29-2 
PROL^ESSOTt C. J. .TOLY 0\ QlLVTl-TvXIONS AXD PRO-TECTIVE GEOMETRY. 
As in Art. 103, we deduce the identity 
{/'(ra) — la ; J''{rl>) — (h ; f' {re) — tc ; f'{rd) — td) 
= {/" (ar) — ta ; f" {hr) — th ; f" {cr) — tc ; f" {dr) — td) = 0 . (428'); 
and tlie result of dividing by {abed) may 1)e written in the form 
K (r) - fK' (r) + d/r (r) - dli"' (r) + A.(429). 
and the latent quartic of /'(rq) or(qr) (functions of q) is obtained Iqv equating tliis 
to zeiv). 
The scheme of tlie .Tacohians is now complete, the six fundamental functions of 
Art. 102 having l)een enqdfwed. 
The points r'q of (424) may he said to he dK Jacobian correspondents, and p and 
r" are IK correspondents. 
AVlien f {]></) is pei nmtative, the JI\ and IK types unite and I coincides with 
w hen f{p<i) is self-conjugate witli respect to p, K coincides with /, and the JK and 
Id correspondences coalesce. 
It readily appears from (410) that wdien the function is doubly self-conjugate it is 
also permutaUe, and when it is permutable and self-conjugate to one element it is 
likewdse self-conjugate to the other. In this case tlie three .lacobians coincide with 
the Hessian of the cubic surfixee 
= ^ .(^30). 
SETHI ON XYll. 
Thu Foru-SvsTEM of Ltneau FrycTruxs. 
Ah Krample of the Use of the Bdinear Function. 
Art. Eage 
107. Method of employing the Idliiiear fnnotioii. A function iletennined hy a point p. . 293 
108. An arhitrary point f/) i.s a united point of a determinate function, jn'ovided it iloes 
not lie on a critical curve, iq ( 7 ) = 0 of the tenth order.293 
109. When 7q (y) = 0, the point 7 is a united point of a definite two-system of functions . 294 
110. The conjugate system and reciprocal properties.294 
111. The tetrads of united points. The rpnartic tran.sforniationp = 7^ ( 7 ).295 
112. When the united point 7 describes a line, the point 75 describes a qnartie curve. . . 295 
113. A right line breaks oft’ this quartic for every intersection of the line and the critical 
curve.295 
114. The various ways in which the functions of a five-system may comlune to destroy a 
line point by point. The twenty singular functions of a four-system.290 
115. The sextic surface described by 7 ) when 7 lies in a jdane.29G 
IIG. The double curve of the sextic .surface ..296 
117. The triple point on the surface. 2 ftT 
