48 Motion of a System of Bodies. 
+ 19 2 rat 12\ p02 9) rot 9 pilld 9 ae 9 y'dz'~2'dy' 
2!2\g? 4 (a!? +y'? \r?—2Qy'2! qr—22!z' pr —22'y' .pq-+ re 
2'dz’ —2'dz' a dy’ — y'dz' dz'* +-dy'? + dz? 
p4a( a )ete( ge 
ady — yd: 
Now (18) are easily changed to d. sm (| =dt. Sm(Qz— 
, dx — xdz\ dz —zd: 
Py), .Sn(- : i) EN asi (P2—Rz), 2.8m ("7 g ee dt. 
Sm(Ry — Qz), (23); put S(y'*+ 2'?)m = A, S(x'?+ 2’?)m = B, 
S(e'2+y'2)m=C, Sy'z’m=D, S2'z'm=H, Szx'y'm=F, (k’), also put 
Ap — Er — Fq =p’, Bg — Dr — Fp = q,, Cr — Dg — Ep=r’, 
xdy' —ydv\ Z'dz' —2'dz' ydz' — ul 
Sn : =) 2h, Sm( = ae ni oe 
C, (l’). By substituting the values of z and y from (a’) and those 
dx d 
of - a from (h'), in the first of (23), we shall have (since a, 8, 
&c., are the same for all the bodies;) d. [(c’—cb').Sm(Ny'— Mz’) 
+(a’'e—ac’) . Sm(Liz'— Nz’) + (ab’— ba’) .Sm(Mc'—Ly’')] = dt. 
Sm(Qzr —Py,) (24). 
Put dt.Sm(Qzr—Py)=dN", dt.Sm(Pz—Rz)=dN”, dt.Sm(Ry— 
Qz)=dN’, (m’); by substituting the values of L, M, N, from (f’), 
we have Sm(Ny' —Mz’)=p'+,C, Sm(Lz'— Nz’)=q'+ B, Sm( Mz’ 
— Ly')=r'+,A, (n’), by substituting these values, and those of bc’ — 
cb’, &c., from (gz), and using dN", (24) becomes d.[{a"( p’+,C)+ 
b’(q'+,B)+e'(r'+,A) ]=dN”, in like manner the second and third 
of (23) will give, d.[a'( p’+ C) + 6(q'+,B) + ¢(r'+ A) J=dN’, 
d. [a( p'+,C)+6(q¢'+,B)+e(r'+,A) ] = dN’, (25) ; the two last of 
these are easily derived from the first by making some very obvious 
changes in (24). 
By taking the differentials indicated in (25), then multiplying the 
resulting equations by a’, a’, a respectively, and cake the products, 
we shall have by (jf) and (d’), after dividing by dé, a a wal 
a’dN””’ 'dN” dN’ 
g(r’ +,A) —r(q/+,B)= a anions , and in a similar way 
igecas) wd" bd’ BAN’ 
oe =) +r + © ©) 5 pr A) "di = 
d.(r a c’dN’ +c'dN” + cdN’ 
if Ay Ds oly Bae ap oye di » (26) 
Put the right hand members of (26) equalto N,, N,,, N,,, respec- 
