f f 
 by the means of a straight line inflexible and without 
 mass, which is supposed to unite the two points. Con- 
 ceive this line to be actuated by two equal and opposite 
 forces p and p, the force p will destroy in the body 
 m a force equal to p 9 arid the force p of the right line 
 will be communicated entirely to the body m 1 . This 
 loss of force in ?T?, occasioned by its action upon m' f , is 
 what is called the re-action of m' ; thus in the commu- 
 nication of motions, the re-action is always equal and 
 oootrary to the action. Fiom observation it appears 
 
LAPLACE'S MECHANICS. 
 
 that this principle has place in al! the forces of na- 
 ture. 
 
 Let us suppose two h ?zvy bodies m and m 1 attached 
 to the extremities of nn horizontal right line inflexible 
 and without mass, which can turn freely about one of 
 its points. 
 
 To conceive the action of the bodies upon each other 
 when they produce the state of equilibrium, it is neces- 
 sary to suppose an indefinitely small bend in the right 
 line at its fixed point ; we shall then have two right 
 lines making 1 at this point an angle, which differs from 
 tAvo right angles by an indefinitely small quantity u. Let 
 fund f represent the distances m and m' from this 
 fixed point: by resolving the weight of m into two 
 forces, one acting upon the fixed point, and the other 
 directed towards m 1 : this last force \vill be represented 
 
 by^ : r 3 g being the force of gravity*. The 
 
 * Let mCm' (Jig. 8.) represent the lever, which is sup. 
 posed to be bent from (he horizontal right line mCn at (7, so 
 that the angle iw'Cwa;, then mC /jw'C' /', the line m r n 
 perpendicular lo the line mC continued ~uj ' nearly, and 
 the line mm 1 joining the bodies m and m' rz:/-f-jf ' nearly. 
 Complete the rectangular parallelogram mnrn'o, and suppose 
 mo or its equal nm 1 to represent the weight of m; this force 
 may be resolved into two others, represented in quantity 
 and direction by the lines mm r and m' o ; we shall (hen have 
 mo~nm' (uj 1 ) to mm 1 (f+f), as mg (the weight of m) 
 
 is to ~ .** - or the force with vhich m acts upon the 
 
 */ 
 
 body m 1 . The force with which m 1 acts upon m may be 
 found in a similar manner. 
 
LAPLACR'S MECHANICS. /,:> 
 
 action of m' upon m will in like manner be /. 
 
 *i/ 
 
 by equalling these two forces, in consequence of their 
 equilibrium, we shall have vnf=.m'f ; ttiis gives the 
 known law of the equilibrium of the kver, and at the 
 same time enables us to conceive the reciprocal action 
 of parallel forces. 
 
 Let us now consider the equilibrium of a system of 
 points 772, m' 9 m" 9 &c. solicited by any forces whatever, 
 and re-acting upon each other. Let /be the distance 
 of m from m' ; /' the distance of m from m" ; and/ /y 
 the distance of m 1 from m", &e. ; again, let p be the 
 reciprocal action of m upon m' ; p' that of m upon m" : 
 //that of m' upon ra"&c., lastly, let mS,m'S f ) m"S",&c. 
 be the forces which solicit m, m 1 , m", &c. ; and s , .v', 
 s", &c. the right lines drawn from their origins unto the 
 bodies m, m'^ m", &c. 
 
 This being agreed upon, the point m may be con- 
 sidered as perfectly free and in cquilibrio, in conse- 
 quence of the force mS, and the forces which the bodies 
 m,m' 9 m H 9 &c. communicate to it : but if it were sub- 
 jected to move upon a surface or a curve, it would be 
 necessary to add the re-action of the surface or of the 
 curve to these forces. Let $s be the variation of s, and 
 let ^/denote the variation off taken by regarding m 1 as 
 fixed. Denoting in like manner, by S 7 f 7 , the variation 
 off ,taken by regarding m 11 as fixed, &c. Let R and 
 R' represent the re-actions of two surfaces, which 
 form by their intersection the curve upon which the 
 point m is forced to move, and Sr, $r' the variations of 
 the directions of these last forces. The equation (d) 
 of No. 3, will give 
 
74 LAPLACE'S MECHANICS. 
 
 In like mannrr, m 1 may be considered as a point 
 which is perfectly free and in cquilibrio, in consequence 
 of the force m'S', of the actions of the bodies m, m", 
 &e., and of the re-actions of the surfaces upon \yhich 
 it is obliged to move ; which re-actions we shall denote 
 by R" and R" 1 . Let $s' be therefore the variation of 
 s ' 5 ^///*h e variation of'/*, taken by regarding m as 
 fixed ; S^'the variation of/ 77 , taken by regarding m' J 
 as fixed, &c. Moreover let W and or 11 ' be the varia- 
 tions of the directions of R 11 and R" 1 ; the equilibrium 
 of m' will jjive 
 Qm'S' fc4-p.yifpU/'4-&c. . 4-j? /7 .V 7 -f 72 77 V. 
 
 if we form similar equations relative to the equilib- 
 rium of 77Z 77 , W 7 ', &c. ; by adding them together and 
 observing that* 
 
 '+ V ; *c. 
 
 * The ninth diagram will serve to render this more evi. 
 dent. Suppose that the line joining the bodies m and m 1 h 
 represented by/y that the point m' being immoveable, the 
 point m passes over the indefinitely small space mn. Join 
 nm' ; from the point n let fall the perpendicular na upon the 
 line mm' , then ma will represent the projection of the line 
 mn upon the line mm' , see notes No. 2, and we shall have 
 
 ma~mm'(f) am 1 , but am' 2 m'n 2 na z , and as na 1 is an 
 indefinitely small quantify of the second order, it may be 
 neglected, therefore am'nm' nearly, consequently ma 
 mm' (f) wm/zz^/. Again, let m 1 be supposed to pass 
 over the indefinitely short space m'n 1 \ whilst m remains im- 
 moveable, join mn 1 , then mm 1 mw'zz^/. If m and m 1 be 
 
LAPLACE'S MECHANICS. 75 
 
 $/,&/', &c. being the whole variations of/, /', &c. 
 we shall have 
 
 0=2. m. S.^+2.p.5/+S. ft.Sr, f A'J 
 an equation in which the variations of the co-ordinates 
 of the different bodies of the system are entirely arbi- 
 trary. It should here be observed that instead of 
 m*S.$s, it is possible in consequence of the equation 
 (a) of No. 2, to substitute the sum of the products of 
 all the partial forces by which m is actuated, multiplied 
 by the variations of their respective directions. It is 
 the same with the products m'SUs', w/'SW, &c. 
 
 If the bodies w,m', m' 1 , &c. are invariably connected 
 with each other, the distances/,/', /", &o. will be con- 
 stant, and weshall have for the condition of the connec- 
 tion of the partsof the system, &/=0,S/':zr.O,&f ''=(), &c. 
 The variations of the co-ordinates in the equation (k) 
 being arbitrary, we may subject them to satisfy these 
 last equations, and then flit- forces ;?, p', p' 1 , &c. which 
 depend upon the reciprocal action of the bodies of the 
 system, will disappear from this equation : we can also 
 cause the terms /?.Sr, 7 J '.S>', &e. to disappear, by sub- 
 jecting the variations of the co-ordinates to satisfy ihe 
 equations of the surfaces upon whrrh the bodies are 
 forced to move, the equation (k) flni* becomes 
 
 Q=2.m.S.**; (I) 
 
 from which it follows, that in the case of equilibrium, 
 the sum of the variations of the products of the iorccs 
 
 supposed to vary at thp same timp, and move respectively to 
 n atul w', let n and n' b< j joiftfd, tlu n VVP shall have mm 1 
 nn'$f$ l f-\-$ li f, by neglecting iucltfiaitelv small quantities 
 of higher orders than the first. 
 
76 
 
 by the elements of their directions is nothing, in what- 
 ever manner we make the position of the system to 
 vary, provided that the conditions of the connection of 
 its parts be observed, 
 
 This theorem which we have obtained on the parti- 
 cular supposition of a system of bodies invariably con- 
 nected together, is general, whatever may be the con- 
 ditions of the connection of the parts of the system. To 
 demonstrate this, it is sufficient to shew, that by sub- 
 jecting the variations of the co-ordinates to these con- 
 ditions, we shall have in the equation (k) 
 
 but it is evident that r, r', &c. are nothing, in con- 
 sequence of these conditions ; it is therefore only re- 
 quired to prove that we have OzzzS.p.S/, by subjecting 
 the variations of the co-ordinates to the same conditions. 
 Let us imagine the system to be acted upon by the sole 
 forces p, p' y p", &c. and let us suppose that the bodies 
 are obliged to move upon the curves which they would 
 describe in consequence of the same conditions. Then 
 these forces may be resolved into others, one part q, 
 q'l q", &c. directed along the lines /, /S/", &c. which 
 would mutually destroy each other, without producing 
 any action upon the curves described ; another part T, 
 T' 9 T", &c. perpendiculars to the curves described, 
 laitly, the remaining part tangents to these curves, in 
 consequence of which the system will be moved*. But 
 
 * la (Jig. 10) where only two bodies m and ml are con. 
 sidered, mm 1 is the line joining the bodies, AmB the curve 
 upon which m is forced to remain, mp the force p that acts 
 upon m in the direction of the line mm' or/ 5 rp or q that 
 
LAPLACE'S MECHANICS. If 
 
 it is easy to perceive that these last forces ought to be 
 nothing ; for the system being supposed to obey them 
 freely, they are not able to produce either pressure 
 upon the curves described, or re-action of the bodies 
 upon each other ; they cannot therefore make equilib- 
 rium to the forces p, />', p", &c. q, <?', q", &c. 
 7", T', T", &c. ; it is consequently necessary that 
 they should be nothing, and that the system should be 
 in equilibrio by means of the sole forces p, p f , 
 p", &c. q,q',q", &c. TjT^T^&c. Let*/, &', &c. 
 represent the variations of the directions of the forces 
 T, T', &c. ; we shall then have in consequence of the 
 equation (k) 
 
 but the system being supposed to be in equilibrio by 
 means of the sole forces <?, q', &c. without any action 
 resulting upon the curves described ; the equation (k) 
 again gives 0=2. .y, which reduces the above to the 
 following 
 
 If we subject the variations of the co-ordinates to an- 
 swer to the equations of the described curves, we shall 
 have ^/zzziO, Jt v =0, &c. and the above equation be- 
 comes 
 
 as the curves described are themselves arbitrary, and 
 
 part of it which is destroyed by the mutual action of the 
 the bodies without producing any effect upon the curve 
 AmB) mr the remaining force of ;;, which is resolved into 
 the force wT, or T that acts perpendicularly to the curve, 
 and Tr which acts in the direction of a tangent to it. 
 
75 LAPLACE'S MECHANICS. 
 
 only subjected to the conditions of the connection of 
 flie parts of the system ; the preceding equation !ms 
 place, provided that these conditions be fulfilled, and 
 then the equation (k) will be changed into the equa- 
 tion (I). This equation is the analytical traduction of 
 the following principle, known under the name of the 
 principle of virtual velocities. 
 
 " If we make an indefinitely small variation in the 
 position of a system of bodies, which are subjected to 
 the conditions that it ought to fulfil : the sum of the 
 forces which solicit it, cnch multiplied by the space 
 that the body to which it is applied moves along its di- 
 rection, should be equal to nothing in the case of the 
 equilibrium of the system." * 
 
 * The following are proofs of the truth of the principle of 
 virtual velocities in the cases of the lever, the inclined 
 plane, and the wedge. 
 
 First, with respect to the lever; let mCnt (Jig. 11.J re- 
 present a straight lever in equilibrio upon the fulcrum G\ by 
 means of the forces S and S' acting in the respective direc- 
 tions of the lines mS~s, and m'S'ms' ; if this lever be S'.ID- 
 posed to be disturbed in an indefinitely sn all degree, so that 
 m and m 1 describe the arcs mn and m'ti 1 respectively, and 
 perpendiculars na and n' b be drawn from n and n' , upon 
 the directions of the forces S and $'', we shall have mn~ 
 c$, and m' 5*'. Let the perpendiculars Cc arid Cd be 
 drawn to the directions of the forces S and S 1 from the ful. 
 erum 67, then as the indefinitely small arcs mn and m f t? may 
 be supposed rectilinear, and the angles Cmn and Cm'n' right 
 angles, we shall have ang. awzwzzang. mCc, and ang dCm* 
 cziang. n'm' 6 ? and consequently the rectangular triangles 
 
LAPLACE'S MECHANICS. 79 
 
 This eqnn'ion lias not only place in the case of equi- 
 librium, but it assures its existence. Suppose that the 
 
 amn, mCc, and dCm' , ri m'b respectively similar, therefore 
 by proportion 
 
 ma : mn : : Cc : Cm, 
 also 
 
 m'b : m'n' : : Cd : Cm' 
 
 which give wzflizr Cc. and m'b r Cd, consenuent. 
 (J m C m 1 
 
 ly 3s = .Cc, and s'~ -. Cd. Let these va. 
 J Cm Cm 1 
 
 of cte and ^' be substituted in the equation of virtual 
 
 velocities 
 
 observing that rz 77- ~r i and we shall find that S.Cc 
 6 m C in 
 
 S'.Cd; which h a well. known property of the lever. 
 
 In the case of the inclined plane, let the twelfth diagram 
 be supposed to represent a section formed by a plane passing 
 through the centres of gravity m and m ; , of two weights in, 
 equilibrio upon two inclined planes AB and L'C, which 
 have the common altitude ?JD, the weights being connected 
 by a s'ring passing over the pulley P. Let the position of 
 the weights be changed so that their centres of gravity m and 
 m 1 may pass through the indefinitely small spaces mn and 
 m'n 1 respectively; from m and in 1 let the lines fwzz$, and 
 m'S's 1 be drawn in the direction of gravity, and suppose 
 the weight of the body resting upon AB to be represented 
 by S and tbat of the body resting upon BC by S 1 : from n 
 and ' draw the perpendiculars na and n'b respectively to 
 the lines mS arid m'S' , or their continuations. Then as the 
 lines forming the triangle nma&re respectively parallel to (he 
 lines forming the triangle ABD, the triangles are similar : 
 
80 LAPLACE'S MECHANICS. 
 
 equation (I) having place, the points wz, m' &c. take 
 the velocities v, tf, &c. in consequence of the forces 
 
 fora like reason the triangles m' n' b and BCD are similar : 
 we hare therefore 
 
 AB : BD : : mn : ma, 
 
 EC : BD : : m' ri : m' b, 
 
 consequently ma-.mn and m 1 - ? ,m' ri . If in the 
 Ait Jj L 
 
 equation of virtual velocities 
 
 T? JTJ 
 
 as $s=ma and ^'zr m' b, their respective values -- . mn 
 
 A13 
 
 and .m'n' are substituted, we shall, as mn m' n' ; 
 
 easily obtain the following equation S.BC~S' .AB, which 
 shews that the weights have the same ratio to each other as 
 the lengths of the planes upon which they rest have f , this is 
 well known from other principles. 
 
 Lastly, in the case of the wedge, let ABC (Jig. 13), re- 
 present the section of a wedge, and the plane MN upon 
 which it rests in equilibrio, from the perpendicular pressures 
 of the forces S and S' upon its sides AB and AC^ in the di- 
 rections mS~s and m 1 S'$' respectively. Suppose that in 
 consequence of an indefinitely small variation in the situation 
 of the wedge, it takes the position abc, its sides meeting the 
 directions of the forces or their continuations inw and w', it 
 is evident that the small right lines mn and m' n' will be the 
 spaces passed over by the powers S and *S" ,1-in their respec- 
 tive directions. Join Aa^ and let CA and ba be prolonged 
 until they meet at jP, and from the points A and a let the 
 perpendiculars AH and Ag be drawn to the prolongations, 
 then we shall evidently have iririGct) and mn- 
 
LAPLACE'S MECHANICS* 81 
 
 mS, m'S', &c. which are applied to them. The sys- 
 tem will be in equilibrio by means of these forces, and 
 the forces my, m'v f , &c. denoting by $v 9 cto', &c. 
 the variations of the directions of the new forces, we 
 shall have from the principle of virtual velocities 
 
 but we have by the supposition Q=2,.mS.$s ; we shall 
 therefore have Q=-2,.m.v$v. The variations $y, W 9 
 &c. ought to be subjected to the conditions of the sys- 
 tem, they may therefore be supposed equal to ydt, v'dt 9 
 &c. and we shall have 0=2. ?wt) a , which equation gives 
 < a=0, D'=O, &c. ; therefore the system is in equilibrio 
 in consequence of the sole forces mS, m'S' &c. 
 
 As the lines aF and aA are respectively parallel to the lines 
 AB and BC, the triangles ABC and Fa A are similar, con. 
 sequently 
 
 AB : AC : : Fa : FA ,- 
 
 we have likewise, as the right angled triangles FaG and 
 FAH t by having a common angle at F 9 are similar, 
 
 aG : AH : : Fa : FA, 
 therefore 
 
 AB : AC : : aG : AH. 
 This last equation, as AH~mn~ &s and Gn m' rf. ^/ 3 
 
 A C 
 
 gives $s zn -777. $s' ; If this value of $s be substituted in 
 
 the equation of virtual velocities 
 
 we shall obtain the following, S.AC=S' .AB, ^hich shews 
 that when the wedge is in equilibrio, the powers acting upon 
 
 M 
 
82 LAPLA.CE'S MECHANICS. 
 
 The conditions of the connection of the parts of the 
 system, may at all times bo reduced to certain equa- 
 tions between the co-ordinates of its different bodies. 
 Let u=0, u f =ft, "=0, &c. be these different equa- 
 tions, we shall be enabled by No. 3, to add to the 
 equation (I) the function *..3u-\-\'.$u'-{-* r/ 3u rl -\-8cc., 
 or S. \.oit; X, X', X' 7 , &c. being indeterminate functions 
 of the co-ordinates of the bodies ; the equation will 
 then become 
 
 in this case (he variations of all the co-ordinates are 
 arbitrary, and we may equal their co-efficients to no- 
 thing ; which will give so many equations, by means 
 of which we can determine the functions X, X' &c. If 
 we lastly compare this equation with the equation (k) 
 we shall have 
 
 from which it will be easy to find the reciprocal actions 
 of the bodies w?, m' 9 #c. and the pressures 7?, R 1 ', 
 #c. that they exercise against the surfaces on which 
 they are forced to remain *. 
 
 it are to each other as the sides of the wedge to which they 
 are applied, which is a well known property of it. 
 
 The principle of virtual velocities may readily be proved 
 in the cases of tbe wheel and axle, the pulley, the screw, &c. 
 and holds good in every case of machinery in equilibrio. 
 
 * The following examples, extracted from the Mechanique 
 Analytique of Lagrange, will serve to shew the facility with 
 which the principle of virtual velocities may be applied to 
 Ihe solution of various problems. 
 
LAPLACE'S MECHANICS. 
 
 15. ]f all the bodies of a system are firmly attached 
 together, its position may bo determined by those of 
 
 It may here be premised, that the forces which act upon 
 any point or body will be supposed to be reduced to three, 
 P, Q, and R 9 acting respectively in the directions of the 
 co-ordinates ar, y^ and z 9 and tending to diminish them. 
 The quantities belonging to the different bodies will be dis- 
 tinguished by one, two, three &c. marks according to the 
 order in which they are considered. Thus the sum of the 
 moments of the forces which act upon the bodies will be 
 
 P.fcc + Q.&H-fl.fc-f-F .**' +Q' .*y -f R .$z' -f F'.^"-f-c. 
 To this must be added, the differentials of the equations of 
 condition, each multiplied by an indeterminate quantity. 
 
 Let us first consider the problem of three bodies firmly at- 
 tached to an inextensible thread. In this case, the condi- 
 tions of the problem are, that the distances between the first 
 and second, and between the second and third bodies, will 
 be invariable ; these distances being the lengths of the por- 
 tions of the thread intercepted between them. Let/be the 
 first of these distances, and g the second, we shall then have 
 5/izzO, and c^frzO, for the equations of condition, therefore 
 w / and w' o^, and the general equation of equilibrium 
 will become 
 
 The values of /and g are 
 
 therefore 
 
 5 f _(^ 
 
84? LAPLACE'S MECHANICS; 
 
 three of its points which are not in the same right line ; 
 the position of each of its points depends upon three 
 
 these values being substituted will give the nine following 
 equations for the conditions of the equilibrium of the thread, 
 
 Jl _ 
 
 
 
 M 
 
 =0. 
 
 It now remains to eliminate the two indeterminate quanti 
 ties x and A 1 from these equations, which may be done vari 
 
LAPLACE'S MECHANICS. 85 
 
 co-ordinates, this produces nine indeterminate quan- 
 tities ; but the mutual distances of three points being 
 
 otis ways, each of which will give equations, either different, 
 or presented differently, for the equilibrium of the three 
 bodies. 
 
 It is evident that if we add the three first equations to the 
 three next, and to the three last, we shall obtain the three 
 following equations delivered from the unknown quantities 
 X and A*. 
 
 P+P'+P" 0, 
 
 R+R'+R"=0, 
 
 which shew, that the sum of all the forces parallel to each of 
 the three axes of x 9 #, and s 9 should be nothing. 
 
 There now remains four more equations which it is ne- 
 cessary to discover ; for this purpose, if the three mkldlq 
 equations are respectively added to the three last, the three 
 following will be obtained, which do not contain A' . 
 
 by the extraction of A, the two following will be obtained. 
 
86 
 
 given and invariable, we are enabled by their means 
 to reduce these indeterminates to six others, which 
 
 R4-R" 
 
 x x 
 
 Lastly, considering the three final equations, by extracting 
 from them, we shall have the two following, 
 
 These seven equations contain the conditions necessary for 
 the equilibrium of three bodies, and when joined to the given 
 equations of condition u and u 1 , will be sufficient for deter. 
 mining the position of each of them in space. 
 
 If an inextensible thread be charged with four bodies, 
 acted upon respectively by the forces P, (2 5 R ', P , (2', R 1 ; 
 P" , Q" , R", &c. in the directions of the three axes of x,y y 
 and 2 ; we shall find by similar proceedings, tbe nine follow- 
 ing equations for the equilibrium of these four bodies, 
 
 Q! -f Q"+Q'" (P'+P" + F" J=*, 
 
LAPLACE'S MECHANICS. 87 
 
 substituted in the equation (I), will introduce six ar- 
 bitrary variations ; by equalling their co-efficients to 
 
 TV , ntr i pv/ 
 it + it -fit 
 
 Q"H-Q'" "V"^ 
 
 ^--SzS-^- ' 
 
 ^/// ^/f 
 
 vxirf *** 2 _ 
 
 It would be easy to extend this solution to any number 
 of bodies, or to the case of the funicular or catenarian 
 curve. 
 
 The solution would have been in some respects simplified, 
 if the invariability of the distances f g 9 &c. had been direct. 
 ly introduced into the calculation. 
 
 Thus, confining ourselves to the case of three bodies, and 
 denoting by %J/ an d 4^ the angles which the lines/ and g 
 make with the plane of x and y and by p and <p' the angles 
 which the projections of the same lines upon the same plane 
 make with the axis of ,r, we shall have 
 
 cos.xj,: z' ? jf.sin.%^; 
 
 n' 7 ' f 
 
 Substituting th values of x f , y, 2', cr f; , y", and z" obtained 
 
88 LAPLACE'S MECHANICS. 
 
 nothing, we shall have six equations that will contain 
 all <Iie conditions of the equilibrium of the system ; let 
 us proceed to dev^elope these equations. 
 
 from these equations, in the general formula of the equilib- 
 rium of three bodies 
 
 and simply causing the quantities x^ y, 2, 9, <p', -v]/, 4^, whose 
 variations will remain indeterminate, to vary, and equalling 
 separately to nothing the quantities multiplied by each of 
 these variations, we shall have the seven equations 
 
 P"sin.' Q"cos.<p' 0, 
 
 n.p'sin.4/ ^"cos.^' 0, 
 
 of which the five first coincide with those found before in 
 Che question of tkree bodies connected by an inextensible 
 thread, by the elimination of the indeterminate quantities x 
 and A'; and the two last are readily reduced by eliminating 
 Q' and Q", by means of the fourth and fifth equations. 
 
 But if by this means we have more readily obtained the 
 final equations, it is because we have employed a preliminary 
 tranformation of the variables which coatains the equations 
 
LAPLACE'S MECHANICS. 89 
 
 For this purpose, if.r, y, z be the co-ordinates of m ; 
 #',y, z' those of m 1 ; x' 1 , y" 9 z" those of m 11 , &c, \v 
 shall have 
 
 of condition, instead of immediately employing the equations 
 with indeterminate co-efficients as before, so that the equa- 
 tion is reduced to'a pure mechanism of calculation. More- 
 over, we have by these co-efficients the value of the forces 
 which the rods/ and g ought to sustain from their resistance 
 to extension, as will be shewn hereafter. 
 
 If. the first body is supposed to be fixed, the differentials 
 S#, oy, and $z vanish, and the terms affected by these differ, 
 entials, will disappear of themselves from the-general equa- 
 tion of equilibrium. In this case, the three first equations 
 
 p_ A .iJZf o, <2_ A /I^ o, and R *. Z = o, w iH 
 u u u 
 
 not have place, therefore the equations, P-fP'-fP''-^^. 
 =0, -r-<2'-r-Q"-f&c. 0, tf-f #'-fft"-h&c. 0, will not 
 have place, but all the others will remain the same. In this 
 case, the thread is supposed to be fixed at one of its ex. 
 tremities, 
 
 If the two ends of the thread be fixed, we shall have not 
 only ^r 0, fyzzO, 8 0, but also S^'&c. Q, Sy/'&c.o, 
 $:/ //&c - ; and the terms affected by these six differentials, 
 in the general equation of equilibrium, will consequently 
 disappear, as well as the six particular equations which de- 
 pend upon them. 
 
 In general, if the two extremities of a thread are not en- 
 tirely free, but attached to points which move after gt given 
 law ; this law expressed analytically, will give one or more 
 equations between the differentials &r, //, and 0.5, which re- 
 late to the first body, and the differentials Ix'U&z-, y&c.^ 
 5 ? ///&c. ? which relate to the last ; and it will be necessary to 
 add tkese equations^ each multiplied by a new indeterminate 
 
90 
 
 &c. 
 
 co-efficient, to the general equation of equilibrium found 
 above ; or to substitute in the general equation, the value 
 of one or more of these differentials obtained from the above 
 equations, and lastly to equal to nothing, the co-efficient of 
 each that remains. As this is not attended with any diffi. 
 culty, we shall omit it. 
 
 In order to discover the forces which arise from the re- 
 action of the thread upon the different bodies, we will, in 
 the present case, consider the equations 
 
 ^ ^ > -^ < - 
 
 &c. 
 
 With respect to the first body whose co-ordinates are #, 
 
 $u x l x $u y y .lu z' z 
 
 and *> s -?= - ' r,= - ~r ' and *= - 
 
 we shall therefore have 
 
 Therefore the first body will be acted upon by the other 
 bodies with a force A, the direction of which is perpendi- 
 cular to the surface represented by the equation S / 0, 
 supposing the quantities #, ^, and z to vary ; but it is evi- 
 dent that this surface is that of a sphere, having / for 
 its radius, and x',y,and z' for the co-ordinates of its centre, 
 
9i 
 If we suppose 
 
 consequently the force A will be directed along this same 
 radius, that is, along the thread which joins the first and 
 second bodies. 
 
 With respect to the second body whose co-ordinates are 
 *', y, s', we have 
 
 u _ x r x S u _ y y I u _ z' z 
 
 is7 ^f~ ) %7 7' * *T~i f * 
 ox of oy j o~ / 
 
 therefore 
 
 _ 
 
 from which it follows, that the second body will also receive 
 a force X directed perpendicular to the surface whose equa- 
 tion is ^Mzn^/nrO, supposing a?', y' y and 2' alone to vary. 
 This surface is that of a sphere, having/ for its radius, the 
 co-ordinates #, #, and z of the first body corresponding to 
 its centre ; consequently the force that acts upon the second 
 body, will be also directed along the thread /, which joins 
 this body to the first. 
 
 With respect to the second body, we also have 
 
 &/_ y 7 of 8tt'_ y y y j'_ " ** 
 far-- ~T~' y- r/ W-" "T"/ 
 
 therefore 
 
 The second body will therefore be acted upon by a force 
 qual to A', the direction of which will be perpendicular <e 
 
LAPLACES MECHANICS. 
 
 we shall have ^ 0, 5/'=0, Sf" &c. ; llic.ncces- 
 sary conditions will therefore be fulfilled, anJ in con- 
 
 the surface represented by the equation S'/~0, by making 
 x', y, and z' to vary. This surface is spherical, having g 
 for its radius ; therefore the direction of the force V will be 
 along this radius, that is, along the line which joins the se- 
 cond and third bodies. 
 
 Similar conclusions may be drawn with respect to the other 
 bodies. 
 
 It is evident that the force X which acts upon the first 
 body, along the direction of the thread which joins it to the 
 next, and the equal but directly contrary force A, which acts 
 upon the second body along the direction of the same 
 thread, are merely the forces resulting from the re-actioa of 
 this thread upon the two bodies, that is, from the tension of 
 that portion of the thread which is included between the 
 first and second bodies ; therefore the co-efficieut A will ex. 
 press the force of this tension. In like manner, the co-effi- 
 cient A 7 will express the tension of that part of the thread 
 which is intercepted between the second and third bodies, 
 and so on with the rest. 
 
 It has been supposed in the solution of this problem, that 
 each portion of the thread was not only inextensible, but 
 likewise incompressible, so that it always preserved the same 
 length, consequently the forces A, A', &c. only express the 
 tensions when they are positive, and their actions incline 
 the bodies towards each other ; but if they are negative, and 
 tend to make them separate to a greater distance from each 
 other, they rather express the resistances which the thread 
 opposes to the bodies by means of its stiffness or iiicom. 
 pressibility. 
 
 To confirm what has been demonstrated, and at the same 
 time to give a new application of these methods, w will sup-. 
 
LAPLACE'S MECHANICS. 93 
 
 sequence of the equation (I) the following may bs 
 obtained 
 
 pose that the thread to which the bodies are attached is elastic, 
 and susceptible of extension and contraction, and that jP, ?, 
 c. are the forces of contraction of the portions/, g-, &c. of 
 the thread intercepted between the first and the second bodies, 
 and between the second and the third &c. 
 
 It is evident from what has preceded, that the forces F^G 
 &c. will give the moments F.y+G.$<r-{-&c. or X.^M-fV.^' 
 -f &c. 
 
 It is therefore necessary to add these moments to those 
 which arise from the action of the forces which are repre. 
 setited by the formula P. frr-f-Q.fy-f- ??. <i*-{-P .^.v ! -%-Q ! .$y' --]- 
 R'.$z' + P".3x"-}-&c. ; and as there are no other particular 
 conditions to fulfil relative to the situation of the bodies, 
 the general equation of equilibrium will be as follows P. fa 
 
 c.r=0. 
 
 By substituting the values of S/, 5^, &c. found above, and 
 equalling to nothing the sum of the terms affected by each of 
 the differentials Jr, 2[y, &c. we shall have the following equa. 
 tionsfor the equilibrium of the thread in the present case, 
 
 R- 1 
 
 P+ 
 
 
94 LAPLACE'S MECHANICS. 
 
 0=2.mS. I ~ ); 
 
 \ J 
 
 !-); Cm) 
 
 
 These equations are analogous to those of the case ia which 
 the thread is inextensible, and give by comparison xrzF, 
 X'm6?, &c. 
 
 It therefore appears that the quantities F 9 6r, &c. which 
 here express the forces of threads supposed elastic, are the 
 same as those which have been found before to express the 
 forces of the same threads, on the supposition that they were 
 inextensible. 
 
 Let us resume the case of an inextensible thread charged 
 with three bodies, supposing at the same time that the second 
 body is moveable along the thread ; in this case the condi- 
 tion of the problem will be, that the sum of the distances be- 
 tween the first and second bodies, and between the second and 
 third is constant : denoting as before, by / and , thest 
 
LAPLACE'S MECHANICS. 95 
 
 which are three of the six equations that contain the 
 conditions of the equilibrium of the system. The se- 
 
 distances, we shall have/-f g- equal to a constant quantity, 
 and consequently eJ/'+^zzO. 
 
 In this case $f+$g=z$u, and consequently *($f+$g ) or 
 A&M must be added to the general equation of equilibrium, 
 which will become 
 
 If the values of $f and $g are substituted, and the sum of the 
 terms which are multiplied by the differentials $x 9 2^, &c. 
 equalled to nothing, the following equations, which are suf- 
 ficient for the equilibrium of the thread, will be obtained, 
 
 
 R A. rr^zrO, 
 
96 LAPLACE'S MECHANICS. 
 
 cond members of these equations are the sums of the 
 forces of the system resolved parallel to the three axes 
 
 Z H _ J 
 
 #"+x. - -=0. 
 g 
 
 It is only necessary to extract the indeterminate quantity X 
 from these equations. 
 
 From the above examples it is easy to perceive how we may 
 extend the question to a greater number of bodies, of which 
 the end ones may be supposed fixed, and the others moveable 
 along the thread. 
 
 Let us now suppose that the three bodies are united by in- 
 flexible rods, and obliged always to remain at the same dis. 
 tance from each other ; in this case, if h be supposed to 
 denote the distance between the first and third bodies, we 
 shall have S/znO, %zr 0, and S^zzO ; consequently by having 
 three indeterminate co-efficients, the general equation of 
 equilibrium will become 
 
 u+*! M +*".W= o. 
 The values of /, and ,g-, or "Su and $u' have been given be. 
 fore ; that of SA, as 
 
 will be 
 
 ^_(^-^)(^ ; - 
 
 By making these substitutions, and equalling to nothing <h 
 sum of the terms affected by each of the differentials $x 9 ty 
 &c. we shall obtain the nine following particular equations 
 
LAPLACE'S MECHANICS. 
 
 .r, y, and s : each of \vhich suras should be equal (o 
 nothing in (ho case of equilibrium. 
 
 K'+x. ~.x'-^i 0, 
 
 It will be necessary to extract from them the three unknown 
 indeterminate quantities A, X' , and x", by which means six 
 equations will be obtained to determine the conditions of 
 equilibrium. 
 
 It is evident from the form of these equations, that by add. 
 ing respectively the three first to the three next, and after. 
 wards to the three last, the three following equations will be 
 obtained, which are free from the quantities X, x', andx", 
 
 = 0. 
 Is would be very easy to find three other equations by 
 
98 
 
 The equations *f=0, y=0, */"=0, &c. will be 
 also satisfied, if -we suppose 3, s', s", &c. invariable 
 quantities, and if we make 
 
 extraction of A, A', and x" ; but this may be done in a much 
 readier and more general manner, by deducing these nine 
 equations from those given above. 
 
 / 
 
 S 
 
 These equations are evidently analogous to the primitive 
 ones, and give in the same manner, by addition, the three 
 following equations, 
 
 p^Qa+py _Q V + P////_ Q" ^ =0, 
 PzRx + P' ^H V+P V Jt^ 0, 
 
LAPLACE'S MECHANICS. 
 
 &c. &c. 
 
 The three first equations shew, that the sum of the forces 
 parallel to each of the three axes should be nothing, and the 
 three last contain the known principle of moments (under- 
 standing by that term the product of the power by its dis- 
 tance, as in the case of the lever), from which it appears, 
 that the sum of the moments of all the forces to make the 
 system turn about each of the three axes, is likewise nothing. 
 
 If the first body be fixed, the differentials &e, Sj/, and 5s 
 will vanish, and the three first of the nine equations first 
 given will not have place ; we shall in this case have only six 
 equations, which by the extraction of the three indeter- 
 minates A, V, aud A", maybe reduced to three. 
 
 In order to obtain these three equations we may proceed 
 in a manner analogous to that made use of to discover the 
 three last equations of the preceding question; provided 
 we take care that the transformed ones do not contain the in- 
 determinates X, and V, which enter into the three first, of 
 which it is necessary to make abstraction. This will be ob- 
 tained by the following combinations. 
 
 s 
 
 \ /.v,' r\f*!i ->t\ 
 
 ) \* ^ /\* ~ ) 
 
 _ ~_ =0 , 
 
 
100 LAPLACE'S MECHANICS. 
 
 Ssr being any variation whatever. By substituting 
 these values in the equation (I), we shall have 
 
 0=2.mS. ? r 
 
 If we add the three first of these equations to the three last, 
 the three following will be obtained, 
 
 0, 
 
 These will always have place, whatever may- be the state of 
 the first body, as they are independent of the equations rela. 
 tive to it. These equations contain the principle of mo- 
 ments, with respect to the axes passing through the first 
 body. 
 
 Let us suppose a fourth body attached to the same in. 
 flexible rod, having ,.?'% y" 1 9 and z'" for its three rectangular 
 co-ordinates, and P' /; , Q //f , acd R" 1 for the three forces pa- 
 rallel to these co-ordinates. 
 
 It will in this case, be necessary to add the quantity 
 P"'M"+@". S/'-fJi'".^" to the sum of the moments of 
 the forces. As the distances between all the bodies ought 
 to remain constant, we shall have by the conditions of the 
 problem, not only 5/rrO, ^rzO, tMizzOj as in the preceding 
 case, put also SfcO, S^zzO, and JwzzO ; naming the dis. 
 tancvs of the fourth body from the three others 1 9 m, and n. 
 The general equation of equilibrium will in this case be. 
 come 
 
 *.^ 
 
101 
 
 If, is evident that we may in this equation change either 
 the co-ordinates #, #', x", &c. or y^y'^y"^ &c. into 
 
 The values of 5/, , and oA, are the same as before, and 
 (hose of SJ, a??!, and Sra. or 5 v/ , &'*, and Sw v , as 
 
 are 
 
 
 m 
 
 By making these substitutions, and equalling to nothing the 
 sum of the terms of each of the differentials &*?, ^//, &c,, we 
 shall find twelve particular equations ; (he nine first of which 
 which will be the same as those in the case of three bodies 
 if the following quantities were respectively added to their 
 fir.st members. 
 
 fa ii^ i' y~y 'n ~ ! '_ z , 
 
 A . , \ !l / J * S 
 
 x ffl x ' ty / ? //y ' z > 
 
 9 
 
 -XV . 
 
 and tbe three last will be 
 
102 I/AFLACE'S MECHANICS. 
 
 , %', z", &c. which will give two other equations that 
 re-united to the preceding, will form the following 
 system 
 
 ; (n) 
 
 / 
 
 As there are twelve equations in all, and six indeter- 
 minate quantities x, A', ^", ^ 7/ 5 A iv 5 x v , to eliminate, there 
 will only remain six final equations for the conditions of 
 equilibrium, as in the case of three bodies ; and we shall find 
 by a method similar to one given before, these six equations 
 analogous to those found in that case, 
 
 P+P'+P+P f "=0, 
 Q+Q'+Q''+Q'=0, 
 
 QW+pny Q"i x > 
 =0, 
 
 =0. 
 
 Instead of the three last, the three following equations 
 may be substituted, which can be found by a method given 
 before. As they are independent of the equations relative 
 to the first body, they possess the advantage of always having 
 place, whatever may be the state of this body. 
 
LAPLACE'S^MECHANICS. 103 
 
 the function S.wS.y. ( ^- )Iis by No. 3, the sum of 
 
 \o X J 
 
 the moments of all the forces parallel to the axis of #, 
 
 Q"(s'z)R'"(y'yy=S). 
 
 Let us now consider the case of three bodies joined by a 
 rod which is elastic at the point where the second body is 
 situated, the distances between it and the other bodies being 
 constant, but the angles which the lines form variable. Let 
 us suppose that the force of elasticity, which tends to augment 
 the angle formed by the lines which join the second body to 
 the two others, is represented by J3, and the exterior angle, 
 formed by one of these sides and the prolongation of the 
 other, by e ; then the moment of the force E ought to be re- 
 presented by E3e 9 or its equal rt'.W ; therefore the sum of 
 the moments of all the forces of the system, as o/ 5 ^gzzO, 
 will be 
 
 It is now only required to substitute the values of Sw, */-, 
 and lu" : those of JM and Sw' are the same as in the first 
 question, but with respect to that of u" or Se, it may be 
 observed, that in the triangle, of which the three sides are 
 jf, g, and 7i, or the distance of the first body from the third, 
 180 e is the angle opposite to the side h ; therefore by 
 
104 LAPLACE'S MECHANICS. 
 
 which would cause the system to revolve about the axis 
 
 of s. In like manner, the function 2. mS.x.(~} is 
 
 V,yy 
 
 f a -f a-2 _ /jZ 
 
 trigonometry cos.d - -^ - j which by differentia. 
 
 tion gives the value of $eor ou" : as by the conditions of the 
 problem 2/ 0, SginO, it will be sufficient to make e and k 
 
 vary, we shall therefore have oe ^ 7/ ~ -- -' . This 
 
 Jg-sin.e 
 
 value being substituted in the preceding equation, it 
 evidently become of -(he same form as the general equation 
 of equilibrium given in the case of three bodies joined to. 
 gether by an inflexible rod ; by supposing in it that V'zz 
 
 Eh 
 
 - : - : the particular equations will necessarily be the 
 fg sm.e 
 
 same in the two cases, with this sole difference, that in the 
 rase above mentioned, the quantity A" is indeterminate, and 
 consequently ought to be eliminated ; but in (he present 
 case it is known, and there are only two quantities A and A 1 
 to eliminate, consequently there will be seven final equations 
 instead of six. But whether the quantity X" is known or 
 not, it may be eliminated along with the two others, X, and 
 M ; we shall therefore have, in the present case, the same 
 equations as were found in the case of three bodies at. 
 tached to each other by an inflexible rod : to find the 
 seventh equation it will only be necessary to eliminate /- 
 from the three first, or M from the three last of the nine par. 
 ticular equations of the above case, and to substitute for 
 
 1?7i 
 
 t*u jt value -v. 
 jg siu. e 
 
 If S/ and <? had not been supposed to vanish in the value 
 
LAPLACE'S MECHANICS. 105 
 
 the sum of the moments of all the forces parallel to the 
 axis of ^/, which would cause the system to turn about 
 the axis of % 9 but in a different direction to the first 
 forces; the first of the equations (n) consequently in- 
 dicates, that the sum of the moments of the forces is 
 
 of &?, it would have been of this form !$e nr 7 : -- 4- 
 
 Jgsm. e 
 
 , A and B being functions of/, g, h, and sin. e ; 
 in this case, the three terms X.&M+X' .SM'+X".^'' of the ge- 
 neral equation, would become (EA-\-x). JM-J- (EB-\-*.'J 
 
 . u ' - : --- ,5A : but A and V being two indeterminate 
 /T sin. e 
 
 quantities, it is evident that x E^. and x' EB may be 
 substituted for them, by which means the quantity treated 
 
 upon will become x.&j-f A'.J^' - - .5 A, the same as 
 
 fg sm.e 
 
 when /and 5- did not vary in the expression of Hie. 
 
 If many bodies be supposed to be joined together by elastic 
 rods, we shall find, in the same manner, the proper equa. 
 tions for the equilibrium of these bodies. The above me- 
 thods will always give with the same facility, the conditions 
 of the equilibrium of a system of bodies connected together in 
 any manner, and acted upon by any exterior forces whatever. 
 The proceedings are always similar, which ought to be re. 
 gardedasone of the principal advantages of this method. 
 
 The following question, and several others, are likewise 
 solved in the Mechanique Analytique of Lagrartge. To 
 find the equilibrium of a thread, all the points of which 
 are acted upon by any forces whatever, and which is sup. 
 posed perfectly flexible or inflexible, extensible or inexteo* 
 *ible ? elastic or inelastic. 
 
 p 
 
106 
 
 nothing with respect to the axis of s*. The second and 
 the third of these equations indicate in a similar man- 
 
 * If the system be at liberty to turn in any direction about 
 a point taken for the origin of the co-ordinates, the instan. 
 taneous rotations about the three axes, may be considered 
 in the following manner ; which will give three equations of 
 rotation with respect to these axes, similar to those of 
 Laplace. 
 
 Let p represent the projection of a line drawn from the 
 origin of the co-ordinates to the body m, / that drawn with 
 respect to the body m' 9 &c., also <p the angle which the 
 line p makes with the axis of x 9 <p' that which f makes with 
 the same axis, &c. we shall have the following equations 
 orrzrp. cos.tp, f/zrp. sin. 9, a//. cos. p', #'/. sin. <p', &c. 
 which by differentiation and substitution, if p, p ; , p" &c. be 
 supposed constant, will give 
 
 S<p', Sp' 7 , &c. being each of them supposed equal top, as the 
 bodies m, m 1 , m a , &c. are imagined to be invariably con. 
 nected. 
 
 These variations of #,#, # r , y' 9 &c. are owing to the ele- 
 mentary rotation p about the axis ofz. 
 
 In a similar manner, if \J/, 4/ , -^"^ &c. represent the an. 
 gles which the projections upon the plane yz of lines drawn 
 from the centre of the co-ordinates to the bodies m, m' 9 m n r , 
 &c. respectively make with the axis 3^, the variations of#, ay 
 y, s', &c. arising from the elementary rotation 4/.about the 
 axis of a? may be obtained, which will give the following 
 equations, 
 
 fy= ~.^, *z=#.N', V= ~-W> ^=^'.^, &c. 
 
 Likewise if <v, y, w", &c. represent the angles which the 
 projections upon the plane xz of lines drawn from the centre 
 of the co-ordinates to the bodies m, 
 
LAPLACE'S MECHANICS. 107 
 
 ner, that the sum of the moments of the forces is no- 
 thing, either with respect to the axis of ?/ 9 or to the axis 
 
 with the axis 2, the variations of a?, 2, x r y s', &c., arising 
 from the elementary rotation w about the axis of ^, will 
 give 
 
 te= ff.Sw, tezs.So/, fc/zz o/.S*/, ^'-zzs'.S*;, & c . 
 If the three rotations take place at the same time, the whole 
 variations of the co-ordinates #, y, 2, #', y, 2', &c. will b 
 equal to the sums of the partial variations belonging to each 
 of these rotations, consequently we shall have the following 
 equations, 
 
 ^ ; s'.Sw -y.^, sy ic'.Jtp s'.^, ^'y.^ ^.So/. 
 
 If these values be substituted in the general formula of equi. 
 librium Z.S.cfciO, we shall obtain the terms belonging to 
 the rotations (>, $, and 5x}/ about the the three axes of 2, 
 ^j and a? ; which ought to be separately equal to nothing > 
 when the system has liberty to turn in any direction about a 
 point placed at the origin of the co-ordinates. The equation 
 2.*$*.^? 0, by substitution, gives the following 
 
 in which 
 
 The co-efficients of the instantaneous rotations \J/, &w, and 
 5<p, are the moments relative to the axes of these rotations, 
 and are respectively equal to nothing in the case of equilib- 
 rium, when the system has liberty to turn about the origia 
 of theco-ordinates. 
 
108 LAPLACE'S MECHANICS. 
 
 of x. By uniting these three conditions <o those in 
 
 If at any point of the system the co-ordinates #, y, and z 
 be respectively proportional to ^, w, and &>, we shall 
 have there 
 
 x3tac-v$$) x.$$zM, ^.5\I/zrj?.5w, 
 and consequently SarmO, ^i=0, and 02 0. 
 This point and all others which have the same property, \\ill 
 consequently be immoveable during the instant that the 
 system describes the three angles eNJ/, 5o>, and <^>, by turning 
 at the same time about the three axes of a?, y and *. 1C 
 may be easily proved that all the points which have this 
 property ar* in a right line passing through the origin of the 
 co-ordinates. The co-sines of the angles A 5 /* ? and v which 
 it makes with the axes of a?, y and z, are 
 
 that by substitution will respectively become 
 
 -/ ^o> 
 
 an 
 
 ^4,_j_^_j_;- 
 This right line is the instantaneous axis of the composed 
 
 rotation. 
 
 If we suppose S0:=v'( r & < 4' a -{-&*+&<?*) we shall have 
 
 .., ../A, .., 
 
 which, by substitution in the general expressions of Jar 
 and S^j will give 
 
 ^n:(^. cos. //,/. cos. y)S0, 
 
 These values being substituted in the expression 
 
 Js a , which is the square of the indefinitely short space passed 
 
LAPLACE'S MECHANICS. 109 
 
 which the sum of the forces parallel to these axis are 
 
 over by any point whatever, it will be changed into the 
 
 following 
 
 {(zcos.p < ?/cos.v) 3 -}-( rcos ' v 2COS.A) 2 -f-(ycos.A o 
 
 as cos. 2 A-{~cos. 2 /A-fcos. 2 v 1. 
 
 It may readily be proved that orcos.A-f-^cos.^-f-scos.y 0, is 
 the equation to a plane passsing through the origin of the 
 co-ordinates, in a direction perpendicular to the right line 
 which makes the angles A, ;*, and v, with the axes of #, i/ 9 
 and s, conse.quently the short space described by any point 
 
 of this plane will be <$Q\/ x z -f-#*-f-2. As the axis of rota- 
 tion is perpendicular to this plane, S0 will represent the an. 
 gle of rotation about it composed of the three partial velo- 
 cities 4/,c^and p, about the three axes of the co-ordinates. 
 It therefore follows, that any instantaneous rotations 4/, 
 Sw, and $<p about three axes which cut each other at right 
 angles at the same point, may be composed into one Sfl- 
 about au axis passing through the same 
 
 point of intersection, and making with them the angles A, 
 p and 9. so that 
 
 Inversely, any rotation $9 about a given axis may be resolved 
 into three partial rotations, denoted by COS.A.^, cos. /.,#, 
 and cos.v.^Q, about three axes, which cut each other per- 
 pendicularly in a point of the given axis, and make with it 
 the angles A, /*, and . The above enable us in a very easy 
 manner to compose and to resolve the instantaneous move. 
 merits or velocities of rotation. 
 
 Let three other rectangular axes be taken which make with 
 the axis of rotation $4/ the angles A',*/ 7 , and A" 7 , with the axis 
 of rotation ^ the angles y! jf* 77 ,/*'' 7 . and with the axis of rota- 
 
110 LAPLACE'S MECHANICS; 
 
 nothing with retpect to tach of them ; we shall have 
 
 tion Sp the angles v', ", and v ;// ; the rotation S>J, may be 
 resolved into three rotations cos.A'.$4/, cos.A".^, and cos.' 
 V'.cNj/ about these new axes, the rotation Sw may likewise 
 be resolved into three rotations cos. n'.'Sw, cos. /^ // .w ) and 
 cos.j/'.Jw, and the rotation p into three rotations cos.v'Jp, 
 cos./.^, and cos. ' /; .Sp about the same axes. By adding 
 together the rotations about the same axis, if we name S0 T , 
 0", and W the complete rotations about the three new 
 axes we shall have 
 
 "' cos.x".5vI/-J-cosV.Sw4- c 
 
 The rotations 5>J/, 5w> and <p, are by this means reduced to 
 three rotations S0', 50", and 50"' about three other rectan, 
 gular axes, which should consequently give the same rota, 
 tion $9 that results from the rotations S>J/, 5a/, and ^, we 
 shall therefore have 
 
 S0 2 ;n S0' z + S0"* -f ^"/ 2 H 2 H- $*>* -f 5 ?> 2 ; 
 as this equation is identic, by substituting for^ 2 , $0 //z , 
 S0"" 2 their values given above, the following conditional 
 equations \\ill be obtained, 
 
 cos. 2 A'+cos. 2 *"-fcos. 2 A'/ ; 1, 
 
 COS.V 
 
 cos.V.cos.y'-f cos.x /7 .cos.v v 4-cos x'/'.cos.,/' 7 0, 
 
 COS.// 
 
 The three first are the respective co-efficients of S4/, eta, 
 and ^, which must each of them be equal to unity, and the 
 three last the respective co-efficients of 2&0'.<ty, 2$0M*i 
 and 25S //; .^, and consequently should each of them vanish 
 that the equation may be identic. 
 
LAPLACE'S MECHANIC*. Ill 
 
 the six conditions of the equilibrium of a system of 
 bodies invariably connected together. 
 
 By means of these relations, the values of ^, Ja/, and tip 
 may be obtained, in terms of 80', S0", and W', by adding 
 together the values of $ff , W, and W" 9 multiplied succes- 
 sively by cos, A', cos.?/ 7 , cos.x'", cos.///, cos.//,", &c., which 
 will give 
 
 If the angles which the composed rotation S0 makes with the 
 axes of the three partial rotations 80', $0", and S0 ;// are de- 
 noted by TT, TT', and V 7 , we shall have 
 
 5fl'=cos. w '-.80, Sfc:cos.90, and S0 ;// :z=cos.7r".S0, 
 and if in the before given values of $0', S0' 7 , and S^ 7 ', there 
 are substituted for ^4/, w, and p, their values cos. A 9, 
 cos./x,.S9, and cos.v,9 ; the comparison of these different ex- 
 pressions of S9', S9" , and W will give, when divided by 9, 
 the following new conditional equations, which may be geo- 
 metrically demonstrated. Vide No. 29 Notes. 
 
 cos.7r / zi:cos.A < cos.^'-|-cos.^.cos./!*'-j- cos '''.cos.>', 
 
 COS.-Tr^ZZCOS.A.COS.^-f-COS./X.COS.^-j-COS.V.COS.y' 7 , 
 
 The above proof shews that the compositions and the resolu- 
 tions of the movements of rotation are analogous to those of 
 rectilinear motions. For, if upon the three axes of the ro- 
 tations of 4") <ta, and p, from their point of intersection, 
 three lines be taken respectively proportional to <ty, $u 9 
 and &p, and a rectangular parallelepiped be constructed 
 upon them, it is evident that the diagonal of this parallele- 
 piped will be the axis of the composed rotation 9, and will 
 be at the same time proportional to this rotation. 
 
 From this, and from the consideration that the rotations 
 about the same axis may be added to or subtracted from each 
 
If the origin of the co-ordinates is fixed and attached 
 
 other, according as they are made in the same or in oppo. 
 site directions, it may bo concluded in general, that the 
 composition and the resolution of the movements of rotation 
 follow similar laws to those of the composition and tie reso- 
 lution of rectilinear motions, by substituting for the move. 
 ments of rotation, rectilinear motions along the directions of 
 the axes of rotation. Vide the Mechanique Analytique of 
 Lagrange, from which the greater part of the above has been 
 extracted. " 
 
 Let SO', 59 /; , SO'", &c. represent any indefinite number of 
 rotations about their respective axes, these may be com- 
 posed into one o9, about an instantaneous* axis of rotation : 
 for if from the point where all the axes cut each other, three 
 rectangular axes be taken, each of the rotations may be 
 resolved into three about these axes, and by adding or sub- 
 tracting, as the rotations are in the same or in contrary 
 directions, there will be three resulting rotations uhich 
 may be composed into one about an instantaneous axis of 
 rotation. Thus if ^'f*>' 9 andSp' represent the three partial 
 rotations about three rectangular co-ordinates, into which 
 the rotation S3' has been resolved, $W 9 ow v , and p v , those 
 about the same axis into which the rotation <^ /; has been re- 
 solved, &c. the following equation may be obtained, 
 -T- &c.) 2 -f (Sw'-r-Sft/'-f $ 
 
 If in the formula Z/.84/+M.&ft/ + JY.<?, uhich contains the 
 terms due to the rotations J^, ow,^and ^?> in the general for. 
 mulaS.^S'.Ss'-f^.V-f&c., the values of H, ^, and 
 op found above be substituted, it will be changed, into the 
 following, 
 
 4 (L.cos.x" +M.cos.j* // +iyr.cos.v' / )*6* 
 +(L cos.*'"-f M cosV + A'.cos.v'" )Jfi' /; . 
 The co-efficients of the elementary angles 0' ? JO", and 
 
LAPLACE'S MECHANICS. 113 
 
 invariably to the system, it will destroy the forces pa- 
 
 express the suras of the moments relative to the axes of the 
 rotations 9', 59'', and $0'". From the above it appears that 
 moments equal to L, M 9 and N, relative to three rectan- 
 gular axes will give the moments 
 
 relative to three other rectangular axes, which respectively 
 
 '" 
 
 make with these the angles/, ^ ' ; A", 
 
 A geometrical demonstration of this theorem is given by 
 Euler in the seventh vol. of the Nova acta of the Academy 
 of Petersburg* 
 
 If the rotations $4,, $, and $p are supposed to be pro- 
 portional to JL, M, and N 9 and we make 
 
 the following equations will have place 
 
 Llf.cos.x, MH.cos.p,, A^ f,cos. y , 
 
 and the three moments will be reduced to this simple form 
 
 //.COS.TT', //.COS.TT", //.cos.w'". 
 
 But TT', 7T 7 , and w "' are the angles which the axes of the ro. 
 tationsSS', 59 /y , and ^ //; form with the axis of the composed 
 rotation $0, if therefore we make the axis of rotation $ff- 
 coincide with the axis of rotation 59, then Tr'nzO, and K" and 
 of" are each equal to a right angle, consequently the moment 
 about this axis will be H 9 and those about the two other 
 axes perpendicular to it will be nothing. We may there. 
 fore conclude that moments respectively equal to L M 
 and A 7 , and relative to three rectangular axes #, y, and * 
 may bo composed into one, //, equal to \/(L 2 -{-M 2 -)- N z ) 
 relative to an axis which makes with them the angles A, p. 
 and v ? so that 
 
 L M N 
 
 COS.A=:~, cos. ^zz , cos. =jf. 
 
 The sum of the moments relative to this axis is a maximum 
 
 Q 
 
114: LAPLACE'S MECHANICS. 
 
 rallel to the three axes ; and the conditions of the equi- 
 
 the tangent of the angle that it makes with the plane x i/ is 
 
 N 
 ~* --- -, ami the tangent of the angle which the projec- 
 
 tion of the axis upon that plane makes with the axis of #, is 
 
 M 
 
 equal to -^r-. 
 Ju 
 
 It is evident from the above that the composition of mo. 
 ments follows the same laws as that of rectilinear motions. 
 It may be immediately deduced from the composition of in. 
 stantaneous' rotations, by substituting the moments for the 
 rotations which they produce, in the same manner as forces? 
 can be substituted for right lined motions. Vide the Me. 
 chanique Analytique of Lagrange. 
 
 Those who are desirous of further information respecting 
 the composition and the resolution of moments, may consult 
 the writings of Euler, Prony, Poisson, &c. also a memoir 
 by Poinsot, in the 13th Cahier of the Journal de 1'Ecole 
 Polytechnique. 
 
 If the moments of the forces which act upon a system be 
 taken directly with respect to a point at the origin of the 
 co-ordinates, they will follow the same laws with respect to 
 different planes, as the projections of areas upon them, thus 
 
 for instance, %. ( ^ J r. i ^ J may No. 3 be 
 
 posed into a single moment with respect to the origin of the 
 co-ordinates. This moment will evidently be the product 
 of the projection of the force S upon that plane multiplied 
 by the perpendicular drawn from the origin of the co-ordi- 
 nates to its direction, and may therefore be represented by 
 an are& equal to twice the area of a triangle, having the pro- 
 jection of a line representing in quantity and direction the 
 force S for its base, and the origin of the co-ordinates for 
 its summit. 
 
 It therefore follows, that the properties of moments with 
 rtspect to a fixed point are similar to those of plane surfaces. 
 
 com. 
 
LAPLACE'S MECHANICS. 115 
 
 librium of a system about this origin, will be reduced 
 
 I shall mention a few circumstances concerning them, re- 
 ferring the reader to No. 21, and the notes accompanying 
 it, which may be read independent of the other parts of the 
 work, from which the following properties may be deduced. 
 Suppose a number of areas represented by A^ A', A", &c. 
 ure in a plane passing through the origin of the co-ordinates, 
 let Z>, &', 6 7 , &e. represent these areas projected upon three 
 rectangular planes passing through the origin of the co-or- 
 dinates, and c, c'j and c' 1 represent the projections of the 
 areas upon three other rectangular planes passing through 
 the same point, then by No. 21 
 
 2 + fr/a + V'zc* -{- C l * -f C //2 , 
 
 consequently 
 
 When b' and b" vanish, the value of b is evidently a maxi- 
 mum, and the line which is perpendicular to it at the origin 
 of the co-ordinates may be found from the following equa- 
 tions, in which a, /3, and y represent the respective angles 
 that it makes with the rectangular co-ordinates x, y, and 
 of the planes containing the areas <?, c'j and c". 
 
 c" 
 
 -/(c+e/' 
 
 The absolute position of the plane of the greatest sum of th 
 projections of the areas is indeterminate in space, as the 
 projections are the same upon all the plauesthat are parallel 
 to each other. 
 
 The sum of the projections of the areas are the same for 
 very plane which is equally inclined to that of the greatest 
 projection, for if / denote the angle which any plan* having 
 
116 LAPLACE'S MECHANICS. 
 
 to the following, that the sum of the moments of the 
 
 the sum of the projections upon it represented by a makes 
 with that of the greatest projection, we shall have 
 
 a\/(^ -f c' 2 -|- c" 2 ) . cos./, 
 therefore if/ be invariable a is also. 
 
 If D represent the sum of the areas upon any plane, E , ' 5 
 and f 7 ' the angles which a perpendicular to it at the origin of 
 the co-ordinates makes with the axes s, y, and x respectively, 
 and A, By and C the sums of the projections of the areas 
 upon the three planes a^r, xz 9 and yz of the co-ordinates ; 
 then the following equation may be easily demonstrated. 
 
 If the areas A, A', A"^ Sec. are supposed to be respectively 
 double the triangles which have lines representing in direc- 
 tion and magnitude, the forces 5, #', S", &c. for their bases, 
 and the origin of the co-ordinates of their points of appli- 
 cation for their common vertex, then if N denote the mo- 
 ments of the forces projected upon the plane xy, it will re- 
 present also the sum of the projections of the areas A^ A ', 
 A", &c. upon the same plane. In like manner, if M andjL 
 represent the moments of the forces projected upon the 
 planes xz and yz, they will also represent the projections of 
 the areas A, A r , A", &c. upon these planes. It is therefore 
 evident that the three quantities!,, 3f, and JV, and analogous 
 quantities relative to the same system of forces, and to other 
 planes, will have the same properties as the sums of the pro- 
 jections upon those planes. 
 
 In the above the origin of the co-ordinates, or the centre 
 of moments, is supposed to be invariable, and the forces S 9 
 &', S 1 ', &c. to be resolved in directions parallel to there, 
 spective planes, and to be moved parallel to themselves to 
 these planes, and to act along them. 
 
 It therefore follows, that if the sums of the moments of 
 the forces S 9 ", S H 9 &c. resolved along the three planes of 
 the co-ordinates be known, the sum of the moments of the 
 same forces resolved along any other plane passing through 
 
117 
 
 forces which would make it turn about the three axes, 
 be nothing relative lo each of them *. 
 
 the centre of moments will be known from the following 
 equation 
 
 D .V.cos.s-f M.cos.s' + L.cos.s", 
 
 in which D represents the sum sought, and s, s', and z fl the 
 angles which a perpendicular to any plane from the centre of 
 the co-ordinates respeciively makes with the axes of the co. 
 ordinates ~, y^ and x, 
 
 The sum of the moments with respect to the plane of the 
 greatest sum of the moments is represented bj y'/f 2 -j- -U 2 -f N*~ 
 and that of any plane making the ange /with it is equal to* 
 V'L* r M* + JVXcos./, if the angle / be a right angle then 
 cos.fciO, and the sum of the moments relative to this plane 
 vanishes. 
 
 If , jS, and y represent the angles which the perpendicular 
 to the plane of the greatest sum of the moments makes with 
 the axes of the co-ordinates #, j/, and 3, its position will be 
 determined by the following equations 
 
 L 
 
 M 
 
 s '~ 
 
 If lines be taken upon the perpendiculars to each plane 
 from the centre of moments proportional to the sums of the 
 moments upon these planes, the line representing the great- 
 est sum will be the diagonal of a parallelepiped constructed 
 upon the lines representing the the three sums L, M and N. 
 
 The composition of moments, theiefore follows tiie sasne 
 laws as that of forces, the greatest sum and the perpendi- 
 cular to its plane having place instead of the resultant and 
 its direction. 
 
 * If the forces are all supposed parallel to each o(her and 
 their directions to make the angles # ; j3, and y \uh the 
 
LAPLACE'S MECHANICS. 
 
 Let it be supposed that the bodies ra, m' 9 m", Sec. are 
 only acted upon by the force of gravity : as its action 
 
 co-ordinates #, y and z respectively; the equations (n) may 
 be changed into the following 
 
 cos.a. 
 
 cos. y. .?,$ j/ cos.fi.'L.mSzj 
 
 .the third of which is a consequence of the two first; but a 
 by trigonometry cos. 2 a-f-cos. 2 <3 + cos.*y 1, we may deter. 
 mine from these equations the angles x 3 /3, and y. By sup. 
 posing, for abridgment, 
 
 m"S"y' -f &c. M, 
 Z.mSs=:mSz + m'S' z 1 + m"Sz f/ + &c.=N, 
 the following equations will readily be found, 
 
 L M 
 
 N 
 
 COS.y - 
 
 The position of the bodies being given with respect to 
 the three axes, it is necessary in order that all motion of 
 rotation may be destroyed, that the system should be 
 placed relative to the direction of the forces, in such a 
 position as to cause the direction to make with the three 
 axes the angles determined above. 
 
 If the quantities .L, j!f, andW vanish, theangles a, /S, and 
 y will remain indeterminate, and the system will be in 
 equilibrio in any position. Therefore, if the sum of the 
 products of parallel forces by their distances from three planes 
 perpendicular to each other be nothing with respect to each 
 of these planes, the effect of the forces to turn the body 
 about the common point of intersection of these planes wUl 
 be nothing. 
 
LAPLACE'S MECHANICS. 
 
 is the .same upon all the bodies, arid as the directions of 
 gravity may be conceited to be the same in all the ex- 
 <ent of the system, we shall have 
 
 the three equations (n) will be satisfied, whatever may 
 be the direction of s, or of gravity, by means of the 
 three following 
 
 0=2.7w.r; 0=2,fny/ 0=2.mz. (o) 
 The origin of the co-ordinates being supposed fixed, 
 it will destroy parallel to the three axes, the forces 
 
 , re- 
 
 ipectively; by composing these three forces, we strati 
 have a resultant equal to S.S.my that is, equal to the 
 weight of the system. 
 
 This origin of the co-ordinates, about which we Here 
 suppose the system to be in equilibrio, is a very re- 
 markable point in it, on this account, that being sus- 
 tained, if the system is supposed to be only acted upon 
 by the force of gravity, it will remain in equilibrio, 
 whatever situation we may give it about this point ; 
 which is called the centre of gravity o'f the system * Its 
 position is determined by the condition, that if wemake 
 any plane whatever pass by this point, the sum of the 
 products of each body by its distance from the plane is 
 nothing ; for the distance is a linear function of the co- 
 
120 LAPLACE'S MECHANICS. 
 
 ordinatcs x^ y and s of the body* ; if it be therefore 
 multiplied by the mass of the body, the sum of these 
 products will be nothing in consequence of the equa- 
 tions (o). 
 
 In order to fix the position of the centre of gravity, 
 let X* Y, and j^bs its three co-ordinates relative to a 
 
 7 7 
 
 given point ; let #, y and z be the co-ordinates of m 
 relative to the same point, x 1 y'^ and z 1 those of m' y and 
 so on ; the equations (o) will then give 
 Q=2.m.(z X) ; 
 
 but we have 2.mX=X.'2,.m 9 2.m being the entire 
 mass of the system ; we shall have therefore 
 
 X.m * 
 We shall have in like manner 
 
 v 2,.m?/ S.wzz 
 
 ~~ %.m ' ~ 2.w 
 
 but as the co-ordinates X 9 Y, and ^determine only one 
 point, it is evident that the centre of gravity of a body 
 is only one point. 
 
 The three preceeding equations give 
 
 
 * Let Ax 1 ^By' -f CV zzO be the equation to a plane pass- 
 ing through the centre of gravity, which is supposed to be 
 the origin of the co-ordinates ; then if x 9 ?/, and z are th 
 co-ordinates of the body, its distance from that plane will 
 
 be - 7-^70 which is a linear function of the co.. 
 
 ordinates #, y and z of the body. 
 
fcfiKfctlow rniy !r; altered to this- form* 
 
 Ih:? finite intent 
 
 t)*} expresses the sum of all the products,, similar to 
 
 tfnt which is contained under the- cbaractemttc %? 
 
 f n order tcr render this evident it wfll be sufficient to 
 at? example, in which only th ree bodies m^mf, and wr^are 
 considered with respect to the co-ordinates of x. In this case 
 
 s equal to - ' . ' .. - and 
 m+mf >\-m >l 
 
 x)*} 
 
 - 
 
 if both the namerator and 
 
 denominator of the first quantity are multiplied by 2.* or 
 
 m n , it will become 
 
 which, by subtracting the last quantity, gives 
 m z x*+Z 
 
 (Z.mx)* 
 
LAPLACE'S MECHANICS. 
 
 which we are able to form, from considering by two 
 and two all the bodies of the system. We shall thus 
 have the distance of the centre of gravity from any fixed 
 point whatever, by means of the distances of the bodies 
 of the system from the same fixed point, and their mu- 
 tual distances. By determining in this manner, the 
 distances of the centre of gravity from any three fixed 
 points whatever, we shall have its position in space, 
 which is a new method of determining it*. 
 
 We have extended the denomination of centre of 
 gravity to a point of any system whatever of bodies, 
 either having or not having weight, determined by the 
 three co-ordinates, X 9 Y, and J2Tt. 
 
 * As the last term of the second member of the equation 
 is independent of ihe given point, if the values of. the first 
 term he determined with respect to three given points not 
 in the same straight line taken either within or without the 
 system, we shall have the distances of its centre of gravity 
 from these points, and consequently its position with re- 
 spect to them. If the bodies were in the same plane two 
 points would have been sufficient, and if in the same line, 
 one. If the given points be taken in the bodies of the sys. 
 tern, the position of its centre of gravity will be given solely 
 by the masses and their respective distances. This method 
 of finding the centre of gravity is independent of the consi. 
 deration of three planes. 
 
 i It is evident from the principle of virtual velocities, 
 that the" centre of gravity of a system of bodies connected 
 together in any manner, is generally the highest or the low. 
 est possible when the system is in equilibrio. 
 
 Let m. m', m 11 , &c. be the centres of gravity of a number 
 of bodies connected together, Mhose weights are denoted by 
 
LAPLACES MECHANICS. 
 16. It is easy to apply the preceding results to the 
 
 the powers 5, S 7 , S", &c. acting respectively upon these cen. 
 tres, and let *, $', $ 7 ,c. represent lines respectively drawn 
 from them to any horizontal plane. If the position of the 
 system he disturbed in an indefinitely small degree, we shall 
 have, in the case of equilibrium, the eqtiaiion of virtual 
 velocities S.$s + S'.M -f&.W-f&c. ; 
 
 the quantity Ss-\-S's' + S fl !>"-\-&.c. is therefore either a max. 
 imum or a minimum. .If the sum of the weights S, S' , S", 
 &c. be represmted by 6r, and .the distance of the centre of 
 gravity of the system from the horizontal plane by g-, we 
 shall have the following equation 
 
 S.s+S'.s'+S".t''+ &c.= G.g. 
 
 As the first member of this equation is either a maximum or 
 a minimum, the second is also, consequently the distance 
 of the centre of gravity from the horizontal plane is either a 
 maximum or a minimum when the system is in a state of 
 equilibrium. 
 
 WhcMi the distance of the centre of gravity from an hori- 
 zontal plane is a maximum, the equilibrium of the system of 
 heavy bodies is unstable, and if moved in an indefinitely 
 small degree would not return to its former state; on the 
 contrary, when the distance of the centre of gravity is a 
 minimum, the system if moved from the state of equilibrium, 
 would, after oscillating some time, return to it. 
 
 This may be exemplified in the case of a cylinder with an 
 elliptical base, which, wheu placed upon an horizontal plane 
 with the edge of contact in the line passing along an extre- 
 mity of the major axis, will have the distance of its centre 
 of gravity from the plane a maximum and its position unsta- 
 ble, and the contrary when placed with the edge of contact 
 in a line passing through an extremity of the minor axis. 
 The above are the only positions in which there can be an 
 equilibrium. 
 
T.AP& AC*S -MECHANICS* 
 
 equilibrium of a solid bcdy having 1 any figure what- 
 ever, foy supposing it formed of an indefinite number of 
 points in variably connected -with each other. Let dm 
 represent one of these points, or an indefinitely small 
 molecule of the body, and let ^r,#, and z be the rectan- 
 gular co-ordinates of this molecule; again, let P, Q, 
 and R be the forces by which it is actuated parallel to 
 the axes of .r, ^, and r ; the equations ( m) and (n) of 
 the preceding No. will be changed into the following; 
 Q=fP.dm; 
 
 the integral sign/ is relative to the molecule e//w, and 
 ought to be extended to the whole mass of the solid*. 
 
 If the body could only turn about the origin of th 
 co-ordinates, the three last equations would be suffici- 
 ent for its equilibrium. 
 
 * It is easy to perceive that in the case of a solid body, 
 which may be supposed to be composed of an indefinitely 
 great number of points invariably connected together, 
 
 the quantity 2. mSl j I becomes JP.dm^ for 51 r 1 is 
 equivalent to P, and f dm to S.m ; in like manner 2,mS 
 
 ~** becomes 
 
 ( vT )& m * s equivalent to/P^.Jm, and 2.l r - l. m to 
 JQx.dm. 
 
CHAP. IV. 
 
 Of the equilibrium of fluids. 
 
 17, 1 o have the laws of the equilibrium and of the 
 motion of each of the fluid molecules, it is necessary to 
 know their figure, which is impossible; but we have 
 no occasion to determine these laws except for the fluids 
 considered in a mass, and then the knowledge of the 
 forms of their molecules becomes useless. Whatever 
 may be these figures and the dispositions which result 
 in the integral molecules ; all the fluids taken in the 
 mass ought to offer the same phenomena in their equi- 
 librium, and in their motions, so that the observation of 
 these phenomena does not enable us to learn any thing 
 respecting the configuration of the fluid molecules. 
 These general phenomena are founded upou the per- 
 fect mobility of the molecules, which are thus able to 
 give way to the slightest effort. This mobility is the 
 characteristic property of fluids : it distinguishes them 
 from solid bodies, and serves to define them. From 
 hence it results, that for the equilibrium of a fluid 
 mast; each molecule ought to be in equilibrio, in con- 
 
126 LAPLACE'S MECHANICS. 
 
 sequence of the forces which solicit if, and the pressure 
 which it sustains from the surrounding molecules. Let 
 
 us clevelopc the equations which result from this pro- 
 perty. 
 
 In order to accomplish if, we will consider a system 
 of fluid molecules forming an indefinitely small rectan- 
 gular parallelepiped. Let #, y, and z be the three 
 recfangkir co-ordinates of the an:le of this parallele- 
 piped the nearest to the origin of the co-ordinates. Let 
 dx, dy, and dz be the three dimensions of this parallel- 
 epiped ; let p represent the mean of all the pressures 
 which the different points of the side dy, dz of the par- 
 allelepiped that is nearest the origin of the co-ordinates 
 experiences, and p 1 the same quantity relative to the 
 opposite side. The parallelepiped, in consequence of 
 the pressure which acts upon it, will be solicited parallel 
 to the axis of x by a force equal to (p p'^.dy.dz; p' p 
 is the differential of p taken by making x alone to vary; 
 for although the pressure of p' acts in a different direc- 
 tion to that of p, nevertheless the pressure which a point 
 of the fluid experiences being the same in all directions; 
 p' p may be considered as the difference of two forces 
 indefinitely near and acting in the same direction ; we 
 
 shall therefore have p' p=-dr; and (p p' ) 
 dy. dz^= - .dr. 
 
 * In (fig. 14) let AX, AY, and A% represent the 
 axes of x,y, and z respectively, and ah a molecule of the 
 in the form of a rectangular parallelepiped, whose fa- 
 
K7 
 
 Let P 9 Q 9 and R \w three accelerating forces which 
 also act upon the fluid molecule, parallel to the respec- 
 tive axes of ^5^5 and z : if the density of the parallel- 
 epiped is named p 9 its mass will be p.dx.dy.d~, and the 
 product of the force P by this mass, will b^ the en- 
 tire resulting force which moves it ; this mass will con- 
 sequently be solicited parallel to the axis of #, by the 
 
 force J pP | -J- J > .dx.dy.dz. It will in like man- 
 ner be solicited parallel to the axes of y and s, by the 
 forces $ P Q - (J) ^ .dr.dy.&,an.l ?R-(Jj) \ 
 
 .dx.dy.dz ; we shall therefore have, in consequent 
 of the equation (b) of No. 3, 
 
 = \ > p - (=) 
 {'-(')}>*> 
 
 ces bh, ag, and ad are respectively parallel to the planes 
 YAX, ZAX) and YAZ. Suppose that the co-ordinates of 
 the angular point b of the molecule are x, y, and ~, and 
 that ^zzf/or, bdmdy^ and ba~dz ; also, let mo represent 
 the quantity and direction of the mean of all the forces act- 
 ing upon the face dy, dz of the parallelogram, or (he force 
 p, and nq the mean of all the forces acting upon the oppo- 
 site face /g- of the molecule, or the force p'. , 
 
 In this case p is supposed to be a function of the ro-ordi- 
 nates a?, ?/, and z 9 consequently for the opposite side of the 
 parallelepiped to that formed by dy and dz, as x becomes 
 
 x + dx, the pressure p is changed into jp-f( - j.dx which - 
 
LAPLACE'S MECHANICS. 
 
 The second member of this equation ought to bean 
 exact variation like the first; which gives the following 
 equations of partial differentials, 
 
 (d.pP\_(d. P Q\ f (d.pP\fd. P R 
 \~d~J \d -~ 
 
 dz 
 from which we may obtain 
 
 for that side must necessarily act in an oppbsite direction te 
 
 . It therefore follows that I ]dx* the difference of 
 
 \dxj 
 
 the two pressures, when multiplied by dy and f/^, gives the 
 whole force arising from pressure that acts upon the paral. 
 lelepiped in the direction of the co-ordinates, which should 
 be taken negatively, as it tends to diminish them, and, in 
 the case of equilibrium, must be equal to the moving force 
 fP.dx.dy.dz.tbxt acts in an opposite direction. 
 * The equation 
 
 by transposition, becomes 
 
 s 
 
 j Page 45, therefore Jp=r 
 
129 
 
 + The proof that if f [P.$x+Q.ty+R3z} is an exact 
 differential, the equations 
 
 fP\ fd. f Q\ /d. P P\ 
 
 have place, may be given as follows. Suppose u 
 
 then dupdx -\-qdy is an exact differential, if/? I I 
 
 and ginl ) : by differentiating^ alone in the first, and 
 
 ^* c/ -S 
 
 x alone in the second of the two last equations, we shall have 
 
 diiJ~\dxdijJ \dxjt \dydafj' 
 
 dacdy 
 
 In like matiner, if u^nf(x^y^z)^ by differentiation du 
 
 $dx+qdy+rdZ) in which equation p~( j, <7=z| | 
 and rma I ; let z be supposed constant, then du~pdx 
 
 -f^rf/y, which gives! Jrzii 1 ; also if y and x are sup. 
 posed alternately constant, the resulting equations will re. 
 spectively give ( J)=( J)and( J)=(|)- I" (he 
 above pP may be substituted for p, pQ f r (7? and p /t for r. 
 By differentiating the equations 
 
 dy J V dx 
 
 multiplying the first by R, the second by Q, and the third 
 by P : (hey will give by adding together, asp disappears, the 
 
 s 
 
J50 
 
 This equation expresses the relation which should 
 exist amongst the forces P, Q and 7?, that the equili- 
 brium may be possible*. 
 
 following equation 0= P - 
 
 * Suppose an incompressible fluid not acted upon by the 
 force of gravity to be contained in a vessel that has a number 
 of cylinders attached to the sides of it, to which a number of 
 inoveable pistons are adapted. Let the areas or bases of 
 the cylinders or pistons be represented by A, A 1 , A", &c. 
 also suppose $, S' 9 S 11 , &c. to denote the powers applied 
 to the pistons having the bases A, A', A", &c. respectively, 
 and that these powers, which act upon each other by the 
 intervention of the fluid, are in. equilibrio. Let p repre- 
 sent, in this case, the pressure upon the area denoted by 
 unity of the surface of the vessel or the base of the pistons, 
 then pA, pA' 9 pA f/ 9 &c. will denote the respective pressures 
 of the fluid upon the bases of the pistons, but these pressures 
 are equal to the forces which act upon the pistons, therefore 
 S~pA 9 S~pA f , S~pA", &c. Let a part of the pistons 
 be pushed downwards, then it is evident that the other part 
 of the pistons must be elevated by an equal quantity of water 
 to that depressed, so that if $s 9 Ss', 3s' 1 9 &c. represent the 
 depressions or the elevations of the respective pistons whose 
 bases are A, A' 9 A" 9 &c. we shall have the equation 
 
 regarding the spaces through which the pistons were depressed 
 as positive, and the spaces through which they were elevated 
 as negative. Let this equation be multiplied by p 9 then 
 
 or by substitution 
 S.$s 
 which is the equation of virtual velocities. 
 
LA.PLA6&'S MECHANICS. 131 
 
 If the fluid be free at its surface, or in certain parts 
 
 Suppose that T, 7", T 77 , &c. represent the different pow- 
 ers which act upon a molecule whose co-ordinates are #, y, 
 and s these powers being directed to certain fixed centres, 
 the distance of which from the molecule solicited aie re. 
 spectively /, t', t" , &c. 
 
 Let the co-ordinates of these fixed centres referred to the 
 origin of the co-ordinates x, y, and s, and respectively 
 parallel to them be , , c; a', &', c', &c. we shall then have 
 
 T// 
 
 &c. & c . 
 
 As the equilibrium is possible, when the fluid molecules are 
 solicited by forces directed towards fixed centres, which are 
 functions of the distances of the points of application from 
 these centres ; we may substitute the above values of P, Q, 
 and R, in the equation 
 
 7= (P< 
 
 which then becomes 
 
 _.{ (x 
 
 -f&c. 
 and is equivalent to 
 
139 LAPLACE'S MECHANICS. 
 
 of its surface, the value of p will be nothing in those 
 parts ; \ve shall therefore have $p=0*, provided that 
 we subject the variations ^r, ^j/, and, $z to appertain 
 
 The sum L (ty) taken throughout the whole extent of any 
 indefinitely narrow canal, which either re-enters into 
 itself, or is terminated at two points of the exterior surface 
 of the fluid mass, is always nothing ; on the supposition that 
 the resistance of the sides, if the fluid be contained in a vessel 
 is regarded, and that the canal is imagined in this case to 
 have one of its extremities terminated at a point of its side. 
 It may therefore be concluded, that for all the cases of the 
 equilibrium of a fluid, the following equation has place 
 throughout the whole extent of the mass, 
 
 In this equation the products of pT, p7", &c. are propor- 
 tional to the moving forces with which each power acts upon 
 the molecule. Let 5, 5"'. S", &c. represent the moving 
 forces which are the resultants of the powers which re. 
 spectively act upon each fluid molecule, and s 9 s f . s !l 9 &c. 
 the lines drawn respectively in the directions of the forces 
 S 9 S' 9 S", &c. from each fluid molecule; then the above 
 equation is equivalent to the following 
 
 This equation is similar to those deduced from the principal 
 of virtual velocities for the equilibrium of a point or a 
 system. 
 
 * This will be the case not only when jpzzO, hot likewise 
 when p is a constant quantity, which also gives fy0. For 
 instance when the atmosphere presses equally upon the sur- 
 face of the fluid. 
 
LAPLACE'S MECHANICS. 133 
 
 to this surface : by fulfilling these conditions we shall 
 consequently have 
 
 * Suppose for example, a fluid mass to be acted upon by 
 a force S tending to the centre of the co-ordinates, and let 
 one of its molecules be placed at the distance r from that 
 centre, having #, y^ and s, for its rectangular co-ordinates, 
 then rzzy/ a; 2^.^2_|_2"2 : the force S resolved parallel to these 
 
 Sx Si/ Sz 
 
 co-ordinates gives 'and for the forces m their re- 
 r r r 
 
 spective directions. These forces, taken negatively as they 
 tend to diminish the co-ordinates, should be substitated for 
 their respective values P, Q, and .R, in tho preceding 
 equations. When they are substituted in the equation 
 P.^x+Q.ty+R.fc they will give, by the suppression of the 
 
 S 
 
 common factor -- , 
 ' 
 
 which is an exact differential, therefore the equilibrium is 
 possible. This equation when integrated becomes # 2 -{- t y 2 + 
 S 2 zzc 2 , which is the equation to a sphere, consequently, the 
 fluid will assume a spherical form. If r is very great the 
 surface of the fluid may be regarded as a plane, as is the 
 case with the surface of a fluid in equilibrio in a vessel when 
 only acted upon by the force of gravity. 
 
 Let the force S be supposed to vary as the nth power of 
 its distance from the centre of the co-ordinates, and to be 
 represented by Ar y also, let/? represent the pressure upon 
 an area of the surface denoted by unity, then the equation 
 
 P=f{ p -*x+Q'ty+R"te} will, by a proper substitution, 
 be changed into the following 
 
 but *to-f jf}y-ftiT$f 9 therefore 
 
134 LAPLACE'S MECHANICS. 
 
 Jf tezd) be the differential equation of the surface, we 
 *hall have P.$x-\-Q.ty-\-R.Sz=X.$tf, x being a func- 
 tion of #,y, and z; from which it follows, by No. 3, that 
 
 TMs is the value of the pressure referred to unity of (he sur- 
 face which acts upon the molecule that has #, y 9 and z for its 
 co-ordinates. 
 
 The equation of equilibrium may he used to find the form 
 which a fluid retains when it has an uniform rotatory motion 
 round a fixed axis, by adding the centrifugarforce to the 
 given accelerating forces which act upon the molecules. 
 Let the axis of z be that of rotation, n the angular velocity 
 common to all the points of the fluid mass, and r \/ '# -\-g* 
 the distance of any point of it from the.axis of rotation ; 
 then, as the centrifugal force of the point is equal to the 
 square of its velocity divided by its distance from the axis of 
 9 it will be represented by rc 2 ;*, which when multiplied by 
 the variation of its direction gives n*r$i n 2 .r.$r--n2yty 
 If this value be added to the formula P.&e + Q%-f U.5s, it 
 will not prevent it from being an exact differential, for the 
 centrifugal force of a point may be considered as a force of 
 repulsion, the intensity of which is a function of the distance 
 of the point from the axis of rotation : we shall therefore 
 have the equation 
 
 for the differential equation of the surface of the laminae and 
 of the free surface of the fluid. That the Telocity n may 
 be uniform it is requisite that the forces P, Q, and R should 
 arise from the mutual attraction of the molecules, or from 
 attractions in the directions of lines joining the molecules 
 and the axis of rotation, or from forces acting towards 
 points which have the same motion as the fluid mass. 
 
LAPLACE'S MECHANICS. 135 
 
 (he resultant of the forces P, Q, and J?, ought to be 
 perpendicular to those parts of the surface where the 
 fluid is free. 
 
 Let us suppose that the variation P.&r-j- Q.ty-\-R.$s 
 is exact; which circumstance has place by No. 2, when 
 the forces P, <?, and R are the results of attractive 
 forces. 
 
 Naming this variation op, we shall have $p=p.ty ; 
 p therefore should be a function of p and of p, and as 
 by integrating this differential equation, we have p a 
 function of p ; we shall havep a function of p. The 
 
 By way of example, let us find the form of the surface of 
 a quantity of water contained in a cylindrical vessel open at 
 the top, having a rotatory motion round its axis which is 
 vertical. As water is an incompressible and homogeneous 
 fluid, the above equation will be sufficient, let therefore the 
 origin of the co-ordinates be at the centre of the base of the 
 vessel having z for the vertical axis of rotation, and let g* 
 represent the force of gravity, then we shall have PzzO, 
 QzzQ and Jlur g*, consequently the equation of the surface 
 of the^fluid becomes by substitution 
 
 which by integration gives 
 
 In this case it is evident from the equation that the upper 
 part of the fluid will form the surface of a paraboloid of 
 which the solid content is given, as it will be equal ti> one 
 half that of the water contained in the vessel. The equatioa 
 
 2o- 
 of the generating parabola is y*~~;z as appears from 
 
 making # to vanish. 
 
136 
 
 pressure p is therefore the same for all the molecules of 
 which the density is the s:ime ; therefore %p is nothing 
 relative to the surfaces of the laminae of the fluid mass 
 in which the density is constant ; and we shall have by 
 relation to these surfaces 
 
 It therefore follows, that the resultant of the forces 
 which act upon each fluid molecule, is in the slate of 
 equilibrium perpendicular to the surfaces of these 
 lamina ; which have been named on that account 
 (couches de niveau,) laminae of level. This condition 
 is always fulfilled if the fluid is homogeneous and in- 
 compressible ; because in this case the laminae to which 
 this resultant is perpendicular are all of the same 
 density*. 
 
 * In the case of the equilibrium of an elastic fluid, the 
 pressure is found by experiment to vary as the density, 
 consequently p may be supposed equal to f . If/>zzff then 
 
 f , let this value of ^ be substituted in the equation dpzz 
 and it will give adfpzzjp^p, consequently d.\og.p-.d<p. 
 
 If the fluid be homogeneal and of the same temperature, a 
 is constant and the equation possible. By integration we 
 
 9 
 
 ai 
 
 shall have \og.p-+lov.c,orpc,e 5 c being a constant 
 
 quantity, and e the base of the system of logarithms which 
 has unity for its modulus. As this value of p and conse- 
 
 quently that of f -are functions of the sole variable <p, the 
 
LAVLACE'S MECHANICS. 137 
 
 For the equilibrium of an homogeneous fluid mass, 
 the exterior surface of which is free, and covers a solid 
 mass fixed and of any figure whatever; it is therefore 
 
 pressure and the density will be the same for all the extent 
 of each lamina of level as in the case of heterogeneal fluids ; 
 the densities of the different laminae are determined by the 
 equation 
 
 9 
 
 t= P -=^. 
 a a 
 
 If in an homogeneal fluid the temperature be not the same 
 throughout the mass, the quantity a will not be con- 
 stant ; let t denote the temperature, then a will be a function 
 of t, but a, if variable, must be a function of <p, conse. 
 quently t must be a function of <p ; it is therefore necessary 
 in the case of equilibrium that the temperature be uniform 
 throughout each lamina of level, as well as the pressure and 
 the density, which are likewise functions of <p. The tern. 
 perature may vary according to any law in passing from one 
 lamina to another, but this law being given the laws of the 
 pressure and the density will be determined by the following 
 aquations, 
 
 being a constant quantity. 
 
 In the case of the equilibrium of the atmospheric air, let 
 it be supposed that a small vertical cylindrical column of it 
 is continued from the surface of the earth to an indefinite 
 height, which must be supposed in equilibrio independent 
 of the surrounding air that may be conceived to become im- 
 moveable. The force of gravity can without sensible error 
 be supposed to act in the direction of the cylinder along its 
 
1S8 LAPLACE'S MECHANICS. 
 
 necessary and it is sufficient, first, that the differential 
 P.r+<?.^+^?.^ be exact; secondly, that the re- 
 sultant of the forces at the exterior surface be directed 
 perpendicularly to this surface. 
 
 whole extent, and in the case of equilibrium, the density, 
 the pressure, and the temperature may be considered as in. 
 variable throughout the whole mass of an indefinitely thin 
 horizontal lamina. Let z represent the distance of any 
 lamina whateTer from the surface of the earth, ^ its density, 
 r the radius of the earth, g the force of gravity at its surface, 
 
 g'r-^- the force of gravity at that altitude, and p its 
 
 elastic force, , g 1 and p being functions of s, dz the breadth 
 of the lamina and a the area of its base. Then if p be the 
 pressure upon a portion of the surface represented by unity, 
 that upon the higher part of the lamina will be Ap and that 
 upon the lower A(p-\-dp)^ but the excess Adp of the first 
 pressure above the second is equal to the weight of the lamina 
 or Aggdz, therefore $p~ &?'&& This equation may be 
 obtained by proper substitution from the general equation 
 
 in which P and Q will vanish and R be equal to g'fe. 
 
& 4 PL A CX'S MECHANICS. 139 
 
 CHAP. V. 
 
 General principles of motion of a system of bodies* 
 
 IS. E have in No. 7 reduced the laws of the motion 
 
 of a point to those of its equilibrium, by resolving the 
 instantaneous motion into two others, of which one 
 remains, and the other is destroyed by the forces which 
 solicit the point : the equilibrium between these forces 
 and the motion lost by the body, has given us the dif- 
 ferential equations of its motion. We now proceed to 
 make u^e of the same method to determine the motion 
 of a system of bodies m 9 m' 9 m fl 9 &c. Let therefore 
 mP, mQ, and mR, be the forces which solicit m paral- 
 lel to the axes of its respective rectangular co-ordinates 
 #, y, and zy let m'P\ m'Q' 9 and m'R' be the forces 
 which solicit m f parallel to the same axes, and in a 
 similar manner with the rest ; and let the time be repre- 
 
 sented by t. The partial forces W'-ri m '~j^> anc * 
 
 m. ~ of the body m at any instant whatever will become 
 dt 
 
140 
 
 in the followin 
 
 dx . .dx 7 dx . n 
 
 m.- \-m.d-- -- m.d. -\-mP.dt ; 
 
 di/ , , dy , du . 
 
 ' 
 
 dz . , dz , dz 
 
 m.+m.d. m.d. 
 
 and as the sole forces 
 
 dx . ,dx dii . du dz . , d% 
 
 remain ; the forces 
 
 -m.d.^j+mP.dt; -m.d.^f+mQ.dt; 
 
 m.d.- \-mR.dt ; 
 
 are destroyed*. 
 
 * The forces which are destroyed during the motion of the 
 system at any instant) will evidently form an equation of 
 equilibrium for it at that instant. If in this equation of 
 equilibrium the bodies undergo an indefinitely small change 
 in their position, the moments of the forces according to the 
 principle of virtual velocities will be equal to nothing; the 
 the forces destroyed are mP, mQ, mR, m'P', &c. m. 
 
 d\y d*z f 
 
 m. -* m. , m . - , c. whose moments 
 dt* dt-' dt*' 
 
 are mP.fa, -mQ.ty, mlOz, &c. ~w.-.^, ~~ m 'jty> 
 
 d z z 
 
 w , ~. & c . the general formula of equilibrium is there- 
 
 fore, when multiplied by 1, as follows 
 
LAPLACE'S MECHANICS. 141 
 
 By distinguishing in these two expressions, (he let- 
 ters ro, x, y, s, P, Q, aid R successively by one, two, 
 Sec. marks, we shall have the forces destroyed in the 
 bodies m 1 , m", &c. This being premised ; if we mul- 
 tiply these forces respectively by the variations ^ Sy, 
 52, So:', &c. of their directions; the principle of virtual 
 velocities explained in No. 14, will give, by supposing 
 dt constant, the following equation ; 
 
 &c*. 
 
 In the equation of equilibrium of the forces destroyed, in 
 order that they may be equal to nothing, either the forces 
 
 d*x d*y 
 
 rc.-r 1 -, &c. or the forces #zP, mQ, &c. 
 
 taken negatively, although in the motion of the system they 
 may tend to increase the co-ordinates. 
 
 * The expression d*x.&>rk&0.foj-<pzjz is independent 
 of the position of the axes of the co-ordinates x, t/ y and z 
 as may be proved in the following manner. 
 
 Let the rectangular co-ordinates #,, y n and z, be substi- 
 tuted for those above mentioned, having the same origin 
 but referred to other axes ; then it may easily be' demon, 
 strated that 
 
142 LAPLACE'S MECHANICS. 
 
 We can extract from this equation by means of the 
 particular conditions of the system, as many of the 
 
 the co-efficients a, /$, y, a'. &c. being constant quantities, 
 and only dependent upon the respective positions of the two 
 systems of co-ordinates. The co-ordinates 47, y 9 and z are 
 relative to the same points as the co-ordinates # /5 y n and z l9 
 consequently z z +y 2 -\-z z ~x, z -}-y l --\-z l z ; this expression 
 gives the six following conditional equations, 
 
 0+a'jS' -f '/#' = 0, y 
 
 from which, it appears, that three of the nine co-efficients 
 
 are indeterminate quantities. 
 
 If the expressions of a?, y, and z are twice differentiated., 
 they will become 
 
 &z=La"d*x, -f &"d*y, -f 7^2,, 
 the following variations may likewise be obtained 
 
 S^a^-f/-fy^ 
 
 By substituting these values and regarding the equations of 
 condition between a, ft, 7) a ' ? & c . we shall find that 
 d**$x-)-d*yty 4- d^zdixfix;, -f c? 2 ^ / ^ / -f d*zfiz,* 
 If the same substitutions are made in the expressions for the 
 right lined, distances between the different bodies of the 
 system. represented by/, /, //, &c. the quantities a, /3, y, 
 *', &c. will equally disappear, and the transformed ex- 
 pressions will retain the same form. Thus 3 9 y, and s being 
 tlie. co-ordinates of the body ro, and x 1 , y 1 , and s ; those of 
 the body ?', their distance/ will be equal 
 
 If the axes are changed, the first co. 
 
 ordinates will become x t y t and z l9 and the second a? 
 
LAPLACE'S MECHANICS. 143 
 
 variations as we have conditions ; by afterwards equal- 
 ling separately to nothing the remaining co-efficients of 
 
 and 2 also 
 
 By substitution and haying regard to the before mentioned 
 equations of condition, we shall have /zzy//^ / _ x.*-L. 
 
 (#/ #,J a -Ks/ Z;)* ; a similar proof may be given for the 
 quantities//, /", &c. 
 
 It follows from the above, that if the system is only acted 
 by forces which are proportional to some functions of the 
 distances/, /'. / /y , &c. between the bodies ; and the equa. 
 tions of condition of the system solely depend upon the 
 mutual situation of the bodies or the lines/, /', / /7 , &c. 
 the general formula of dynamic will be the same for the 
 transformed co-ordinates # y fl and z t , as for the original 
 ones #, y, and z. If, therefore, the different values of #, 
 y, and z for each body are found with respect to the time 
 by integration of the different equations [deduced from this 
 formula, and those values are taken a? /s y^ and z {9 we shall 
 have these more general values for x^ y, and z. 
 
 in which the nine co- efficients a , 0. 7) a ' , &c. contain three 
 indeterminate quantities, as there are only six conditional 
 equations amongst them. 
 
 If the values of x fl y^ and z t contain all the constant quan- 
 tities necessary to complete the different integrals ; the three 
 indeterminate quantities will be mixed with some of the 
 other six constant quantities, they will also make up those that 
 
the variations, we shall have all (he necessary equation^ 
 for determining the motions of the different bodies of 
 the system*. 
 
 19. The equation (P) contains nnny general prin- 
 ciples of motion, which we shall proceed to develope. 
 The variations &r, ty, $z 9 &e', &c. will be evidently 
 
 are wanting, without which the solution would be incom- 
 plete. Thus hy means of these three new constant quantities 
 which may be introduced after the calculation, it will be pos- 
 sible to suppose the same number of the other constant 
 quantities equal to nothing, or to some determinate quan- 
 tities; which will often much facilitate and simplify the calcu- 
 lation. Vide the Mechanique Analytique of Lagrange. 
 
 * Al though the effects of the forces of impulsion or percus- 
 sion may be calculated in the same manner as those of 
 accelerating forces, yet when the whole impressed velocity 
 only is required, its successive increments can be neglected, 
 and the forces of impulsion considered in what follows as 
 equivalent to the impressed motions. 
 
 Let therefore S 9 ', S" 9 &c. represent the forces of im- 
 pulsion applied to any body m of the system in the directions 
 of the lines *, *', s", &c. and suppose that the velocity given 
 to this body may be resolved into three velocities, represented 
 by x\ y 9 and z in the direction of the co-ordinates x, y y 
 and z 9 we shall have by changing the accelerating forces 
 
 ^, ^ and - into the velocities x 9 y 9 and z 9 th 
 
 general equafion 
 
 S.mfkr+^+s"**; Z(S.*s+S'.*8'+S a .'*s l >) + &c. 0. 
 This equation will give as many particular ones, as we shall 
 have independent variations, after they have been reduced 
 to the smallest number possible by means of the conditional 
 equations belonging to the system. 
 
LAPLACE'S MECHANICS. H5 
 
 subjected to all the conditions of tbe connection of the 
 parts of the system, if they are supposed equal to the 
 differentials dx, dy, dz, dx', &c*. This supposition is 
 
 If the system is continuous and of an invariable figure 
 as a solid body, or variable as flexible bodies and fluids ; by 
 denoting its whole mass by m and any one of its molecules 
 by dm, it may be considered as an assemblage or system of 
 an indefinitely great number of molecules ; each represented 
 by dm and acted upon by the accelerating forces &. &', S", 
 &c. ; and it will be sufficient in the general equation to sub. 
 stitute dm for m, and for the sign 2, S or the sign of inte- 
 gration relative to the whole extent of the body, that is, to 
 the instantaneous position of all its molecules, but indepen- 
 dent of the successive positions of each molecule. For a 
 fuller detail respecting solid bodies, I refer the reader to 
 the seventh chapter of this work. 
 
 * It is necessary in this case that the equations of condi- 
 tion should not contain the time , which sometimes happens, 
 as for instance, if one of the bodies be forced to move upon 
 a surface which is itself moving according to some given law, 
 there will then be an equation of condition of the co- 
 ordinates and the time /, for the equation of the surface at 
 any instant, which may be represented as follow s, 
 
 In the equation of equilibrium of a system formed by those 
 forces which are supposed equal to nothing when it is in 
 motion, it is necessary, in order that the indefinitely small 
 change in its position according to the principle of virtual 
 velocities may be proper, that the co-ordinates of th f e bodies 
 in the new position of the system when substituted should 
 satisfy that equation. These co-ordinates are x-}-$x 9 ^"4"^> 
 and s-J-Ss for the body m, s'+So/', #'+&/, and 2/-fcb' for 
 the body m'. &c. which should satisfy the above conditional 
 
LAPLACE'S MECHANICS, 
 
 therefore permitted, consequently the equation (P) 
 will give by integration 
 
 dz); (Q) 
 
 c being a constant quantity introduced by the integra- 
 
 tion. 
 
 If the forces P, Q, and R are the results of attracting 
 forces directed towards fixed points, and of attracting 
 
 equation when respectively substituted for #, # 3 z, x' 9 y' 9 
 &c. ; the differential of the function F, 
 
 will then be equal to nothing, t being regarded as constant 
 and the variations of the co-ordinates x, y^ z; #', y 1 , s'; &c. 
 denoted by the characteristic $. But as the co-ordinates of 
 the bodies are functions of the time, the complete differentia. 
 tion of jP with respect to /, x 9 #, 2, #', &c. being regarded 
 as functions of t, will be equal to nothing. We shall there- 
 fore have the following equation, 
 
 >+a>+ (SMS) 
 
 , 0. 
 
 T.dt being the differential of F taken with respect to the 
 time which is contained explicitly in this function. If T.dt 
 be equal to nothing it is evident that the former equation 
 will coincide with this, by taking ^zzc/^, ^zzti^, &c. 
 
 From the above it appears, that when the time is not ex- 
 plicitly contained in any of the equations of condition, 
 the virtual velocities of the moving bodies along the axes of 
 their co-ordinates may be supposed equal to the differentials 
 of these co-ordinates, or the spaces passed over by their 
 projections upon these axes during the time dt. 
 
LAPLACE'S MECHANICS. 147 
 
 forces of the bodies one towards another, the function 
 2.fm.(P*dx-}-Q.dy-\-R.dx} is an exact integral. In 
 fact, the parts of this function relative to attracting 
 forcei directed towards fixed points, are by No. 8, 
 exact integrals. This is equally true with respect to 
 the parts which depend upon the mufual attractions of 
 the bodies of the system ; for if the distance between 
 m and m' is called /, and the attraction of m' for m, 
 m'F, the part of m.(P.dx+Q.dy-{-R.dz) relative to 
 the attraction of m 1 for m, will be, by the above cited 
 No. equal to mm'.Fdf, the differential df being taken 
 by only making the co-ordinates x, y, and 2 to vary*. 
 
 * As m'F is the accelerating force of m arising from the 
 attraction of m 1 which acts along the line /, its components 
 
 x < _ x 
 in the directions of the axes of a?, #, and x are m'F. , 
 
 m ' p t ^HM^ and w'JF.^^, we shall therefore have the 
 
 J J 
 
 following equation with respect to this force alone 
 
 In a similar manner, P', Q', and R 1 denote the components 
 of the accelerating forces which act upon m' parallel to the 
 same axes, we shall have relative to the force' mF the 
 equation 
 
 mF 
 
 If, after having multiplied the first of these equations by m 
 and the second by m', we add them together, they will in. 
 troduce into the expression ?<m(Pdx+Qdy+Rdz) the term 
 
148 LAPLACE'S MECHANICS. 
 
 But the re-action being equal and contrary to the 
 action, the part of m'.(P.'dx'-\-Q f .dy'-}-R t .dz') rela- 
 tive to the attraction of m torn', is equal to mm'.Fdf, 
 if the co-ordinates .r', y, and %' alone vary in f\ the 
 part of the function ^.m(Pdx-\-Qd^Rdz) relative to 
 the reciprocal attraction of m and of m' 9 is therefore 
 mm'.Fdf; the whole being supposed to vary in/. This 
 quantity is an exact differential when / is a function of 
 .F, or when the attraction varies as a function of the 
 distance, which we shall always suppose; the function 
 Z.m.(Pdj;-\-Qdy-\-Rdz) is therefore an exact differ, 
 ential, whenever the forces which act upon the bodies 
 of the system, are the result of their mutual attraction 
 or of attracting forces directed towards certain fixed 
 points. Let d$ represent this differential and o the 
 velocity of m 9 uf that of m' 9 Sec. ; we shall then have 
 
 2.mv a =c+2<p. (R) 
 
 This equation is analogous to the equation (g) of No. 8; 
 it is the analytical traduction of the principle of the 
 preservation of the living forces. The product of the 
 mass of a body by the square of its velocity, is called 
 
 + (z'-z)(dz'-dz)}. 
 
 If the equation f 2 =(x-x')* + (yy')*+(z<-z')* be 
 completely differentiated it will give 
 
 fdf= (x x')(dx-dx l ) + (ytf)(dydj/) -f (z z') 
 
 (dz-dz 1 ) ; 
 
 the preceding term by substitution therefore becomes 
 mm'Fdf. 
 
LAPLACE'S MECHANICS. J4 
 
 its living force. The principle upon which we are treat* 
 ing consists in this, that the sum of the living forces, 
 or the whole living force of the system is constant, if 
 the system be not solicited by any forces ; and if the 
 bodies be solicited by any forces whatever, the sum of 
 the increments of the whole living force is the same, 
 whatever may be the curves described by each of the 
 bodies, provided, that their points of departure and 
 arrival are the same. 
 
 This principle has place only in the cases in which 
 the motions of the bodies change by insensible grada- 
 tions. If these motions experience sudden changes, the 
 living force is diminished by a quantity which we shall 
 determine in the following manner, The analysis 
 which has conducted us to the equation (P) of the 
 preceding number, gives in this case instead of that 
 equation, the following*, 
 
 * The equation i.. -P.^+-Q^ 
 
 (d*z \ > fdP-s: 
 
 -fat R ) ^ C o f No. 19 is equivalent to Z.m.l -r-j 
 
 or 
 
 .~, . 
 
 ut (It 
 
 which equation if the differences A.~, A., and A. 
 
 substituted for the differen 
 changed into the following, 
 
 substituted for the differentials d. , rf.^t and d ft 
 
 dC dt y 'dt* 
 
150 LAPLACE'S MECHANICS. 
 
 A.~, A.^, and A,^ being the differentials of^, ~ 
 
 and ^r from one instant to another ; which differen- 
 
 tials become finite when the motions of the bodies 
 receive finite alterations in an instant. We may sup- 
 pose in this equation 
 
 *#=(?#+ A. <te; Sysrzdy+A.^ ; $*=cfe+A.<fe ; 
 because the values of dx. dy, and dz are changed in the 
 following instant into r-}-A.<r, */y-j-A.^, and dz-\- 
 A.c?2, these values of dx 9 dy^ and ?s satisfy the 
 conditions of the connection of the parts of the system ; 
 we shall therefore have 
 
 This equation should be integrated as an equation of 
 finite differences relative to the time f, of which the 
 variations are indefinitely small, as well as the varia- 
 tions of X) i/, z, x') &c. Let us denote by 2 X the finite 
 integrals resulting from this integration, to distinguish 
 them from the preceding finite integrals, relative to all 
 the bodies of the system. The integral of mP.(dx-\- 
 A.drJ, is evidently the same as fmP.dx; we shall 
 therefore have* 
 
 * The integral of mP.(dx-\-&.dx) is fm.P.dx, for 
 =$ff, but/w.P.^e is equivalent tofm.P.dx, therefor* 
 
LAPLACE'S MECHANICS. 151 
 
 constant =2 ,, 
 
 if v, ?/, i/ 7 , &c. denote the velocities of m, w', w ff , &c. 
 we shall have 
 
 The quantity contained under the sign 2, being neces. 
 sarily positive, we may perceive that the living force 
 of the system is diminished by the mutual action of the 
 bodies, whenever during the motion some of the varia- 
 
 tions A.-^, A'-jp &c. are finite. The preceding 
 
 equation moreover offers a simple means for determining 
 this diminution. 
 
 At each sudden variation in the motion of the system, 
 it is possible to suppose the velocity of m resolved into 
 
 .(cfjr4 p AcT#) is equivalent iofmP.dx. In this equation 
 
 dx dx 1 dx* 
 
 the integral of . A. -r- is supposed to be-r.-r-r. 
 at at 2 dt z 
 
 Those who wish to obtain information respecting finite 
 differences and their integrals, may find it in the two w,orks 
 of S. F. Lacroix, upon the Calcul Differentiel and Calcul 
 Integral, in the Traite De Calcul Differentiel et Calcul 
 Integral, par J. A. J. Cousin, and in the 19th Legon of 
 the Legons sur Le Calcul Des Fonctions 3 par J. L. 
 Lagrange, &c. &e. 
 
J5f LAPLACE'S MECHANICS. 
 
 two others, one of which v remains during the following 
 instant; the other of which V is destroyed by the action 
 of the other bodies; but the velocity of m being 
 
 \/(dx* + dy* + dz*) 
 
 - - -~ - before this resolution, and changing 
 
 afterwards into /( (d*+*.dx)*+(dy+*.4y)* 
 
 dt 
 A.ffc) 2 } 
 
 -- ' it is easy to perceive that we have 
 
 the preceding equation ought therefore to be put into 
 this form 
 
 =: const. 'Z. 
 
 * The equation 2. wi? 2 :n:c-f;2<p, when differentiated with 
 
 dv dq> dv , , dv' 
 
 respect to *, becomes 2.mu.-r-i=^, or mu. -fm 'v '. 
 
 +^.^.'+&c.^.| + S'.|' 4 ST. g'+ftc. which equa. 
 
 lion has place for a system of bodies connected with each 
 other in any manner whatever, which reciprocally attract or 
 repel each other, or are attracted towards or repelled from 
 fixed centres by any forces S 9 S f , S f/ $ &c. ; naming the mu- 
 toai distances of the bodies which attract or repel each other, 
 or tbeir distances from fixed centres of attraction or repulsion 
 *, /, s", &c.; taking the quantities S, S ! , 5"'', &c. which 
 represent the forces, positively or negatively, according as 
 tliesc forces are repulsive or attractive ; as the first tend to 
 increase and the second to decrease the distances s, s' 9 6 /y , 
 &c. This principle also has place in the movement of in- 
 elastic fluids so long as they form a coatinuous mass and 
 there is no impact amongst their molecules. 
 
LAPLACE'S MECHANICS. 153 
 
 . If in the equation (P) of No. IS, we suppose 
 
 If the forces , 5', 5", &c. are respectively functions of 
 the distances s 9 s' 9 s", &c. along which they act, which may 
 always be supposed when these forces are independent, of 
 each other ; or in general if the quantities S, *$", S f/ 9 &c. 
 
 are such functions of*, s', s" 9 &c. that the quantity .--{- 
 
 gjj fiJi 
 
 $> " +$.- +&c. is the differential of a function of s, 
 ut dt 
 
 *', s" 9 &c. which can be denoted by F(s, s', s" 9 &c .) 9 the 
 integral of the above equation will be 
 
 iz>*+roV*+ifi V a +&c.=c+2F(>, s r , *'/, &c.;, 
 c being a constant quantity. In this case the forces S 9 S f 9 
 S f/ , &c. which act along the lines s, /, s r/ 9 &c. will be re. 
 
 ^ d.Fc*, *', * v , &c.; d.jFx*, *', ", &c .; 
 
 presented by- -, ^. >&c , 
 
 respectively. 
 
 Let a, a', a' 7 , &c. be the respective values of s, s r , s", &c. 
 and F, V 9 V", &c. the respective velocities of m 9 m' 9 m ff 9 
 &c. at a given instant; the preceding equation referred to 
 this same instant will give 
 
 m ^ + m /r /2 4-m^^4-&c.z=c-h2Ffa ? ', a", &c J ; 
 consequently c^nmV z + m'V' z +m"V f ' 2 +&c.-~ %F(a, a', a", 
 &cj, therefore, by substitution, we have the following 
 general equation, 
 
 < l v'"' + &c.=imV 2 +m t V'* + m f/ F f/2 + &c. + 
 7 , &c.J 2F(, ', a", &c J. 
 
 This is the general equation of the preservation pf the 
 living forces, from which it is evident, that the whole living 
 force of the system depends upon the active forces, such as 
 the forces of attraction or repulsion, or springs, &c. ; and 
 upon the position of the bodies relative to the centre of 
 
154 LAPLACE'S MECHANICS 
 
 z+ 
 
 &c. 
 
 these forces ; therefore if at two instants the bodies are at 
 the same distances from these centres, the sum of their 
 living forces will be the same. 
 
 If the bodies should strike each other, or meet with ob. 
 stacles which cause a sudden alteration in their motions, the 
 above formula may be applied to the bodies during these 
 alterations, however short they may be ; thus denoting by 
 F, V 9 F /x , &c. their velocities at the commencement of the 
 sudden change, and by v, v' 9 v f/ 9 &c. their velocities at its 
 termination, also by a, a', a" 9 e. the values of the distances 
 5, *', s" 9 &c. at the beginning, and by A, A' 9 A" 9 &c. their 
 values at the end of the same action, the following equation 
 will have place, 
 
 m' V*+mV"* + &c.mv* m'v'* m"v'^ &c. = 
 rt, a', a", &c.) 2F(A 9 A', A 1 , Sec.). Which shews 
 that the difference of the living forces at the commencement, 
 and at the end of the action will be 2-Ffa, a', a" 9 &c J - 
 %F(A 9 A , A (f 9 ike.). This expression may have any finite 
 value whatever, however small the difference between the 
 respective quantities a, a', a fr , &c. and A 9 A' ', A ir 9 &c. 
 
 When perfectly elastic bodies strike each other, either 
 directly or by the intervention of levers or any machines 
 -whatever, the compression and the restitution of the shapes 
 of the bodies follow the same law, and the action is supposed 
 to continue until the bodies are restored unto the same re. 
 spective positions in which they were when the compression 
 commenced. In this case we shall have a=A, a'~A' 9 
 a"~A" 9 &c. and consequently F.(a 9 a', a" 9 &c.)=:F.(A 9 
 A' 9 A lf , &c.) ; therefore the living force will be the same 
 after as before the shock. 
 
 The following proof in the case of two elastic bodies im- 
 pinging upon each other, is derived from the lawi of tht 
 
LAPLACE'8 MECHANICS. 155 
 
 by substituting these variations in the expressions of 
 the variations $/, $/', S/ 7 , &c. of the mutual distances 
 
 motion of elastic bodies. Let Fand V be the velocities of 
 two elastic bodies m and m! before the shock, v and v' their 
 velocities after, also let u represent their common velocity at 
 the time of contact ; then as may be seen in the elementary 
 treatises upon Mechanics 
 
 V+m'V 
 
 ., 9 and the quantity 
 
 by substitution becomes m(% 
 
 la the shock of in. elastic bodies, the action is only sup. 
 posed to continue, until the bodies have acquired the 
 velocities which hinder their acting upon each other any 
 longer. As therefore the effect of these velocities upon the 
 mutual action of the bodies is nothing, if we had impressed 
 these same velocities before the action it would have been 
 the same, in consequence of the velocities composed of these, 
 and of the velocities properly belonging to the bodies. 
 
 Again therefore, it would be the same if the velocities 
 impressed were equal and directly contrary to those above 
 mentioned ; for the action will not be varied by supposing 
 that these impressed velocities were destroyed by the opposite 
 velocities. It consequently follows, that in the shock of 
 hard bodies the velocities v 9 v f , D", &c. after the shock, 
 ought to be such, that if we give these same velocities 
 to the bodies m, ', i 7/ , &c. in a contrary direction, 
 the equation mV 2 + m'V'*+ &c. mv* m'v'* m'V 72 &c. 
 = 2F(a, a'X, c.) 2F(^, A', A" 9 &c.) given above 
 will equally have place. But the terms which compose 
 the second member, as they depend upon the mutual 
 action of the bodies, will necessarily remain the same; 
 
156 LAPLACE'S MECHANICS. 
 
 of the bodies of the system, of which we have given 
 the values in No. 15 ; we shall find that the variations 
 
 therefore the value of wF 2 -f wz r F'*-f i^F //8 +&c. mv* 
 m'v' 2 wV /2 c. will not be changed by composing the 
 velocities F, F', F", &c. and the velocities v 9 v' 9 t>", &c. 
 with the velocities v, u', c", c. respectively. If 
 therefore the velocity composed of Fand v is represented 
 by M 9 the velocity composed of F' and v' by M' 9 &c. the 
 following equation will have p!ace, 
 
 M 2 -f m'M'* + m W*-f-&c. 
 
 as the velocities 2 v 9 vf 1?' ? u' 7 v fl 9 Sic. vu;-*sh. 
 
 Becaase F, F% F//, &c, are the velocities before the 
 shock, and v 9 v' 9 v r/ 9 &c. the velocities after iiie sair.e, it is 
 evident that M, M' 9 M" 9 &c. will be the velocities lest by 
 the shock; therefore mM*+m'M'*+m lf M l/ *+ &c. will be 
 the living force which results from these velocities, conse- 
 quently this conclusion may be obtained. That in the shock 
 of hard bodies, there is a loss of living forces equal to the 
 living force which the same bodies would have had, if each 
 of tliem should have been actuated by the velocity which it 
 lost by the shock. Vide the Principes Fondamentaux de 
 I/Equilibre et du Movement, par L. M. N. Carnot; and 
 the Theorie des Fonctions Analytiques, par J. L. Lagrange. 
 
 It is evident from what has been said, that when the bodies 
 of a system move in a resisting medium, or are subjected to 
 friction from fixed obstacles, the living forces are constantly 
 diminished and would at length entirely cease if the bodies 
 should not be kept in motion by other forces. The formula 
 Z.J'm(P.djC-\-Q.dy-{-R.dz) in these cases would not be an 
 exact integral. 
 
 In the equation given in the notes at page 144, we may 
 suppose that the variations &c, 2[y, and 5s are proportional 
 
LAPLACE'S MECHANICS. 15T 
 
 &r, 5y, and S. "will disappear from these expressions.* 
 If the system be free, that is if it have no one of Hs 
 parts connected 'with foreign Uodies ; the conditions 
 relative to the mutual connection of the bodies depend- 
 ing only upon their mutual distances, the variations 
 &F, tyj and 5s will be independent of these conditions ; 
 from which it follows, that by substituting for jr', $/, 
 J2 ; , jjc", &c. their preceding values in the equation 
 (PJ> we ought to equal separately to nothing the co- 
 
 to the velocities x, y, and z which the bodies have received 
 by impulsion. We shall then have the equation 
 
 in which Swz(^ 2 -|-j/ 2 -J-5 2 ) represents the whole living force 
 of the system. 
 
 * At No. 15, /= 
 
 &c. &c. 
 
 consequently by differentiation 
 
 ?/= 
 
 V IV*~ 
 
 &c. &c. 
 
 If otf + cta/ be substituted foretr', 3(jf-H&f/ f r %/j an ^ S- 
 for $3, in the preceding value of 5/, it will become 
 
 !/=^= 
 
 contain either &c, Sj/, or Ss. In like manner by making the 
 proper substitutions, the same quantities will disappear from 
 the values of $/', $/", &c. 
 
158 LAPLACE'S MECHANICS. 
 
 efficients of the variations &r, \y^ and $2, which gives 
 these three equations, 
 
 *=*.-(-*) 
 
 Let us suppose that X, F, and Z are the three co- 
 ordinates of the centre of gravity of the system; we 
 shall have by No. 15, 
 
 j T/K13T' * , Tfi It j-i T/Yl % 
 
 \7 ? nr . 
 
 ^* - r v> 3 y - v> S -^> - ~^ 3 
 
 2.??2 2.WZ ' 2.?72 
 
 consequently the following equations may be obtained, 
 _5^L. o 5^- 
 rf 3 ^ S.^J? 
 
 (he centre of gravity of the system therefore moves in 
 the same manner, as if, all the bodies m, m', &c. were 
 united at this centre, and all the forces which solicit 
 the system applied to it. 
 
 If the system be only submitted to the mutual action 
 
 (rf s a? "\ d z x 
 
 -fa t P I? then S.w. =:2.wP, but 
 
 w- ~2^~~9 therefore -^ 
 
 -^ ; in like manner it may be proved that -rr -r 
 
 t7t 
 
 _ 
 > aT1 *"" 
 
159 
 
 f the bodies which compose it, and to their reciprocal 
 attractions, \ve shall hare 
 
 0=2. mP ; 0=2. Q ; Q=2.mR ; 
 for, from expressing by p the reciprocal action of m 
 and m' whatever may be its nature, and denoting by/ 
 the mutual distance of these two bodies, we shall have 
 in consequence of this sole action 
 
 
 +, + J 
 
 from which we may obtain 
 
 Q=mP+i'P't ; Q=mQ+m'Q'; = m#-|-m'72' ; 
 and it is evident that these equations have place also 
 in the case in which the bodies exercise upon each 
 other a finite action in an instant. Their reciprocal 
 action will therefore make the integrals 2,.mP, 2,.mQ 
 and 2.mR disappear, consequently they are nothing 
 
 * If from one extremity of the line /a line be drawn pa. 
 rallel to the axis of #, and from the other extremity another 
 perpendicular to it, we shall have by No. 1. notes,/ : # x ' 
 
 p(xx')- 
 : : p : - - , or the resolved force acting in a direction^ 
 
 parallel to the axis of x, therefore P ^-- in like 
 
 ffianoer mQ= 
 
160 LAPLACE'S MECHANICS. 
 
 when (he system is not solicited by forces unconnected 
 with it. In this case we have 
 
 t* dt* 
 and by integrating 
 
 X=za+bt ; Y=a'+Vt ; =a fl 
 , , a', b'y a fi , b", being constant quantities. By ex- 
 tracting the time , we shall have an equation of the 
 first order either between X and F, or between X and 
 ; from which it follows, that the motion of the centre 
 of gravity is rectilinear. Moreover as its velocity is 
 
 equal to ^f , or to 
 
 ) it is constant, arid the motion is uniform. 
 It is evident from the preceding analysis, that this 
 inalterability of the motion of the centre of gravity of a 
 system of bodies, whatever may be their mutual actions, 
 subsists also in the case in which some of the bodies 
 lose during an instant by this action, a finite quantity 
 of motion*. 
 
 * T/..T. V ' V " * <y ' 
 
 * If the equations JT= . Frz -. and Z~ 
 
 2m. S.m' s.wz 
 
 are differentiated with respect to the time they will give the 
 following 
 
 dX dx dY dy dZ d* 
 
 S.w. -zzS.7w.-r-, S.TW.- ZZ2.W.-J 2,m.=2.m. -. 
 dt dt dt dt dt dt 
 
 As <he simultaneous impact of a part of the bodies will 
 change in general the velocities of all the bodies on account 
 of their mutual connection, let , 6, and c represent the 
 velocities of the body m in the respective directions of the 
 
 so-ordinates, or the values of ^p -^, and ^- immediately 
 
LAPLACE'S MECHANICS. 161 
 
 . If we make 
 
 the variation &r will again disappear from the expres- 
 
 before the impact ', b' 9 and c' those of ml &c. and ^, Z?, 
 and Cthe values of these velocities immediately after for the 
 body m 9 A', B', and C 1 the values for the body m r , &c. in 
 this case before the impact we shall have 
 
 v dX dY _ _ d 
 
 zm.2,ma 9 2w. zz2m& 3 2/w. -7- 
 
 and after 
 
 _ 
 
 . ,.-, Zm. 
 e/^ 
 
 The quantities of motion lost by al! the bodies at the in- 
 stant of impact should be such as would cause an equilibrium 
 in the system ; these forces in the respective directions of the 
 co-ordinates x, /, and z are ma mA 9 mb mB, and me 
 mC 9 which for all the points of the system are 2> ma 2,.mA, 
 "2,.mb S.mB, &c. These quantities, as the system is not 
 supposed to contain any fixed point, are respectively equal 
 to nothing, consequently S.mn:S.m^, X.:^.wJ5, 
 
 &c. therefore the values of -7-, -T-, &c. are the same after 
 
 dt j dt 7 
 
 as before the impact, and the velocity of the centre of gravity 
 f the system is the same in quantity and direction. 
 
 The following is a proof of its truth in the simple case of 
 the impact of two non. elastic bodies obtained in a very 
 different manner. 
 
162 LAPLACE'S MECHANICS. 
 
 sions S/, $f, /', &c. ; by supposing therefore (he 
 system free, as the conditions relative to the connection 
 
 Take upon the right line which the bodies describe any 
 point whatever for the origin of the spaces described, let x and 
 x' represent the distances of two non. elastic bodies m and ml 
 respectively from that point at the end of the time tf, and X that 
 of their centre of gravity, then by the known laws of the 
 motion of non. elastic bodies given in elementary treatises 
 
 mx -f- m' x' 
 
 upon Mechanics JC -- ; if v and v be the respective 
 m-\-m' 
 
 velocities of the bodies m and m 1 before, and F their common. 
 velocity or the velocity of their centre of gravity after the 
 
 shock then Fzz -- r , as is well known ; but the velo- 
 m -f m 
 
 7 V 
 
 city of their centre of gravity before the shock is 
 
 dx . dx' 
 m. -- \-m. . , dx , dx 
 
 dt ^ dt , therefore as n: y. and t/. the ve- 
 ' dt dt 
 
 locity of their centre of gravity is the same before as after 
 the shock. 
 
 Let the bodies which compose a free system be supposed 
 to be acted upon only by impulses, and in the equation given 
 in the notes page 144, let&e+r/, ^-f %//, &c. be substituted 
 for ^', $/, &c. and the forces S, S' , S ff , &c. be reduced 
 to the rectangular forces P, <2, R, P', &c. then the follow. 
 ing equations may easily be proved from what has preceded, 
 
 0=2.(mi P), OzzS.CmjJ-Q), 0=2.(mz R). 
 If the co-ordinates #, ^/, and 2 are referred to the centre of 
 gravity of the system we shall have 
 
 which being differentiated relative to *, by making (/JTzz 
 Xdt, dY=YM, dZZdt; dv=xdt, dy^ydt, dz~zdt, 
 dx'~x'dt, &c. we shall have 
 
LAPLACE'S MECHANICS. 163 
 
 of the pads of the system, only influence the variations 
 /, /', &c. the variation &a? is independent and arbi- 
 trary ; therefore if we substitute in the equation (P) 
 of No. 18, in the places of $#', &&", &c. ; fy, ty', //, 
 &c. their preceding values; we ought to equal sepa- 
 rately to nothing the co-efficient of #, \vhich gives 
 
 from which we shall obtain by integrating with respect 
 to the time t, 
 
 c being a constant quantity. 
 
 We may in this integral change the co-ordinates y^ y'^ 
 &c. into z 9 s', &c. provided that we substitute instead 
 of the forces <?, (?', &c. parallel to the axis of y, the 
 forces Jf2, R', &c. parallel to the axis of 2 ; which gives 
 xdz zdx 
 
 c f being a new constant quantity. We shall have in 
 like manner 
 
 being a third constant quantity. 
 
 and consequently 
 
 JSm SP0, YI,m 2Qzz 0, Z2m 2/2=0.' 
 These equations shew that the velocities given to the centre 
 of gravity are the same as would be given if all the bodies 
 of the system were united at Hand received at the same time 
 the impulses 2P 5 2Q, and 2R. 
 
164 
 
 Let us suppose that the bodies of the system are only 
 submitted to their mutual action, and to a force di- 
 rected towards the origin of the co-ordinates. If we 
 name, as above, p the reciprocal action of m and of m', 
 we shall have in consequence of this sole action 
 0=m.CP#Qat)+m'.(P'y'Q l x l ) *; 
 thus the mutual actions of the bodies will disappear 
 from the finite integral ^,.m(Py Qx). Let S be the 
 force which attracts m towards the origin of the co- 
 ordin^tes; we shall have in consequence of this sole 
 force, 
 
 S.x S.y 
 jp . _ f O y . 
 
 -yV-f^-f a* -Va,4#a+s* ' 
 
 the force S will therefore disappear from the expression 
 Py Qx; consequently in the case in which the 
 
 * The notes given to the preceding number, render it 
 unnecessary to say any thing respecting the equations given 
 in the first part of this, as they may be proyed in a similar 
 manner. The equations *2.m(Py Q#J 0, &c. are evi- 
 dently true, as 2.mPzz:0, Z.wQmO, and S.mEzzO. See 
 the last number. 
 
 f If a radius vector is drawn from the body m to the 
 origin of the co-ordinates, it will be equal to v / Y# 2 -f-# 2 + 
 z 2 ) ; by drawing a perpendicular from m to the axis of x 
 we shall have from resolving the force S, which must be 
 taken negatively as it tends to diminish the co-ordinates, 
 into -two others one acting in the direction of the axis and 
 the other perpendicular to it, the following proportion, 
 -^+s 2 -) : % :: g : P, therefore 
 
LAPLACE'S MECHANICS. 163 
 
 different bodies of the system are only solicited by their 
 action, and their mutual attraction, and by forces 
 directed towards the origin of the co-ordinates we 
 shall have 
 
 If we project the body m upon the plane of .r 
 the differential -~^ -- will be the area, which the 
 
 radius vector drawn from the origin of the co-ordinates 
 to the projection of m traces during the time dt* ; 
 
 * Suppose that A (Jig. 15,) represents the origin of the 
 co-ordinates x and y which are measured along the rectan. 
 gular lines A X and AY 9 and N the place of a body or its 
 projection, which is supposed to move along the curve 
 ACN 9 having AP=iy and PN=:AX~xfor its co-ordinates ; 
 then the area ACN described by the radius vector AP drawn 
 from the centre A to the point A', is equal to the area AC 
 NP minus the area ANP, but the area ACNPfxdy+c, 
 
 and the area ANP=:, therefore the zrea.ACNi=fxdy+c 
 -- and consequently its differential ~xdy d.-^-~ 
 
 _ xdy_ ydx _ xdyydx 
 .__ ___ _ 
 
 If # f.cos.w and# ^.sin.'a-, see page 21 ; by substitution 
 and neglecting indefinitely small quantities of the second 
 
 y ydx cd-n 
 order, -- - -- ___ ? uhich is the area of the sector 
 
 described by AN during the time dt. 
 
166 LAPLACE'S MECHANICS. 
 
 the sum of these areas multiplied respectively by the 
 masses of the bodies, is therefore proportional to the 
 element of the time; from which it follows, that in a 
 finite time it is proportional to the time. It is this 
 which constitutes the principle of the preservation of 
 areas*. 
 
 The fixed plane of x and y being arbitrary, this 
 principle has place for any plane whatever; and if the 
 force S is nothing, that is to say, if the bodies are only 
 subjected to their action and mutual attraction, the 
 origin of the co-ordinates is arbitrary, and we are able 
 to place that fixed point at will. Lastly, it is easy to 
 perceive by what precedes, that this principle holds 
 good in the case in which by the mutual action of the 
 bodies of the system, sudden changes take place in 
 their motions t. 
 
 * The product of the mass of a body by the area described 
 by the projection of its radius vector during an interval of 
 time denoted by unity, is equal to the projection of the 
 entire force of this body multiplied by the perpendicular let 
 fall from the fixed point upon the direction of the force thus 
 projected. This last product is the moment of the force 
 which wonld make the system turn about an axis passing 
 through the fixed point in a perpendicular direction to the 
 plane of projection. The principle of the preservation of 
 areas may therefore be reduced to this ; the sum of the mo. 
 ments of the finite forces that would make the system turn 
 about any axis whatever, which is nothing in the case of 
 quiiil)rium, is constant in that of motion. In this point of 
 view the principle answers to all the possible laws between 
 the force and the velocity. 
 
 It appears from what has been said, that the law of the 
 motion of the centre of gravity and that of the areas described 
 
LAPLACE'S MECHANICS. 167 
 
 There is a plane with respect to which c f and c f/ are 
 nothing which, for this reason, it is interesting to 
 know ; for it is evident that the equality of c 1 and c' 1 
 to nothing, ought to introduce the greatest simplicity 
 into the research of the motion of a system of bodies. 
 In order to determine this plane, it is necessary to refer 
 the co-ordinates x^y^ and z to three other axes, having 
 the same origin as the preceding. Let therefore Q re- 
 present the inclination of the plane sought that is formed 
 by two of these new axes, to the plane of as and y ; and 
 4 the angle which the axis of x forms with the intersec- 
 
 9F 
 
 tion of these two planes; so that - may be the in- 
 clination of the third new axis to the plane of x andy r 
 and - 4- may be the angle which its projection upon 
 
 the same plane makes with the axis of #, TT being the 
 semi-circumference*. 
 
 being proportional to the times, (which last "will again be 
 noticed in this number) are independent of the mutual action 
 of the bodies of the system ; these two laws Have therefore 
 a more universal application than the law of the preservatioiji 
 of living forces, which is oniy independent of those passive 
 resisting forces such as the pressures of the bodies, the ten, 
 sions of the threads or rods connecting them, Sec. whiclj 
 hinder the conditions of the system from being disturbed, or 
 la general all those forces which can bs expressed by (qua. 
 tions between the different co-ordinates of a body, whick 
 are independent of the time. 
 
 * In (Jig. 16, ) let the lines Ax, Ay, and Az represent 
 the axes of the three rectangular co-ordinates a?, ^r, and z. ? 
 
168 LAPLACE'S MECHANICS. 
 
 In order to fix our ideas, let us suppose that the 
 origin of the co-ordinates is at the centre of the earth ; 
 that the plane of x and^/ is that of the ecliptic, and 
 that the axis of z is a line drawn from the centre of the 
 earth to the north pole of the ecliptic ; let us also sup- 
 pose that the plane sought is that of the equator, and 
 that the third new axis is that of the rotation of the 
 earth directed towards the north pole ; will then be 
 the obliquity of the ecliptic, and %}/ the longitude of 
 the fixed axis of x relative to the moveable equinox of 
 spring. The two first new axes will be in the plane of 
 the equator ; and by naming cp the angular distance of 
 the first of these axes from this equinox, (p will repre- 
 sent the rotation of the earth reckoned from the same 
 
 equinox, and 2 + ^ w ^ ^ e tne an g u ^ ar distance of the 
 
 the line Ae the common section of the planes x y and 
 and let another plane be supposed to pass through the point 
 A perpendicular to Ae the common intersection of the planes, 
 cutting the plane x' y' in the line Ad and the plane ocy in the 
 line aAb; the.n as every line drawn from the point A per- 
 pendicular to the common section Ae must be in the perpen. 
 dicular plane, the axes Az and Az' of the ordinates z and z 1 
 are in it, consequently as the angle dAb or is the inclina. 
 tion of the two planes and the angle dAz' is a right angle, 
 the angle z'Aa formed by the ordinate z' and the plane xy is 
 
 it 
 represented by 9. Also the angle xAe formed by the 
 
 intersection of the planes xy and x' y 1 and the ordinate x is 
 represented by 4/, and the angle xAa, as the angle aAe is a 
 
 right angle by - %}/. 
 
LAPLACE'S MECHANICS. 169 
 
 second of these axes from the same equinox : let these 
 three axes be named principal axes. This being- agreed 
 upon, suppose x n y,^ and z f to be the co-ordinates of 
 m relative, first, to the line drawn from the origin of 
 the co-ordinates to the equinox of spring ; x, being 
 taken positive on the side of this equinox ; secondly, 
 to the projection of the third principal axis upon the 
 plane of x and y\ and thirdly, to the axis of z; we 
 shall then have* 
 
 * In (fig. 17,3 let A be the centre of the co-ordinates, 
 the perpendicular lines AX , and AY f the axes of the co- 
 ordinates x, and y n and the perpendicular lines AXand. AY 
 the axes of the co-ordinates x and y making with the former 
 lines the angle 4/, also suppose the line AM to represent 
 the projection of the line drawn from the origin of the co- 
 ordinates to the body m upon the plane of the x^s and y's^ 
 from the point 3/draw the lines Mx f and My t perpendicular 
 to the lines AX t and AY n and the lines MX and My perpen. 
 dicular to the lines ^LYand AY^ then we shall have Mx f ~y j9 
 My l ~Aif fzzx D Mx~y and My yfr x ? also the angle 
 X t AX~-^) and the angle Aax t formed by the intersection 
 of the lines AX and Mx^QO ^' ^ n the right angled tri- 
 angle aAx t by trigonometry cos. -^ rad. (1) : 
 
 Aa^n. -, also rad.(l) : sin.-vj/ : : ^ 
 
 sin.^ sin.^- _ 
 
 COM; ; then as Jfcfj: /=y/> **=*'Ht5i*' Intheri 5 ht 
 
 angled triangle M#, as the angle Me* 
 
 Y 
 
170 LAPLACE'S MECHANICS. 
 
 Let x,,) y lt ) and z a be co-ordinates referred, first, to 
 the line of the equinox of spring ; secondly, to the per- 
 pendicular to this line in the plane of the equator ; and 
 thirdly, to the third principal axis, then we shall have 
 
 Lastly, let x lln ?/ //l9 and z in be the co-ordinates of m 
 referred to the first, to the second, and to the third 
 principal axis respectively ; then 
 
 .r,, r //y .cos,<p ^///.sin.p ; 
 
 From which it is easy to conclude, that 
 
 .sn.^; 
 i ^=.r /// .{cos.0.cos.4/.sin.(p sin, 4.. cos. <p} 
 
 -\~ijin .{ cos. 9 . cos . -4". cos. (p-[-sin . 4 . sin . < }-\-z IJ/ . sin. 
 
 .COS.4/ ; 
 
 the angle aMx^i an ^ by trigonometry rad.(l) : sin. 
 
 (sin.4/ \ 
 y-*<-~^j ' 
 
 sin. 2 4/ 
 .: = * r cos. 4- +y,. 
 
 in in the triangle aMx, by trigonometry rad. (1) 
 
 cos. 
 
 gin. 4 / 
 
 In a similar manner the co-ordinates relative to the other 
 axes may be found. 
 
LAPLACE'S MECHANICS. 171 
 
 By multiplying these values of a?, y, and 2 respect- 
 ively by the co-efficients of x n , in these values, we shall 
 have from adding them together 
 
 -{-y.{cos.0.cos.4,.sin.<p sin.\J/.cos.(p} s.sin.Q.sin.p. 
 
 By multiplying in like manner the values of x, y^ 
 and z respectively by the co-efficients of y tll in these 
 values and afterwards by the co-efficients of z nn we 
 shall have 
 ^ /// n=,r.{cos.6.sin.v},.cos.(?) cos. >J/, sin.?)} 
 
 .,.sin.( z. sin. 9. cos. <. 
 
 These different transformations of the co-ordinates 
 will be very useful to us as we proceed. If we place 
 one, two, &c. marks above the co-ordinates #, y^ z, 
 x fln y ttn and z lln we shall have the co-ordinates cor- 
 responding to the bodies m 1 , m 1 *, &c. 
 
 From the above \i is easy to conclude by substituting 
 
 xdy ydx xdz 
 
 c, c f , and c" in the places of S.TTZ. - ^- , S.m, 
 
 zdx ydzzdy 
 
 - - , and S.m. 9 that 
 
 sin.-^; 
 
 . 
 
 cos. 9. cos. \|/. cos. 
 
 cos.9.cos,-4/.8in.(p} 
 
172 LAPLACE'S MECHANICS. 
 
 If we determine 4> and so that we may have sin 
 
 c /y c ' 
 
 which give 
 
 COS. < 
 
 we shall have 
 
 
 
 * Let a, , and y represent the angles which a perpendi- 
 cular to the invariable plane respectively makes with the 
 axes #, y, and z we shall then have the following equations, 
 cos.zz sin. 0. sin. 4^ 
 cos./3:nsin.0.cos.4', 
 
 COS.y^ZCOS.0. 
 
 In order to prove that cos.arzsin.0.sin,-4/ ? let C (Jig 18) be 
 supposed the centre of the co-ordinates, CX the axis of #, 
 CF the line of intersection of the planes xy and tf//^///, CA 
 the axis of z a/ which is perpendicular to the line CF from 
 any point A of the line CA let fall the perpendicular AB 
 upon the plane xy, join CB, then CB will be the projection 
 of the line CA upon that plane; let the angle of the inclina- 
 tion of the planes xy and x ja y Ul be represented by 0, then 
 
 9T 
 
 its complement the angle ACE will be 9, TT being the 
 
 semi. circumference of a circle the radius of which is unity. 
 From A draw AD perpendicular to CX and join BD 9 let 
 
LAPLACE'S MECHANICS* ITS' 
 
 the values of c' andc" are therefore nothing with respect 
 to the plane of x ui and y,,, determined in this manner. 
 There is only one plane which possesses this property, 
 for supposing it is that of x andy, we shall have 
 
 
 the angle FCX be denoted by 4/, then its complement the 
 angle XCB will be -- %!/. If AC is supposed to repre- 
 
 sent the radius equal to unity of a circle, then BCzzsin.5. 
 and, as the triangle BDC is right angled, by trigonometry 
 we have 
 
 rad.(l) : sin, 4, : : JBC(sinJ) : Cflzzsin.O.sin.^ ; 
 but CD is the cosine of the angle ACD of inclination of the 
 axes z tll and #, therefore cos.a sin.0.sin.-4/. 
 
 From the centre C of the co-ordinates draw the line CY 
 perpendicular to CX or the axis of #, then CY will repre- 
 sent the axis of y, from A let fall the perpendicular AY 
 upon CK, join BY. In the right angled triangle UFCwe 
 have 
 
 rad.(l) : cos.^ : : O?(sin.d) : CFsin^.cos.^ ; 
 but CFis the cosine of the angle ACY or /S, therefore cos. 
 
 The angle formed by the axes z and z ni is equal to that 
 formed by the planes xy and x ia y, H consequently cos, yrrcos.fi. 
 We have therefore the following equations, 
 
 cos.a:nsm.0.sin.4/ m 
 
174 LAPLACE'S MECHANICS. 
 
 By equalling these two functions to nothing, we shall 
 Lave sin.SrrrO ; that is the plane of x, n and y Hl then co- 
 incides with that of x and y. The value of 2.m. 
 
 being equal to /^j^ 7 ^^", whatever 
 
 may be the plane of x and y ; it results that the quan- 
 tity c*-{-c /2 +c //2 is the same, whatever this plane may 
 be, and that the plane of x ltt and y in determined by 
 what precedes, is the plane relative to which the 
 
 function S.m.'i' j s the greatest* ; the plane 
 
 from which the position of the plane may be readily found. 
 It appears preferable to take c' S.w.- -- - instead of 
 
 2.w. - - - in which case cos. /3 would be affirmative. 
 at 
 
 As the quantities c, c', and c" are constant the position of 
 the plane is invariable. 
 
 * As the quantity c*-}-c!*+c f/ * * s invariable whatever may 
 be the plane of x and ^, let a?', z/', and z' represent the co- 
 ordinates of any other system of rectangular axes about the 
 same point, as those of x, ?/, and z, then if a, d , and a" 
 have the same relation to the planes formed by these co- 
 ordinates as c, c', and c" have to those formed by the 
 co-ordinates x, #, and z, we shall have z -{-a' 2 + a//2 c2 "f 
 
 zo therefore 1/c 2 -f </a_fc" 2 a' 2 o^ ; conse- 
 quently, when / and a" vanish is a maximum and equal 
 to t/cTfc^+c**. 
 
 We may at any instant find the position of the invariable 
 plane relative to any determinate point in space, if we know 
 
175 
 
 we are treating about therefore possesses these remark- 
 able properties, first, that the sum of the areas traced 
 by the projections of the radii vectores of the bodies 
 and multiplied respectively by their masses, is the 
 greatest possible ; secondly, that the same sum relative 
 to any plane whatever which is perpendicular to it is 
 nothing, because the angle <p remains indeterminate. 
 AVe shall be able by means of these properties to find 
 this plane at any instant, whatever may have been the 
 variations induced by the mutual action of the bodies 
 in their respective positions, the same as we are able 
 easily to find at all times the position of the centre of 
 gravity of the system ; and for this reason it is as natu- 
 
 at this instant the velocities and the co-ordinates of all the 
 moving bodies of the system, as the position of this plane 
 with respect to the centre of the co-ordinates depends upon 
 the three quantities c, c', and c" which are respectively equal 
 
 
 
 to 2..- . S.. -nd S.. y 
 
 ~ z.-~ I. These quantities are evidently composed of 
 
 the co-ordinates of the moving bodies and the components 
 of the velocities parallel to their axes. 
 
 It appears from the above that c is equivalent to the sum 
 of the moments taken with reference to the centre of the co- 
 ordinates and projected upon the plane xy of the quantities 
 of motion of the bodies m^ m', &c. at any instant ; c' and c" 
 are the sums of the moments of these forces taken with re. 
 spect to the same point and projected upon the planes of xz 
 and yZ) consequently the invariable plane coincides with the 
 principal plane of these moments ; seepage 114. 
 
176* LAPLACE'S MECHANICS, 
 
 ral to refer the x-s and ys to this plane, as to refer the 
 origin of the co-ordinates to the centre of gravity of the 
 system*. 
 
 * If two or more bodies of <he system strike each other, 
 the position of the invariable plane will not be altered either 
 with respect to a fixed point, if the system turns about it, 
 or with respect to any point whatever, if the system moves 
 freely in space. In order to prove this, it will be necessary 
 to shew that the values of c, c', and c" are the same after as 
 before the impact, for which purpose the denominations 
 given page 160 notes, will be made use of. This proof 
 follows from the quantities of motion lost by the impact pro. 
 ducing equilibrium, which will give the following equations, 
 
 C)z(b B)} 0, 
 which are part of the six general equations of equilibrium 
 No. 15, and are equivalent to the three following 
 
 2m (x b ya)~ 2m (x B yA ) , 
 
 2m (xc za)-=2,m(xC zA), 
 
 '2m(yc zb)^=2,m(yC zB). 
 
 It is evident that the first members of these equations are 
 the values of c, c', and c 11 immediately before the impact and 
 the second members their velocities immediately after, there. 
 fore these values are the same after as before the impact. 
 
 It therefore appears that the sums of the areas denoted by 
 e, c'ty aiidc^l are notaitered by the mutual impact of the 
 bodies of the system ; these areas will consequently always 
 be proportional to the time employed to. describe them, 
 although in the interval of that time sudden changes may 
 have been produced in the velocities of the bodies from their 
 mutual impact. The impact of a body not belonging to the 
 system will in general change the values of these areas and 
 consequently the Direction of the invariable plane. 
 
LAPLACE'S MECHANICS. 177 
 
 22. The principles of the preservation of living 
 forces and of areas have place also, when the origin of 
 
 When a system, at liberty to turn in any direction about 
 a fixed point, receives a number of impulses, after they have 
 been reduced to three P, Q, and R in the directions of the 
 
 &T, y. and s, the accelerating forces > , and - 
 ;.. dt* dt 2 dt z 
 
 be changed into the velocities a?, y, and z and we shall 
 have by substitution (see the beginning of this number) the 
 following equations, 
 
 '-3/<0 + 2>< 
 
 for the first instant of the motion produced by the impulses. 
 If the system is entirely free, the point may be taken any 
 where in space and the above equations will hold true. This 
 will also be the case if there is no fixed point and the system 
 turns about its centre of gravity. 
 
 If there are no accelerating forces, the effect of the im- 
 pulses will be continued, the terms which depend upon the 
 impulses P, Q, and R being regarded as constant. For, as 
 ff, y^ and sare the velocities in directions respectively paral- 
 lel to the axes of x 9 #, and z, we have dx~xdt, dyydt^ 
 
 dzzdt, &c. and the above equations will be changed into 
 the following, 
 
 consequently 
 
 c =2.(Qx 
 
 '=?,. (RxPz), 
 
178 LAPLACE'S MECHANICS, 
 
 the co-ordinates is supposed to have a rectilinear and 
 uniform motion in space. To demonstrate it, let X 9 
 I 7 , and Z be named Ihe co-ordinates of this origin, 
 supposed in motion, referred to a fixed point, and let 
 
 &c. &c. &c. 
 
 MI! y v , s t , .I'/j &c. will be the co-ordinates of w/, m', 
 &c. relative to the moving origin. We shall ha vg by 
 the hypothesis 
 
 but we have by the nature of the centre of gravity when 
 the system is free 
 
 0=2^1^ X+d 2 ^} Z.m.P.dt*-, 
 
 the equation ( Pj of No. 18, will also become by sub- 
 stituting aX+fcr,, 5I4-^> &c. instead of ^, Jy, &c. ; 
 
 -P 
 
 an equation which is exactly of the same form as the 
 equation (P} 9 if the forces P, Q, and /^ only depend 
 upon the co-ordinates ^ O y, ? s,, ^/ &c. By applying 
 
 The values of the constant quantities c, c', and c" may 
 therefore be expressed by the initial impulses given to each 
 body, and it has been shewn that these values are the sums 
 of the moments of these impulses with respect to the axes of 
 x, y, and s. 
 
LAPLACE'S MECHANICS. 179 
 
 io it the preceding analysis, we shall obtain the prin- 
 ciples of the preservation of living forces and of areas 
 with respect to the moving origin of the co-ordinates. 
 If the sys-em be not acted upon by any forces uncon- 
 nected with it, its centre of gravity will have a right 
 lined and uniform motion in space, as we have seen at 
 No. 20; by fixing therefore the origin of the co- 
 ordinates X) ?/ 9 and s at this centre, these principles 
 will always have place, X, Y 9 and Z being in this 
 case the co-ordinates of the centre of gravity, we shall 
 have by the nature of this point, 
 
 0=S.m,^; 0=2. m.y^; 
 which equations give 
 
 * If X+Xv F-f^, Z+s,, &c. be substituted for x> y, 
 z, &c. in the equations 
 
 and regard be had to the equations 
 
 7-t ~\7" 
 
 --. 
 
180 LAPLACE'S MECHANICS. 
 
 Thus the quantities resulting from the preceding 
 principles are composed, first, of quantities -which 
 
 
 we shall have the following transformations, 
 
 which are similar to the original equations. If the quanti- 
 ties PytQx^ &c. disappear, we shall by integration have* 
 the following equations, 
 
 at 
 
 
 These equations are similar to those given in the last number 
 and the same consequences may be deduced from them. 
 In like manner the equation 
 
 may by similar substitutions be changed into the following, 
 
 which only differs from it in having the co-ordinates referred 
 
LAPLACE'S MECHANICS. 181 
 
 wonll have place if all the bodies of the system were 
 united at their common centre of gravity ; secondly, of 
 quantities relative to the centre of gravity supposed 
 immoveable; and as the first of these quantities are 
 constant, we may see the reason why the preceding 
 principles have place with respect to the centre of 
 gravity. By fixing therefore at this point the origin of 
 the co-ordinates #, y, z, x f , &c. of the equations (Z) 
 of the preceding No. they will always have place, from 
 which it results, that the plane passing constantly 
 through this centre and relative to \\hich the function 
 
 xduydx , 
 S.m. is a maximum, remains always parallel 
 
 to itself during the motion of the system, and that the 
 same function relative to every other plane which is 
 perpendicular to it is nothing. 
 
 The principles of the preservation of areas and of 
 living forces, may be reduced to certain relations 
 amongst the co-ordinates of the mutual distances of the 
 bodies of the system. In fact the origin of the ,r's, of 
 the ys, and of the s's being always supposed at the 
 
 to the centre of gravity instead of a fixed point. By inte. 
 gration, if the quantity 2*.m.(Pdx l -\-Qdy l -i- Rdz^ be in. 
 tegrable and supposed equal to c/<p, we shall have the foil owing 
 equation for the preservation of living forces with respect to 
 the centre of gravity, 
 
 c being a constant quantity. 
 
LAPLACE'S MECHANICS. 
 
 centre of gravity, the equations (Z) of the preceding 
 No. may be changed into the following forms, 
 
 (dx'dx) > 
 
 .2. M=2 .^'. \ 
 
 (dx'dx) > 
 
 It may be observed that the second members of these 
 equations multiplied by dt, express the sum of the pro- 
 jections of the elementary areas traced by each right 
 line that joins two bodies of the system, of which one 
 is supposed to move about the other that is considered 
 as immoveable, each area being multiplied by the pro- 
 duct of the two masses which the right line joins. 
 
 If we apply the analysis of No. 21, to the preceding 
 equations, we shall perceive that the plane which passes 
 constantly through any one of the bodies of the system, 
 
 and relative to which the function 2mm f . 
 
 f . 5 (x 1 - 
 
 _ ls a maximum, re- 
 
 mains always parallel to itself in the motion of the 
 system; and that this plane is parallel to the plane 
 passing through the centre of gravity relative to which 
 
 the function 2.m.-- is a maximum. We shall 
 perceive also, that the second members of the preceding 
 
LAPLACE'S MECHANICS. 185 
 
 equations are nothing relative to every plane which 
 passes through the same body, and is perpendicular to 
 the plane of which we have treated*. 
 
 * The positions of the invariable planes of the same system 
 relative to two different points in space may be compared as 
 follows -, let , /3, and y represent the co-ordinates of a new 
 point in space, and C, C' 9 and C" the values of c, c', and <P 
 when the origin of the co-ordinates is placed at this point ; 
 we shall then have, for instance, 
 
 C du dx~) 
 
 If the sum of the masses of all the bodies be denoted by M 
 and the co-ordinates of the centre of gravity of the system by 
 X, Y, and Z, then 
 
 d dY dx dX 
 
 therefore 
 
 dii dx\ du dx 
 
 ax d 
 
 iu-*- 
 
 In like manner 
 
 / a 
 
 \?-iu-*-d; 
 
 ax 
 
 It is evident from the above, that if the values of ? J3, and 
 7 be such as to render the three quantities 
 
 dX dY dX dZ dY dZ * 
 
 MS^IP y "di-~*-dt' * m T?r+'Tp 
 
 respectively equal to nothing, we shall have Czrc, (7zrc', 
 and C H c f/ 9 consequently the invariable planes relative to 
 the two centres of co-ordinates will be parallel to each other, 
 
184 LAPLACE'S MECHANICS. 
 
 The equation Q of No. 19 may be put info the form 
 
 dy?-\~(dz'-dz)**} 
 ~- -- ! -- 1 const 
 at* 
 
 an equation relative solely to the co-ordinates of the 
 mutual distances of the bodies, in which the first 
 member expresses the sum of the squares of the relative 
 velocities of the bodies of the system about each other, 
 considering them two and two and supposing one of 
 the two immoveable, each square being multiplied by 
 the product of the two masses which we have con- 
 sidered. 
 
 23. By resuming the equation (/?) of No. 19, and 
 differentiating it with respect to the characteristic 5, we 
 shall have 
 
 As in this case the centre of gravity of the system mores 
 
 dX dY 
 uniformly in a right line, its relative velocities -y-> r- and 
 
 dZ 
 
 ~ are constant, therefore the three above mentioned quan. 
 
 tides will be eqnal to nothing when (he line joining the two 
 centres is parallel to that described by the centre of gravity 
 of the system. The invariable planes relative to all the 
 points of any line parallel to that described by the centre of 
 gravity of the system are therefore parallel to each other ; 
 and the direction of the invariable plane does not change 
 but when it is passing from one parallel to another. It is 
 also evident that the invariable plane relative to the centre 
 of gravity of the system always remains parallel to itself 
 during the motion of that centre. 
 
LAPLACE'S MECHANIC*. 18$ 
 
 the equation (P) of No. 18, then becomes 
 
 .x.d.~-- t d.--^z.d.l t.mdt.vdv. 
 
 Let ds be (he element of the curve described by m , 
 ds r the element of that described by m' y &c. ; we shall 
 have 
 
 vdt=ds ; i/flfer^' ; &c. 
 
 z* ; &c. 
 
 from which we may obtain by following the analyst* 
 of No. 8, 
 
 By integrating this equation with respect to the differ- 
 ential characteristic </, and extending (he integrals to 
 the entire curves described by the bodies m y m' y &c. 
 we shall have 
 
 dx.fa + dy . ty+dz 5s 
 
 the variations JJT, S;y, Jg, &c. being thus but the constant 
 quantity of the second member of this equation, relative 
 to the extreme points of the curves described by m 9 
 in', &c. 
 
 It follows from the above, that if these points ar 
 supposed invariable, we shall have 
 S.S./wwfo; 
 which shews that the function S.fmvds is a minimum*. 
 
 * As ds~vdt, the function Z.fmvds which is a maximum 
 or a minimum may be made to assume the form S.m/ 2 <# or 
 fdl.E.mv*) in which Z.mv* denotes the living force of the 
 whole of the system at any instant whatever* Therefor* th* 
 
186 LAPLACE'S MECHANICS. 
 
 it is in this that fhe principle of the least action in the 
 motion of a system of bodies consists ; a principle which, 
 as we have seen, is only a mathematical result from the 
 
 principle treated upon may be properly reduced to this, that 
 the sain of the instantaneous living forces of all the bodies 
 from the time that they depart from certain given points 
 until they arrive at other given points, is either a maximum 
 or a minimum. Lagrange therefore proposes to call it the 
 principle of the greatest or the least living force, as con- 
 sidered in this manner it has the advantage of being general 
 as well for the state of motion as for that of equilibrium ; for 
 it may be proved, that the living force of a system is always 
 the greatest or the least in the state of equilibrium, from tne 
 equation 
 
 In this equation if Z.mv 2 is a maximum or a minimum the 
 f auction <p is, generally speaking, a maximum or a minimum, 
 consequently 
 
 This is the equation of the equilibrium of a system when the 
 forces P, Q, and R are respectively functions of the lines 
 X 9 y y and z ; it gives the following principle, first made 
 known by M. de Courtivron, The situation in which a 
 system has the greatest or the least living force is that in 
 which, if it were placed, it would remain in equilibrio. 
 
 The equilibrium would be stable if the sum of the living 
 forces should be a maximum and unstable if a minimum; for 
 the bodies of the system when moved from the situation of 
 equilibrium would tend to return to it if the equilibrium 
 were stable, their velocities would therefore diminish as they 
 receded from It, consequently the living force would be a 
 maximum in this position ; but it would be a minimum if the 
 equilibrium were unstable, as the bodies in Preceding from 
 
LAPLACE'S MECHANICS. 187 
 
 primitive laws of the equilibrium and of the motion of 
 matter. We have seen at the same time, that this 
 principle combined with that of living forces, gives 
 
 their position relative to this state would tend to recede 
 from it still farther by reason of the increase of their velo- 
 cities. 
 
 In a system of bodies let g denote the force of gravity, M 
 the sum of the masses of the bodies, anc- z the co-ordinate 
 of the centre of gravity of the system, the axis of z being 
 supposed vertical and in the direction of gravity, the equa- 
 tion of living forces relative to this system will be as follows, 
 
 2 , m v 2 c -^-gMz, 
 
 which shews that 2. mv z will be a maximum or a minimum 
 at the same time as z. 
 
 When the moving bodies are not acted upon by any 
 accelerating forces, the sum of the living forces at each in. 
 fctant is constant, consequently the sum of the living forces 
 for any time whatever is proportional to that time, from 
 which it follows, that the system passes from one position to 
 another in the least time possible. 
 
 In the case of perfectly hard or of perfectly elastic bodies, 
 the sum of the products of each mass by the square of the 
 difference between its velocities before and after the impact 
 is a minimum. This answers to the principle of the least 
 action. 
 
 The three equations (a), page 177, combined with that 
 of living forces 
 
 (see page 157) give a property de maximis et minimis relative 
 to a line about which the system turns iq the first instant 
 when it has received any impulse whatever, which line may 
 be called the axis of spontaneous rotation. 
 
JS8 LAPLACE'S MECHANICS. 
 
 the equation (P) of No. 18, which contains all that is 
 necessary for the determination of the motions of the 
 system. 
 
 Let a, 6, and c represent the parts of the velocities x r y y 
 and 2 which depend upon the change of position of the 
 bodies of the system with respect to each other ; when they 
 are added to the velocities resulting from the rotations page 
 
 107, the complete values of x, y, and 3 will be expressed 
 as follows 
 
 If these equations are differentiated, 4/, u } and p being re. 
 garded as variable, we shall have 
 
 The three equations (a) being multiplied respectively by 
 
 * * 
 
 Sp, Sw, and J4> and added together, making the variationg 
 J^, S^j and ^4/, which are the same for all the bodies, pasg 
 under the sign 2, will give by the substitution of the pre. 
 ceding values 
 
 The above equation of living forces, being differentiated 
 with respect to , gives 
 
 By the comparison of these equations it is evident that 
 
 -L.mfxix 4yty -f *J=0, 
 
 consequently 
 
 l..mjf a fy ft +*V=0 
 
 This equation shews that the living force which the system 
 
 acquires by impulsion, is always either a maximum or amin. 
 imum with respect to the rotations relative to three axes; 
 and as these rotations may be composed into one about the 
 
LAPLACE'S MECHANICS. 189 
 
 Lastly we have seen at No. 22, that this principle 
 has place also, when the origin of the co-ordinates is 
 in motion ; provided that its motion be right lined and 
 uniform and that the system be free. 
 
 axis of spontaneous rotation it follows, that this axis is in 
 such a position as to have the living force of all the system 
 the greatest or the least with respect to it. 
 
 This property of the axis of rotation with respect to solid 
 bodies of any form was discovered by Euler, and extended 
 by Lagrange to any system of bodies either invariably con. 
 nected together or not, when these bodies receive any 
 impulses whatever. 
 
LAPLACE'S MECHANICS. 
 
 CHAP. VI. 
 
 Of the laws of the motion of a system of bodies in all 
 
 the possible mathematical relations between 
 
 the force and the velocity. 
 
 24. K have before observed at No. 5, that there 
 are an infinite number of ways of expressing the force 
 by the velocity, which do not imply a contradiction. 
 The simplest of them is that of the force being propor- 
 tional to the velocity, which as we have seen, is the 
 law of nature. It is according to this law, that we 
 have explained in the preceding chapter the differential 
 equations of the motion of a system of bodies ; but it is 
 easy to extend the analysis of which we have made use, 
 to all the mathematical laws possible between the velo- 
 city and the force, and thus to present under a new 
 point of view, the general principles of motion. For 
 this purpose, let us suppose that F being the force and 
 v the- velocity, we have F=Q( v) ; (p(v) being any 
 function whatever of v : let us denote by <p f (v) the 
 differential of q( v) divided by dv. The denominations 
 of the preceding numbers always remaining, the body 
 
LAPLACE'S MECHANICS. 
 m will be acted upon parallel to the axis of x by the force 
 . *. In the following instant this force will be- 
 
 CIS 
 
 dx . dx 
 
 come t 
 
 y 1 , because y =0. Moreover P 5 9, and 72 being 
 
 the forces which act upon the body m parallel to the 
 axes of the co-ordinates : the system will be by No. 18, 
 in equilibrio in consequence of these forces and of the 
 
 differentials d . 
 
 \ dt -o 
 
 taken with a contrary sign ; we shall have therefore 
 instead of the equation (P) of the same number the 
 following; 
 
 which only differs in this respect, that , -^-, and --- 
 
 are multiplied by the function - ; which may be 
 
 supposed equal to unity in the case Of the force being 
 proportional to the velocity. But this difference ren* 
 ders the solution of the problems of mechanics very 
 difficult. Notwithstanding we can obtain from the 
 equation (SJ, certain principles analogous to those of 
 the preservation of living forces, of areas, and of th 
 centre of gravity. 
 
 If we change for into dx 9 ly into rfy, and $a into tfsr, 
 &c. we shall have 
 
192 LAPLACE'S MECHANICS. 
 
 and consequently 
 
 By supposing !L.m>(Pdx-\-Qdy-{-Rdz) an exact 
 differential equal to d\, \ve shall have 
 
 2.fmvdv.q>'(v)=const.-{-\ ; (T) 
 an equation analogous to the equation (R) of No. 19, 
 and which chanes into it in the case of nature where 
 
 The principle of the preservation of living forces has 
 place therefore, in all the mathematical laws possible 
 between the force and the velocity, provided, that we 
 understand by the living force of a body, the product 
 of its mass by double the integral of its velocity multi- 
 plied by the differential of the function of the velocity 
 which denotes the force. 
 
 If we suppose in the equation (S), fa^=j,r-f-*.r/ ; 
 ty'=*y+ty/ ; 9z'=*z+*z/ ; *^==*ff-B*/ r ; &c.; we 
 shall have by equalling separately to nothing the co- 
 efficients of &r, Sy, and J, 
 
 These three equations are analogous to those of No. 
 20, from which we have deduced the preservation of 
 the motion of the centre of gravity in the case of nature, 
 where the system is only subjected to the action and 
 mutual attraction of the bodies of the system. In this 
 
LAPLACE'S MECHANICS. 193 
 
 case S.raP, S.mQ, and 2.mR are nothing and we 
 have 
 
 v dx <p(v) dy <p(v) 
 
 const.=2.m.-r-. ; const. =2. m. . ; 
 
 at v at v 
 
 dz e>(v) 
 const. =2.^.---. ; 
 dt v 
 
 m . . is equal to mQ('o). ) and this last quantity 
 
 is the finite force of a body resolved parallel to the axis 
 of x\ the force of a body being the product of its mass 
 by the function of the velocity which expresses the force. 
 Therefore the sum of the finite forces of a system 
 resolved parallel to any axis whatever, is in this case 
 constant whatever may be the relation of the force to 
 the velocity ; and what distinguishes the state of motion 
 from that of rest is, that in the last state this same sum 
 is nothing. These results are common to all the 
 mathematical laws possible between the force and the 
 velocity ; but it is only in the law of nature that the 
 centre of gravity moves with an uniform and rectilinear 
 motion. 
 Again, let us suppose in the equation ( S} 9 
 
 the variation Ix will disappear from the variations of 
 the mutual distances /, /', &c. of the bodies of the 
 system and of the forces which depend upon*these 
 quantities. If the system is free from obstacles inde- 
 pendent of it, we shall have by equalling to nothing 
 the co-efficient of $#, 
 
 9m 
 
194 LAPLACE'S MECHANICS. 
 
 from which \ve may obtain by integration 
 
 We shall in like manner have 
 
 m 
 
 c, e', and c* being constant quantities. 
 
 If the system is only subjected to the mutual action 
 of its parts, we have by No. 21, 'Z.m.(Py--Qx)=Q: > 
 ?,.m.(PzRx)=; 2.m.(Qz %;=0; also, 
 
 i #. - u. V is the moment of the finite 
 \ dt ^ dt/ v 
 
 force by which the body is actuated, resolved parallel 
 to the pjane of x and y^ to make the system turn about 
 
 the axis of ,; the finite integral 2..<^fa^.?^ 
 
 dt v 
 
 is therefore the sum of the moments of all the finite 
 forces of the bodies of the system, which are exerted 
 to make it turn about the same axis ; this sum is there- 
 fore constant. It is nothing in the state of equilibrium ; 
 we have here therefore, the same difference between 
 these two states, but relatively with respect to the sum 
 of the forces parallel to any axis whatever. In the law 
 of nature this property indicates, that the sum of the 
 areas described about a fixed point by the projections 
 of the radii vectores of the bodies, is always the same 
 in equal times ; but the areas described are constant, 
 only in the law of nature, 
 
LAPLACE'S MECHANICS. 195 
 
 If we differentiate, with respect to the characteristic 
 , the function 2.fm.q>(v).ds ; we shall have 
 
 but we have 
 
 ftfc^^l*^!^?*-!. 5 * AW.+. d. 
 
 ds v dt dt 
 
 we shall therefore have from integrating by parts, 
 
 .=Ky. \ ,.^,,.+|. 
 
 If the extreme points of the curves described by the 
 bodies of the system are supposed to be fixed, the terra 
 without the sign / will disappear from this equation; 
 we shall therefore have in consequence of the equa- 
 tion ( S) 9 
 
 but the equation (T) differentiated with respect to 
 gives 
 
 we have therefore 
 
 This equation answers to the principle of the least 
 action in the law of nature, m.^(v) is the entire force 
 of the body ?w, therefore the principle comes to this, 
 that the sum of the integrals of the finite forces of the 
 bodies of the system, multiplied respectively by the 
 elements of their directions is a minimum: presented 
 
196 LAPLACE'S MECHANICS. 
 
 in this manner it answers to all the mathematical laws 
 possible between the force and the velocity. In the 
 state of equilibrium, the sum of the forces multiplied 
 by the elements of their directions is nothing in conse- 
 quence of the principle of virtual velocities; what 
 therefore distinguishes in this respect, the state of 
 equilibrium from that of motion, is, that the same 
 differential function which is nothing in the case of 
 equilibrium, on being integrated gives a minimum in 
 that of motion. 
 
LAPLACE'S MECHANICS. 
 
 CHAP. VII. 
 
 Of the motions of a solid body ofanyfgure whatever. 
 
 25. THE differential equations of the motions of 
 translation and rotation of a solid body, may be easily 
 deduced from those which we have given in Chap. V ; 
 but their importance in the theorj of the system of the 
 world, induces us to devclope them to a greater extent. 
 Let us suppose a solid body, all the parts of which 
 are solicited by any forces whatever. Let x^ y, and * 
 be the orthogonal co-ordinates of its centre of gravity ; 
 x-\-x', y-\-y'i and z -f- *' the co-ordinates of any 
 molecule dm of the body, then x'^ y', and z f will be 
 the co-ordinates of this molecule with respect to the 
 centre of gravity of the body. Let moreover P, Q 9 
 and R be the forces which solicit the molecule parallel 
 to the axes of x y y, and z. The forces destroyed at 
 each instant in the molecule dm parallel to these^axes 
 will be by No. 18, if the element dt of the time is sup- 
 posed constant, 
 
198 LAPLACE'S MECHANICS. 
 
 It follows therefore that all the molecules acted upon 
 by similar forces should mutually cause an equilibrium. 
 We have seen at No. 15, that for this purpose it is 
 necessary that the sum of the forces parallel to the 
 same axis should be nothing, which gives the three 
 following equations ; 
 
 the letter S being here a sign of integration relative to 
 the molecule dm, which ought to be extended to the 
 whole mass of the body. The variables x, y, and % 
 are the same for all the molecules, we may therefore 
 suppose them independent of the sign S j thus denoting 
 the mass of the body by m, we shall have 
 
 , d z x , d*x ^ d* 
 
 S.-.dm=m.-; S. 
 
 ... 
 
 We have moreover by the nature of the centre of 
 gravity 
 
 S.x'.dm=0 ; S.y'.dmQ ; S.z'.dm=0 ; 
 which equations give 
 
 we shall therefore obtain 
 
LAPLACF/8 MECHANICS. 199 
 
 ,72 r > 
 
 m.^-S.Pdm; 
 
 d 2 z 
 
 these three equations determine the motion of the centre 
 of gravity of a body, and answer to the equations of 
 No. 20, relative to the centre of gravity of a system of 
 bodiesl* 
 
 We have seen at No. 15, that for the equilibrium 
 of a solid body, the sum of the forces parallel to the 
 axis of x multiplied respectively by their distances 
 from the axis of z 9 minus the sum of the forces parallel 
 to the axis of y multiplied by their distances from the 
 axis of % 9 is equal to nothing ; we shall therefore have 
 
 * As the equations ( A) do not contain the co-ordinates 
 a/, x" 9 &c. of the different molecules of the body, they are 
 independent of them and only indicate the motion of the 
 centre of gravity of the body. This motion is not influ- 
 enced by the mutual actions of the molecules upon each 
 other, but solely by the accelerating forces which solicit 
 them. 
 
 It is evident from the above that the centre of gravity of 
 any free body whatever, like that of a system, has always 
 the same motion as if this body were all concentrated into 
 one point and acted upon by the same accelerating forces as 
 the parts of the body were, when in their natural state. 
 This accords with what has been given at No, 20. 
 
200 LAPLACE'S MECHANICS, 
 
 c 
 s. 
 
 S.{(x+x').Q(y+y').P}.dm ; (I) 
 but we have 
 
 and in like manner 
 
 S. ( QxPy).dm=x. S. Qdmy. S. Pdm ; 
 lastly we have 
 
 d*x.S.y'dm-\-x.S.d*y'.dmy.S.d*x'.dm', 
 and by the nature of the centre of gravity each of the 
 terras of the second member of this equation is nothing; 
 the equation (J) will therefore become in consequence 
 of the equations (A), 
 
 S * (--^- -).dm=S.(Qx'-Py').dm ; 
 
 by integrating this equation with respect to the time f ? 
 we shall have 
 
 S- .dmS.f( Qx'-Py').dt.dm ; 
 
 the sign /of integration being relative to the time t. 
 
 From the above it is easy to conclude, that if we 
 make 
 
 S.f( Qx'Py').dt.dm=N; 
 
 S.f( Rx'Pz') . dt. dm= N 1 ; 
 
 we shall have the three following equationi 
 
LAPLACE'S MECHANICS. 201 
 
 these three equations contain the principle of the pre- 
 servation of areas; they arc sufficient for determining 
 the motion of the rotation of a body about its centre of 
 gravity* ; united to the equations (A) they completely 
 determine the motions of the translation and of the 
 rotation of a body. 
 
 If the body be forced to turn about a fixed point ; it 
 results from No. 15, that the equations (R) are suffi- 
 cient for this purpose; but it is then necessary to fix 
 the origin of the co-ordinates x'^y'^ and z 1 at this point. 
 
 26. Let us particularly consider these equations and 
 suppose the origin fixed at any point whatever, differ- 
 ent or not from the centre of gravity. Let us refer the 
 position of each molecule to three axes perpendicular 
 to each other and fixed in the body, but moveable in 
 space. Let be the inclination of the plane formed by 
 the two first axes upon the plane of x and y ; let <p be 
 the angle formed by the line of intersection of these 
 two planes and the first axis ; lastly, let ^ be the cotn- 
 
 * As the equations (B) do not contain the co-ordinates of 
 the centre of gravity of the system, they only shew the 
 different positions of the body with respect to three axes 
 which have their origin at that centre, consequently the 
 motion of rotation which the equations (B) determine, is 
 the same as if the centre of gravity were at rest. 
 
 A body acted upon by accelerating forces may therefore 
 have two motions ; one that of translation, the same as if 
 the body were concentrated into one point at its centre of 
 gravity and acted upon by all the forces parallel to their 
 directions, and the other that of rotation about the centre 
 ef gravity the same as if this point wer fixed. 
 
02 
 
 plement of the angle which the projection of the third 
 axis upon the plane of x and y makes with the axis of #. 
 We will name these three new axes principal axes, and 
 we shall denote by x"> y a > and z" fhe three co-ordinates 
 of the molecule dm referred to these axes ; then, by 
 No, 21, the following equations will have place, 
 
 cos.-^.sin.tpj-J-^.sin. 
 
 ( y==# // .{cos.9.cos.\l'.sin.<p sin. -4/. cos. <p} 
 
 -\-y"*{ cos . 0. cos . \j/ cos . <p-(-sin . ^ . sin . <p }-\-z" . sin . 
 
 z r =z".cos.Q y.sin.Q.c 
 
 By means of these equations we shall be enabled to 
 develope the first members of the equations ( R) in 
 functions of 0, -4/ ? and cp, and of their differentials. 
 But we may considerably simplify the calculations, by 
 observing that the position of the three principal axes 
 depends upon three constant quantities, which can 
 always be determined so as to satisfy the three equations 
 
 Suppose then 
 
 S. ry* +"* ) .dm=A ; S. (aP+z**) .dm B ; 
 
 and for abridgment let 
 
 The equations (B) will, after all the reductions, be 
 changed into the three following, 
 
LAPLACE'S MECHANICS. 
 
 203 
 
 -2V,- 
 
 cos. 4'.{^g f .cos.0.sin.<p-|- Br. cos. 9. cos. (p 4" 
 
 Cp.sin.8} 
 
 cos.-4/.{jBr.sin.(p Aq.cos.q*} 
 
 sin. -vK{^. cos. 0. sin. <>-[- JBr.cos.Q.cos.p 
 
 these three equations give by differentiating them and 
 
 supposing ^=0 after the differentiations, which is 
 
 equivalent to taking the axis of the jr's indefinitely 
 
 near to the line of intersection of the plane of x 1 and y* 
 
 with that of x 11 and y 1 ^ 
 
 ^.cos.Q.fjBr.cos.^-j-^^'Sin.^J-j-sin.d.^.f JBr.cos.<p-[- 
 
 Aq.sin>q>)d.(Cp.cos.Q) = dN; 
 
 d^.(Br.sin.<p Aq.cos.Q) dQ.sm.Q.( Br.cos,Q-\-Aq. 
 
 sin.<p)-\-cos.Q.d.(Br.cos.q>-\-Aq.sin.q>) -|- d.(Cp.s'm.Q) 
 
 = dN 1 ; 
 
 d.(Br.s'm.q> ^.cos.<p.J d^,. cos. Q.(Br>c 
 
 If we make 
 
 Cp=p'; Aq=q>; Br=r' ; 
 these three differential equations will give the following 
 
 n _ n 
 
 dr'+ 
 
204 LAPLACE'S MECHANICS. 
 
 these equations are very convenient for determining the 
 motion of rotation of a body when it turns very nearly 
 about one of its principal axes, which is the case of the 
 celestial bodies. 
 
 27. The three principal axes to \vhich we have re- 
 ferred the angles 0, 4/, and (p, deserve particular atten- 
 tion. Let us proceed to determine their position in 
 any solid whatever. 
 
 The values of #', y, anclofV the preceding number, 
 give by No. 21, the following equations, 
 
 sn.(p; 
 y / :=r / .(''cos.0.sin.4,.cos.p cos. 4,. sin 
 
 cos.p; 
 
 ^z.r'.sin.Q.si 
 From which may be obtained 
 
 Suppose 
 
 then 
 
 *.Q.S.xs a .dm sin.tp, S.yz tt .dm= ( a* 
 
 4/ A. sin. 
 
LAPLACE'S ML.HANICS. 205 
 
 By equalling to nothing the second members of these 
 two equations, we shall have 
 
 ~Y 2 6 2 ;.sin.4,.cos.4/4-/.(cos. 2 v}/ sin. 
 
 t ^ 
 
 * 
 
 but we have 
 
 by equalling these values of \ fang. 20, and substituting 
 in the last instead of tang. its preceding value in ^, 
 and then making for abridgment tang.^n^w, we shall 
 obtain, after all the reductions, the following equation 
 of the third degree ; 
 
 0=(gu+h) . (hu g) 2 
 
 As this equation has at least one real root, it is evidently 
 always possible to render equal to nothing at the same 
 time, the two quantities 
 
 cos. (p. S.x"z".dm sin.Q.S.y r/ z f/ ,dm '; 
 
 and consequently the sum of their squares, ( S x !l z".dm) 
 -\-(S.y"z ll .dm)*) which requires that we should have 
 separately 
 
 S.a?V,dfw=0 ; S.y ll z'i.dm=Q. 
 The value of u gives that of the angle 4,, and conse- 
 quently that of the tang. 0, and of the angle 9. It is 
 yet required to determine the angle p, which may be 
 done by means of the condition S.z n y*.&iiK~& 9 which 
 remains to be fulfilled. For this purpose it may be 
 observed, that if we substitute in S-^'/.dm for x" and 
 i/ 1 their preceding values, that function will be changed 
 into the form, /f.sin.9(p-|~L-cos.2^, H and L being: 
 
206 LAPLACE'S MECHANICS. 
 
 functiqns of the angles and 4/, and of the constant 
 quantities a 2 , & 2 , c 2 , /, g, h^; by equalling this ex- 
 pression to nothing, we shall have 
 
 The three axes determined by means of the preceding 
 values of 0, 4,, and tp, satisfy the th/ee equations 
 
 =Q ;S.z f/ z.dm=.Q ; S.V'.feziiOt ; 
 
 * If F be supposed equal to tf'cos.-^ ^'sin.4/ and 6r to 
 a/cos.0.sin.4/-f ^/'cos.Q.cos.-v}/ s'sin.d, then .y//- R 
 G.sin.^ and ^G. cos. p F.sin.^; consequently 
 
 therefore 
 S.x"ii'dm sin. < 
 
 If the second member of this equation be equalled to nothing, 
 as 2sin.9.cos.p sin.2p and cos, 2 <p sin. 2 9cos.2p, we 
 shall have 
 
 sin ^O 
 
 If H=S.(G*F*).dm, L=:2.S.FG.dm and - -zrtan. 
 
 cos. 2<p 
 
 2<p, by substitution, the above equation may be changed 
 into the following 
 
 L 
 
 t These calculations which would be found very tedious 
 in practice are much facilitated by the knowledge of one of 
 the principal axes of rotation. Thus for instance, let the 
 position of the axis x" be known in the body, as the situa- 
 tion of (he three rectangular axes #', y, andz' are arbitrary, 
 at" may be supposed to coincide with x 1 which will cause the 
 angles <p and %]/ and consequently their sines to vanish, their 
 
LAPLACE'S MECHANICS. 207 
 
 The equation of the third degree in u seems to indicate 
 three systems of principal axes similar to the preceding ; 
 but it ought to be observed, that u is the tangent of the 
 angle formed by the axis of x 1 and the intersection of 
 
 co- sines will then he equal to unity, also the quantities/ 
 and g will be equal to nothing, as is evident from substituting 
 in them fory and z' their respective values. The equation 
 
 T tang. 2 ca a 2.sin. 2 4/ 6 2 .cos. 2 4/ 2/.sin.^.cos.4 
 
 2/*' 
 therefore be transformed into the following tan.20'ji: , 2 _ , 2 , 
 
 in which h', c', and b j are the respective values of h, c, and 
 b when x" coincides with d\ It appears from this expres- 
 sion, as tan.Sfl'rrtang.Sf&'-f-SOj, that the other two axes 
 must be taken in the plane of y' s', one making the angle 
 & and the other the angle 0'-f 90 with the axis of y or the 
 plane of x" y'. 
 
 If VrzO and b'~c' the angle 6 becomes indeterminate, 
 therefore every line'in the plane of y z ( passing through the 
 origin of the axes is a perpendicular axis. 
 
 When the bodies are symmetrically formed, the axis of 
 the figure is always a principal axis and the others may be 
 found by this rule. Thus for instance in the case of the 
 ellipsoid which has three unequal conjugate diameters, let 
 them betaken for the axes of #, #, and z, then the body will 
 be divided into similar and equal parts by each of the planes of 
 the co-ordinates ; therefore each molecule dm above the plane 
 of x y having the co-ordinates #, y, and z ^vill have another 
 corresponding and equal one below that plane having x , y, 
 and z for its co-ordinates ; consequently the indefinitely 
 small elements of the integrals S.xz.dm and S.yz.dm corres- 
 ponding to these molecules will be xz.dm und xz.dm, 
 yz.dm and yz.dm, therefore the integrals will vanish as 
 
208 LAPLACE'S MECHANICS. 
 
 the plane of x f and y 1 with that of x" and y" ; but it is 
 evident that it is possible to change into each other the 
 three axes of x" of y" and of /y , as the three preceding 
 equations will be always satisfied ; the equation in u 
 ought therefore equally to determine the tangent of the 
 angle formed by the axis of x 1 and the intersection 
 of the plane of x 1 and y f } either with the plane of x" 
 and y n , or with the plane of x" and z", or with the 
 plane of y 11 and z". Thus the three roots of the equa- 
 tion in u are real and appertain to the same system of 
 axes. 
 
 It follows from the above that, generally, a solid has 
 only one system of axes which possess the property 
 treated upon. These axes have been named the prin- 
 cipal axes of rotation, by reason of a property that is 
 peculiar to them, which will be noticed in the course 
 of this work*. 
 
 they may he supposed to consist of an indefinitely great 
 Mumber of indefinitely small quantities, which from their 
 contrary signs destroy each other. The integral S.xy.dm 
 may in a similar manner be proved equal to nothing, con. 
 aequently the three axes of the ellipsoid are principal axes. 
 
 * These axes are called the principal axes of rotation on 
 account of the property which the body possesses, if not 
 acted upon by any accelerating force, of always turning round 
 any one of them, if once put in motion about it in conse- 
 quence of an initial impulse. This property may be demon- 
 strated in the following manner. 
 
 Suppose a body, not acted upon by any accelerating force, 
 to turn about a fixed axis in consequence of an impulse which 
 has been given it. Let one of the three rectangular co, 
 rdinates to which the molecules of the body are referred, 
 
LAPLACE'S MECHANICS. 09 
 
 The sum of the products of each molecule of a body 
 into the square of its distance from an axis, is called its 
 moment of inertia with respect to this axis. Thus the 
 
 that of z for instance be supposed to pass along this axis, 
 conceive r to represent the distance of a molecule dm from 
 it, having its angular velocity, \vhich is common to all (he 
 points of the body, denoted by . The centrifugal force of 
 the molecule dm will be represented by r w z , consequently 
 the moving force with which the molecule acts upon the fixed 
 axis in a perpendicular direction to its length is r u *.dm. 
 The resultant or the two resultants of the whole nun:b?r of 
 forces which act upon the axis, shew the pressure which it 
 sustains. 
 
 To find this pressure let us suppose that the force ru*.dm 
 is applied directly to the axis of z where its direction meets 
 it. The force may be resolved at the point where it acts 
 upon the axis into two others parallel to the axes of x and y 9 
 the cosines of the angles that the direction of the force ru z . 
 
 dm makes with the axes x and y are - and -, consequently 
 
 the components of this force parallel to these axes are x u z . 
 dm and 3/w z .t/m, therefore the resultant or sum of all the 
 components parallel to the axis of x is the integral S.x^.dm 
 oru 2 S.x.dm; this quantify, if M represent the mass of the 
 body and X the value of x at its centre of gravity, is equal 
 to u 1 MX. In like manner the resultant of the forces paral. 
 lei to the axis of y directed in the plane of yz is 2 AfF, Y 
 representing the value of y at the centre of gravity of the 
 body. Let z' and z" denote the distances of these resultants 
 from the plane of x and y, then by the theory of moments 
 we shall have the following equations 
 
 MXz'S.xz.dm, MYzi=S.yz.dm. 
 which will give the values of z 1 and z". 
 
210 LAPLACE'S MECHANICS. 
 
 quantities A 9 J3, and C are the moments of inertia of 
 the solid which \ve have considered, with respect to 
 
 If z r and z" are equal, the forces uMX and ^ 
 applied at the same point, and may consequently be reduced 
 to one u z .M\/X 2 -}~Y*, which represents the action upon 
 the fixed axis arising from the centrifugal force of the body. 
 Let the axis of z be one of three principal axes which have 
 their origin at the centre of gravity of the body at the distance 
 a from the origin of z upon that axis ; in this case JTn:0 and 
 V- 0; let the origin of the co-ordinates be moved without 
 changing their directions from its first point to the centre of 
 gravity of the body, then the co-ordinates of dm will be #, 
 #, and z a. As z is one of the principal axes, we shall 
 have the following equations. 
 
 S.x(z a 
 
 S,y(z a 
 
 therefore because S .xdmMXQ and 
 it is evident that S*xz.dm-=Q and S.ys. cfazzzO, consequently 
 u*S.xz.dm=;Q and u*S.yz.dm~Q. As the resultant u*.M Xof 
 forces directed in the plane of xs 9 and the sum w z S.xz.dm 
 of their moments are nothing, these forces will be in equi. 
 librio independent of the fixed axis. The same is evidently 
 the case with respect to the forces in the plane of y and z $ 
 therefore in this case the centrifugal forces of the different 
 molecules of the body do not act upon the axis of rotation 
 and the same motion would be continued about it s if it were 
 not fixed. 
 
 If the fixed axis z is a principal one of rotation that does 
 not passothrough the centre of gravity of the body X and Y 
 will not vanish, but as S.xz.dm and S.yz.dm are equal to 
 nothing, the distances z' and z 11 must consequently be equal 
 to nothing, therefore the fixed axis will be acted upon by 
 the force U ~~.M\/X 2 +* 2 applied at the origin of the co- 
 ordinates 5 if this point be fixed the pressure arising from 
 
LAPLACE'S MECHANICS. 
 
 the axes x fl 9 y" 9 and z".* Let us name O the mo- 
 ment of inertia of the same solid with respect to the 
 
 the centrifugal force will be destroyed, and if the axis of 
 rotation were free to move about this point the rotatory 
 motion with respect to it would still be continued* 
 
 It is evident therefore, that if any point in a body be 
 take^ as a fixed one ; there will be three axes passing through 
 it, round which the body would move uniformly without 
 acting upon them. 
 
 If the body were forced to turn about any other axis 
 passing through that point, the action of the centrifugal 
 forces upon it would not take place at that point and it 
 would consequently be displaced. Vide the Mechanique 
 Philosophique of Prony, a similar proof is also given by 
 Poisson. 
 
 * Suppose for example that MA (Jig. %) is a parallele- 
 piped it is required to find its moment of inertia with respect 
 to the axis MB. Let MG=:a 9 ME=b, and M B=. c and 
 suppose the axes of the co-ordinates #, y, and z to be taken 
 from their origin at M in the respective directions of these 
 lines, then if the uniform density of the parallelepiped is 
 represented by f and dx dy dz is the volume of a molecule 
 of the body, its inertia will be denoted by the integral SSS. 
 (x*+y*)dxdyds t This quantity when integrated with re- 
 spect to z 9 from szzrO to sure, gives 
 
 This expression when integrated with respect to y 9 from 
 b 9 becomes 
 
 from which by integrating with respect to x 9 from #zz:0 to 
 tfzzfl the result 
 
 (a*b ab* 
 T+T 
 
2J2 LAPLACE'S MECHANICS, 
 
 axis of 2', and we shall find by means of the values of 
 x 1 and of i/ of the preceding number 
 
 may be obtained, which is the moment of inertia of the 
 parallelepiped MA with respect to the axis MB. , The mo- 
 ments with respect to the other axes may be obtained by 
 properly changing amongst themselves the letters a, 6, and 
 c; let JVf f.a&c, then the moments with respect to the 
 three axes MG, ME, and MB will be 
 
 yfft' + cO, ^rH-c'>>, ~ f 
 
 The moments of inertia of an homogeneous ellipsoid with 
 respect to the three principal diameters, may be readily 
 found from the general equation to its surface 
 a 2 6 2 2 2 + a*c z y* -f 6 2 c 2 # 2 :ira 2 & 2 c z , 
 
 a, 6, and c representing the lengths of the three rectangular 
 semi-diameters, which are respectively in the directions of 
 the three co-ordinates #, #, and z. The inertia with re- 
 spect to the axis of z is 
 
 t> representing the density of the body. 
 
 This expression when integrated with respect to the entire 
 ellipsoid gives 
 
 .abctvjtf+b*) 
 1 o 
 
 it denoting the ratio of the circumference to the diameter of 
 a circle. By changing the letters a and c into each other 
 the moment of inertia 
 
 McFtj&W) 
 
 15 
 
 will be obtained with respect to the axis of x. In like man. 
 ner by changing a and b into each other the moment of 
 inertia 
 
 
LAPLACE'S MECHANICS. S13 
 
 The quantifies sin. 2 0.sin.*<p, sin 2 0.cos. 2 ;p, and cos. a d 
 are the squares of the cosines of the angles which 
 the axes of x n , y", and z" make with the axis of z 1 
 from which it follows in general, that if we multiply 
 the moment of inertia relative to each principal axis of 
 rotation, by the square of the cosine of the angle that 
 it makes with any axis whatever, the sum of these three 
 products will be the moment of inertia of the solid rela- 
 tive to this last axis. 
 
 The quantity O is less than the greatest and greater 
 
 ill be obtained with respect to the axis of y. The content 
 
 4<7T 
 
 of the ellipsoid is represented by Mbc, and its mass M by 
 
 .abc: by substitution therefore the following will be 
 3 
 
 the moments of inertia of the ellipsoid with respect to the 
 axes x, y, and z, 
 
 The greatest moment of inertia is about the shortest and 
 the least about the longest axis. 
 
 In the case of the spheroid bc and the moments with 
 
 M 
 respect to the axes of x and y are .b 2 and (a i -\-b 1 ). 
 
 O O 
 
 In the case of the sphere azz&mc, and the moments of 
 inertia with respect to any axis passing through its centre is 
 
 STT 
 -.&.'. 
 
LAPLACE'S MECHANICS. 
 
 than the least of the three quantities A, B, and C* ; 
 the greatest and the least moments of inertia therefore 
 appertain to the principal axest. 
 
 * Suppose A to be the greatest and C the least of the 
 quantities A, B, and C, and a, /3, and y to denote the 
 three angles which the axis z' respectively makes with the 
 axes x !/ 9 y") and z ff 9 then by observing that cos. 2 a-j~cos. 2 j3 
 cos. 2 yzz:l, notes No. 2, the preceding equation may be 
 made to assume the following forms, 
 
 which shew that G is less than the greatest and greater 
 than the least of the three quantities A, B, and C. 
 
 t A method of finding one of the principal axes of rotation 
 of a body may be derived from the properties which two of 
 the axes possess, of having the moments of inertia with re- 
 spect to one of them a maximum and to the other a minimum. 
 Thus let x" be an axis of rotation passing through the origin. 
 of the co-ordinates which it is required to find, and suppose 
 it makes the angle Q with the plane x'i/' 9 and that its projec- 
 tion upon the plane x'y' makes the complement of the angle 
 4/ with the axis of x 1 . Suppose also the axis of y" to be 
 drawn perpendicular to x 11 in the plane of x' y 1 r , and the axis 
 of z" to be perpendicular to the plane x" y, all the co- 
 ordinates having the same origin, then the following values 
 ofx f/ 9 y 7 , and rJ 1 may be obtained, 
 
 Let these values of y" and z" be substituted in the expres- 
 sion S(y" 2 -}-z" z )dm which may be denoted by L, then if the 
 angles Q and 4- are supposed variable we shall find 
 
LAPLACE'S MECHANICS. 215 
 
 Let X, F, and Z be the co-ordinates of the centre 
 of gravity of the solid referred to the origin of the co- 
 ordinates, which we shall fix at the point about which 
 the body is forced to turn, if it be not free ; x' X^ 
 y 1 Fand s' Z will be the co-ordinates of the molecule 
 dm of the body relative to its centre of gravity : ihe 
 moment of inertia relative to an axis parallel to the axis 
 of s' and passing through the centre of gravity, will 
 therefore be 
 
 but we hare by the nature of the centre of gravity, 
 S.x'dm=mX; S.y'dm=mY; the preceding moment 
 will therefore be reduced to 
 
 We shall consequent!}' have the moments of inertia of 
 a solid relative to the axes which pass by any point 
 whatever, when these moments shall be known relative 
 to the axes which pass through the centre of gravity. 
 It is evident at the same time, that the least of all the 
 
 f-V- 
 
 \dtj- 
 
 r~v~ 
 
 \<t*)- 
 
 If these values be equalled to nothing they will give L 
 Either a maximum or a minimum : the angles an J > may 
 then be obtained from them which will shew the position of 
 the axis #", from which the other two may be readily found. 
 This method like that of Laplace requires yery tedious cal. 
 culations. 
 
LAPLACE'S MECHANICS. 
 
 moments of inertia has place with respect to one of the 
 three principal axes which pass through this centre*. 
 
 * The moment of inertia with respect to any axis s f is 
 S.(x fz -t-y' 2 ).dm) but for any other axis parallel to z 1 which 
 has the lines a and b for the co-ordinates of any one of its 
 points, this expression becomes 
 
 Let r be equal to the distance between the axes or \/a z -\-b 2 
 then if the axis of 3' passes through the centre of gravity of 
 the body, as S.x'dm0 and S.y'dm~0, No. 15, the ex. 
 pression will become 
 
 therefore the moment of inertia with respect to an axis 
 passing through the centre of gravity of a body, is less than 
 for any other axis, by the square of the distance of the two 
 axes multiplied into the mass of the body, consequently the 
 minimum minimorum of the moments of inertia of a body 
 belongs to one of the principal axes which passes through its 
 centre of gravity. 
 
 To find those points of a body, if there he any, about 
 which all the moments of inertia are equal. 
 
 Let , by and c denote the co-ordinates of one of these 
 points, the centre of gravity of the body being taken for the 
 origin of the co-ordinates which are supposed to be in the 
 directions of the principal axes passing through it, then a: , 
 y b 9 and z c will be the co-ordinates of any molecule 
 with respect to the point sought ; now from the nature of 
 the question every straight line passing through this point 
 must be a principal axis, consequently we have the following 
 equations 
 S.(x a)(y b 
 
LAPLACE'S MECHANICS. 217 
 
 If it be supposed that by the nature of the body, 
 the two moments of inertia A and B are equal, we 
 shall have 
 
 S.(x a)(z c).dm~S.xz.dm aS.z.dm c.S.x.dm-\-ac. 
 
 S.dmQ, 
 
 S,(yb)(z c),dmS.yz.dm b.S z dm c.S.y.dm + bc. 
 
 S.dmQ; 
 
 but S.xy.dm, S.xz.dm, S.yz.dm, S.x.dm, S.y.dm, and 
 
 S.z.dm are respectively equal to nothing, therefore the 
 
 above are reduced to these 
 
 fl^TwrzO, acm~0, bcm~0. 
 
 It is evident, that if the point sought exist, as from the 
 last equations two of the quantities a, 6, and c are equal to 
 nothing, it must be upon one of the principal axes belonging 
 to the centre of gravity of the body. Suppose & c 0, then. 
 a the distance of the point sought from the centre of gravity 
 is indeterminate and upon the axis of x. The moment of 
 inertia of this point with respect to the axis of x is A, but 
 with respect to axes parallel to those of^ and z it is B-\-m 
 a 1 and C'-j-wa 2 . The problem requires that we should have 
 
 13 -j- wza 2 nrC -f ma* A. 
 These equations are impossible unless I? (7, which gives 
 
 a _^~ g 
 m ' 
 
 therefore 
 
 "J^c 
 
 m 
 
 consequently a has two values which, if A be greater than 
 C, are real and upon the axis of x at equal distances on each 
 side of the centre of gravity. 
 
 It appears from the above that there cannot be any point 
 in a body about which all the moments of inertia are equal. 
 
SIS LAPLACE'S MECHANICS. 
 
 by making 6 equal to a right angle, which will cause 
 the axis of z' to be perpendicular to that of s /7 , the 
 equation will give C=sl. The moments of inertia 
 relative to all the axes situated in the plane perpedicu- 
 lar to the axis of z", will be therefore equal to each 
 other. But it is easy to be assured that, in this case, 
 we shall have for the system of the axis of 2 /y and of any 
 two axes perpendicular to it and to each other 
 
 for, from denoting by x 11 and y" the co-ordinates of a 
 molecule dm of a body referred to the two principal 
 axes taken in the plane perpendicular to the axis of z", 
 with respect to which the moments of inertia are sup- 
 posed equal, we shall have 
 
 S. (x f i*+z fiz ) . dm=S. fy /8 +s >iz ) . dm ; 
 
 if the quantities A^ B, and C belonging to the centre of 
 gravity of the body are unequal ; if one of the three quanti. 
 ties A, By and C is greater than either of the others, and 
 the others equal, in this case, there are two points with 
 respect to which all the moments of inertia are equal upon 
 the principal axis to which the greatest of the moments A^ 
 J?, and C belongs. If the three moments A, /?, and Care 
 are equal, the centre of gravity of the body is the only point 
 about which all the moments of inertia are equal. 
 
 For example, ia the oblate spheroid the points are upon 
 the minor axis of the generating ellipse, at the distance of 
 the square root of the fifth part of the difference between the 
 squares of the semi-major and semi. minor axes on each side 
 from the centre of the spheroid. This last problem was first 
 solved by M. Binet, and afterwards in a manner similar td 
 the above by S. D. Poisson. 
 
LAPLACE'S MECHANICS, 219 
 
 or simply S.z*.dm=S.y ll *.dm ; but by naming s the 
 angle which the axis of x 1 makes with the axis of x f/ , 
 we have 
 
 .r'uz^.cos.e-j-^.sin.e ; 
 
 consequently 
 
 dm . s i n . s . cos . enrzO . 
 
 We shall find in like manner S.x's v .dm=0 ; S.y'z"dm 
 -0 ; all the axes perpendicular to that of z" are there- 
 fore principal axes, and in this case the solid has an 
 infinite number of principal axes. 
 
 If at the same time ^=B=C', we shall have gen- 
 erally C'=A ; that is to say, all the moments of inertia 
 of the solid are equal ; but then we have generally 
 
 S.x'i/'.dm=0; S.z'z'.dm=Q; S.y*'.di=aO; 
 whatever may be the position of the plane of x 1 andy, 
 so that all the axes are principal axes. This is the 
 case of the sphere : we shall find in the course of the 
 Mechanique Celeste that this property belongs to an 
 infinite number of other solids of which the general 
 equation will be given. 
 
 28. The quantities p, q, and r which we have intro- 
 duced into the equations (C) of No. 26, have this 
 remarkable property, that they determine the position 
 of the real and instantaneous axis of rotation of a body 
 with respect to the principal axes. In fact, we have 
 relative to the points situated in the axis of rotation 
 dx'Q, dy'=Q, and cfc' 0; by differentiating the 
 values of x' 9 y'^ and z j of No. 26, and making the sine 
 4cmO after the differentiation, which may be done, 
 because we are able to fix at will the position of the 
 axis of x 1 upon the plane of x 1 and y, we shall have 
 
LAPLACE'S MECHANICS. 
 
 oF.<p} -f- 
 
 os.0.hin.<p e$.sin.0.cos.<p} 
 +s".<?0.cos.fi= ; 
 
 cos. 0. cos. <p. <$p.sin.0.sin.(p} s/'.f/Q.si 
 If we multiply the first of these equations by sin.<p, 
 the second by cos. 0. cos. <>, and the third by sin.0. 
 cos,<p ; we shall have from adding them together, 
 
 =px fl qz". 
 
 If we multiply the first of the same equations by cos.<>, 
 the second by cos. 0. sin. <p, and the third by sin.O. 
 sin.(p ; we shall have from adding them together, 
 
 JLastly, if we multiply the second of the same equations 
 by sin. 0, and the third by cos.Q, we shall have from 
 adding them together, 
 
 Qz=zqy fl rx". 
 
 This last equation evidently results from the two pre- 
 ceding ones ; thus the three equations dx'=.0, dy'=.0, 
 and dz'=iO are reduced to these two equations which 
 belong to a right line forming with the axes ofx f/ 9 y h ', 
 and z r/ , angles which have for their cosines 
 
 r p * 
 
 * 
 
 Let the right Hue be represented by 
 
 then by letting fall a perpendicular from the end of it upon 
 
 the axis of x", we shall have by trigonometry \/x" z -+y l/2> -f z"* 
 
 x 
 : afl : : rad. (1) : _= or the cosine of the angle 
 
LAPLACE'S MECHANICS. 
 
 This right line is therefore at rest and forms the real 
 axis of rotation of the body. 
 
 In order (o have the velocity of rotation of the body, 
 let us consider that point in the axis of ' 7 which is a 
 a distance equal to unity from the origin of the co- 
 ordinates. We shall have its v iociti^s parallel to the 
 axes of x 1 , y', and s', by making z n =fi 9 ?/ a =Q 9 and 
 %"-=l in the } receding expressions of cfo*', dy' 9 and dz' 9 
 and dividing them by dt ; which gives lor these partial 
 velocities 
 
 
 the whole velocity of the point is therefore - 
 
 - or vV-j-r z . If we divide this velocity by the 
 distance of the point from the instantaneous axis of 
 rotation, we shall have the angular velocity of rotation 
 of the body; but this distance is evidently equal to the 
 sine of the angle which the real axis of rotation makes 
 with the axis of z" 9 the cosine of which angle is 
 
 which the line makes with the axis of x !l ; by substituting 
 
 rx' 1 px" 
 for v a and z f/ their respective values and , this ex-o 
 
 pression becomes / x " z .r 2 x ffz p*x" z or 1//> 2 -f ? 3 + r 2 . 
 
 v "ir^^r 
 
 The cosines of the other angles may be found in a similar 
 manner. 
 
LAPLACE'S MECHANICS. 
 
 nave 1/ a2 --r z for 
 
 4/ a_L allr? we sa 
 
 the angular velocity of rotation*. 
 
 It appears from the above, that whatever may be 
 the movement of rotation of a body, either about a fixed 
 point, or one considered as such ; this movement can 
 only be regarded as one of rotation about an axis fixed 
 during one instant, but which may vary from one in- 
 stant to another. 
 
 The position of this axis with respect to the three 
 principal axes and the angular velocity of rotation, 
 depend upon the variables p, q ) and r; the deiermin- 
 
 * To find the angular velocity about the immoveable axis 
 of rotation ; from the distance equal to unity upon Ihe axis 
 of z a let fall a perpendicular upon the axis of rotation ; the 
 perpendicular will represent the sine of the angle which 
 this axis makes with z 11 , and is consequently equal to 
 
 : the angular T elo. 
 
 city about the axis of rotation at a distance represented by 
 nity, may therefore be foand by (he following proportion 
 
 If the quantities p, gr, and r are constant the axis of rotation 
 will remain fixed in the body, the angular velocity will also 
 be invariable ; but the converse of this is not equally true, 
 for the axis of rotation may change its position in the body 
 and the angular velocity remain the same, that is, the quan. 
 tity V fP + cf + r* may continue constant although p } <jr, 
 and r vary. 
 
LAPLACE;S MECHANICS. 223 
 
 ation of which is very important in these researches, 
 and which by expressing quantities independent of the 
 situation of the plane of x 1 and y' 9 are themselves in- 
 dependent of this situation. 
 
 29. Let us determine these variables in functions of 
 the time, in the case in which the body is not solicited 
 by any exterior forces. For this purpose let the equa- 
 tions ( D) be resumed of No. 26, containing the varia- 
 bles p') q', and r' which are in a constant ratio to the 
 preceding. The differentials rfZV, dN' 9 and dN" are 
 in this case nothing, and these equations give by being 
 added together after they have been respectively multi- 
 plied by p', <?', and r' 
 
 Q=p f dp f +q f dq f +r f dr r , 
 which becomes from integration 
 
 k being a constant quantity. 
 
 If the equations (D) are multiplied respectively 
 AB*p r , BC.q', SindAC.r', and afterwards added to- 
 gether, they will give by integrating their sum 
 
 H being a constant quantity ; this equation contains 
 the principle of the preservation of living forces. From 
 these two integrals the following equations may be 
 obtained, 
 
 , 2 _AC.k*-H* + A.(B-C).p'* . 
 C.(A-B) 
 
 12 -ff* BC.k* B.(A C).p'~ . 
 
 C.(A-B) 
 
 thus q' and r 1 will be known in functions of the time t, 
 when p' shall have been determined : but the first of 
 the equations ( D) gives 
 
 AB.dp> 
 
LAPLACE'S MECHANICS. 
 which, by substituting the values of q' and r', becomes 
 
 Aj^dp^ ^ 
 
 ~ V AC. k z - llA\ir^'C' ll z BC.k*^K 
 
 an equation that is only inferrable in one of the three 
 following cases, B=A, BC, or A=*. 
 
 * In the cases in which this equation can be integrated it 
 may be made to assume ihe following forms, in which a and 
 b are substituted for the constant quantities. 
 
 a z clp' 
 First. If A=B, <//&. - and * b. (circular arc 
 
 having a for radius and jV for tangent)-|-const. 
 
 dp' 
 Second. If ^ C, dtb .,===== and ^^z&. hyp. log. 
 
 2 -f;7*7-f const. 
 
 IfU C, dfc:&, /= ; ^T : and '=* (circular 
 Va /'* 
 
 re having for radius and|>' for sine) -f const. 
 
 todpf 
 
 -Fourth. liAC.k*H*) dtb ~^-==== and t=b. hyp. 
 
 BC.k*, dtb. 
 
 Hh* , 
 
 log ------- r^rkrr^ -- ~! Const/ 
 
LAPLACE'S MECHANICS, 3z5 
 
 The determination of the three quantities p', q r , and 
 r', requires three constant quantities, // a , % and that 
 which is introduced by the integration of the preceding 
 equation. But these quantities only give the position 
 of the instantaneous axis of rotation of the body upon 
 its surface, or relative to the three principal axes and 
 its angular velocity of rotation. To have the real 
 movement of the body about a fixed point, it is necessary 
 also to know the position of the principal axes in space ; 
 this should introduce three new constant quantities rela- 
 tive to the primitive position of these axes, and requires 
 three new integrals, which when joined to the pre- 
 ceding will give the complete solution of the problem. 
 The equations (C) of No. 26 contain the three constant 
 quantities N, N r , and N"; but they are not entirely 
 distinct from the constant quantities H and k. In fact, 
 if we add together the squares of the first members of 
 the equations (C) 9 we shall have 
 
 which gives k*=N*+N'*+N a *. 
 
 The constant quantities N 9 N' 9 and JV", answer to 
 the constant quantities c, e', and C H of No. 21, and the 
 
 function |JvV 2 +<? /2 + expresses the sum of the areas 
 described during the time t by the projections of each 
 molecule of the body upon the plane relative to which 
 this sum is a maximum. N' and N" are nothing rela- 
 tive to this plane ; by therefore equalling to nothing 
 their values found in No. 26, we shall have 
 sin.^ Aq.cos.Q ; 
 
 from vfhich may be obtained 
 
LAPLACE'S MECHANICS. 
 
 - 
 sin. 0. sin. 0= .- -- _ : ; 
 
 ' 
 
 sin 0.cos.<p= ,==. -- 
 
 By means of these equations, \ve shall know the values 
 ol & and <p in functions of the time relative to the fixed 
 
 * If from the centre of the co-ordinates a perpendicular 
 be erected to the invariable plane, and a, , and y denote 
 the respective angles which it makes with the co-ordinates 
 * /7 , y, and z fl , then it may be readily proved, No. 21 notes, 
 that cos.azz sin.0.sin.p, cos./Srr sin. 0. cos. 9, and cos.y 
 :z:cos.d, consequently we have the following equations 
 
 cos - 0= ./^T^T^' 
 
 COS.yn: ,== = 
 
 The position of the invariable plane with respect to three 
 axes fixed in space may be found in the following manner. 
 Let O (Jig. 19) represent the centre of the co-ordinates or 
 the point about which the body turns, Ox ff 9 Oy", and Oz a 
 the three principal axes to which the co-ordinates x", y ; , 
 and z f/ are referred, Om the perpendicular to the invariable 
 plane, and Ox one of the three rectangular fixed axes be. 
 longing *o a:, y 9 and z to which the ordinates x are referred. 
 From any point a? in the axis Ox let the right line xm be 
 drawn cutting the line Om at z, then in the triangle xmO 
 
 The co-ordinates of the poiat x with respect to the axes of 
 
I/APLACE'S MECHANICS. 27 
 
 plane that we have considered. It only remains to find 
 the angle 4*, which the intersection of this plane and 
 that of the two principal axes makes with the axis of #'> 
 which requires a new integration. 
 The values of q and r of No. ^6 give 
 
 from which may be obtained 
 
 x a , y H ) and z ff are xO.cos.xOx", xO.co^.xOy'^ and xO cos. 
 xO^" and those of m to the same co-ordinals are mO cos. 
 mOx", mO.cos.mOy", and mO.cos.mOz", we therefore have, 
 page 8, 
 xm z ~(xO.cos.xOx fl mO. cos. m Ox" )*-{-(: O cos.xOj/ 11 m 
 
 From these two values of xm 2 we shall find, by making the 
 co-efficients of 0# a , Om* and Ox.Om in the equations equal 
 to each other, that those of Ox.Om give 
 cos.mOtf cos.tfCXc^.cos.wOji/ 7 -f- cos. t rCy / .cos.mOy / -f cos. 
 xOz".cos.mOz". In a similar manner it may be proved that 
 cos.m0^rzcos.3/O^ // .cos.mOa?''-f-cos. < yOy / .cos.wOy / -f- cos. 
 gOz".cos.mOz", and cos.mOz cos.sC^.cos-mO^-f-cos.sO 
 y".cos.mOy fl + cos.zOz f/ .cos.mOz f/ . (See page 111). We 
 therefore evidently have the following equations 
 
 p'.cos.xOz"-)- o'.cos xOxH + r'.cos.zOu 11 
 cos.mOx =.- ---- - --- --- ZL 
 
 Vp' i + q i +r'* 
 
 p'.cos.yOz"-}- o'.cos.wO^-f r'cos.vOy'* 
 eos . - - - ^ 
 
 y.cos.202 /7 -f T'.cos.sOa/' 
 os. mw - 
 
LAPLACE'S MECHANICS. 
 
 but by what precedes 
 
 c 
 
 we shall therefore have 
 
 If we substitute instead of dt its value found above ; wte 
 shall have the value of %}/ in a function ofp f ; the three 
 angles 0, <p ? and %]/ will thus be determined in functions 
 of the variables p', q', and r', which arc themselves 
 determined in functions of the time t. We shall there- 
 fore know at any instant whatever the values of these 
 angles with respect to the plane of x 1 and y , which we 
 have considered ; and it will be easy by the formulas of 
 spherical trigonometry to find the values of the same 
 angles relative to any other plane ; this will introduce 
 two new constant quantities, which united to the four 
 preceding ones will form the six constant quantities, 
 that ought to give the complete solution of the problem 
 about which we have treated. But it is evident that 
 the consideration of the plane above mentioned simpli- 
 fies this problem. 
 
 The position of the three principal axes upon the 
 surface of the body being supposed to be known ; if at 
 any instant whatever we are acquainted with the posi- 
 tion of the real axis of rotation upon this surface, and 
 the angular velocity of rotation ; we shall have at this 
 instant the values of p, <?, and r, because these values 
 divided by the angular velocity of rotation express the 
 cosines ef the angles which the real axis of rotation 
 forms with the three principal axes : we shall therefore 
 have the values of p', #', and r' ; but these last values 
 are proportional to the sines of the angles which the 
 
229 
 
 three principal axes form with the plane 
 relative to which the sum of the areas of the projections 
 of the molecules of the body multiplied respectively by 
 these molecules, is a maximum ; we shall therefore be 
 able to determine at all times the intersection of the sur- 
 face of the body by this invariable plane, and conse- 
 quently to find the position of this plane by the actual 
 conditions of the movement of the body. 
 
 Let us suppose that the movement of rotation of a 
 body is owing to an initial impulse, which does not 
 pass through its centre of gravity. It results from what 
 Las been demonstrated in numbers 20 and 22, that the 
 centre of gravity will take the same motion as- if this 
 impulse was immediately applied to it, and that the 
 body will take the same movement of rotation about 
 this centre as if it were immoveable. The sum of the 
 areas described about this point by the radius vector of 
 each molecule projected upon a fixed plane and mul- 
 tiplied respectively by these molecules, will be pro- 
 portional to the moment of the initial force projected 
 upon the same plane, but this moment is the greatest 
 relative to the plane which passes by its direction and 
 by the centre of gravity ; that plant; is therefore the 
 invariable one. If the distance of the initial impulse 
 from the centre of gravity is denoted by/, and the ve- 
 locity which it impresses upon this point by v; m 
 representing the mass of the body, mfv will be the 
 moment of this impulse, which being multiplied by | 
 will give a product equal to the sum of the areas de- 
 scribed during the time t; but this sum by what pre- 
 cedes is f.vV 2 -|-<?' 2 -f-^ /z ; we have therefore 
 
 If we had known at the commence ruent of the move- 
 
ment, the position of tbe principal axes relative to the 
 invariable plane, or the angles B and p; we should also 
 have known at this commencement, the values of p' y 
 q', and r' ', and consequently those of/7, q, and r; we 
 shall therefore have at any instant whatever the value* 
 of these same quantities*. 
 
 * The diagram (Jig. 20 J may serve to assist the learner 
 in the readier understanding of this number. 
 
 Let the body be supposed to be put in motion about the 
 point O situated upon the principal axis z", by an impulse 
 acting upon the point B of the surface in the direction of the 
 line AB. Let MBM' be a section of the body made by a 
 plane passing by the point O and the right line AB : this 
 plane is the invariable one ; let Om be a perpendicular to 
 it. If we know the section MBM' of the body at the be- 
 ginning of the motion, we shall know the angles which its 
 perpendicular makes with the three principal axes. Let Ox n y 
 Oy, and Oz f/ represent the three principal axes and OI the 
 instantaneous axis of rotation of the body, then at the be. 
 ginning of the motion 
 
 cos./O*= -= 
 
 cos.XV = , . 
 
 V> 2 -H 2 -f-r a 
 
 Let MOM' be the section of the planes MBM 1 and x y Otf' ; 
 then one of the constant quantities which belong to the values 
 of t and \J/ in functions of p will depend upon the time when 
 t commenced, and the other upon the line taken arbitrarily 
 in the plane MBM' from which the angle ^ commenced. 
 
LAPLACE'S MECHANIC*. 231 
 
 This theory may serve to explain the two motions of 
 the rotation and revolution of the planets, by one 
 sole initial impulse. Let us suppose that a planet is 
 an homogeneous sphere having a radius 7?, and that 
 it turns about the sun with an angular yelocity U:_ 
 
 In the case of the rotatory movement of a solid body not 
 acted upon by any accelerating forces we evidently have, 
 (see page 107 5 ) 
 
 dxzdu yd<p, 
 
 If the three equations (Z) No. '21, are respectively multi- 
 
 (/A du d-^/ 
 
 plied by -j- , , and , which are made to pass under the 
 
 $ign 2, and added together, when , -~-, a nd -^ are 
 
 at at at 
 
 substituted for their values, they will give 
 
 ^ + ^+^ ^ ^ ^ 
 
 at* at dt^ <U 
 
 but as the equation (Q) No. 19, when ^zz^), givet 
 
 we shall have 
 
 In this equation c, c', and c /x represent the initial forces of 
 impulsion, and C a constant quantity which is necessarily 
 positive. 
 
 If a.cos.a, a.cos.jS, and a.cos.y are respectively substi- 
 
 dQ 
 *utd for c U ) d, and c, (se notes No. 21) and -T-.COS.A 
 
LAPLACE'S MECHANICS. 
 
 r being imagined to denote Us distance from the sun ; 
 we shall then have v=rUi moreover if we suppose 
 that the planet moves in consequence of an initial im- 
 pulse, the direction of which took place at a distance/ 
 
 dQ d9 d-^ du dq> 
 
 .cos./*,, and .cos.v for ? > and (see notes 
 
 page 108) the above equation will be changed into the 
 following, 
 
 dQ ( \ 2C 
 
 I COS.a COS.A-f COS.j3.CO5/x-f COS.y CO C v I . 
 
 In this equation , /3, and y are the ang'es which the per. 
 pendicular axis to the invariable plane makes with the fixed 
 axes of .r, ^, and z; and X, />/,, and v are the angles which 
 the instantaneous axis of the composed rotation makes with 
 
 dQ 
 the same axes, - being the velocity of rotation. Let a- re- 
 
 present the angle which the instantaneous axis of rotation 
 makes with the perpendicular axis to the invariable plane, 
 then, (notes page 227) 
 
 cos.<m:cos.a.cos.X-J-cos./3,cos./x,4" C()S 'y cos ' v I 
 
 dQ <2C 2C 
 
 consequently -r-.cos.od: ? being a constant quantity 
 dt a a 
 
 which depends upon the initial movement of the hody. We 
 therefore have a ratio, independent of the form of the hody. 
 between the real velocity of rotation at each-insfant and the 
 position of the axis of rotation relative to the invariable 
 plane. This curious property was discovered by Lagrange. 
 If the plane of x y be taken by the centre of the body and 
 the right line in the direction of which the impulse is given, 
 the con'stant quantities c' and c" will vanish, and the general 
 
 r/<p 
 equation found above will be reduced to c. zz2C; which 
 
 shews that the velocity of rotation with respect to the axis 
 of z ? that is parallel to the plane of the impulse, is invariable. 
 
LAPLACE'S MECHANICS. 233 
 
 from its centre, it is evident that it will revolve about 
 an axis perpendicular to the invariable plane ; by there- 
 fore considering this axis as the third principal axis, 
 we shall have $=.0, and consequently ^'zzzO, r'=0; 
 we shall therefore have p'^^mfv or Cp=.mfrU, 
 
 2 
 But is well known that in the sphere we have C=- 
 
 o 
 
 mR* 9 consequently 
 
 f^-T'V ' 
 
 which gives the distance /of the direction of the initial 
 impulse from the centre of the planet, and answers to 
 the relation observed between the angular velocity p of 
 rotation, and the angular velocity U of revolution 
 
 about the sun. Relative to the earth we have 77= 
 
 T> 
 
 .166,25638; the parallax of the sun gives = 0,0000 
 
 42665 and consequently f=.--. J?, very nearly. 
 
 As the planets are not homogeneous, they may be 
 considered as formed of spherical and concentricai la 
 minae of unequal densities. Let g represent the density 
 of one of these lamina? of which the radius is jR, being 
 
 a function of R; we shall then have 
 
 
 
 C~ 
 
 * In order to find the value of C in the case where the 
 density p of each spherical lamina varies as some function of 
 its radius, let us suppose (Jig. <ll) that ACBD la the section 
 
 ** 
 
LAPLACE'S MECHANICS. 
 
 m being the entire mass of the planet, and the integrals 
 taken from .#=0, to its value at the surface ; we shall 
 therefore have 
 
 If, as it is natural to suppose, the laminae be densest 
 
 f /?* // 7? 
 
 nearest the centre, /' ' in will be less than %R* ; the 
 j,R*.dK 
 
 value of / will therefore be less than in the case of 
 homogeneity. 
 
 of a sphere made by a plane passing through its centre O and 
 axis of rotation CD, let the radius QA be drawn perpen. 
 dicular to CD or axis of z 9 and RS parallel to it or cutting 
 the circumference of the circle JBCD in R and S ; draw 
 the radius OR meeting the line SR at R, suppose ORR 9 
 
 OPV'x*+y 2 =u, then PR~V~R* w 2 , and if p 3.14159, 
 &c. the surface of the cylinder generated by the revolution 
 
 of SR about CD is 4puR t u z 9 therefore 
 is the differential of the solid generated by the revolution of 
 the plane CRSD about CD, consequently 4pfu*du\/J&Iu* 
 is the integral of that solid, when it has each of its molecules 
 multiplied into the square of its distance from the axis CD ; 
 this integral may be readily found by supposing R* u 2 2 
 or u 2 ~R z TO 2 , then M^zzU 4 SK^+^S consequently 
 
 + w*d& and 4ptt J dttV^ a ~** = 4p.( Jfc* 
 ) which by integration becomes 4p.( R 2 w* 
 when u~Q this integral should vanish but 
 
 R 5 
 az^U in this case 4p.( ?R 5 ^ --f-Cor, zzO, there- 
 
LAPLACE'S MECHANICS. 235 
 
 30. Let us now determine the oscillations of a body in 
 the case in which it turns very nearly about the third 
 principal axis. It is possible to deduce them from the 
 integral in the preceding number, but it is simpler to 
 obtain them directly from the differential equations 
 (D) of No. 26. The body not being solicited by any 
 forces, these equations will become by substituting in 
 the places of p', q', and r' their values Cp, Aq> and J3r, 
 
 Si _ TJ 
 
 dq -J -- --.rp.dt=0 ; 
 
 The solid being supposed to turn very nearly about 
 
 fore, as when wzzjR, zs:0 the ahove integral for the sphere 
 whose radius K becomes -&pR s . If R is supposed to be 
 yariable in this last expression, its differential will be -fp-R 4 
 <//?, which is the value of an indefinitely small lamina of a 
 sphere at the distance R from the centre, which has each of 
 its molecules multiplied into the square of its distance from 
 an axis passing through that centre; if p=:f.(R) represent 
 the density of the lamina, then ^pf>R*dR will denote the 
 number of its molecules each multiplied into the square of 
 its distance from the axis, therefore C=z%pfpR*dR. Now 
 the mass m of the sphere is equal to 4pf?R 2 dR, therefore 
 
 the value of p is .j, which by substitution gives 
 
 3 *fpR*.dK * 
 
LAPLACES MECHANICS. 
 
 third principal axis, q and r are very small quantities*, 
 the squares and (he products of which may be neglected: 
 this gives dp=.Q, and consequently p a constant quan- 
 tity. If in the two other equations we suppose 
 
 we shall hav6 
 
 D _ 
 
 p ' AB V B.(C-B)' 
 
 M and y being two constant quantities. The angular 
 
 velocity of rotation will be ^p* + q*-\-r 2 9 or simply p y 
 by neglecting the squares of q and r ; this velocity will 
 
 * If the angle lOz" (Jig. 20) is very small the angles lOx" 
 and /Oy will be very nearly right angles, therefore their 
 
 q r 
 
 cosines =. -- and . --- =. and consequently 
 
 ' 
 
 the quantities q and r will be very small. 
 
 + By substituting M.sin.fwf-j-y) for q and jl/'.cos.fw* 
 -f y) for r in these equations, they will be changed into the 
 following, 
 
 /" _ TJ 
 
 M.cos.(nt-\-y)ndt+ .M'.cos. 
 A 
 
 A _ Q 
 
 M'.sin.(nt + 7 )ndt-] -- .M.sin. 
 Consequently 
 
 M'n -f M. 
 
 from which n=p y ^- and Jlf = . 
 
 ~~A) m |j e rea c(iiy obtained. 
 B(CB) 
 
LAPLACE'S MECHANICS. 237 
 
 therefore be very nearly constant. Lastly, the sine of 
 the angle formed by the real axis of rotation and by 
 
 the third principal axis will be ^ ... 
 
 P 
 
 If at the origin of the movement we have q-=:0 and 
 and r Q ? that is to say, if the real axis of rotation co- 
 incides at this instant with the third principal axis, we 
 shall have M=D and M'-=$ ; q and r will be there- 
 fore always nothing, and the axis of rotation will al- 
 ways coincide with the third principal axis; from 
 which it follows, that if the body begins to turn about 
 one of the principal axes, it will continue to turn uni- 
 formly about the same axis. This remarkable property 
 of the principal axes, has caused them to be called the 
 principal axes of rotation; it belongs exclusively to 
 them ; for if the real axis of rotation is invariable at 
 the surface of the body, we have cfy?:z=:0, dq^=Q, and 
 cfrzzzO ; the preceding values of these quantities there- 
 fore give 
 
 B-A C-B A-C 
 
 -0.r0=0; -.rp=0; ~-.pq=0. 
 
 In the general case whereof, B, and Care unequal, 
 two of the three quantities p^ q, and r are nothing in 
 consequence of these equations, which implies, that 
 the real axis of rotation coincides with one of the prin- 
 cipal axes. 
 
 If two of the three quantities^, JE>, and Care equal, 
 for example, if we have^fczB; the three preceding 
 equations will be reduced to these rpi=.0 and pq=0> 
 and they may be satisfied by supposing p 0. The 
 axis of rotation is then in a plane perpendicular to the 
 third principal axis ; but we have seen, No. 27, that- 
 all the axes situated in this plane are principal axes. 
 
238 LAPLACE'S MECHANICS. 
 
 Lastly, if we have at the same time 4=zB=:C 9 the 
 three preceding equations will be satisfied whatever 
 may be p, q, and r, but then by No. 27, all the axes 
 of the body are principal axes. 
 
 It follows from the above, that the principal axes 
 alone have the property of being invariable axes of ro- 
 tation ; but they do not all of them possess it in the 
 same manner. The movement of rotation about that 
 of which the moment of inertia is between the moments 
 of inertia of the two other axes, may be troubled in a 
 sensible manner by the slightest cause; so that there 
 is no stability in this movement. 
 
 That state of a system of bodies is called stable 
 in which, when the system undergoes an indefinitely 
 small alteration, it will vary in an indefinitely small 
 degree by making continual oscillations about this 
 state. This being understood, let us suppose that the 
 real axis of rotation is at an indefinitely small distance 
 from the third principal axis; in this case the constant 
 quantities Jkf and M 1 are indefinitely small ; if n be a 
 real quantity the values of q and r will always remain 
 indefinitely small, and the real axis of rotation will 
 only make oscillations of the same order about the third 
 principal axis. But if n be imaginary, sin.(nt^-y} 
 and cos.(nt~\-y) will be changed into exponentials; 
 consequently the expressions of q and r may augment 
 indefinitely, and eventually cease to be indefinitely 
 small quantities* ; there is not therefore any stability 
 
 * By the rules of trigonometry jl/.sin. (w/ -}- y) zz 
 
 9 in this expression e 
 
LAPLACE'S MECHANICS. 239 
 
 in the movement of rotation of a body about the third 
 principal axis*. - The value of n is real if C is the 
 
 represents the number of which the hyperbolical logarithm 
 is unity. If n is an impossible quantity it may be supposed 
 equal to m\/ 1 ; let this value be substituted for it in the 
 above equation, and we shall have <jrAf.sin.(;ntft/ l 
 
 i ^ _ -yr ^f __ e __ . As the quantity 
 
 mt is real, the value of q may increase as that of t increases, 
 until it ceases to be indefinitely small. It may be shewn in 
 a similar manner that if t increase r will also increase, and 
 at length cease to be indefinitely small. 
 * In the equations 
 
 let the values of p' 9 ', and r 1 be substituted, then if the 
 second, after having divided both its members by AB, be 
 subtracted from the first, the following will be obtained, 
 
 If q and r are very small at the beginning of the movement, 
 
 J/ 2 
 
 the constant quantity k z -- - which may be represented 
 
 by jL, is very small at that time ; the quantities q* and r * 
 will therefore, if the difference A C arid & C have the 
 same sign, always remain very small and have^for their re. 
 
 limits _ and 
 
 If the differences A C and B C have not the same sign 
 and the constant quantity L is supposed very small, the 
 above equation may have place although the values of q and 
 and r increase indefinitely. 
 
540 LAPLACE'S MECHANICS. 
 
 greatest or the least of the three quantities A^ B, and 
 C, for (hen the product (C A).(C B) is positive; 
 but I his product is negative if C is between A and B, 
 and in this case n is imaginary ; thus the movement of 
 rotation is stable about the two principal axes of which 
 the moments of inertia are the greatest and the least; 
 but not so about the other principal axis. 
 
 In order to determine the position of the principal 
 axes in space, let us suppose the third principal axis 
 very nearly perpendicular to the plane of x' and i/ 1 ', so 
 that may be a very small quantity of which the square 
 can be neglected. We shall have by N. 26, 
 
 which gives from integration 
 
 %k=<p -pt-s, 
 s being a constant quantity. If we afterwards make 
 
 the values of q and r of No. S6, will give by extract- 
 ing 
 
 and by integration 
 
 . 
 --pu=r; -+ ps= -q, 
 
 * By differentiation and substitution the equations 
 
 ds du 
 
 pw~r ? |-jW #> 
 
 will become 
 
LAPLACE'S MECHANICS. 
 
 /3 and X being two new constant quantities ; the problem 
 is thus completely resolved, because the values of s 
 and u give the angles 9 and <p in a function of the time, 
 and %|/ is determined in a function of <p and t. If /3 is 
 nothing the plane of x 1 and y 1 becomes the invariable 
 plane to which we have referred in the preceding num- 
 ber the angles 0, <p, and %J/. 
 
 31. If the body is free, the analysis of the preceding 
 numbers will give its movement about its centre of gra- 
 vity ; if the solid is forced to move about a fixed point, 
 it will always shew its movement about this point. It 
 remains for us to consider the movement of a solid sub- 
 jected to turn about a fixed axis. 
 
 Let us suppose that x is the axis which we shall 
 imagiae to be horizontal : in this case the last of the 
 equations ( B) of No. %5 "will be sufficient to determine 
 the movement of the body. Let us suppose also, that 
 the axis of y' is horizontal and that the axis of z' is ver- 
 tical and directed towards the centre of the earth ; let 
 us conceive lastly, that the plane which passes by the 
 axes of y' and s', passes through the centre of gravity 
 of the body, and also that an axis passes constantly 
 
 These equations may be readily integrated. See No. 623 
 of the Traite du Calcul Differentie! et du Calcul Integral of 
 Lacroix, where a general formula is given for equations of 
 this description. 
 
242 
 
 through this centre and the origin of the co-ordinates. 
 Let be the angle which this new axis makes with that 
 of %' ; if we name the co-ordinates referred to this new 
 axis, y 11 and z" we shall have* 
 
 yzszy'.coiJ+^.sin.S; z'=.z fl . cos. 6 y.sin.9; 
 from which may be obtained 
 
 S. dm. ( y lfi -\- %*") is the moment of inertia of the body 
 relative to the axis of x l ; let C be this moment. The 
 last of the equations (B) of No. 25 will give 
 
 Let us suppose that the body is solicited only by the 
 force of gravity : the values of P and Q of No. 25 will 
 be nothing and R will be constant, which gives 
 
 The axis of z a passing through the centre of gravity of 
 the body, we have S.y".dm^=& ; moreover if h repre- 
 sents the distance of the centre of gravity of the body 
 
 * By referring to figure 17 ? and notes page 169, A may 
 be supposed to be the centre of the co-ordinates and point 
 through which the horizontal axis a/ passes, AY the vertical 
 axis of 2', AX the horizontal axis of y, AY t that of z" which 
 passes through the centre of gravity of the body and makes 
 the angle Q with a', and AX, the axis of y 11 . In this case 
 the values of y and z' may be found in the terms of y 11 and 
 s y/ , in a manner similar to that in which the values of x and 
 y were found in the terms of x f and y t in the above men. 
 tioned number. 
 
LAPLACE'S MECHANICS/ 
 
 from the axis of x', we shall have S.z"*dm=:mh, m 
 being the whole mass of the body ; we shall therefore 
 
 have 
 
 dN" 
 
 
 and consequently 
 
 </*0 mh.R.sm.Q 
 
 
 Let us now consider a secpnd body, all the parts of 
 which are united in one sole point at the distance / from 
 the axis of x' ; we shall have relative to this body C= 
 m'P, m' being its mass; moreover h will be equal to /; 
 by equating 
 
 #0 R . . 
 ___... 
 
 These two bodies will therefore have exactly the same 
 movement of oscillation, if their initial angular veloci- 
 ties, when their centres of gravity are in the vertical, 
 
 C 
 
 are the same, and we have /= r- 
 
 mm 
 
 The second body just noticed is the simple pendu- 
 lum, of which we have considered the oscillations at 
 No. 11; we are therefore always able to assign by this 
 formula the length / of the simple pendulum, the oscil- 
 lations of which are isochronous to those of the solid 
 which has here been considered and which forms a 
 compound pendulum. It is thus that the length of 
 the simple pendulum which oscillates seconds, is 
 determined by observations made upon compound 
 pendulums. 
 
244 LAPLACE'S MECHANICS. 
 
 CHAP. VIII. 
 
 Of the motion of Jtuids. 
 
 32. VTE shall make the laws of the motion of fluids 
 depend upon those of their equilibrium, in a similar 
 manner to that by which we have in Chap. 5, deduced 
 the laws of the motion of a system of bodies from those 
 of its equilibrium. Let us therefore resume the general 
 equation of the equilibrium of fluids given in No. 17; 
 
 the characteristic S being only relative to the co- 
 ordinates x, ?/, and s of the molecule, and independent 
 of the time t. When the fluid is in motion, the forces 
 which would retain its molecules in the state of equi- 
 librium are by No. 18, dt being supposed constant, 
 
 '-() <M3> *-()' 
 
 it is therefore necessary to substitute these forces for 
 jP, Q, and R y in the preceding equation of equilibrium. 
 Denoting by W the variation 
 
LAPLACE,'S MECHANICS. 245 
 
 which we will suppose exact* ; we shall have 
 
 >r-f =,,()+.,.()+(> 
 
 this equation is equivalent to three distinct equations, 
 for the variations %x, oy, and Js being independent, we 
 may equal their co-efficients separately to nothing*. 
 
 * As this variation is exact in the cases in which the 
 forces of attraction are directed towards centres that are 
 either fixed or moveable, it comprehends all the forces in 
 nature which can act upon the molecules of a fluid mass, and 
 may therefore be regarded as always exact. 
 
 f In places where an incompressible fluid is supported at 
 one of its sides, the value of p shews the pressure against 
 this side in the direction of a normal to it ; at those parts of 
 the fluid mass Avhich are free this value is nothing. When 
 the value of p is a known function of t, jr, y, and z, it 
 will give, by being equalled to nothing, the equation of the 
 surface of an incompressible fluid during its motion. If / is 
 not contained in this value of />, the surface of the fluid will 
 preserve the same form and the same position in space, on 
 the contrary when p contains t it will change its form or po- 
 sition every instant. 
 
 J In order that the reader may have a correct idea of the 
 corresponding variations of/, a?, y, and s and the total or par- 
 tial variations of a function of them, I shall suppose .F a 
 function of , #, y, and s, and first imagine ,r, t ?/, and z to 
 vary, t remaining constant. In this case the contempora- 
 neous values of Fmay be compared, which at a determinate 
 instant answer to the different points of a system and belong 
 to the different molecules placed at these points at the same 
 instant. 
 
 If on the contrary #, y, and z are supposed constant and 
 t to vary, the values of jP will appertain to the different 
 
246 
 
 The co-ordinates x 9 ?/, and z are functions of the 
 primitive co-ordinates and of the time t ; let , 5, and 
 c be these primitive co-ordinates, we shall then have 
 
 By substituting these values in the equation ( F) 9 
 co-efficients of So, &, and Sc may be equalled sepa- 
 rately to nothing ; which will give three equations of 
 partial differentials between the three co-ordinates x 9 
 
 molecules which during successive instants pass by the same 
 point which has x 9 y, and z for its co-ordinates. 
 
 Lastly ; if we make x 9 y 9 and z to vary either partially or 
 together and suppose t also variable ; the different values of 
 F will belong to the same molecule, and change as it passes 
 in successive instants from one point to another in the system. 
 If the position of the molecule is known at the commence, 
 ment of the motion, the constant quantities belonging to 
 the three equations which give the values of x 9 y, and z in 
 functions of t will be known. The values of #, y^ and z 
 may therefore be found at any instant, which will give the 
 position of the molecule at that instant. 
 
 If the time t be eliminated from the three equations given 
 by the values of x^ y^ and z 9 two equations of the curve 
 described by the molecule will be known. The form and 
 position of the curve will change by passing from one mo- 
 lecule to another : the constant quantities in this case 
 changing their values as the initial position of the molecule 
 changes. 
 
LAPLACE'S MECHANICS. 247 
 
 y, and z of the molecule, its primitive co-ordinates , 
 by c 9 and the time t. 
 
 It remains for us to fulfil the conditions of the con- 
 tinuity of the fluid. For this purpose let us consider 
 -at the origin of the motion, a rectangular fluid paral- 
 lelepiped having for its three dimensions d, d#, and 
 dc. Denoting by (%) the primitive density of this 
 molecule, its mass will be (%).da.db.dc. Let this 
 parallelepiped be represented by (A)*; it is easy to 
 
 * In figure 22 the rectangular parallelepiped A is repre- 
 sented, having da, d&, and dc for its three edges; this 
 parallelepiped is changed after the time t into that given 
 in figure 23, in which from the extremities of the edge fg 
 which is composed of the molecules that formed the edge dc, 
 two planes gn, fo, are supposed to be drawn parallel to the 
 plane of x and y ; by the prolongation of the edges of the 
 parallelepiped gl or (B) to these planes a new one (C) is 
 formed equal to (B), as the parts cut of from (B) and those 
 added to (C) respectively compensate each other. The 
 height fg of (C) 9 as it is independent of the molecules in 
 da and d6, is found by making c alone to vary in differen- 
 tiating the value of z 9 it is therefore equal to t ~ l.dc. In 
 
 figure 24, let tiqrp denote the section (e) having its side $p 
 formed by molecules of the side d&, dc, and its side Hq by 
 molecules of the side da, dc of the parallelepiped (A). 
 From and p the lines $m and pn are supposed to be drawn 
 parallel to the axis of a?, meeting the line qr or its continu. 
 ation in m and w, and consequently forming a new" paral- 
 lelogram (X) which is equal to the former ( E ) ; as it has the 
 same base $p and is between the same parallels. The value 
 of Sp is found by taking the differential of y in making , z 9 
 ^nd t constant, and the value of $m by taking the differential 
 
248 LAPLACE'S MECHANICS. 
 
 perceive that after the time t, it will be changed into an 
 oblique angled parallelepiped; for all the molecules 
 primitively situated upon any side whatever of the 
 parallelepiped (A), will again be in the same plane, 
 at least by neglecting the indefinitely small quantities, 
 of the second order : all the molecules situated upon the 
 parallel edges of (A) will be upon the small right lines 
 equal and parallel to each other. Denoting this new 
 parallelepiped by ( B}, and supposing that by the ex- 
 tremities of the edge formed by the molecules which in 
 the parallelepiped (A) 9 composed the edge de, we 
 draw two planes parallel to that of x and y. By pro- 
 longing the edges of (B) until they meet these two 
 planes, we shall have a new parallelepiped (C) con- 
 tained by them, which is equal to (B) ; for it is evi- 
 dent that as much as is taken from the "parallelepiped 
 (B) by one of the two planes, is added to it by the 
 other. The parallelepiped (C) will have its two 
 bases parallel to the plane of x and^y: its height con- 
 tained between its bases will be evidently equal to the 
 differential of z taken by making c alone to vary ; 
 
 which gives ( -j- J.dc for this altitude. 
 
 of x in supposing y and s constant. These last values mul- 
 tiplied/ogether give the value of the surface of the paral- 
 lelogram (X), or that of its equal ( ) ; which value, when 
 
 multiplied by ( ~T )'^ c ^ ie differential of z, gives the con. 
 tnt of the parallelepiped (C) or (B). 
 
219 
 
 We shall have its base, by observing that it is equal 
 to the section off B) made by a plane parallel to that 
 of x and y ; let this section be denoted by (s). The 
 value of s \vill be the same with respect to the molecules 
 of which it is formed ; and we shall have 
 
 Let $p and <$q be two contiguous sides of the section 
 fej, of which the first is formed by molecules of the 
 side d6.dc of the parallelepiped (A), and the second 
 by molecules of its side da.dc. If by the extremities 
 of the side $p we suppose two right lines parallel to the 
 axis of a.', and" we prolong the side of the parallelogram 
 CO parallel to $p until it meets these linos ; they will 
 intercept between themselves a new parallelogram (\) 
 equal to (s), the base of which will be parallel to the 
 axis of x. The side lp being formed by molecules of 
 the face d&.dc, relative to which the value of z is the 
 same ; it is easy to perceive that the height of the 
 parallelogram pO, is the differential of y taken by- 
 supposing 0, s, and t constant, which gives 
 
 from which may bo obtained 
 
 this is the expression of the height of the parallelogram 
 (K}. Its base is equal to the section of this parallelo- 
 gram made by a plane parallel to the axis of x ; this 
 section is formed of the molecules of the parallelepiped 
 
550 
 
 by relation to which % and y are constant, its 
 length is therefore equal to the differential of x taken 
 by supposing z, y^ and t constant, which gives the 
 three equations 
 
 Suppose for abridgment 
 
 ^(*v*v- w-v 
 
 -V</<J VSJ Ve/c7 Vd<J \dcdb 
 
 (dx\(*L\(te_\ (*L\(W\.(*L \ 
 
 \db) \dc) \daj \dbj \daj \dc / 
 dz\ sdx\ fdy 
 db)-- (do )\di 
 
 we shall have 
 
 /3.da 
 
 dcj \db. 
 
 this is the expression of the base of the parallelogram 
 (X} ; the surface of this parallelogram will therefore be 
 
 This quantity also expresses the surface of 
 dc ' 
 
 the parallelogram fO, ^ we mulitiply it by 
 
 we shall have ;3.d0.d.dc for the magnitude of the 
 parallelepipeds (C) and (E). Let be the density 
 of the parallelepiped (A) after the time*; then its 
 mass will be represented by ? ./3.da.d&.dc, which being 
 equalled with the first mass (;.dfl.d&.dc, gives 
 
LAPLACE'S MECHANICS. 251 
 
 for the equation relative to the continuity of the fluid. 
 33. We may give to the equations (F) and (G) 
 another form more convenient for use in certain cir- 
 cumstances. Let w, v, and v, be the respective velo- 
 cities of a fluid molecule parallel to the axis of tf, y^ 
 and % : \ve shall have 
 
 CD (!)=- ()=< 
 
 By differentiating these equations, and regarding w, v 9 
 and v as functions of the co-ordinates #, y, and z of 
 the molecule, and of the time t ; we shall have 
 
 dv 
 
 d z z\ fdv 
 
 7u')=(^ 
 
 The equation (F) of the preceding number will then 
 become 
 
 +-(!) 
 
 (H) 
 
 Tn order to have the equation relative to the continuity 
 of a fluid, let us snppose that in the value of /3 of the 
 preceding number, a ? b, and c may be equal to x, ?/, 
 
252 JLAPLACE'S MECHANICS. 
 
 and s, and that #, j/-, and z may be equal respectively 
 to x-*\-udi) y-\-vdt) and s-}-vd!f, which is equivalent to 
 taking the first co-ordinates , &, and c indefinitely 
 near to .?, y, and s/ we shall then have 
 
 the equation (G) becomes 
 
 If we consider p as a function of x, y, s, and t 9 we 
 shall have 
 
 the preceding equation will therefore become 
 
 this is the equation relative to the continuity of the 
 fluid, and it is easy to perceive, that it is the differen- 
 tial of the equation (G) of the preceding number, 
 taken relative to the time /t. 
 
 * The equation 
 
 (dp\ ( d.pu 
 rf -9+(^ 
 
 is equivalent to the following 
 
 which' is what the equation (G) becomes when the values of 
 |3 and (?) are substituted. 
 
 + If the fluid be incompressible, as the mass the density 
 and the magnitude of each molecule of the fluid \vill remain 
 invariable, the equation (K) 9 by equalling separately the 
 
LAPLACE'S MECHANICS. 
 
 The equation ( H) is susceptible of integration in a 
 very extensive case, that is when w&r+T&y+v&s is an 
 
 variations of the density and the mass to nothing, will give 
 the two following 
 
 By joining these equations to the three given by that of (H) 
 we shall have five which will enable us to determine the un- 
 known quantities f , p, u, t>, and v in functions of /, ,r, 
 y, and z. 
 
 If the incompressible fluid is homogeneous the density 
 will be a constant quantity ; in this case we shall have only 
 the second of the above equations and the three given by 
 that of (II) to determine the four unknown quantities p, 
 u 9 u, and v. 
 
 If the fluid is elastic we shall have the equation (K) and 
 the three given by that of (H) : if the temperature be the 
 same throughout the mass, the density will be as the pres- 
 sure, which gives p=:k ; therefore there will be only four 
 unknown quantities which the four equations above men. 
 tioned are sufficient to discover. If the temperature be 
 variable and a given function of the time, the quantity k 
 will be a function of these variables, consequently the before 
 mentioned equations will be sufficient to determine the values 
 of p, M, v 9 and v. 
 
 It appears from the above that we shall have in every case 
 as many equations as there are unknown quantities in the 
 problem. But as these are equations of partial differentia. 
 tions of , JT, #, and z 9 they have at present resisted every 
 attempt to integrate them. In some instances they have 
 been simplified and integrated by particular suppositions, 
 
554 LAPLACE'S MECHANICS. 
 
 exact variation of r, y, and z, p being also any func 
 tion whatever of the pressure p. If therefore &p repre 
 sent this variation ; the equation ( H) will give* 
 
 from which we may obtain by integrating it with re 
 spect to S, 
 
 It is necessary to add a constant quantity v hich is a 
 function of t to this integral, but we may suppose that 
 this quantity is contained in the function <p. This last 
 function gives the velocities of the fluid molecules 
 parallel to the axes of x, y^ and s : for we have 
 
 The equation (K) relative to the continuity of the fluicl 
 becomes 
 
 but eren then the greatest difficulty has attended the deter- 
 mination of the arbitrary constant quantities which depend 
 upon the state of the fluid at the commencement of its motion, 
 
 * That the equation (H) gives 
 
 may be rendered evident from considering, for instance, 
 
 which is 
 
LAPLACE'S MECHANICS. 
 
 thus we have relative to homogeneous fluids 
 
 ft may be observed that the function ti 
 is an exact variation of #, y^ and z at all times, if it be 
 during one instant. Let us suppose that at any instant 
 whatever, it is equal to p ; in Ihc following instant we 
 shall have 
 
 it will therefore be an exact variation at this instant, 
 
 tion at the first instant ; but the equation (H) gives at 
 this instant (*).**( )%+(>=^-^ 
 
 this equation is consequently an exact variation in x, 
 
 The integration of the equation 
 
 which presented the greatest difficulties has been fortunately 
 accomplished by Marc Antoine Parseval, a French mathe. 
 matician. Vide the eighth Cahier of the Journal de 1 
 Poljtechnique, 
 
256 
 
 y, and %; thus if the function u.<$a:-\-'0.ty-{-v.'$z ^> e ari 
 exact variation one instant, it will also be one in the 
 nexf, it. is therefore an exact variation at all times. 
 
 When f he motions are very small ; we may neglect 
 the squares and the products of w, v 9 and v; the equa- 
 tion (II) then becomes 
 
 therefore in this case u 3 x-\-v. \y-\-v.lz is an exact va- 
 riation, if, as we have supposed, p be a function off ; 
 by naming this differential t$ 9 we shall have 
 
 and if the fluid be homogeneous, the equation of con- 
 tinuity will become 
 
 These two equations contain the whole of the theory of 
 very small undulations of homogeneous fluids. 
 
 34. Let us consider an homogeneous fluid mass 
 which has an uniform movement of rotation about the 
 
 * In the case of the very small undulations of an homo. 
 
 dp n 
 geneous incompressible heavy fluid, such as water,/ =^-; 
 
 if the axis of z be supposed in the direction of gravity, at 
 
 dp fd<p\ 
 its surface the equation Vf ml -r J is changed into the 
 
 fd\ 
 
 following gszzrl -r 1 : g representing the constant force of 
 gravity. 
 
LAPLACE'S MECHANICS. 25? 
 
 axis of x. Let n represent the angular velocity of ro- 
 tation at a distance from the axis which we will take 
 for the unify of distance ; we shall then have r= nz ; 
 ; the equation (II) of the preceding number 
 also become 
 
 f 
 
 which equation is possible because its two members are 
 exact differentials. The equation ( K) of the same 
 number in like manner will become 
 
 * In figure 25 let A represent the origin of the co. 
 ordinates, AY the axis of y^ AZ that*of 2, and AD the 
 projection upon the plane yz of a line drawn from a molecule 
 in the fluid mass perpendicular to the axis of x from D 
 draw the line DZ perpendicular to the axis AZ 9 then DZ 
 =y, AZ= 2, and AD \/#*+z*. Let a line DE be 
 drawn from D perpendicular to AD, and from any point E 
 in it draw a line EF perpendicular to DZ ; then if DE re. 
 present the velocity of the molecule in the direction perpen- 
 dicular to AD, it may be supposed to be resolved into two 
 others DF and FE in the respective directions of y and z. 
 As the velocity at the distance represented by unity from 
 
 the axis of x is n, that at the distance yV-f-s 2 is nv/J^f? 
 =:DE. From the similarity of the right angled triangles 
 AZU and DFE 
 
 but DFnz is the velocity in the direction of the axis y 9 
 and ought to be taken negatively as it tends to diminish that 
 
258 LAPLACE'S MECHANICS. 
 
 and it is evident that this equation is satisfied if the fluid 
 mass be homogeneous*. The equations of the motion 
 of fluids are then therefore satisfied^ and consequently 
 this movement is possible. 
 
 The centrifugal force at the distance vV*-f-2 2 from 
 the axis of rotation, is equal to the square of the velo- 
 city w 2 .(^ a +2 2 J divided by this distance ; the function 
 n*.(yty-\-z$z) is consequently the product of the cen- 
 trifugal force by the element of its direction I ; therefore 
 by comparing the preceding equation of the movement 
 
 axis. Again 1/3/ 2 +s z V ' ' n\/y*-\-z z : PEny, or the 
 velocity of the molecule in the direction of the axis 2. 
 
 * That the equation (II) is reduced to the value given in 
 this number appears evident from considering, that all the 
 
 terms in the second member vanish except v.f l.tyrz 
 
 n*yty and v.l J.$srr-- 2 2$2 
 
 In the equation (K) the respective values of p I -- ), 
 
 (Ju\ fdv\ 
 
 r- 1 and p.i -j- I, in this case, are evidently equal to 
 
 nothing: also the partial differentiations of p respectively 
 vanish if the fluid is homogeneous. 
 
 t The centrifugal force at the distance 
 
 axis of rotation is equal to 
 
 orn 
 
 value multiplied into the element of its direction or 
 
LAPLACE'S MECHANICS. 259 
 
 of a fluid, with the general equation of the equilibrium 
 of fluids given in No. 17 ; we may perceive that the 
 conditions of movement of which it treats, reduce 
 themselves to those of the equilibrium of a fluid mass 
 solicited by the same forces, and by the centrifugal 
 force due to the movement of rotation : which is other- 
 ways evident*. 
 
 If the exterior surface of a fluid mass is free, we 
 shall have fy?z=0 at this surface, and consequently 
 
 from which it follows, that the resultant of all the 
 forces which act upon each molecule of the exterior 
 surface, should be perpendicular to this surface ; it 
 ought also to be directed towards the interior of the 
 fluid mass* If these conditions be fulfilled an homo- 
 geneous fluid mass will be in equilibrio, supposing at 
 the same time, that it covers a solid body of any figure 
 whatever. 
 
 The case which we have examined is one of those in 
 which the variation u.%z-\-v.ty-}-v.'<iz is not exact : for 
 
 * The general equation given in this number, when the 
 value of Fis substituted, becomes 
 
 f 
 
 is independent of the time, and has n z .(i/ty-\~z$z) for 
 the value of the centrifugal force multiplied into the element 
 of its direction ; it is therefore evident that the conditions 
 of movement are, in this case, the same as those of the 
 equilibrium of a fluid mass solicited by the same forces and 
 by the centrifugal force arising from the rotatory motion of 
 the mass. See notes page 1 34. 
 
260 LAPLACE'S MECHANICS. 
 
 (his variation becomes n.($yTfiz); therefore in 
 the theory of (he flux and reflux of the sea, we cannot 
 suppose that the variation concerned is exact, because 
 it is not in the very simple case in which the sea has no 
 other movement than that of rotation, which is com- 
 mon to it and the earth. 
 
 35. Let us now determine the oscillations of a fluid 
 mass covering a spheroid possessed of a movement of 
 rotation nt about the axis of x ; supposing it to be very 
 little altered from the state of equilibrium by the action 
 of very small forces. At the beginning of the move- 
 ment, let r be the distance of a fluid molecule from the 
 centre of gra\ ity of the spheroid that it covers, which 
 we will suppose immoveable, let be the angle that 
 the radius r forms with the axis of x, and is the angle 
 which the plane that passes by the axis of x and this 
 radius forms with the plane of a; and y. Let us sup- 
 pose that after the time t the radius r is changed into 
 r-\-as y that the angle is changed into 0-|-a, and that 
 the angle OT is changed into nt-\-Tx-{-(xv ; as, aw, and i) 
 being very small quantities of which we may neglect 
 the squares and the products; we shall then have* 
 
 * In figure 18, let C represent the origin of the co- 
 ordinates at the centre of gravity of the spheroid, CB 9 CF 9 
 and CE the respective axes of a?, #, and z, CA the distance 
 of a fluid molecule from C, AD a perpendicular let fall 
 from the molecule to the plane of FCB or xy 9 DB and con. 
 sequently AB^ perpendiculars drawn from D and A to the axis 
 of a?, then C^r + a*, C/fae, BD=y, and AD~z, also the 
 angle ACB made by the radius and axis of tf~#-f"w ? and 
 
LAPLACE'S MECHANICS. 26*1 
 
 . cos. 
 
 s ) . sin . ( --ait . cos, 
 z=(r-\-xs) .sin . ( Q-}-zu ) .sin . 
 If we substitute these values in the equation (F)o( 
 No. 32, we shall have by neglecting the square of a, 
 
 n.'fl.(g)+Sii.8in.8.cos.8.( 
 
 sin. 2 9.( J) | =|-.J. 
 
 At the exterior surface of the fluid %p=.0 ; also in the 
 state of equilibrium 
 
 ($ V) being the value of P" which belongs to this state. 
 
 the angle ABD or the inclination of the planes ACB and 
 tf^zrrwtf+CT + ay. By trigonometry, in the right angled 
 triangle ACB we have 
 rad.(l) : cos.(0-f a ) : : r-f s : a:iz:fr+ a 5>os.(9 + ^), 
 
 and 
 
 rad.(l) : sin.(9-L aM ) : : r-f as : AB (r-f *0.sin.(0-f-*zO j 
 also in the right angled triangle ADB we have 
 rad.(l) : cos.(w#-f-CT-|-ay) : : (r-f .y)a 
 -f ^)sin,(04-w)cos.(/-f CT i at') and 
 rad.(l) : sin.(w/+^+ a y) : : (r-f *4>i. 
 
26 LAPLACE'S MECHANICS. 
 
 Let us suppose the sea to be the fluid treated upon ; 
 the variation (W) will be the product of the gravity 
 multiplied by the element of its direction. Let g re- 
 present the force of gravity, and ay the elevation of a 
 molecule of water at the surface above the surface of 
 equilibrium, which surface we shall regard as the true 
 level (niveau) of the sea. The variation (IV) in the 
 state of movement will by this elevation be increased 
 by the quantity ag.^y, because the force of gravity 
 acts very nearly in the direction of the ay's and towards 
 their origin. Lastly denoting by a$V the part of SF 
 relative to the new forces which in the state of move- 
 ment solicit the molecule, and depend either upon the 
 changes which the attractions of the spheroid and the 
 fluid experience from, this state, or from foreign attrac- 
 tions; we shall have at the surface 
 
 The variation -~3.{(r-\-<x.s).sii\,(Q-}-oiu)}* is increased 
 
 by the quantity a a .$y.r.sin. a 0*, in consequence of the 
 height of the molecule of water above the surface of the 
 sea ; but this quantity may be neglected relative to the 
 
 n 2 r 
 term etgtyy because the ratio of the centrifugal 
 
 * The quantity w 2 .?/.r.sin. 2 may be obtained from the 
 variation .&.{(> -f- x s).$in.(Q-}-au)} 2 by differentiating 
 
 that variation with respect to r, neglecting the quantities 
 s and u, which are relative to time, and supposing that 
 $r is equal 
 
LAPLACE'S MECHANICS. 263 
 
 force at the equator to gravity, is a very small fraction 
 equal to *. Lastly, the radius r is very nearly con- 
 
 28 SP 
 
 stant at the surface of the sea, because it differs very 
 little from a spherical surface ; we may therefore sup. 
 pose Sr equal to nothing. The equation L thus be- 
 comes at the surface of the sea 
 
 du\ 
 J -f 
 
 the variations ty and I V being relative to the two va- 
 riables and CT. 
 
 Let us now consider the equation relative to the 
 continuity of the fluid. For which purpose, we may 
 suppose at the origin of the movement a rectangular 
 parallelepiped, of which the altitude is dr, the breadth 
 rdw.sin.fi and the length rd0. Let r', 0', and w' re- 
 present the values of r, 0, and OT after the time t. By 
 following the reasoning of No. 32, we shall find that 
 after this time, the volume of the fluid molecule is 
 
 * *w*.fy.r.sin. 2 has the same ratio to ag.ty as 
 
 r.sin. 2 
 
 has to 1, but the centrifugal force atiheequa. 
 
 tor is ~ or w 2 r, and is nearly equal to - there. 
 
 ?z 2 .r.sin. 2 
 f ore - _. - ma y |j e ne gi ec ted when compared with I. 
 
264? LAPLACE'S MECHANICS. 
 
 equal to a rectangular parallelepiped of which the 
 
 altitude is .clr and the breadth 
 
 by extracting dr by means of the equation 
 lastly its length is 
 
 - K? 
 
 by extracting dr and dw, by means of the equations 
 
 Supposing therefore 
 
 V- W- "I f -Y(- 
 
 / VMrV V^7 \cirj \d^ 
 
 dr 
 
 the volume of the molecule after the time t \vill be /3'. 
 r /2 .sin.Q.dr.d0.dOT; therefore naming (p) the primitive 
 density of this molecule, and p its density correspond- 
 ing to t ; we shall have by equalling the primitive ex- 
 pression of its mass, to its expression after the time t> 
 
 p.fi'r' 2 .sin.Q'=(p).r*.sm.Q; 
 
 this is the equation of the continuity of the fluid. In 
 the present case 
 
 r 7 r-|-a$ ; V=Q-\-au ; w=/rf+^+ a ^ ; 
 we shall therefore have by neglecting the quantities of 
 the order a 2 
 
LAPLACE'S MECHANICS. 265 
 
 Let us suppose that after the time 9, the primitive den- 
 sity (p) of the fluid is changed into (p J-^-ap'; the pre- 
 ceding equation relative to the continuity of the fluid 
 will give 
 
 (_I | 
 dr j 
 
 36. *Let us apply these results to the oscillations of 
 the sea. Its mass being homogeneous we have '//=0' 
 and consequently 
 
 Let us suppose conformably to what appears to have 
 place in nature, that the depth of the sea is very small 
 relative to the radius r of the terrestrial spheroid ; let 
 this depth be represented by y, 7 being a very small 
 function of 6 and ts which depends upon the law of 
 this depth. If we integrate the preceding equation 
 with respect to r, from the surface of the solid which 
 the sea covers unto the surface of the sea ; we shall 
 
 * As the notes necessary to elucidate this and the follow, 
 ing number satisfactorily would from their very great length 
 too much increase the size of the Work, I shall refer the 
 reader who is desirous of full information respecting them 
 to the fourth book of the Mechanique Celeste, where all 
 the equations are integrated and every particular explained 
 in the fullest manner. 
 
266 
 
 find that the value of 5 is equal to a function of 0, 37, 
 and t independent of r, plus a v-ery small function 
 which will be with respect to u and v of the same small 
 
 order as the function - ; but at the surface of the solid 
 
 which the sea covers, when the angles d and -57 are 
 changed into 6-\-au and nt-\--&-\-av 9 it is easy to per- 
 ceive, that the distance of the molecule of water con- 
 tiguous to this surface from the centre of gravity of the 
 earth, only varies by a very small quantity with respect 
 to au and att, and of the same order as the products of 
 these quantities by the eccentricity of the spheroid 
 covered by the sea : the function independent of r 
 which enters into the expression of s is therefore a 
 very small quantity of the same order ; so that we may 
 generally neglect s in the expressions where n and v are 
 concerned. The equation of the motion of the sea at 
 its surface given in No. 35 therefore becomes 
 
 dt 
 
 (M) 
 
 the equation (L) of the same number relative to any 
 point whatever of the interior of the mass of fluid, 
 gives in the state of equilibrium 
 
 (IV) and ($p) being the values of SP and *p which in 
 the state of equilibrium answer to the quantities r-|-as, 
 Q-j-aw, and tv-\-(x.'n. Suppose that in the state of mo- 
 tion, we have 
 
 the equation (L) will give 
 
LAPLACE'S MECHANICS. 267 
 
 The equation f 3/J shews that wj j is of the same 
 
 yu 
 order as # or s, and consequently of the order ; the 
 
 value of the first member of this equation is therefore of 
 the same order; thus multiplying this value by rfr, 
 and integrating it from the surface of the spheroid 
 which the sea covers unto the surface of the sea, we 
 
 7> J 
 shall have V equal to a very small function of the 
 
 f 
 
 order , plus a function of 0, w, and t independent of 
 
 7-3 which we will denote by, A; having therefore regard 
 in the equation ( L) of No. 35 only to (wo variables 
 6 and w, it will be changed into the equation (M) 9 
 with the sole difference, that the second member will 
 be changed into %K. But * being independent of the 
 depth at which the molecule of water which we are 
 considering is found ; if we suppose this molecule very 
 near the surface, the equation (L) ought evidently to 
 coincide with the equation (M) ; we have therefore 
 ^=sP gty> and consequently 
 
 the value of 5 V in the second member of this equation 
 being relative to the surface of the sea. We shall find 
 in the theory of the flux and reflux of the sea, that this 
 value is nearly the same for all the molecules situated 
 upon the same terrestrial radius, from the surface of the 
 solid which the sea covers to the surface of the sea ; we 
 
266 LAPLACE'S MECHANICS. 
 
 have therefore relative to all these molecules frssgjy : 
 
 which gives p' equal to p .gy plus a function independ- 
 ent of 0, 37, and r : but at the surface of level of the 
 sra, the value of /?' is equal to (he pressure of the small 
 column ay of water which is elevated above this sur- 
 face, and this pressure is equal to a-p*s:y\ we have 
 therefore in all the interior of the fluid n;ass 9 from the 
 surface of the spheroid which the sea covers, to the 
 surface of level of the sea, p'=pg?f ; therefore any 
 point whatever of the surface of the spheroid covered 
 by the sea, is more pressed than in the state of equi- 
 librium, by all the weight of the small column of water 
 comprised between the surface of the sea and the sur- 
 face of level. This excess of pressure becomes negative 
 at the points where the surface of the sea is sunk below 
 the surface of level. 
 
 It follows from what has been said, that if we only 
 have regard to the variations of and of ts ; the equation 
 (L) will be changed into the equation (M} 9 for all 
 the interior molecules of the fluid mass. The values of 
 u and of v relative to all the molecules of the sea situ- 
 ated upon the same terrestrial radius, are therefore de- 
 termined by the same differential equations: therefore by 
 supposing as we shall in the theory of the flux and reflux 
 of the sea, that at the beginning of the motion the values 
 
 ^ u "> \'dt) 9 ^' \/F/' were the same for all the mo- 
 lecules situated upon the same radius these molecules 
 would remain upon the same radius during the oscil- 
 lations of the fluid. The values r, u, and v may 
 therefore be supposed very nearly the same upon the 
 small part of the terrestrial radius comprised between 
 the solid that the sea covers and the surface of the sea: 
 therefore from integrating with relation to r the equation 
 
LAPLACE'S MECHANICS. 26$ 
 
 M.COS.0) 
 
 ImTS' 
 
 we shall have 
 
 (r*s) being the value of r*s at the surface of the sphe- 
 roid covered by the sea. The function r*s^~- (r*s) i? 
 yery nearly equal to r*.{s (s)}-\-2ry(s), (s ) being 
 the value of s at the surface of the spheroid ; the term 
 2ry.(s) may be neglected on account of the smallness 
 of y and (s) ; thus we shall have 
 
 r 5 (r*s)=r*.{s(s)}. 
 
 Moreover the depth of the sea corresponding to the 
 Angles Q-}" aW an d nt-\-Tz-\-a,v is y-|~ a< { 5 ( s )}i ^ we 
 place the origin of the angles 6 and nt-\--& at a point 
 and a meridian fixed upon the surface of the earth, 
 which may be done as we shall forthwith see; this 
 
 (dy\ fdy \ 
 
 ^ J-j-a^.I - 1, plus <he eleva- 
 
 tion y of the fluid molecule of the surface of the sea 
 ^bove the surface of level ; we shall therefore have 
 
 The equation relative to the continuity of the fluid 
 consequently will become 
 
 d,yv \ yu.cos.Q 
 
 (d.yu \ 
 i/r/- 
 
 It may be observed that in this equation, the angles $ 
 and nt-^-vs are reckoned relative to a point and to a 
 meridian fixed upon the earth, and that in the equation 
 (M) these same angles are reckoned relative to the 
 axis of j;, and to a plane which passing through this 
 axis will have a movement of rotation about it equal to 
 n ; but this axis and this plane are not fixed at the sur- 
 
270 LAPLACE'S MECHANICS. 
 
 face of the earth, because the attraction and the pres- 
 sure of the fluid which covers it ought to alter their 
 position a little upon this surface, as well as the move- 
 ment of rotation of the spheroid. But it is easy to 
 perceive, that these alterations are to the values of au 
 and av 9 in the ratio of the mass of the sea to that of the 
 terrestrial spheroid ; thus, in order to refer the angles Q 
 and nt-}- to a point and to a meridian which are in- 
 variable at the surface of this spheriod in the two 
 equations (M) and (N) ; it is sufficient to alter u and 
 
 o by quantities of the order and , which quanti- 
 ties may be neglected ; in these equations therefore, it 
 may be supposed that &u and av are the movements of 
 the fluid in latitude and longitude. 
 
 Again, it may be observed, that the centre of gravity 
 of the spheroid being supposed immoveable, it is ne- 
 cessary to transfer in a different direction to the fluid 
 molecules the forces by which it is actuated in conse- 
 quence of the re-action of the sea ; but as the common 
 centre of gravity of the spheroid and the sea does not 
 change its situation in consequence of this re-action, it 
 is evident that the ratio of these forces to those by 
 which the molecules are impelled from the action of 
 the spheroid, is of the same order as the ratio of the 
 fluid mass to that of the spheroid^ and consequently of 
 
 *y 
 the order - ; they may therefore be neglected in the 
 
 calculation of c V . 
 
 37. Let us consider in the same manner the motions 
 of the atmosphere. We shall in this research neglect 
 the variations of the heat at different latitudes and dif- 
 ferent heights, as well as all irregular causes which 
 
LAPLACE'S MECHANICS. 271 
 
 agitate it, and only have regard to the regular causes 
 which act upon it as upon the ocean. We shall con- 
 sequently suppose the sea covered by an elastic fluid of 
 an uniform temperature ; we will also suppose con form- 
 ably to experience, that the density of this fluid is 
 proportional to its pressure. This supposition gives 
 an indefinite height to the atmosphere, but it is easy to 
 be assured, that at a very small height its density is so 
 trifling that it may be regarded as nothing. 
 
 This being agreed upon, let s', ti' y and v f represent 
 for the molecules of the atmosphere, what s, u^ and 
 signified for the molecules of the sea ; the equation (L) 
 ef No. 35 will then give 
 
 du'\ . 2.sin. 2 
 
 u\ . 2.sn. 
 l)+r 
 
 -2*r. sin.'O. =. 
 
 Let us at present consider the atmosphere in the slate 
 of equilibrium in which s r , u ! 9 and v r are nothing. 
 The preceding equation will give by integration, 
 
 F-/--= constant. 
 * p 
 
 The pressure p being conceived to be proportional to 
 the density, we shall make p=.Lg.p, g being the 
 gravity at a determinate place, which may be supposed 
 to be the equator, and / being a constant quantity that 
 gives the height of the atmosphere, conceived tobeofths 
 same density throughout, as at the surface of the sea : 
 
272 LAPLACE'S MECHANICS. 
 
 (his height is very small when compared with the radiu 
 of the terrestrial spheroid, of which it is not the 720th 
 part. 
 
 The integral/ is equal tol.g.log.p; the preceding 1 
 
 P 
 
 equation of the equilibrium of the atmosphere conse- 
 quently becomes 
 
 n z 
 /g.log.fznconst.-j- V-\ .r. 2 sin.*Q. 
 
 At the surface of the sea, the value of V is the same for 
 a molecule of air as for the molecule of water which is 
 contiguous to it, because the forces which solicit each 
 molecule are the same ; but the conditions of the equi- 
 librium of the sea requires that we should have 
 
 M 2 
 
 V-\ .r 2 .sin. 2 0=consf. ; 
 
 therefore at this surface p is constant, that is to say, the 
 density of the lamina of air next to the sea, is through- 
 out the same in the state of equilibrium. 
 
 If R represent the part of the radius r comprised 
 between the centre of the spheroid and the surface of 
 the sea, and r 1 the part comprised between this sur- 
 face and a molecule of air elevated above it, r' will 
 
 (7Z Z \ 
 _ r t 18 
 
 R 
 
 from the height of this molecule above the surface of 
 the sea: we shall therefore neglect the quantities of 
 this order. The equation between p and r will give 
 
 iin.'fl+ft'.Rr'.sin.'fl; the values of V, and 
 
LAPLACE'S MECHANICS. 273 
 
 'cl i V\ 
 
 -T-- \ being relative to the surface of the sea where 
 
 ^ye have 
 
 sdy\ 
 the quantity I ^- J w 2 7?. sin. 2 9 is the gravity at this 
 
 same surface ; we shall denote it by g f . The function 
 
 d*F\ 
 
 -j z f being multiplied by the very small quantity ?' 2 , 
 
 we may determine it on the supposition that the earth 
 is spherical, and reglect the density of the atmosphere 
 relative to that of the earth; we shall thus have very 
 nearly 
 
 dr 
 
 sd*V\ 
 m denoting the mass of the earth ; by equating I - I 
 
 zzz: zzi-~ ; \ve shall therefore have Ig.log.p ncoust. 
 
 r' 2 
 r tg' -- -g/ . f rom w hich may be obtained 
 
 
 c being the number of which the hyperbolical logarithm 
 is unity, and IT being a constant quantity evidently 
 equal to the density of the air at the surface of the sea. 
 Let/i and h 1 represent the respective lengths of pendu- 
 lums oscillating seconds at the surface of the sea ander 
 the equator, and at the latitude of the molecule of air 
 
 ' h 1 
 
 which has been considered : we shall have -=r, and 
 
 g h> 
 
 consequently 
 
274 LAPLACE'S MECHANICS. 
 
 This expression of the density of the air shews, that the 
 laminae of the same density are throughout equally 
 elevated above the surface of the sea, except by the 
 
 quantity - ^ - nearly ; but in the exact calculation 
 
 of the heights of mountains by the observations of the 
 barometer, this quantity ought not to be neglected. 
 
 Let us now consider the atmosphere in the state of 
 motion, and let us determine the oscillations of a lamina 
 of level, or of the same density in the state of equilib- 
 rium. Let a$ be the elevation of a molecule of air 
 above the surface of level to which it appertains in the 
 state of equilibrium ; it is evident that in consequence 
 of this elevation, the value of SF will be augmented by 
 the differential variation &.$$; we shall therefore 
 have 2 F=f * V) g.ty+* V ; (* V) being the value 
 of ^V which in the state of equilibrium corresponds to 
 the lamina of level and to the angles 0-j-aw and nt-\~ix 
 -\-av; SF ; being the part of F arising from the new 
 forces which in the state of movement agitate the at- 
 mosphere. 
 
 Let f=C/0+a/, (p) being the density of the lamina 
 of surface in the state of equilibrium. If we make 
 
 1 '-= we shall have 
 
 but in the state of equilibrium 
 
 0=!.S.{fr+* S ;.sin.f9+* 
 
 the general equation of the motion of the atmosphere 
 
LAPLACE'S MECHANICS. 275 
 
 will therefore become relative to the laminae of level, 
 with respect to which Sr is very nearly nothing, 
 
 sin. 9 O.(V (*');, 
 
 being the variation of r corresponding in the 
 state of equilibrium to the variatons aw' and at/ of the 
 angles 9 and sr. 
 
 Let us suppose that all the molecules of air situated at 
 first upon the same terrestrial radius, remain constantly 
 upon the same radius in the state of motion, which has 
 place by what precedes in the oscillal ions of the sea; 
 and let us try if this supposition will satisfy the equa- 
 tions of the motion and of the continuity of the atmos- 
 pheric fluid. For this purpose it is necessary, that 
 the values of u 1 and v 1 should be the same for all these 
 molecules; but the value IV is very nearly the same 
 for these molecules, as will be seen when we shall de- 
 termine in the sequel the forces from which this varia- 
 tion results ; it is therefore necessary that the variations 
 Sfp and ty' should be the same for all these molecules, 
 
 and moreover that the quantities 2wr.CT.sin. 2 0.j j, 
 
 and /z 2 r.sin. 2 O.{s' (s 1 )} may be neglected in the 
 preceding equation. 
 
 At the surface of the sea we have $=y, ag> being 
 the elevation of the surface of the sea above its surface 
 of level. Let us examine if the suppositions of (p equal 
 toy, and of y constant for all the molecules of air situ- 
 ated upon the same radius, can subsist with the equation 
 
276 LAPLACE'S MECHANICS. 
 
 of the continuity of the fluid. This equation, by 
 
 No. 35, is 
 
 n 2 $ 
 
 =>> \ 
 
 \, J 
 
 1*&\ 
 
 from which we may obtain 
 
 /-[-$' is equal to the value of r of the surface of level 
 which corresponds to the angles Q-\-&u and t3-\-av, plus 
 the elevation of the molecule of air above this sur- 
 face; the part of as' which depends upon the variation 
 
 of the angles and is being of the order , may be 
 
 neglected in the preceding expression of^', and con- 
 sequently it may be supposed in this expression that 
 
 V -<p ; lastly if we make <p=y, we shall have ( - 1 
 
 zznO, because the value of 9 is then the same relative to 
 all the molecules situated upon the same radius. More- 
 
 n* 
 
 over y is by what precedes of the order /or ; the 
 
 
 
 expression of y will thus become 
 
 u'.cos.Q 
 
 therefore u 1 and ^ being the same for all the molecules 
 situated originally upon the same radius, the value of 
 y 1 will be the same for all these molecules. Moreover 
 it is evident from what has been said, that the quantities 
 
 - and rc 2 r.sin. 2 O .{*' (*')}, may 
 
 be neglected in the preceding equation of the motion of 
 the atmosphere, which can then be satisfied by sup- 
 
LAPLACES MECHANICS. 
 
 posing thai it' and ' are the same for all the molecules 
 of air situated originally upon the same radius : the 
 supposition that all these molecules remain constantly 
 upon the same radius during the oscillations of the fluid, 
 is therefore admissable with the equations of the motion 
 and of the continuity of the atmospheric fluid. In this 
 case the oscillations of divers lamina of level are the 
 same, and may be determined by means of the equations 
 
 -fr'.fo 
 
 > 
 
 \ 
 
 These oscillations of the atmosphere ought to produce 
 analogous oscillations in the altitudes of the barometer. 
 To determine these by means of the first, let us suppose 
 a barometer fixed at any height whatever above the 
 surface of the sea. The altitude of the mercury is pro- 
 portional to the pressure which its surface exposed to 
 that of the air experiences; it may therefore be repre- 
 sented by Ig.p ; but this surface is successively exposed 
 to the action of different laminae of level which elevate 
 and lower themselves like the surface of the sea; thus 
 the value of p at the surface of the mercury varies; 
 first, because it appertains to a lamina of level which 
 in the state of equilibrium was less elevated- by the 
 quantity ay ; secondly, because the density of a lamina 
 
 is augmented in the state of motion by ap r or by j-~. 
 Jn consequence of the first cause the variation of p is 
 
578 LAPLACE'S MECHANICS. 
 
 ^-( T" ) or ~~^~5 the ** a l variation of the den- 
 sity p at the surface of the mercury is therefore a,(p). 
 
 y-\-y r 
 
 7 . It follows from the above, that if the altitude 
 
 of the mercury in the barometer at the state of equilib- 
 rium is denoted by "A: ; its oscillations in the state of 
 
 motion will be expressed by the function ~~T~ L * 
 
 they are therefore similar at all heights above the sur- 
 iace of the sea, and proportional to the altitudes of the 
 barometer. 
 
 It now only remains in order to determine the oscilla- 
 tions of the sea and of the atmosphere, to know the 
 forces which act upon these two fluid masses and to in- 
 tegrate the preceding differential equations ; which will 
 be done in the fouilh book of this work. 
 
LAPLACE'S MECHANICS. 
 
 CHAP. IX*. 
 
 Of the law of universal gravity obtained from 
 phenomena* 
 
 38. AFTER having developed the laws of motion, we 
 will proceed to derive from them and from those of the 
 celestial motions presented in detail in the work entitled, 
 Exposition du Systeme du Monde, the general law of 
 these motions. Of all the phenomena that which seems 
 to be the most proper to discover this law, is the elliptic 
 motion of the planets and of cornets about the sun : let 
 us see what may be derived from it. For this purpose, 
 let x and y represent the rectangular co-ordinates of a 
 planet in the plane of its orbit, having their origin at 
 the centre of the sun ; also let P and Q denote the forces 
 
 * This chapter which forms part of the first chapter of the 
 second book of the Mechanique Celeste, is added in order to 
 afford the reader some idea of the manner in which Laplace 
 applies the rules given in the introductory treatise. 
 
280 LAPLACE'S MECHANICS, 
 
 \vhich act upon this planet parallel to the axes of x and 
 y^ during its relative motion about the sun, these forces 
 being supposed to tend towards the origin of the co- 
 ordinates, lastly, let dt represent the element of the 
 time which we will regard as constant ; we shall 
 have by Chap. 2, 
 
 r72v, 
 
 (0 
 
 (2) 
 
 If we add the first of these equations multiplied by y 
 to the second multiplied by x 9 the following equal ion 
 will be obtained, 
 
 It is evident that xdyydx is equal to twice the area 
 which the radius vector of the planet describes about 
 the sun during the instant dt ; by the first law of Kepler 
 this area is proportional to the element of the time, we 
 shall therefore have 
 
 c being a constant quantity ; the differential of the first 
 member of this equation is equal to nothing, conse- 
 quently 
 
 xQ ?/P=.0. 
 
 It follows from this equation tliat^P has to Q the 
 same ratio as x has to y, and that their resultant passes 
 by the origin of the co-ordinates ; that is by the centre 
 of the sun. This is otherways evident, for the curve de- 
 scribed by the planet is concave towards the sun, con- 
 sequently the force which causes it to be described 
 tends towards that star. 
 
 The law of the proportionality of the areas to the 
 times employed to describe them, therefore conduct? 
 
LAPLACE'S MECHANICS. 2S 
 
 us to this remarkable result ; that the force which soli- 
 cits the planets and the comets is directed towards the 
 centre of the sun. 
 
 39. Let us now determine the law by which the force 
 acts at different distances from this star. It is evident 
 that the planets and the comets alternately approach to 
 and recede from the sun, during each revolution ; the 
 nature of the elliptic motion ought to conduct us to this 
 law. For which purpose resuming the differential 
 equations (1) and (2) of the preceding No., if we add 
 the first multiplied by dx to the second multiplied by 
 dy^ we shall have 
 
 which gives by integration 
 
 the constant quantity being indicated by the sign of 
 integration. If we substitute instead of dt its value 
 
 - - , which is given by the law of u the propor. 
 tionality of the areas to the times, we shall have 
 
 ^ 
 
 l 
 
 For greater simplicity let the co-ordinates x and y 
 be transformed into a radius vector and a traversed angle 
 conformably to astronomical practice. Let r represent 
 the radius drawn from the centre of the sun to that of 
 the planet, or its radius vector, and v the angle which 
 it forms with the axis of .r ; we shall then have 
 
 consequently 
 
2S2 LAPLACE'S MECHANICS. 
 
 If the principal force that acts upon the planet be 
 denoted by <p ; the preceding No. will give 
 
 therefore 
 
 by substitution we shall have 
 
 consequently 
 
 cdr 
 Hn -_ (<% \ 
 
 rVc*Zr 2 fydr ' 
 
 This equation will give by means of quadratures th 
 value of V in r, when the force $ is a known function of 
 r; but this force being unknown, if the nature of the 
 curve which it causes to be described is given, by 
 differentiating the preceding expression of 2/<pc?r, we 
 shall have to determine <p the equation 
 
 dr* 
 
 c/r 
 
 The orbifs of the planets are ellipses, having the 
 centre of the sun at one of the foci ; if in the ellipse w 
 denotes the angle which the major axis makes with the 
 axis of .r, and the origin of x be fixed at the focus, 
 and a represent the semi- major axis, and e the ratio of 
 the eccentricity to the semi-major axis; we shall have 
 
 which equaf ion belongs to a parabola if <r=1, and a be 
 infinite; and to an hyperbola if e surpass unity, and a 
 fee negative. This equation gives 
 dr- 2 
 
LAPLACE'S MECHANICS. 283 
 
 consequently 
 
 therefore, the orbits of the planets being conic sections, 
 the force p is inversely as the square of the distance of 
 th centre of these planets from that of the sun. 
 
 We also perceive that if the force <p is inversely as 
 
 the square of the distance, or expressed by , h being 
 
 a constant coefficient, the preceding equation of conic 
 sections will satisfy the differential equation (k) 
 between r and xi, which gives the expression of <p when 
 
 h c* 
 we change <p into , We shall then have h=i r 
 
 which forms a conditional equation between the two 
 constant quantities a and e of the equation of conic 
 sections ; the three constant quantities , e, and 73 of 
 this equation are therefore reduced to two distinct con- 
 stant quantities, and as the differential equation be- 
 tween r and v is only of the second order, the finite 
 equation of conic sections is the the complete integral 
 
 From the above it follows, that if the curve described 
 be a conic section the force is in the inverse ratio of 
 the square of the distance, and reciprocally if the force 
 be inversely as the square of the distance, the curve 
 described is a conic section. 
 
 40. The intensity of the force (p relative to each pla- 
 net and to each comet depends upon the coefficient 
 
 c 2 
 
 - ; the laws of Kepler likewise give ihej means 
 
 (i e ) 
 
 of determining it. Thus, if T denote the time of the 
 revolution of a planet, the area that its radius vector 
 describes during this time being the surface of the pla- 
 netary ellipse is equal to <araVlt% it being the ratio 
 
LAPLACE'S MECHANICS. 
 
 of the serai-circumference to the radius ; but by what 
 precedes, the area described during the instant dt is - 1 
 cdt : the law of the proportionality of the areas to 
 the times will therefore give (he following proportion, 
 
 icdtl va\ 1 e: Idt: T -, 
 
 therefore 
 
 e 
 
 T" 
 
 Relative to the planets, the law of Kepler, that the 
 squares cf the times of their revolutions are as the cubes 
 of the great axes of their ellipses, gives T^ssJ 9 ^, k 
 being the same for all the planets ; we therefore have 
 
 2flf 1 (?) is the parameter of the orbit, and in different 
 orbits the values of c are as the areas traced by the radii 
 vectores in equal times ; these areas are therefore as the 
 square root of the parameters of the orbits. 
 
 This proportion has equally place relative to the or- 
 bits of comets compared either to each other or to the 
 orbits of the planets; this is one of the fundamental 
 points of their theory which answers so exactly to all 
 i heir observed motions. The major axes of their orbits 
 and the times of their revolutions being unknown, we 
 calculate the motions of these stars in a parabolic orbit, 
 and, expressing by D their perihelion distance, we 
 
 suppose c=. - ; which is equivalent to making 
 
 i 
 equal to unity, and a infinite, in the preceding ex- 
 
 pression of c; we have therefore relative to comets, 2 2 
 r= a a% so that when their revolutions shall be known, 
 the major axes of their orbits can be determined. 
 
LAPLACE'S MECHANICS. 
 
 The expression of c gives 
 c* 
 
 a(l**) k* f 
 we have therefore 
 
 The coefficient ~ being the same for all the planets 
 
 K 
 
 and comets, it results that for each of these bodies, 
 the force p is inversely as the square of the distances 
 from the centre of the sun, and only varies from one 
 body to another by reason of these distances ; from 
 which it follows, that it is the same for all these bodies 
 supposed at equal distances from the sun. 
 
 We are therefore conducted by the beautiful law* of 
 Kepler to regard the centre of the sun as the focus of 
 an attractive force which extends itself infinitely in all 
 directions, decreasing in the ratio of the squares of the 
 distances. The law of the proportionality of the areas 
 described, by the radii vectores to the times employed in 
 describing them, proves to us that the principal force 
 which solicits the planets and the comets is constantly 
 directed towards the centre of the sun; the ellipticitjr 
 of the planetary orbits, and the very nearly parabolical 
 motions of the comets, shew that for each planet and 
 lor each comet this force is inversely as the square of 
 the distance of these stars from the sun; lastly, from 
 the law of the proportionality of the squares of the 
 times of the revolutions to the cubes of the major axes 
 of the orbits, or that of the proportionality of the 
 areas described during the same time by the radii vec- 
 tores in different orbits to the square roots of the para- 
 meters of these orbits, which law contains the preceding 
 and is ext;>nc{?M.l to comets : it results, that this force if 
 
LAPLACE'S MECHANICS. 
 
 tbe same for all the planets and the comets placed at 
 equal distances from the sun, so that in this case, these 
 bodies would be precipitated towards it with the same 
 Telocity. 
 
 Printed by H, Baroett, Long Row, Market-place, Nottingham. 
 
ERRATA. 
 
 Page 11. Notes line 3, for, direc tion "by S. $c. read, 
 
 direction by lines denoted by S. Sfc. 
 Page 56. Line 1, for, dx, read, d*z. 
 Page 135. Notes line 4 from the bottom, instead o', 
 
 to on* half that, read, to that. 
 
LOAN DEPT. 
 
 i . - |v 
 
 . 
 
 MAR 1 8 1980 
 
 ore. 5 
 
 FB 3 1983 
 
 LD 2lA-60m-3 '65 
 (F2336slO)476B 
 
 . General Library 
 
 University of California 
 
 Berkeley 
 
U.C.BERKELEY LIBRARIES