150 PROCEEDINGS OF THE AMERICAN ACADEMY. 



sirable in cases where the analytical processes become too complex, 

 to determine graphically the forms of lines of force or flow due to 

 a combination of two simple elements. This note discusses briefly 

 the conditions under which the ordinary method of procedure is 

 possible. 



Let (a, yS) and (X, /x) be two pairs of orthogonal functions of the 

 two variables (.r, y), so that 



dx dx dy dy ' 



ax a^ _^ ax . a^ ^ ^ _ (2) 



dx dx dy dy ' 



then if (a + X, /? + i^) are to form an orthogonal pair, the equation 



/aa ^ axwa^ ^ a^N fda ^ ^W^ _^ ^\ ^ (3^ 



\dx dx] \dx dx) \dy dy J \dy dy J 



must be identically satisfied, Since (1) and (2) are true, (3) takes 

 the form 



\dx dx dy dy J \dx dx dy dy) 



If /^„, h^, h)^, h^ represent the values of the gradients of a, /8, X, /x, 

 and if the angle at any point between the directions in which X and /3 

 increase most rapidly be denoted by [X, /S], (4) becomes 



hi,- h^- cos [X, P\ + h^ ■ h^ • cos [a, fj] = 0. (5) 



Whatever the sequence of the directions of the gradient vectors 

 might be, the two angles which appear in (5) would be either equal 

 or supplementary, and their cosines would be ecjual in absolute value, 

 but the gradients themselves are intrinsically positive and the sequences 

 must therefore be such that 



hjh^ = hjh^. (0) 



Suppose that in the case of two given pairs of orthogonal functions 

 (s A) {\ H-)i the necessary condition (6) is satisfied, and that the 



