84 PROFESSOR CAYLEY 03 POLYZOMAL CURVES, 



taking a 2 , b 2 positive c 2 , d, will be negative) we have 



a : b : c : d = c 2 (b 2 - d 2 ) : d 2 (c 2 - a 2 ) : - a 2 (b 2 - d 2 ) : - b 2 {c 2 - a 2 ) , 



I 7Tb 7h 



and then the equations ;; + t + -=0, s/l + Jm + Jn = , are satisfied by 



the two sets of values 



*Jl y/m, : *Jn = b 2 — c 2 :c 2 - a 2 :a 2 -b 2 , 



and 



= — b„ — c 



-a - «2 : a i + \ , 



and we have the equations of the same two cubic curves, each equation under a 

 fourfold form, viz., these are 



and 



, - c 2 + d 2 , - d 2 + b 2 , - b 2 + c 2 \ ( ^/A, *JB, JC, */D) - 



On "■"" l^o j • j CI o "~ ~ C'rt j *"~ On l" t'o 



— b 2 + d 2 , a 2 — d 2 , . , — a., + Z> 2 



Un ~~" On j Co "~ "n y CJn "~ On • * 



(first curve) , 



( 



- d 2 + c 2 , 



- & 2 + rf 3 , 



b 2 + c 2 , 



c 2 + tf 2> - »/ 2 + b 2 , ~b 2 - c 2 )( X /A. V B , */C. VD) =° 



a, + ^9> 



Co + a 2 , — «, — &„, 



a 2 + b 2 



(second curve) , 



191. And again AB and CD meet in T, and denoting by a 3 , b 3 . c 3 , d 3 the 

 distances from Tof the four points respectively, so that a 3 b 3 = c 3 d 3 = rad. 2 T , we 

 have 



a : b : c : d = b 3 (c 3 - d s ): - a 3 (r 3 - rf,) : - tf 3 (« 3 - J a ) : c 3 (a 3 - &,) . 



/ . m 



The equations -+_+- = (),,/£ + */«i + Jn = 0, then give for «//, ,/w, Vw 



two sets of values, viz., these are 



J I : */m '• J n = b 3 - ft 



B ' 



'8 - r '3 : «3 - ^3 



and 



= b 3 + f, : - r s - n 3 : rt 3 



*> 3 ; 



and we again obtain the equations of the two cubics, each equation under a four- 

 fold form, viz., these are 



( • > - c 3 + d 3 , - d 3 + b 3 , c 3 - b 3 ) (VA, JB, v/C, */D) = , 



- d 3 + c 3 , . , - a 3 + d 3 , a 3 - c 3 ' 



- J 3 + rf 3 , - rf 8 + ff 3 , 



& 3 - C 3 . C 3 



and 



\ -a, 



l 3 ' 



«8 - & 3 



c 8 -^ 3 > ^3 + h> ~ c ,-h )(-v/A, x/B. s/C, V d ) =°- 



, - a, - d 3 , a s + c 8 



3 ' *! T "3 ] ■ , P 3 3 



ft 3 + r 3 • ~ C S ~ a Z • a Z ~ b Z • 



rf g -C, 



5, — (f, 



