DEMONSTRATION OF A GEOMETRICAL THEOREM OF JACOBIS. 
It only remains therefore to determine the values of l, m, n from the last equation 
but one. The condition which expresses that the first side of this equation divides 
itself into factors is easily reduced to 
ß 2 
T To. T H - 
a 2 + h b 2 + h c 2 + h 
Next, since the equation is identical, write 
.(A). 
a 2 + h ’ 
ß 
y b 2 + h’ 
7 
c 2 + h ’ 
we deduce 
la 
+ mß ^ ny 
a 2 -t h b 2 + h c 2 + h 
(B). 
Again, putting 
l 
a 2 + h 
the whole equation divides by 
V b 2 + h’ 
c 2 + h ’ 
+ 1 > 
IV mm' nn 
K a~ + h b 2 + h + c 2 + h 
a factor whose value is easily seen to be = — 1. And rejecting this, we have 
l 2 
fi 
rn“ 
+ vr 
a 2 + h b 2 + h c 2 + h 
= 0 
(C). 
Thus of the three equations (A), (B), and (C), the first determines h, and the remaining 
two give the ratios l : m : n. It is obvious that 
a 2 + h b 2 + h ' c 2 + h 
is the equation of the surface confocal with the given surface which passes through 
the point (a, ¡3, y). The generating lines at this point are found by combining this 
equation with that of the tangent plane at the same point, viz. 
, fry , l z . 
a 2 + h b 2 + h c 2 + h ’ 
and since these two equations are satisfied by x=a + lr, y = /3 + mr, z — <y + nr, if l, m, n 
are determined by the equations above, it follows that the focal lines of the cone are 
the generating lines of the surface, the theorem which was to be demonstrated. It is 
needless to remark that of the three confocal surfaces, the hyperboloid of one sheet 
has alone real generating lines; this is as it should be, since a cone has six focal 
lines, of which four are always imaginary. 
y 
+ 
= 1