Joi.Y — Homographie Divisions of Planes^ Spheres, and Space. 523 



then using tlie auxiliary vectors o- and t, which have been defined 

 in the 4th Article, 



xa^T + Vao-) + y5(T + V^cr) + zc(t + Fycr) + ivcKj + FScr) = 0, 



and xa\T+Va'(x) + yV{T+r[i'a-) + zc'{T-vry'(j)^ivd'{T^Vh'cr) = 0. 



Operating on these by >S'X and Six, and eliminating x, y, s, and w, the 

 determinant of the fourth order 



= 



a SX (t + Fao-) ^ ^A (t + V/3a-) cSX{t+ Vycr) cl SX (t + FScr) 



aSfi{T+racr) bSix{r+r^(T) cSix{T+Vy(r) dS/Ji^T+VSa-) 



a'SX{T+Va'a) i'SX{T+ VfB'o-) c'SX{t+ Vy'a) d'SX^r+VS'a) 



a'Six{T+Va'<T) b'Sfx^T+V/S'a) c'Six{T+Vy'(r) d' S/x^r+VB'o-) 



is the resultant. 



13. Expanding this by minors from the first and second rows, since 



SX (t + Vaa-) Sfi. (t + V/3o-) - SX{t+ Vacr) S/jl (t + V^(t) 

 = SVX/xV{t+ Fao-)r(T+ Vfia-) = - SXfxa-{ST{a-/3) + Saa/S) 



l)y Art. 5, it is evident that the factor {SX/xa-)'^ is extraneous. This 

 being discarded, the result is 



hca'd' {St {/3-y) + S<7f3y) {Sr^a' - 8') + Saa'S') 



+ cal'd' {St (y - a) + Saya) {St{(3' - 8') + S(t(3'8') 



+ ahc'd'{ST{a-P) + >So-a/5)(6'T(y'-8') + Say'^') 



+ l'c'ad{ST {(3' - y') + So-fS'y') {ST{a - S) + SaaS) 



+ c'a'bd{ST{y'-a') + So-y'a'){ST{ft-8) + Sa/BS) 



+ a'b'cd{ST{a'-j3') + Sa-a'l3'){ST{y-S)+ Sa-yS) - 0. 



The form of this equation shows that the complex is of the second 

 order and of the second class, for o- and t both enter in the second 

 degree. As in Art. 6, putting t = Varp, the equation in cr which 

 results, is that of a quadiic cone containing the lines which pass 

 through p. If SXp = 1 and Sfxp - 1 determine a line of the 

 complex, it is easy to see that cr = VXfx and r = A - /x ; so if A is 

 given and /jl variable the result of substituting these values in the 

 equation of the complex is a quadratic in /x, and represents therefore 

 a curve of the second class in the arbitrary plane SXp = 1. Tljis 

 curve is the conic enveloped by the lines in that plane. 



