668 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The Gradients of Functions op Two Independent Variables. 



Before we consider briefly some of the equations of condition which 

 have just been stated, it will be well to make a few simple statements 

 concerning the gradients * of functions of x and y. 



The gradient of a function may or may not be expressible in terms 

 of the function itself. The gradients of the expressions (x 2 -f y 2 ), 

 (x 2 — y 2 ) illustrate these two cases. 



If the gradient of a function v is equal tof(v), it is possible to form a 



-zT-r, the gradient of which is constant. 



A v ) 



If the gradient of a function v is equal to the constant a, it is possible 



to form two functions of v, namely — and - / f(v) dv, the gradients of 



which are equal, respectively, to the arbitrarily chosen constant b and to 

 the arbitrary function f(y). 



If the gradient of a function v is either constant or expressible in terms 

 of v, the gradient of any differentiate function of v is expressible as a 

 function of v. 



If h v is neither constant nor expressible in terms of v, no function of 

 v exists the gradient of which is expressible in terms of v. 



Since the gradients of two conjugate functions are numerically equal, 

 it is clear that if h v is expressible in terms of v, not all other functions 

 the gradients of which are functions of v, are themselves expressible in 

 terms of v. 



If, for x and y in the expression 



"-8)'+®' 



the quantities X = G (x, y), /x = H (x, y) be substituted, we shall obtain 

 the new expression 



v-^(»'+v(g) , + .fS'.£ + |.S)(S.g) 



(13) 

 and if we write A = (x + yi), /x = (x — y i), 



* Lame, Le?ons sur les coordonnees curvilignes, p. 6; Maxwell, Treatise on 

 Electricity and Magnetism, § 17. 



