552 M. OSTROGRADSKY ON A SINGULAR CASE OF 



As these must vanish in case of equilibrium, we have 



/ (xY — y X) dm =/ (x cos . [ji, — 1/ cos . >^) p d s 



I {yZ — zY) dm = / (y cos . v — 2 cos. i>.)pds . . (2.) 



/ (zX — xZ) dm = / (z cos.X — X cos ft.) pds. 

 Now if we have an integral such as 



J \dx ^ dy^ dz) 

 P, Q, R being functions of x, y, z, and t? w a differential volume, and 

 if this integral is to be taken in the extent of a volume V, we shall have, 

 as is known, 



^^l[:^^-d^)d^=J {^^os.X^Qicos.^^^cos.v)ds. 



The latter integral is taken only for the surface of the volume. We 

 shall have as a consequence of the foregoing formula 



/• , Pd^ 



pds cos . X = # -J — d CO 



/• , /^dP 



pds COS. I/, =1 -J— d w 



/• r'dF 



pdscos.v=f -j^duj 



J(xcosi^-ycosX)pds=J ("^ "J^ " ^ ^) '^ « 



J (y cos.y -z cos. ix.)pds=J (^ "^ " ^ ■^) '^ "» 



yp ( dV dV\ 

 {z cos. X — x cos. X)pds^= I \^~d7'~ ^7l^ I ^'^ 



The equations of equilibrium (1) and (2) will become 



//»rfP 

 Xdm = I -J— d w 



J^dra=J -j^d. 



/pdP , 

 Zdm-=- t -T^ d w 



P /V r/P d?\ , 



JixY--yX)dm=J (^r j^ - y j^) du. 



