178 The Rev. S. Haucuton on the Equilibrium and 
the corresponding radii vectores, and (a, B, y, a’, B’, y’) the directions of (7, 7’). 
cos(p,r’) _ scl de r) 
This expression for 
pr 
“, in terms of (a, B, y; a’, B’, 7’), may 
be proved as follows : the tangent plane to the surface is 
(aw + FY + Ez)a’ + (By + vz+ Frax)y'4+ (ce + Er + dy) = 1; 
therefore, if (A, 4, v) denote the direction of the perpendicular, we shall have 
cosA = pr(Acosa + FcosB + Ecosy) 
cosu = pr(BcosB + Dcosy + Fcosa) 
cosy = pr(ccosy + Ecosa + Dcosf) 
therefore 
cosA cosa’ + cospcosp’ + cosvcosy’ _ cosd’ cosa + cosy’ cos + cosy’ cosy 
pr si pe 
= acosacosa’ + BeosBcosp’ + ccosycosy’ 
+ p(cospcosy’ +cosycosp’ ) +5 ( cosy cosa’ +cosacosy’ )-+-F(cosacosp’ +-cospcosa’ ) ; 
which is equivalent to (41). 
If now we substitute for (&, 7, ¢) their values, 
€=7cosa.cosp, = TCosB.cosd, ¢ = TCOSy.Ccosd, 
o= = +my + nz—vrt) 
in the equations 
apt wien dy 
Tana ee a+ud oP a0 4 av + aw 
if male tet a fi arte ; 
GE yi iartriag 1B iw S + Bu + Bev + Bw 
Tk. Seen G dé dy 2 
qe ae ie Se Ca + RV +73" 
df aes dé dy dg 
i eS nae tu + wv + BW 
rg Um dy d¢ 
Te Ss 2 thay te + 7.0 + MV + aw 
ah dy 
sp —) 
= Tet eg ty 27s of + Bu + ay + aM 
