516 Mr. A. Cay ley on the P or ism of the 



ax 2 + by* + ex* = and # 2 + y 2 + £ 2 = 0. Consequently, in the 

 particular case in which (x v y v z x ) is a point on the conic 

 •r 2 -f y* + z 2 = 0, the point (,r 2 , y 2 , » 2 ) is the point in which this 

 conic is met by the tangent through (x v y } , z x ) to the conic 

 ax* + bi/ + c^ = 0. 



It has already been seen that lx x + my l + nz x = and 

 ew 1 2 + /Syi 2 + 7-i 2 =0 identically ; consequently we have identi- 

 cally lx 2 + »iy 2 + n r 2 = and a ,r 2 2 + /3y 2 2 + y 2f g ? = 0. The latter 

 equation, written under the form 



(axt + byf + czft^ + yf + zf) 



-( x * + y* + ; ;*)(ax 2 * + by* + czf)=0, 

 shows th at if x 2 , y q , z^ are such that ^ 2 2 + y 2 2 + z£ = x x * + yj 2 + ^i 9 , 

 then that also #;r 2 2 + ~by£ + cz£ = axf + byf -f czf. I proceed to 

 determine © so that we may have x^ + y^ + z^ = x x 2 + y x 2 + z l il . 

 We obtain immediately 



gifo" + y a s + «»*) = PV + *V + « 4 ~ , 2 ) («V + /3Vi 2 + y V) 



- (a 2 /V + /3 2 m V + y^i 4 z 4 - 2/3y/tt 2 » 2 yi V - 2y«n 2 / 2 * ,y 



-2a/3/ 2 »i 2 tf,V). 

 Write for a moment ax^ + by* + cz ] 2 =p, xf + yf + z^szq, 

 so that a = p — aq, /3=p—bq, y=p — cq, then 



« V + (3*y , 2 + y V = gp 9 - 2p pq + (a V + £ V + c V) r/ 

 = r / {( fl y + ^ I 2 + ^ 1 2 ) g - J0 2 } 

 = q{b — c l y?z x ~ + c— az?x? + a — b*x x hj x % 

 « 2 /V +/3*m 4 y x 4 + y 4 /i 4 ^ 1 4 -2/Syy«Vy 1 2 ^, 2 -2y«H 2 ^, 2 ^ 1 2 



=p*(l 4 x 4 + m 4 //, 4 + *•*,« - 2?« 2 /i 2 y, V- 2/i 2 / 2 ?> 1 2 -2/ 2 m 2 1 r 1 2 >/ 1 2 ) 

 ~2pq[al 4 x 4 -h bm 4 y x 4 + cn A zf — b + cm*n % y?zf—c + arPPzfz* 



— a + bPm~x x -y x *) 

 + q*(an 4 x 4 + bhn 4 y 4 + c*n% 4 - 2bcrt?r?y?z?-2cannH?x? 



— 2abl' 2 m 2 x x ' 2 y x ' 2 ), 

 the first line of which vanishes in virtue of the equatiou 

 lx x +my x +nz x = 0, we have therefore 



^- 2 (* 2 2 + vi + ~V) + (*, 2 + z/i 2 + ~~! 2 ) = 



(/ 4 *, 2 + m 4 y? + »V) (*-«fyiV + wPtfa* + ^b\\*) 



+ 2(a.r 1 2 + fy, 2 + cz, 2 ) (a/ 4 ^ 4 + Am'ty, 4 + cm**! 4 — 5+cnAtyi V 



— c + uiH^z^x^— a + bPm*x*y*) 



~ (*i 2 + Z/i 2 + *i 2 ) («^V + 6 2 »* V + cV~ , 4 - 2bcmWy 1 %* 



- 2can*Pz l *x 1 * - ZabPnPxfyfl . 



