Motion of a System of Bodies. 49 
tively and we have by (m’), N,==Sm((a’z a Cea 
(ay—a'r)R,) (0'); by (a’), and (s) a'z—a"'y=cy' — bz', ax —az 
=c'y'—b'z', ay— ag b"z'; .°-put cP+¢c'Q+c’R= R, bP-+- 
6'Q+b’"R=Q’ also, aP +a’ Qt0"R= P’, (p’), and we have N=Sm 
(R’y' — Q’z’), in the same way (by making some very obvious chan- 
ges in (0’),)we have N,=Sm(P’z'—R’z’), N_,=Sm(Q’2'—P'y’), (q’),. 
Multiply (2) by m, take the finite integrals relative to all the bodies 
dr? +dy? +-dz? 
of the system, put Sm( : — “ ) =2T =the living force of 
the system, and we have by (1’), (since p, q, r, are the same through- 
A B C 
out the system ;) peta i “i> : )- — Dgr— Epr — Fpq+,Cp 
d lo d 19 dz'2 
+ Bg+,Ar+ Sin( za = t a (r’); by taking the partial 
differential co-efficients of (r’) relative to p, g, r, we have by (J), 
dT dT ay 
laa ) =p’ +,C, S = +.B, iz =r'-+- A, (s’); by substitu- 
| aT 
ting from (s’) and (q’) in (26), they will be changed to d (5) + 
dt 
he ee te a( dT\ =| dT 
a(ae) ~r( aq) =SmRy-a2), a) +7() -P( Ge) =m 
dt 
i a 2 A ae 
(P’z-R’z’), d\ + +0( Gq mae =Sm(Q’z'— P’y'), (27) : also 
fdé: 
ee : dT 
substituting from (s’) in (25), they will be changed to d ((F) 4. 
vl) te) eee (lS) (vet) )mae 
a(« (Os >)+ +0(“) +e ae) =a (28). 
By : v'=ar+a'y +a"'z, y=be+b'y+6"z, 2’ =crteyte'z, 
da' 
(’);-".supposing a, 6, ¢, &c. tobe momentarily constant, we have —— Ga 
adz--a' adz+-a'dy+a''dz dy eo dy + + b"dz dz' _ eda + e'dy + edz 
dt CLS oa ide dt 
dx’ d dz 
(u’); hence, and by (g’), L==>, dp M= _ N=— de’ (v') ; substitu- 
Vou. XXVI.—No. 1. q 
