12* MAXWELL'S LAAY 375 



necting the variable magnitudes u, v, w with the given constants 

 E, a, b, c. We have then from the functional equation 



C = F(u } , Vi, wJFfa, v 2 , w 2 ) F(u N , V N , W N ), 



coupled with the conditions represented by the above four equa- 

 tions, to determine the function F(u, v, w). 



This is done by the known processes of the calculus of varia- 

 tions. If we represent a second system of values which satisfy 

 the equations by the symbols u n + lu M v n + $v nt w n + lw n , where n 

 may represent any integer between 1 and N, then we have also 



, v l + $v i , w i + $Wi) . . . F(u N + Su N , V N + Zv 

 + (Vi +fo l )* + (wi + 2wi) 2 +. . . 



+ (U N + ?O 2 + (% + ^) 2 + (W N + 2w N )*} 

 Na = UL + ciii + . . . + UN + CU N 

 Nb = Vi + $v l + . . . + VN 4- %Vy 

 Nc = Wi + $Wi + . . . +W N + 2w N . 



On subtracting the one system of equations from the other we 

 obtain five equations from which the constants C, E, a, b, c are 

 absent. In these equations we take the variations $u, 8v, w as 

 infinitely small, being justified in this if we choose both systems 

 of values of u t v, iv to be such as to differ infinitely little from 

 each other ; developing, then, the equations in powers of <>u, $v, hv, 

 and neglecting their higher powers, we obtain five equations whose 

 terms are all of the first order. 

 If for shortness we put 



STTT dF(u,,, v n , w n ) , dF(u n , v n , w n ) ? dF(u n , v n , w n ) . 



vJP n jsz- ^ 'VUJ n "T 1 ^ u n T j CW; HJ 



du n dv n dw n 



the first equation becomes 



which, on dividing by the product of all the functions F, we may 

 write in the form 



The single terms of this equation have the meaning 



i_F, = _1 (^ dV, s dF, h 

 F n F,: \du lt dv n dw, " 



