186 



THE ROYAL SOCIETY OF CANADA 



(8) 



A = a-\-3^t-hSyt~-]-ôf = 



1, -3/, 3/-, -^ 



ai, b\, Ci, di 



0-2, bi, Co, di 



as, bs, cs, ds 



2. The invariant ds. The seminvariants P2, P3 are defined by 

 P2 = p2-Pr-Pi', 



P3=pZ-3plp2+2Pi'-p^". 



Substituting the values of pi, po, ps we find 



(9) -~ A^P2=l-2mt-\-nt\ 



4 



(10) ^ A^Ps = 4^l-2am+ (15yl-\-9an)t-\- i2l(37i-\- 15ôl)r~-{- {Uy7t - 

 4 



In order to study the projective properties of the cubic defined by 

 (1) it is necessary first to calculate the invariants, which are not 

 changed by any transformation of the type 



3' = X3', t = ^{t), X = some function of t. 

 The simplest of these invariants is defined by 



2 



Substituting the values of Po, -P3, we obtain, after reduction, 



(11) 



where 



(12) 



— A^d3 = A + Wt+ZCt- + Dt\ 

 20 



A = -{am + fiï), C= -(ym-\-8l), 

 B= —(j3m-\-yl), D= ôm-\-yn. 



3. The condition that equations (1) shall represent a 

 CONIC. A necessary and sufficient condition* that equations (1) shall 

 represent a conic is ^3=^0. That is, ^ =5 = C = Z) = 0. Hence 



am + ^l =0, 

 ^m + yl =0, 

 ym-{-8l =0, 

 ôm-{-yn = 0. 



(13) 



Consider the first three equations 

 l = m = n = or 



a ^ 



= -/ = 0, 



a/3 

 y 8 



It is evident that either 



= -2m = 0, and ^Z =-n = 0. 

 y 



*Wilczynski, Projective Differential Geometry, p. 61. 



