380 vm - NEWTONIAN POTENTIAL FUNCTION. 



Transforming the other two integrals in like manner, 

 L cos (nn ) -f P 2 cos (nn 2 ) + P 3 cos ( 



_ / / { 



But this is equal to 

 But since 



multiplying and dividing the last integrand in 31) by \\li^ we find 

 that since the volume integrals are equal for any volume, the integrands 

 must he equal, or 



If the vector is lamellar, its projections are the partial parameters 

 according to &, &, & of its potential V ( 110), 



Pi = hi /S - ? Pj> = fas) Q - f P a = lln Q - ' 



1 8f X 2 ^?2 3 ^?8 



Equation 32) then becomes 

 33) 



This result for the value of z/F was given by Lame, by means 

 of a laborious direct transformation. The method here used is a 

 modification of one given by Jacobi and Somoff. 1 ) 



In order to prove Green's theorem in its general form, we remark 

 that from the nature of the mixed parameter of the two functions U 

 and F as a geometric product we have 



34) A ( U, F) = PfPf + PfP 2 F + Pf P 3 r 



,, dU dV . 19 dU dV , 72 dU dV 

 = M o h /*; o^ y h ">\ 



Forming the volume integral, and integrating the first term partially 

 according to q we obtain 



1) Lame, Journal de I'Ecole Poly technique , Cahier 23, p. 215, 1833; Legons 

 sur les Coordonnees curvilignes, II. Jacobi, Uber eine particular e Losung der 

 partiellen Differentialgleichung JV=Q, Crelle's Journal, Bd. 36, p. 113. Somoff, 

 Theoretische Mechanik, II. Teil, 51, 52. 



