PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 253 
This suggests a quaternion equation such as 
q = {/q- xf (/ + yf (/ + zf e = / {(/ + ,r) {f + y) [f z)] e . ( 162 ), 
where e is some constant quaternion, as equivalent to the equation of a system of 
generalized confocals 
^2' (/+ q = 0.(163). 
On substitution in the scalar from the quaternion equation the result is 
Se(/+?/)(/+2)e = 0.(164), 
and y and z disappear, provided e is chosen to be one of the eight points satisfying 
= Se/e = ^ef^e =0.(165). 
Thus e is one of the intersections of three known quadrics. 
It is not necessary to dwell on Hamilton’s theory of the umbilicar generatrices, as 
the subject will be resumed in an extended form."^' Accordingly it is sufficient to 
mark that the equation of such a generator is 
= (/ + y) if + = + if + - ■ (166), 
where y is varialile; and the form of this equation shows that when x varies the 
generator sweeps out the developable of which the cusjiidal edge is the curve 
^ = (/+^)*e .(167). 
41. More generally, starting from any two quadrics, 
= 6.(ihs); 
the equation of the system of quadrics inscribed to their common circumscribing 
developable (compare Art. 11) is 
% (/r^ + y = ^ .( 16 i>)- 
This by the principles of Art. 38 may be replaced by 
ififrVy + = 0 .( 176 ); 
and on comparison with (163) and (162) it is manifestly equivalent to the quaternion 
equation 
or, by an application of (158), to 
2 = (/ry; + »=)‘(/rA+</)M/r’/. + U*« .... (m), 
* Compare Arts. 41 and 71. 
