552 M. OSTROGRADSKY ON A SINGULAR CASE OF 
As these must vanish in case of equilibrium, we have 
S @¥—yX)am = [ (xeos.u —ycos.d) pas 
J wt-2X)am = [ (ycos.7— 2 008.4)pas slisy (BA) 
Jf @X—a%) am = [ (200s. — 2 00s 4) pds. 
Now if we have an integral such as 
dP dQ, dR 
salieri eke Tatts Wy 
yo ee ie. 
P, Q, R being functions of x, y,z,,and dw a differential volume, and 
if this integral is to be taken in the extent of a volume V, we shall have, 
as is known, 
dP d dR 
Set : Tatas) deaf Peos. A+Q cos.u+ Reos. vyds. 
The latter integral is taken only for the surface of the volume. We 
shall have as a consequence of the foregoing formula 
d Saag) ¢ *>dP 
pds cos.A= da ¢” 
feces u- y cosa) pds ate ay ae) 
Sves. y—zcos.u) pds =f" (uF Fas “5 ,) de 
Sf eeos. A—acos. pata (ot all dw 
The equations of equilibrium (1) and (2) will become 
Sfxan =f Gets 
Jvinaf Faw 
dP 
StincS ae 
dP 
Jex- yX)dm=f (5, 7 Uae) a 
