• 2 ( 3 :^ 
PROFESSOR C. J. JOEY OX QUATERXIOXS AXD PROJECTIVE GEOMETRY. 
The first quadric is converted into the second by the transformation (218) 
Fo~V/o*) ^his converts the first complex into 
s^).F„Vo r/,Vo 
( 228 ); 
and, on comparison with the second complex, it appear that we must have 
F(,-SF,Fo-* = f/Zo-y/o'y- or R = .... 
(229). 
where we have introduced two new symlDols for greater convenience. 
Equation (229) requires the functions and to have the same latent roots 
(Art. 23) ; or again, Fq“^F^ and i^^ust have the same latent roots, and this is 
the sufficient condition, for it appears, on substituting a united point of F^^ in (229), 
that the function g' must convert the united points of F^^ into those of ; and it 
is always possible to find a function g capable of doing this, because (Art. 12) the 
united points of the two functions are quadrilaterals upon the unit sphere, and a 
function g' always converts this sphere into itself 
I’hus, given two quadrics and two linear complexes, it is possible to transform 
simultaneously one (juadric into the other and one complex into the other whenever the 
latent roots oj the functions fd~f, and Fq“^F^ are proportional. 
55. To find a transformation ivhich shall convert one conic into another. 
The essentials of the problem are contained in the equation 
f{at^ 2ht c) = IV + 2b's + o') 
. (230). 
In order that the rightdiand number may be a quadratic function of q it is necessary 
to have 
.(^ 31 ); 
vt V 
w = {vt + A) h = 
so tliat on equating powers (fiA we obtain, in the usual notation for binary quantics. 
fa = {a'b'c'Xuvf; fb = {a'b'c'fuvXu'd) ; fc = {a'b'c'Xu'v'f .... (232). 
i hese relations are not sufficient to determine the function ; we may arbitrarily assume 
two quaternions d and d! and write/A = d' (Art. 3). The function thus determined 
involves eleven arbitrary constants, the four u, u', v, v' which regulate the corre¬ 
spondence of point to point on the conics, and the seven (eight less one) involved in the 
two quaternions d and df, for multiplying these by a common factor is without effect. 
56. In order to transform simultcmeously two given conics into two other conics, 
a single relation must exist connecting the conics. 
Affixing numeral suffixes, 1, 2, to the various symbols in (232), we obtain the 
system of six equations which tlie functicai /’must satisfy. Any six quaternions are 
connected by two relations, and the equations 
