332 VIII. NEWTONIAN POTENTIAL FUNCTION. 



where the sign + is to be taken if V and q increase in the same, 

 - if in opposite directions. 



Suppose now that V = f(q l} q%, q 3 , ) 



d V d V q^ d V d q% d V 3 q s 



and if h , h 2 , . . . denote the parameters of q ly # 2 , . . . the above 

 theorem gives 



T> /T^ N 2V 1 /7 \ , 2F 7 n \ 



P cos (Ps) = o h* cos (his) + o ^9 cos (hs) H 



q l v q% 



dV 



Now + -o hi is the parameter of V, considered as a function 

 of fa, and we may call it the partial parameter P l} and since P/ 

 and hi have the same sign if -g > 0, opposite signs if g- < 0, we 

 have in either case 



-o hi cos (his) = Pi cos (PiS). 

 0% 



Pcos (Ps) = P! cos (P^) + P 2 cos (P 2 s) H 



This formula holds for am/ direction s and therefore shows that 

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

 parameters, 



T> ~p I -p I 



Thus we have the rule for finding the parameter of any function 

 of several point -functions. If we know the parameters h lt h%, . . . of 



the functions q l9 q 2 , . . . and the partial derivatives g > g > we 

 lay off the partial parameters 



Q Y*- 



in the directions h lf h 2 , . . . or their opposites, according as ^ > 0, 

 or the opposite, and find the resultant of P lf P 2 , . . . 



If the functions q lt q 2 , ... are three in number, and form an 

 orthogonal system, the equation 



gives for the modulus, or numerical value of the parameter 



Examples. (1) in 108. Let the distance of M in the given 

 direction from the plane be u. 4V=4u = ~- > where a is the 



cos a 

 angle between the given direction and the given plane. 



p_ 



An cos a cos a 



