50 Motion of a System of Bodies. 
‘d aes Ti , 
ting from (v’) in (n’) we have Sm . —= =| =p’ +,C=(by (s’),) 
dT ‘2'da' —x'dz! dT z'dy’ — y'dz' 
(Gp) Sule) =a B= | Zoi oh as 
dT 
+A=(); (w’), see Mec. Anal. Vol. 2. p. 364, &c. If the sys- 
tem is not affected by any forces except the mutual actions of the 
bodies which compose it, (whether the bodies are acted on by foices 
in the directions of straight lines drawn to the origin of the codér- 
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 
d ih 2d: = dz dz — d' 
members we have a io = NG —— = id A eed _ or 
— 
(29), A’ B’C’ being the arbitrary constants ; but it is evident from the 
ady — yd. 
method of obtaining (25), that aay =a" (p'+,C)+6"(q7'+,B)+ 
dT dT dT an 
er(r'+A) =(by (w)) a (7) 40"(T)te"(5-),-0"(ae 
8 ae ea zh) x0 dT 
b” i a +e ae in the same manner a i. + ( a +¢’ 
dT dT dT dT 
( ) =B’,a a ney a) +e 7 =C’, (30); by adding the squares 
of (30), we have by nla) a e yl os ae oe eae 
C’?=const. (31). 
If we suppose the forces to be as before, and besides that a, 6, c, 
&c. are invariable, then we shall have p’,q’,r’, each=0; .'.Sm 
/ ‘d ‘dd, fs ‘dz! ‘dz! un et 
eed =a, sn(* =) =B,-Sn(! : at) = 
\4 
at dt dt 
/C, and ,A, ,B, ,C, will each be constant; and we have a’”,C-+6”,B 
+e” A=A’, aC +6" B+c A=B’, a,C+b,B+c,A=C;, (32), hence 
,A?+,B2?+,C?=A?+B?+C”, (33); multiply (32) by a”, a’, a, 
respectively, add the products, and we have ,C=a/’A’+a’/B'+aC’, 
in like manner ,B=6/A’+0/B’/+8C’, ,A=c’’A/+c/B’+cCr (34). 
Now since the position of ae plane 2’, y/ is ue let it be so 
ae umed that .¢”%= 
V A’? +B? 4.6" TRO" Te ee 
/ 
eee ee a 
