496 



Prof. Helmholtz on Integrals expressing Vortex-motion 



d*L d*L d*L . 

 dx* + % 2 + dz* ~ *' 





dm dm dm 



dx* + dy 2 + dz 2 V ' 



. . . . ( 



d 2 N d 2 N d 2 N 

 .<&* + df + dz*~ *' 



* ^ 



d*P ( d 2 P d*? 



dx* + rf y « + <fe* - u - J 





(5) 



The method of integrating these equations is known. L, M, N 

 are the potential functions of imaginary magnetic matter distri- 

 buted through the space S 2 with the densities 



2tt' 2tt' 2tt' 



P the potential of masses external to the space S. If we denot 

 by r the distance of a point a, b, c from x } y, z, and by f , rj , fr 

 the values of %, rj, £ at that point, we have 



M: 



N: 





da db dc, 



oa 



*? da ^ tffc, 



the integration extending to the whole space S 2 , and 

 P= \\i~dadbdc, 



where /; is an arbitrary function of a, b, c; and the integration 

 extends through all space exterior to S. The arbitrary function 

 k must be taken so as to satisfy the conditions at the bounding 

 surface, a problem whose difficulty resembles that of magnetic 

 and electric distribution. That the values of u, v, and w in (4) 

 satisfy the conditions in (1) 4 is proved by differentiation, with 

 attention to the fourth of equations (5). 



We also find by differentiation of (4), attending to the first 

 three of equations (5), that 



^ 1 ^ - 9£_ — (^h + ^M + —} 

 dz dy ~~ dx \dx dy dz /' 



^_ fk =2 d ( dL dM —) 



dx dz dy \dx dy dz / 



du_cfo_ d_(dL dM dW\ 



dx~~ dz \dx dy dz J' 



dy 



