176 The Rev. 8S. Haucuton on the Equilibrium and 
dered, we have at present nothing to do with equations (36) or (37), or with the 
triple integrals in (38), but only with the double integrals in equation (38), pro- 
duced, asin (19), by integrating the right hand member by parts, according to 
the rules of the Calculus of Variations. 
It is easy to see that the condition at the limits will be 
Kh =.= A= 0, 
A’, 4” arising from the two parts of (38), and having the form of A in equation 
(32). This condition, expressed at full length, is evidently equivalent to the 
equations 
(di , du GUN du du jp DSi. 2 
(Nig 1 iy og ee —(x de)? fg.’ © aa =0 
ar) = du lu Tu 
(3 oe , a “4 / - ad ! ("Gti ety oY by!’ = 0 
stash (eet ee 
But as the motion of the molecules which constitute the surface f(a, y,2) = 0 
is the same, whichever medium they are supposed to belong to, we shall have 
evidently #& =&’, 4 = 9’, ¢ = ¢"’; and, expressing these in the equations just 
found, and equating to zero the coefficients of the arbitrary variations, we shall 
obtain finally the three following conditions, to be satisfied at the surface which 
separates two contiguous bodies. 
" ty!) du du _ 
(NN + (3) dy + (1a'— mT = 
1 
(2 —3'\ Stee se iy FS Se Ee (39) 
/ ny Wt 1_ys/! = , ni du _ 
ea neies 7 a eel a= 
As an example of the formule, we shall suppose the bodies to be in contact 
along an indefinite plane, which we shall take for the plane (2, y). The surface 
u = f(a, Yy,z) =O becomes for this case z = 0, and consequently we shall have 
to introduce the conditions, 
