PEIRCE. — LINES OP CERTAIN PLANE VECTORS. 669 



c/A dfj. 



from this last equation it is evident that if the gradient of v vanishes, 

 v is either a function of x -f- y i or a function of a; — y i. 



It is often convenient in dealing with differential equations which 

 involve the gradients of functions, to use the independent variables of 

 equation (14) and we may note that u and v, two functions of A and /x, 

 are conjugate if, and only if, 



Qu . 9v 9u . Sv . . 



5a 5a 9/x dfx. 



If u and v are orthogonal functions, 



9u 9v 9u 9v _ 

 3a 5/a cV 3a 



If the gradients of « and v, two real functions of x and y, are every- 

 where equal while the directions of their gradient vectors arc different, 



9(u — t;) 9(u + v) 9(u — v) 9(u + v) __ Q 

 9x 9x 9y 9y 



and the functions (?< — r) and (•« + v) are orthogonal. The converse 

 of this statement is true. If two orthogonal functions have equal gra- 

 dients these functions are conjugate. 



If the gradient vectors of two funptions have the same direction at 

 every point of the xy plane, one of these functions is expressible in terms 

 of the other. 



The quantities u = cos (b x — y), v = sin (by + x) illustrate the fact 

 that the gradient of each of two orthogonal functions may be expressible 

 in terms of the function itself. 



The quantities u = x 2 + y' 2 , v = tan -1 [ '-) illustrate the fact that 



the gradients of both of two orthogonal functions may be expressible in 

 terms of one of the functions. 



If the gradient of v, one of two orthogonal functions (u, v) is expressi- 

 ble in terms of u, or is constant, no other but a linear function of v has 

 a gradient expressible in terms of u. 



If the gradient of each of two orthogonal functions (it, v) is expressi- 

 ble as a product of a function of u and a function of v, so that 



