286 PROFESSOK C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
{q, dy, d~q, h ^p) = 0, or xq y dq z d^-q + ivh '^p> = 0 . . (388) 
is the equation of a geodesic. 
Operate with Sdji;, S/ny, S/nIp and by (364), (365), 
z^pd~q + iv^plr^p = 0 ; ySd^^ dq + zSdp d-p + iv^diplr^p = 0 ; 
x^qhq + y^qh dq + z^qh dhq = d ) x^qh dq + y^dqh dq-\-z^dqli d'q = Q . ( 389 ). 
Introducing the function /' and eliminating the scalars xyziv, we find 
^dph~^p _ _ y' ^dqf dq eSd^'/’d^g' 
^ph~^p z^dqj" dq 
_ _ Sdp/dnj' . ^qhq^dqh d^ q — ^q li dq^qh d~q /qQo\. 
'^dq fdq '^qhq^dqh dq ~ [^qh dqY ' ' ' 
and this, when the surface is a quadric so that f is constant, immediately integrates, 
and gives 
S 2 :)h~''^ 2 ^Sdqf dq = u {SqhqSdqhdq — (Sqhdq)-) .... (391), 
where is the constant of integration. 
SECTION XV. 
The Analogue of Hamilton’s Operator V. 
Art. 
94. The operator D. If dQ = Spch, then j; = DQ. Symbolical equation of definition 
involving four arbitrary differentials. 
95. The form of the operator in special cases. 
96. Examples of the efiect of the ojjerator and analogues of Laplace’s equation 
97. Method of forming polars and analogy to Aronhold’s notation. 
Page 
286 
287 
287 
288 
94. In applications of quaternions to jirojective geometry an operator analogous to 
Hamilton s V is occasionally useful. I define it by the equation (compare Art. 85) 
DQ=p when dQ = Sj^dy.( 392 ). 
To render this operator available for use, take any four independent differentials of 
q and write down the identity 
j/i (diy d'p d'^^ d'”q) = [d'(/ d''q d'"q~\ S^ dc^ — [di/ d"(2 d'''q~\ d'ly 
+ [dij d'q d'"q~\ Sp d''q — [dg d'l^ d'^j-] Sp d %7 . (393), 
which suggests the symljolical equation 
D = ^ [dy d”q d"'q'] d 
(d(2 d'lj' d"q d"'q) 
where the summation refers to the four symbols d. 
( 394 ), 
