MAXIMA AND MINIMA 
185 
the variables being connected by p equations : 
^ >(1) (^i, —, 0 = °, • • • , •••, o= o, 
is now obvious. The relations which correspond to (6) are : 
F k 
F^n—p+i 
• F 
n 
(8) 
<e P+ i • 
. 
^ n 
.. *(p) 
7. Conclusion; Critique. In the first case considered, § 5, it was 
tacitly assumed that the functions F (x, y, z) and $ (x, y, z) are con 
tinuous, together with their first partial derivatives, in the neighbor 
hood of a point («„, y 0 , z 0 ) whose coordinates satisfy equations (2), 
(5), and (6). But this is not enough. The equation (2) must deter 
mine such a functions of x and y that equations (3) can have a meaning. 
This will surely be the case if <f> 3 (a: 0 , 2/o> z 0 )^0. Moreover, this is 
also precisely the condition which we need in Lagrange’s method, in 
order that equation (9) may have a meaning. It is, of course, im 
material whether we solve equation (2) for z or for one of the other 
letters. We see, then, that Lagrange’s method will apply if at least 
one of the numbers <& k (x 0 , y 0 , z 0 ), k = 1, 2, 3, is different from 0. 
In § 6 the situation is similar. It is enough, ‘in addition to the 
continuity of the functions F, <i>, (together with that of their first 
partial derivatives) in the neighborhood of a point whose coordinates 
satisfy equations (2) and (6), that at least one of the two-rowed 
determinants 
* ; 
yp. 'I' ’ 
t 1 
where i and j are two distinct numbers chosen from the set 1, 2, 3, 
be different from zero. 
The extension to the general case is now obvious. At least one 
p-rowed determinant from the matrix made up of the last p rows of 
the determinant (8) must be different from zero; — at least, this is 
sufficient, in order that u be stationary. The student must have a 
firm hold on the theory of Linear Dependence; cf. Bocher, Algebra, 
Chaps. 3, 4.