Motion of a System of Bodies. 285 
given solid: then 1,9,¢,can be found so as to satisfy the equations 
Sz'y/dm=0, Sx’z/dm=0, Sy/2’dm=0, (p), S being the sign of j ‘integra- 
dz’ 
tion relative to the mass of the solid. Let (= ia), (aa ) de is the 
partial differential coéfficients of 2’, relative to J, 4, respaetivel’s 
then by (e), (0), (A), (Kk), 2’=cx+c'y+c'z=(z sin. L+y cos. L). 
d2ty:.% «4 
sin. +2 cos. 4, 2’ cos.9—y’ sin. p=2 cos. }—y sin, }= aus a 
sin. 6 
‘sin. oy cos. 9=(z siu. }+y cos, +) cos.é—z sin. (5), (q)3 
tS dz'\ cos.o /dz’\ * dz'\ sin.» {dz’ 
hence z’=sin. » a He (a) =COS. © (a) - ae %) 
neg sin. 2p ¢ [dz Iz'\2\— cos.29 /dz 
which give vy =55 en (es ) sitas:$, Oy a) i sin. 6 (F)- 
dz! sin.g (d.2/*\ | cos. /d.2’ COS. (“F 
(zi) #¥= 2 ( di )+5an a\ ar) allies dak OF ) 
Si ae Soh Wc ) 
ame ae 7 “fi (7). By assuming Sz’z’dm=0, Sy2/dm=0, the sec- 
'g 42 
ond and third of (r) give ( mona =0, {= A Seen =0, (, 
which are the conditions requisite to make Sz’?dm= to a maximum or 
minimum, ae : and { only to vary. By (q), 2/2 +y'?# +2! 
=r? +y? +2 .S(2/2 + y'2)dm = §L?2dm—Sz/2dm, but 
SL?dm=const..’ ae ‘yen as maximum when Sz’?dm =a 
minimum, and reciprocally ; but S (2’? +-y'* )dm=the moment of in- 
ertia relative to the axis of z’,..the second and third of (p) require 
(generally,) this moment to be a maximum or minimum. Put 
Sz?dm=g, Sy?dm=h, Sz*dm=k, Szydm=g’, Szzdm=h’, Syzdm 
=k’; then by (q) Sz/?dm=sin. ? 6 (g sin.? J+A cos. ? $+2g's sin. 
1 cos. L)+cos. ? ¢4+2 sin. é cos. 6 (h’ sin. } +’ cos. 1), (t). 
By (s), making the partial differential coéfficients of Sz/? dm rela- 
live to é and J separately =0; we have sin. é cos. 4 (g sin.?)+ 
heos.?.. + 2g’ sin.) cos.) — ) + (cos.2¢—sin.24).(h/ sin.) +k’ cos.) 
=0, ((g—h) sin. | cos. L+-g’(cos.?. —sin.?-L)) sin. 6+ (h’ cos. L — 
k’sin..L) cos. =0, (uw) 3 substituting the value of cos.é from the sec- 
ond of these in the first, [(g — —h) sin. J cos. b+" (cos. *-)—sin.?))] 
X[(g sin.?2 +h cos.?-) +20” sin. 1 cos. 1 —).(k’ sin. | —’ cos. 1) 
+((g —;h) sin. J cos. +8 (cos.?.) —sin.?)). aia ++ cos.)] 
