304 PROFESSOR C. J. JOEY OX QUATERXIOXS AXD PROJECTIVE C4SOMETRY. 
120 points on the critical curve; and thus tLcentij-four functions of the system have 
four equal latent roots and four coalesced united points. 
130. Again, suppose that two roots of (488) are zero and that the remaining two 
arc equal. In this case 
8 J( 7 )^- 4 j(q)r'(^>) + 4 r(p)-r"(^>)' = o . . . (492); 
and this equation, combined with (489), gives a curve locus of order 36 (= 8 X 12 
— 2 X 3 X 10), wliicli is the locus of v.nited points of functions ivhose roots are equal 
in j)airs. 
We have now outlined the general theory of the four-system, hut in a later section 
some supplementary remarks will be made on this subject. 
SFX!TION XV111. 
The Quadratic Transformatiox of Puixts in Space. 
Ihe Second Example of the Use of the Bilinear Function. 
Art. Page 
131. The quadratic transformation 7 ; =/q/ 7 ). The cctads of points .304 
132. A line transforms generally into a conic, but into a line if it is a connector of points 
of an octad, or (what is equivalent) of Jacobian correspondents / ( 7 ) 7 ) = 0. 
Harmonic properties .305 
133. The limiting points into which Jacobian correspondents transform.305 
134. The arrangement of connectors and Jacobian corresjjondeuts in a plane.306 
135. The points of an octad and the twenty-eight connectors.306 
136. A plane transforms into a Stefner’s quartic.307 
137. Geometrical relations. The conics of ring-contact.307 
138. The focal pcn'nts on a ray of the congruency of connectors.308 
139. The orders of complementary loci and of the loci into which they transform . . . 309 
140. The complementary of the Jacobian is the focal surface of the congruenc}' of 
connectors.310 
111. The focal surface of the transformed connectors is the transformed Jacobian and the 
reciprocal of a symmetrical Jacobian.310 
142. The nnmorical characteristics of the two congruencies.• . . 311 
131. Tlie general (piadratic transformation in space is represented bv the equation 
p=f{q,jf) . (493), 
in which it i.s olndously permissible to regard tlie bilinear function as pernmtahle, or 
the four linear functions (409) as self-conjugate. The transformation involves 40 
constants. 
