154 



PROCEEDINGS OF TUB AMERICAN ACADEMY. 



0. (2G) 



If a and A happen to be harmonic, the elements of the last column 

 vanish and the equation is satisfied, as it should be. 



It is possible to factor the determinant, after it has been reduced, 

 and if 



a-a . ax _ a^ . aa d'a , ax _ a=^x , da 



dx' dx dx' dx dx ' dy dy dx ' dy dy' 



d'a dX 



dx ■ dy dx dx ' dy dx dy' dy dy' dy ' 



31 = 



N^^[V^(a) 



d'X 



da d'a dx 



a^X da 



ax 



dx 



-^(-(^))-£-|(-«)| 



ax 



(27) 



the condition of (2G) demands either that a and X satisfy the 

 equation 



n rt /In 



L 



M 



V\X) 



N 



= 0, 



(28) 



