PEIRCE. — DERIVATIVE OF A SCALAR POINT FUNCTION. 343 



If u is the distance (r) to a point on a curve (s) from a fixed point 

 outside the curve, 



dr _ . \ d f l\ _ cos (s, r) 



^ = + cos (s,r), l(}) = - 



Any function of the complex variable (ax + by + izs/a? + b 2 ) has a 

 gradient identically equal to zero, but every differentiable real point 

 function has a gradient in general different from zero. The gradient of 

 a function may be constant throughout a region of space : if the 

 gradient of u is constant, the surfaces upon each of which u is constant 

 form a parallel system. If the gradient of a function, u, is either con- 

 stant or expressible in terms of u, any differentiable function of u has 

 a gradient either constant or expressible in terms of u. If the gradient 

 of u is expressible in terms of u alone [h u =f(u)], it is possible to form 



jj-\ , of u the gradient of which shall be constant. If 



h u is neither constant nor expressible in terms of ti, no function of u 

 exists the gradient of which is expressible in terms of u. The functions 

 u = sin (x + y + z), v = sin (x + 2y — 3z), w = sin (5x — £y — z) 

 illustrate the fact that the gradient of each of three orthogonal point 

 functions may be expressible in terms of the function itself. 



If the gradient of each of two orthogonal point functions, u, v, were 

 expressible as the product of a function of u and a function of v, so that 

 h u = U\ • Vi, and h v = 17% • V 2 , it would be possible to form two func- 

 tions / jy-, J -==■ J of u alone and of « alone, respectively, the 



gradient of each of which would be expressible in terms of the other. 

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

 every point of space, one of these functions is expressible in terms of 

 the other. If the gradients of two real functions, u, v, are everywhere 

 equal while the directions of their gradient vectors are different, 



d(u - + v) d(u — v) d(u + v) d(u — v) d(u + v) d(u — v) _ . . 

 dx dx dy dy dz dz ' 



and the functions [u + v], [tt — v] are orthogonal, as are F{u + v), 

 f{u — v), where F and /are any differentiable functions. If n and v 

 are orthogonal functions, the functions \_F(ii) +/(«)], [F(u) —f(v)] 

 have gradients numerically equal to each other at every point. 



Two scalar point functions, the level surfaces of which are neither 

 coincident nor orthogonal, may have gradients each of which is ex- 



