156 The Rev. 8. Haveuton on the Equilibrium and 
p= Az ag cos’a — °° sp “2 “cos! *y-+ u cosBcosy + vcosacosy-+- weosueosp (6) 
Substituting this value of p’ in (5), we obtain 
_ ‘ dG ae 
= SG Grn(E cet Mente + Beaty + 
ucosf cosy -+- vcosa cosy + wcosa cosp p’sinddpdédo; 
or, since, 
cosa = cospsiné, cosB = sing siné@, cosy = cosé, 
dy d 
eee a bese + bw bey EN, 
1. €. V, is a homogeneous function of the first order of the six quantities, 
dé dn d 
da’ d ay 7° Boel 
where 
A, = SSS F,. cos*psin°6 p>dpdodd, D, = SSS F,.sindsin*é cos p>dpdédd, 
B, = Si) r,.sin’psin®d p*dpdod@, E, = §§§ F,. cospsin°é cos6 p'dpdéd@, 
Cc, = Si r,.cos*0siné.p’>dpdod@, F, = SSS F,.sin¢cosd.sin’é p*dpdod@; 
but, since in fluids, the structure is not crystalline, F, will not be a function of 
(0, @), but only of p; and by integrating twice with respect to @ and ¢, we 
obtain, 
DE 0, 
AT, aie 
Ay = BSC, =p=5) FE, padp; 
hence we deduce, finally, for the value of v,, the expression 
pe Ue Oe tC 
“= P(Get +a te). sy 
Substituting this expression for v, in the general equation of fluids (5), we obtain 
by the calculus of variations, the following result, 
SS (x85 ++ ven + 28¢)dm = SS peed. "SS + §§ pindxdz + SS pegdady 
1 
SiN (feet +4 on +2 ee da dy dz, (8) 
ay 
