252 PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
38. Except in the case in luliich the primitive has loci of united points, the square 
roots are all commutative with one another and with the primitive, for they j^ossess a 
common system of united points.* 
Moreover, for a definite square root, 
(/+ (/+ yf = ((/+ x) {/+ y)f .(155), 
with liberty to change the order of the factors. This follows most easily by operating- 
on the united points. 
In general also, for any two functions and f.^, and a definite square root, 
because 
. . . . 
(y?/Ai-T =/i%/r* =/y/y/r‘ • 
and in particular a relation which is occasionally useful is 
(/rVs + = /r' (/r%/r* +«) A* • • 
(156) , 
(157) ; 
(158) . 
39. It is evident from the foregoing that the square roots of a function and of its 
conjugate are conjugate when they have the same latent roots. 
Thus we may write 
(/'-)'=P .(159), 
to signify that the conjugate of a square root is the corresponding square root of the 
conjugate function. 
In particular, taking the conjugate of (158), 
+ o' + tff- 
(160). 
SECTION VI. 
The SQ[rARE Hoot of a Function in e,elation to the Geometry of 
Quadrics. 
Page 
40. The quaternion equation of generalized confocals, q= d{{f+x) {f+y) {f+z)\e . . . 252 
41. The general case of quadrics inscril)ed to a common developable.253 
42. The quaternion equation (/= + ii^tersection of any two quadrics. . 254 
40. The transformation 
.(HH) 
Converts the quadric '^eqfq — 0 into the unit sphere Sp" = 0,/’ being a self-conjugate 
function. 
Compare ‘ Elements of (Quaternions,’ New Ed., vol. ii., A|)pendix, p. 364. 
