16] VARIABLES AND FUNCTIONS. 23 



will result, simply the parameter, denoted by P or P F . The 

 above theorem may then be stated by saying that the derivative 

 in any direction is the projection of the vector parameter in that 

 direction. The theorem shows that the parameter gives the 

 direction of the fastest increase of the function V. 



If V is a function of a point-function q, V =/(#), its level 

 surfaces are those of q, and 



~dn ~ dq Sn~ dn' 



and if S ^ = h,P^f(q).h, 



where the sign -f is to be taken if V and q increase in the 

 same, if in opposite directions. 



Suppose now that V=f(q 1 , q z , q 3 ...... ) 



= __ + _ + 



ds 9^ ds dq 2 ds dq s ds 



and if hi, h z , ...... denote the parameters of q lt q 2 , ...... the above 



theorem gives 



P cos (Ps) = ^h 1 cos (hjs) + ^h 2 cos 



dV 



Now ^ hi is the parameter of F, considered as a function of 



qi, and we may call it the partial parameter Pi, and since Pi and hi 

 have the same sign if ^ > 0, opposite signs if ^ < 0, we have in 

 either case 



^ hi cos (^s) = Pi cos (Pi s), 

 and P cos (Ps) = P x cos (P^s) + P 2 cos (P 2 s) + ....... 



This formula holds for any direction s and shows that the 

 parameter P is the geometrical sum, or resultant, of the partial 

 parameters, 



Hence we have the rule for finding the parameter of any 

 function of several point-functions. If we know the parameters 



