164 
Proceedings of Royal Society of Edinburgh. [sess. 
its separate areas are superposed) one of tlie minima. This mini- 
24 
mum is surrounded by a detached part of u 2 = ~ , while the rest 
surrounds the other two minima (u 2 — 1); and the double points of 
these contours are the minimaxes. 
A general idea of their forms may be gathered from their ortho- 
gonal projections on the principal planes, as shown in figs. 1, 2, 3 
of the Plate. These projections are curves of the 4tli order : — hut 
g 
u 2 = - (dashed) splits into two equal ellipses on the xy plane, and 
24 
hyperbolas on that of xz ; while u 2 = — (dotted) gives ellipses on 
yz and hyperbolas on xz. Pig. 4 gives, on a fourfold scale, the 
region near the z pole of the projection on yz , of which the details 
cannot be shown on the smaller figure. 
The curves were traced from their equations. One example 
must suffice. Thus 
(3a 2 + 2 y 2 + z 2 ) 2 
$x 2 + 4 y 2 + z 2 
gives, eliminating z by the condition 
x 2 + y 2 + z 2 = 1 , 
(2x 2 + y 2 + l) 2 = ?(8^ 2 + 3y 2 + 1) or 
2a; 2 + 2/ 2 -^) = ^x 2 , 
The forms of these curves depend only on the ratios g 1 : g 9 : g 3 , 
so that I have appended fig. 5, in which we have 5 : 4 : 3, for com- 
parison with fig. 3 where we have 3:2:1. 
