90 



NEWSON: LINEAR GEOMETRY. 



(2). Three quadratic covariants, viz: the first polar P with respect 

 to C, Q,; the first of P with respect to G^, Qo; and the Hessian of 

 the cubic, H^. These written in full are 



Qj=i=(ab J — ajb).\2-^2(bbj — a^c)xy-f-(b,c — ajd)y2. 

 Q2=(a2bjd — 3abbjC-;-2b3bj — aajbd — ajb2c-|-2aajC-)x2 

 -|-2(abbjd-|-b-b,c — 2abjC2-j-aa,cd-(-ajbc' — 2ajb2d)xy 

 -|-(2b~bjd — abjCd — bbjC^ — 3a jbcd-(-aa jd^ — 2ajC3)y2. 

 Hx=(ac— b2)x'- + (ad— bc)xy-f (bd— c2)y2. 

 (3). Four linear covariants, viz: the point P; the second polar of 

 P with respect to C (which is identical with the polar of P with respect 

 to Qi), Pj; the second polar of P with respect to Gx (which is iden- 

 tical with the polar of P with respect to Q.,), Po; and the polar of P 

 with respect to H^. These written in full are 



P=a,x+b,y; P,=^(ab -_2a,b,b+a^c)x+ (b^b— 2a,b,c + a\l)y. 



2 2** 



p2=(i*^b d— 3abb c — 2b3b' — 2aa,bb,d — 2ajb2b,c -]- 4aa jbjC^ 



-j-2a b^d — a'bc^ — aa'cd)x4-(abb d-fb-b c — 2ab"c2 — 4a, b- b, d 



-]-2aa,bjCd-j-2a,bb,c"-[-3a''bcd — aa d^— 2a c^))'. 



P3 = (2ab,c — 2bjb2-f-aa,d — aa,c)x — (ab,d — bbjC-|-2a,bd — 2ajC2)y 



In calculating the foregoing polars it is convenient to make use of 

 the following theorem: the first polar of the point, P:^a,x-fb,y, with 

 respect to any binary quantic U is identical with the Jacobian of P 

 and U. 



This theorem is easily proved as follows: the polar of any point 

 (x, y) with respect to any quantic U is given by the operation 



r d d K, I x,d d 1 



/>d^+^''dy I ^''"' [.7d7+d7J ^- ^I't if the equation of the 



X, b, 

 point (x,, yj) is P^ajX-f bjy=o, then, — ^= -• 



Substituting this value of -7^- in the above operation for the polar, we 



have 



-b,d 



dx ' 'dy J 

 the last equation gives 



itt: 1 L— o. 



dP 



But a ^i^j— , and b,=; 

 ' dx ' 



dP 

 d7 



whence 



dP dU 



dx dv 



dPdU 



dy dx 



=0; 



which is the Jacobian of P and U. All the above polars of P with 

 respect to the various quantics are readily calculated as Jacobians. 

 When the equation of the cubic is written in the full form 



