174 The Rev. S. Haucuton on the Equilibrium and 
tion that they must take place along a given surface. Let the limiting surface 
F(% y 2) = 0, 
then the condition to be added to A = 0, 1s 
be 
1 1 
fee + Pa + Lig = 0; 
and, since (cosA, cosu, cosy) are proportional to (Z 7 +) we shall obtain, 
finally, for the conditions at the limits, 
(w 43 + mf Pe+(25 a G4 Pin + (a ae Foi ie= 0 
\ Nie ( dz du nk 
(33) 
Yee 4 Vin Ug = 
date t Gy + Gz 6 = © 
which, by eliminating any one of the variations, will give ultimately the ¢wo con- 
ditions to be satisfied. It is evident, a priori, that in this case there must be two 
conditions at the limits, as the equation of the external surface is a function of 
two independent variables. 
2nd. If particular forces (=, ®, ¥) act at the external surface, then the equa- 
tion of condition at the limits will be 
A = (§ (Ete + by + FE) 0, 
or, 
= = ncosA + dcosn + cosy 
® = dcosu + d5cosy + Jcosr (34) 
¥ = jcosy + w2cosA + dcosu 
which are the same as equations (25). 
3rd. If the limiting surface be perfectly free, then (&é, &, @¢) are independent 
of each other, and the general condition A = 0 will give the three conditions, 
ncosA + dcosu + w~cosy = 0 
acosu + dcosy + 5cosA = 0 (35) 
jycosy + pcosA+ 5cosu = O 
or, if w= f(a, y, z, t) be the external surface, 
