Motions of a System of Bodies. 45 



If the bodies are affected by no forces but their mutual actions, 

 and forces directed towards the origin of the coordinates ; then in 

 the use of (7), their right hand members are each = 0; .'.dSmc = 0, 

 dSm,c=0, dSm„c=^0, whose integrals are Smc=A, Sm/=/A, 

 Sm^/;=„A, (11); A, ,k, „A, being the arbitrary constants, also the 

 same results are true in the use of (8). (11) are evidently the same 

 that they would be if the bodies did not act upon each other ; but in 

 this case each of them would manifestly describe a plane curve 

 around the center of force situated at the origin of the coordinates; 



put^=the area described by the radius vector of m in a unit of 



time, -^=the area described by the radius vector of m' in the same 



time and so on ; let a, b, c, a', I', c', &fc. denote the angles which 

 the first, second, &ic. of th^se planes make with the planes x, y, x, 

 z, 3^, 2r, severally ; then evidently wzD cos. a + ??iD' cos. a'+ &£C. = A, 

 wjD cos. 6+m'D' COS. h' -\- Stc. =/A, mD cos. c+w'D' cos. c'+&ic. 

 =//A, (12); it is also evident that equations analogous to (12) will 

 exist for any other rectangular coordinates, ,x, ,y, ,z, whose origin is 

 the same as that of x, y,z ; for the position of the coordinates is ar- 

 bitrary, although they are to be considered as fixed during the mo- 

 tion of the system: hence mD cos. /«+m''D'cos. X+ &c. =B, 

 mD COS. ,b-{-m'D' COS. ,b'-{- he. =/B, mDcos. ,c-\-m'D' cos.,c'-\- 

 hc. =„B, (13); where B, ,B, ^^B, are what A, ^A, ^^A, become 

 when the planes, x, y, x, z, y, z, are changed to the planes ,x, 

 fy, ,x, ,z, ,yi./Z, severally, and ,a^ ,b, ,c, ^c. are what a, b, c, fyc. 

 become respectively. Let I, m, n denote the angles made by the 

 plane ,x, y, with the planes x, y, x, z, y, z; and V, m', n' the cor- 

 responding angles for the plane ,x, ,z ; also I", m", n" those for the 

 plane ,y, ,z, •'. cos. ,a= cos. a cos. Z+ cos. icos. to+ cos. c cos. 

 w, COS. ,a' = cos. a' cos. l-\- cos. b' cos,. m-{- cos C cos. w, ^c. ; hence 

 by (12) and (13) Acos. i+^Acos. w-fz/Acos. n = B, also A cos. 

 Z'+/A cos. m' + //A COS. w' = zB, A cos. l"-\-,A cos. m''-{-„A cos. 

 n'''=//B, (14). By adding the squares of (14) there results A^+zA^ 

 +//A2=B-+,B2-f„B% (15); since cos. ^Z+ cos. -l'+ cos. H''= 

 1, cos. ~m-{- cos. ^m'-t" cos. -m''=l, cos. ^w+ cos. ^n'-\- cos. ^n'^ 

 = 1, cos. Zcos. m-\- COS. V cos. m'+ cos. I" cos. w" = 0, he. ; now 

 since the position of the plane ,x, ,y is arbitrary let it be so assumed 



