22 



VARIABLES AND FUNCTIONS. 



[INT. ii. 



virtue of continuity, when the distance MM' is infinitesimal, 

 V - V = A F is also. The ratio 



F'-F_AF 



MM' ' As 



is finite, and as M. M ' = As approaches 0, the direction of MM' 

 being given, the limit 



,. AF 9F 



lim -r = -5- 

 As=0 As 9s 



is defined as the derivative of V in the direction s. We may lay off 

 on a line through M in the direction of s a length MQ = ^- and 



p as we give s successively all possible 



directions, we may find the surface 

 that is the locus of Q. 



Let MN be the direction of the nor- 

 mal to the level surface at M, and let 

 v MP represent the derivative in that 

 direction. Let M' and N be the inter- 

 sections of the same neighbouring level surface, for which F= F', 

 with MQ and MP. Then 



AF _ AF MN 

 MNMM" 



MM' 

 As MM' approaches zero, we have 



r AF 9F 



lim -r = -=-, 

 s =o &s vs 



MN 

 MM' = COS 



Hence 



9F 9F D ,. n 

 - = -cosPifQ, 



that is, the derivative in any direction at any point is equal to the 

 ' {projection on that direction of the derivative in the direction of 

 (the normal to the level surface at that point. Accordingly all 

 | points Q lie on a sphere whose diameter is MP. 



The derivative in the direction of the normal to the level 

 surface was called by Lame'* the first differential parameter of the 

 function F, and since it has not only magnitude but direction, we 

 shall call it the vector differential parameter, or where no ambiguity 



* G. Lam6. Legons sur les coordonnees curvilignes et leurs diverses applications. 

 Paris, 1859, p. 6. 



