PEIHCE. LINES OF CERTAIN PLANE VECTORS. 669 



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

 V is either a function of a; + y z or a function oi x — 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 X and y., 

 are conjugate if, and only if, 



9u . 9v 9u . dv 



r^. 





- iV • (15) 



c/A c/A 5/A C7X 



If II and V are orthogonal functions, 



5« 9v . 9u 9v 

 9i\ 9/M 9fji 5a 



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

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



^(^ — ^) S(u + v) ^ 9(u — v) ^ 9(u + r) ^ Q .j^. 



9x ' 9x c)y ' 9y 



and the functions (it — v) and (« + v) are orthogonal. The converse 

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

 dients these functions are conjugate. 



If the gradient vectors of two functions 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. 



Tlie quantities u = ur + y'', v = tan~^ ( - j 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 it, 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 (u, v) is expressi- 

 ble as a product of a I'unction of ic and a function of r, so that 



