C. IX. 
6 
[562 
562] ON A THEOREM IN MAXIMA AND MINIMA. 41 
A THEOREM IN 
that is, 
dP dQ dQ_ dP 
dy dx ’ dy dx ’ 
and passing thence to the second differential coefficients, we may write 
dP _dQ _ L dP _ dQ _ ^ 
dx dy ’ dy dx 
d 2 P d 2 Q d 2 Q 
dx dy dx 2 dy 1 * 
d 2 Q d 2 P d 2 P b 
dx dy dx 2 dy 2 
so that we have 
8P = L8x + M8y, SQ = — MSx + L8y, 
8 2 P = (b, a, — b\8x, 8y) 2 , 8 2 Q = (— a, b, a][Sx, 8y) 2 . 
atics, vol. x. (1870), 
Hence, for the maximum or minimum elevation of the path, we have 0 = 8P, where 
L 2 + M 2 
8Q = 0 ; that is, 0 = —-y— 8x, and therefore L 2 4- M 2 = 0 ; that is, L = 0, M = 0 ; and 
is u, v, io: (so that if 
at any such point 8z = 0, that is, there is a crux of the surface z =P ; and 8Q = 0, 
that is, there is a node of the curve Q = 0. Moreover the crucial directions for the 
surface z = P are given by the equation (b, a, — b\8x, 8y) 2 = 0, or these are at right 
angles to each other ; and the nodal directions for the curve Q = 0 are given by 
(— a, b, a$8x, 8y) 2 = 0 ; or these are likewise at right angles to each other. 
sntial coefficients of P, Q 
r minima; but only level 
Ly summits or imits, but 
level) directions intersect 
icent to a crux of the 
Q = 0 at the node are 
tion of the surface z = P 
node at the crux ; or say 
s at right angles, and are 
1 directions; that is to 
ling, each way from the 
the path is then only a 
¡t; and that at any crux 
way from the crux.