374 GENERAL PROPERTIES OF CERTAIN PARTIAL DIFFERENTIAL EQUATIONS 



and write 



ir * de * J_ ^ de * ^ dd » 



K - *, ,,,, + <I> 2 r/ - +.... + *„_, g£ • 



Multiply each of these equations by A u , A n , . . . . A ln respectively and add: this 

 gives 



A u A\ + A l2 Iu + .... + A ln K„ = (51) 



a necessary condition, as is easily seen; for if we multiply each iTas given in (49) 

 by the minor A having the same member for its second subscript and add, there 

 results 



A n K 1 + A 12 K 2 + .... + A ln K n = % £A U A lk ( jgf - =£*) . (52) 



An interchange of.y with k in this should not alter the value of the right-hand 

 member of the equation, but it obviously does alter it to the extent of changing its 

 sign, and the quantity is therefore equal zero, or 



A n K x + A-a K 2 + . . . . A ln K„ = 0. 



For the determination of the functions * we may proceed as follows. Denote the 

 minors of A lh with respect to the element in the i + 1 th row and k tb column by A"), ; 

 then the quantity 



« = n — 1 M 



represents the determinant found from A a by changing the column of elements con- 

 taining differential coefficients with respect to x k into a column containing differential 

 coefficients with respect to Xj. In general then this determinant must vanish, as 

 there will exist in it two columns containing the derivatives of the functions 9 with 

 respect to Xj. If j = k the determinant becomes simply A lh . In case j = h the 

 determinant contains no derivatives with respect to Xj and for that case 



2 ^At,=-A ll: 

 1 = 1 j 

 since 



<1Aj _ _ dAk 

 d de L d <ie i - 



d,, 



k 



