40 
[562 
562] 
that is, 
and passing the 
562. 
so that we have 
[ADDITION TO MR. WALTON’S PAPER “ON A THEOREM IN 
MAXIMA AND MINIMA.’’] 
Hence, for the 
SQ = 0; that is 
at any such p( 
that is, there i 
surface z = P a 
angles to each 
(— a, b, affx, 8; 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. X. (1870), 
pp. 262, 263.] 
In what follows I write x, y, z in place of Mr Walton’s u, v, w: (so that if 
i = 1), as usual, we have 
f(x + iy) = P + iQ) : 
and I attend exclusively to the case where the second differential coefficients of P, Q 
do not vanish. 
There are not on the surface z = P any proper maxima or minima; but only level 
points, such as at the top of a pass: say there are not any summits or imits, but 
only cruxes; and moreover at any crux, the two crucial (or level) directions intersect 
at right angles. Every node of the curve Q = 0 is subjacent to a crux of the 
surface z = P; and moreover the two directions of the curve Q = 0 at the node are 
at right angles to each other; hence, considering the intersection of the surface z = P 
by the cylinder Q = 0, the path Q = 0 on the surface has a node at the crux ; or say 
there are at the crux two directions of the path ; these cross at right angles, and are 
consequently separated the one from the other by the crucial directions; that is to 
say, there is one path ascending, and another path descending, each way from the 
crux. And the complete statement is; that the elevation of the path is then only a 
maximum or minimum when the path passes through a crux; and that at any crux 
there are two paths, one ascending, the other descending, each way from the crux. 
The analytical demonstration is exceeding simple; we have 
C. IX.