SIMILAR TO THOSE OF HYDRODYNAMICS. 375 



In all cases, then, when the above determinant does not vanish, it is equal to either 



dA 

 A lh or — A lk . Multiplying it then by -^ and summing for h and k, from h = 1 to 



/* = u and from k = 1 to k = u we have 



Now the function A^ may be written in the form 



/V=2^$-2.A^ (a) 



fc = l * ft=l 



This differs from the second member of the last equation merely in sign. This equa- 

 tion may be written in the form 



1 = 1 <■ k = \ A = l 



Now since . 



2 <h 2 2 i£ AfJ = - Kj. (p) 



"- i H | "^12 I | a - a -\n n 



the quantity in brackets can be written 



k = n $ t = n \ 



2^2 AhAfA (y) 



k = l Wi = l 



Now as we have for the determination of the functions * a set of equations of the 

 form 



Now observing the equations (a) and (/3) we have at once by comparing the coef- 

 ficients of ^r 



*•=-,?, 2^& (53) 



or, on taking (y) into account 



*, = - 2 f7 c 2 4*4T» (54) 



fe = l ' * (a = i ) 



Now since 



■^■lA — J <10 { 



d 



it is obvious that the quantity in brackets is the sum of the differential coefficients of 

 2 q with respect to ^ and therefore finally 



