Motion of a System of Bodies. AT 
the same for the parallel plane as for that of x, y, as is evident by 
supposing the system to be reduced orthographically to the planes. 
Now since the plane 2, y is either at rest or moves parallel to itself, 
its parallel plane will always be parallel to itself as the system moves: 
it is also manifest that the first members of (22) will each =0, rela- 
tive to any plane drawn through the same body (as before,) perpen- 
dicular to the parallel plane ; Mec. Cel. Vol. I, p. 63. 
Again, let, (for brevity,) any body of the system be indefinitely de- 
noted by m, z, y, z, being its rectangular codrdinates; also suppose 
(as in (d), (e), &c.) that 2’, y’, 2’, are the rectangular codrdinates of 
m, when referred to any other system of rectangular axes, which 
have the same origin as the axes of z, y, z; then denoting by a, , 
&c. the same things as in at) (e,) &c., we have by (d),7=ar’ + by’ 
fez’, y=a'r' Oy’ + C2’, z= ale’ + b"’y’+ c"z', (a): supposing 
ee e ° e Y ae 
the quantities in (a’) to be functions of the time, we have 7 
mums eae, ete Ue + bdy' + cdz' aaa ada’ +- a'da’ + y'db’ 1 z'de’ 
a dt dt heen dt 
a'da’+ b'dy’ te'de' dz _ da’ +-y'db" +-2'de” adr’ +b" dy’! + c'dz' 
dt deny dt rae ie 
(b’). Put cdb+c'db'+-c''db"=pdt, ade+-a'de' + a''de" = qdt, bda+- 
b’da’+-b"da"= rdt; then by (f) edb + c'db’+c’db" = — bde—b'de' 
—b'de"= pdt, ade+a'de'+a!de" = —cda—c'da' —c''da'= qdt, bda 
+6'da'+ b"da"” = —adb — a'db'—a'db" = rdt, (d'); by substituting 
the values of a, 6, &c. from (0) in (d’), we have sin. 9 sin. dd) — 
cos. pdi=pdt, a g sin. éd.p+sin. etal dp — cos. eel fe). 
Put gz’ Lyf eeL, rx’ —pz' + Sry, py’ ee ‘<N, Chit: 
then multiply (0’) by a, a’, a”, respectively, add the products, and 
dx+a'd ‘dz 
wehave by (f), (d),(f), "5" 
bdx--b'dy + b'dz cdz—+-e’ oy ‘de 
di =M, dt =N, (g’)5 multiply (g’) by 
=L, in like manner 
d. 
a, b, c severally, add the products, and we have “jaa to +N, 
dy dz 
and in like manner — =a Lb! M-+c'N, = =a"L+b'M+e'N, (f’). 
Vat 
: dz? +dy?+d 
By adding the squares of. (R’), we have by (f) sa eee sels 
dx? dy2 dz? 
+M?-+N?; hence by (f’), z as = (y? 4+ 2/2) p24 (22 
