522 PROFESSOR CHRYSTAL ON 
(18) along with (4) gives 
fie gm 
. TG a ae =1, 
and we find as the condition that the dw» in (11) and (138) be the same 
al Be ae 
Pe vs ee =? 
which is obviously’ satisfied by any pair of points on the two focal conics. The 
system of rays is, therefore, doubly infinite. 
Returning to the alternatives in (10) and (12), we shall find them included 
in the general case. 
Consider all the points in the plane of xy not lying on the conic (8) or the 
axis of &. For them z=0 and the biquadratic has two roots =0, and two =o. 
The former case, nse (5) and the last of (2), gives \=0 p=O0, whence 
=0 =r 
ry 
sae Ze igen sian) Aeunseep ie ea 
The latter gives h = © p=o ee ie 
wo g+h” 
whence =O ais 
bp Woxmath : 
Me eg ‘ (15) 
In both these cases, therefore, we get rays lying in the plane xy and passing 
through points 7==tg on the axis of 4, 7.¢., passing through the vertices of the 
focal conic in the plane of xz. 
Similarly, we may shew that all rays passing through points in the plane of 
xz not lying on the focal conic, lie wholly on that plane and pass through the 
vertices of the focal conic in the plane of zy. 
We get, therefore, no addition to the rays intersecting the two conics. 
An examination of the rays passing through points on the axis of a, for 
which y=0 and z=0, so that (7) is apparently indeterminate, leads also to 
nothing new, for we get the axis of x itself and rays in the planes of xy and xz 
passing through the points v= +g= +h. 
Hence we are led to our original conclusion, that the single resultants in 
MINDING’s system consists solely of the congruency of lines intersecting the two 
focal conics (8) and (9). 
