154 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



shall be positive. For a maximum the determinants must be alternately negative 



d 2 F 

 and positive, beginning with A = T-^ negative. 



7.230 Maxima and Minima with Conditions. If F(XI, x 2 , ...... , #) is to 



be made a maximum or minimum subject to the conditions, 



I. 



,Xz, .-. . . . ,X n ) =0 



where k<n, the necessary conditions are, 

 dF 



^ 



where the X's are & undetermined multipliers. The n equations (2) together with 

 the k equations of condition (i) furnish k + n equations to determine the k + n 

 quantities, xi, x 2 , ..... , x n , Xi, X 2 , ..... , X&. 



Example : 



To find the axes of the ellipsoid, referred to its center as origin, 



Denoting the radius vector to the surface by r, and its direction-cosines by 

 /, m, n, so that x = Ir, y = mr, z = nr, it is necessary to find the maxima and 

 minima of 



an + #22 w -f 

 subject to the condition 



$(/, m, n) = I 2 + w 2 + n 2 - i = o. 

 This is the same as finding the minima and maxima of 



F(l, m, n) = dul 2 + 

 Equation (2) gives: 



(#11 + X)/ + #1 2 W + #13^ = O, 

 Gut + (#22 + X)W + #23^ = O, 

 #13^ + #23W + (#33 + X) = O. 



Multiplying these 3 equations by /, m, n respectively and adding, 



