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PROCEEDINGS OF THE AMERICAN ACADEMY. 



two other point functions, v, tv, are constant ; that is, along the line 



v — V P , W — Wp. If 



L = 



(8) 



and if R 2 = L 2 + M 2 + N 2 — which is equal to h v 2 • h w 2 , if v and w 

 are orthogonal — this direction is defined by the cosines L/R, M/R, 

 N/R, and the derivative required is 



1 / T du , n r du , „ du 



(9) 



If the maxima and minima of the function u =f(a, y, z) are to be 

 found under the condition that the functions v, w shall have given 

 numerical values, the derivative of u taken in the direction in which v 

 and iv are constant must be made to vanish. Thus, if 



u = x 2 + y 2 + z 2 , 



and if the conditions are 



xyz = c 3 and x + y = d, 



equation (9) yields immediately the required relation 



(•'■y + z 2 ) (y - x) = 0. 



When/' (u) is positive, the direction of the gradient vector off(n) 

 coincides with that of the gradient vector of u itself : these directions 

 are opposed when/' (ii) is negative. The tensors of both vectors are 

 always positive. If 



iv =/(«), k w 2 = [/' (u)] 2 • h u 2 , and cos (w, s) = cos (u, s) : 

 in particular, when 



w = 1/u, h w = h u /u 2 and cos (w, s) = — cos (u, s), 



so that 



ds \uj ds u 2 



