152 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and, so far as the relation (12) is concerned, we may use either the 

 upper signs or the lower signs, but if (4) is to be identically satisfied, 

 the same sign must be used in (9) and (13) and the sign opposite to 

 this in (11) and (14). Equation (6), then, together with the proper 

 choice of sequence of directions for the gradient vectors which corre- 

 sponds to the convention with regard to signs just made, will ensure 

 the orthogonality of a + X, ^ + /a. For practical purposes, however, 

 it is well to approach the problem from another side. 



If (a, /3) and (A, /x) are given pairs of orthogonal functions, and if 

 we denote the given scalar point functions obtained by dividing dfi/dx 

 by da/dy, and by dividing dfx/dx by dX/dy, by t, and rj, the equations 

 (l) and (2) can be written in the forms 



and 



or 



and 



W%-Wi^=^ 





a^ ^ dy' dy ^ dx' ^ ^ 



dfi dX dfi dX 



d:v = '' dy' ry = -'^ dTr- ^^^^ 



If the values of the derivatives of ft and /a given in (17) and (18) 

 be substituted in (4) this equation becomes 



and if (a + X, /3 + /x) are to be orthogonal, a and X must be such as 

 to satisfy it. If A were expressible as a function of a, and /x as a 

 function of /8, the second factor would vanish, but this case is of no 

 practical interest and (10) demands in general that C and rj shall be 

 identical, so that 



