50 Motion of a System of Bodies. 



y'dz'-z'dy' 

 ting from (v') in («') we have Sml -r J =^'4- / C = (by (s') t ) 



(dT\ l j z' dx' - x' dz>\ j. fdT\ ■ Ix'dy'-y'dx' , 



[T P h Sm [—dr-)=<i'+ B <={T q )> ^^af-H* 



j- 1 , (k/), see Mec- Anal. Vol. 2. p. 364, &c. If the sy*- 



tem is not affected by any forces except the mutual actions of the 

 bodies which compose it, (whether the bodies are acted on by foice$ 

 in the directions of straight lines drawn to the origin of the coor- 

 dinates or not,) then, by what has been before shown, the second 

 members of (23) will each=0,.\by taking the integrals of their first 



xdy—ydx zdz — xdz ydz—zdy 

 members we have -r- =A', ir =B', — -r. = C' 



(29), A' B' C being the arbitrary constants ; but it is evident from the 



xdv ydx 



method of obtaining (25), that — -jf— == a"(p'+ / C)+6"(9'+ / B)-h 



»v ,. , „ (dT\ (dT\ (dT\ /(TT, 



+ 



_ /dT\ (dT\ . . . /rfT\ idT\ 



dT\ ^ fdT\ JdT\ (dT x 



^) =B >l^j+M^j +c l^) =c ''( 3 °)5 b y addin s thes q uares 



C"=const. (31). 



If we suppose the forces to be as before, and besides that a, b, c 7 

 Uc. are invariable, then we shall have //, q', r', each = 0; .'.Sm 



tfdy'-ydx'\ lz'dx>-x>dz'\ rt ' [y'dz'-z'dy' 

 ^— )=,A, »m[ 3P— ]= /B , fa^-y-f 



,C, and ,A, ,B, ,0, will each be constant; and we have fl" / C + ft" / B 

 +«*A*A', a^C + ^B + c^B', fl / C+i / B + c < A=C, (32), hence 

 / A 2 + / B J -KC 2 =A' 2 +B' 3 +C' 2 , (33); multiply (32) by a", a', a, 

 respectively, add the products, and we have / C = a"A / +tt'B'+aC / , 

 in like manner JB^'A'+ft'B'+AC, ,A=c''A'+c'B'+cO (34). 

 Now since the position of the plane a?', y 7 is arbitrary, let it be so 



A' B' 



assumed that c"» -->-—= == £•■ — -g s, c'=— ;===-=■ =., c 



v^+B^+C"' a/B 



C 



-/A 



+ B' 2 +C' 2 =A / V.by(33), 



