198 
Proceedings of the Eoyal Society 
application in the same direction, and the following general process 
is applicable to all the problems I have seen treated by the authors 
above referred to. 
Let p = a + x/3 , 
where T/2 — 1 , 
be any ray of a complex ; and let the scalar condition determining 
a point on it be 
F( P ) = 0. 
This determines x in terms of a, ft ; and the number of independent 
scalar variables is reduced by two additional data, such as a relation 
between a and j3 ( i.e. } S.a/? = 0), or a relation among the values of 
x (i.e.j f(x lt x 2> . . . .) = 0), according to the nature of the complex. 
We have now to make Ta a maximum or minimum, subject to 
the additional condition that 
Ua = constant. 
This gives rise to three scalar equations which may be written 
S./3/? = 0, 
S . /hq = 0 , 
S . pv 2 = 0 , 
or, finally, 
S./ftqv 2 = 0, 
i.e., /3 is coplanar with iq, v 2 , which are usually normals to surfaces 
at the points of intersection with a ray of the complex. This is 
one of the chief points of Mannheim’s treatment of the subject. 
When, as is often the case, the surface on which the complex is 
made to depend is an ellipsoid 
S.p<£p = 1 ; 
the last written equation usually takes the form 
P + ycf>/3 + za = 0 , 
or 
P z(2/0 + l)"‘a, 
whence 
— 1 =2 2 S.a(y<£+ l)" 2 a. 
In this equation y and z depend upon Ta, so that the space-locus 
is closely connected with Tresnel’s wave-surface, whose equation is 
capable of a very remarkable series of transformations, depending 
on the properties of the expression 
S . a(<£ + g)~ 1 a . 
