210 Mr. A. Cayley on the Homographic Transformation of 



simple, that it is, I think, worth while to reproduce it here, 

 although for several reasons I prefer exhibiting the final result 

 in relation to the form x^ -\- 1/ -\- z^ + iv^ = Q oi the equation of the 

 surface of the second order. I consider then the surface xy — ziv = 0, 

 andItake(«,/9,7,S), {a.',0,y',8') for the coordinates of the poles 

 of the two sections ^, 4>, and (^i, y^, c„ ?<',), (^2,?/2> ~2' "'2) as the 

 coordinates of the points P, Q. AVc have of course w^yi—ZiWi=0, 

 a-g'/g— 2'2"'2=0- The generating lines through P are obtained 

 by combining the equation xy—zw — of the surface with the 

 equation xyi + yxi — zw^—wzi = of the tangent plane at P. 

 Eliminating x from these equations, and replacing in the result 



X, bv its value ^^^—^, we have the equation 



We may if we please take y^i— 2-^1 = 0, xyy + yx^—zw^—wz^ — Q 

 as the equations of the line PA ; this leads to 



yz^-zy^^O \ yw^-wy^=0 \ 



xyi + 7/Xi—zWi—wZi=:0J xy^-^ yx^—zw2 — wZci = 0J 

 for the equations of the lines PA, QA respectively ; and we have 

 therefore the coordinates of the point A, coordinates which must 

 satisfy the equation /3x + ay — Bz — <yw =0 of the plane 6. This 

 gives rise to the equation 



2/2(«yi-^^i) -"^2(7^1 -^^i) = 0. 

 We have in like manner 



yiv^—y^w = 1 y?i—^yci = \ 



xy^+yXi — zw^ — ivZi^O J xyc^ + yx^—zwci—tvz2=0J 



for the equations of the lines PB, QB respectively ; and we may 

 thence find the coordinates of the point B, coordinates which 

 must satisfy the equation ^'x + m'y — B'z—^iv = of the plane (f). 

 This gives rise to the equation 



y^yi -7''«i) -^2(^Vi -0'(>i) • 



It is easy by means of these two equations and the equation 

 z^^—z^w^=Q to form the system 



y«= (7i/i -/S2^i)(SVi -l^'wi) 

 *8= (72/1 -^-j)(*Vi ~7'«'i) 



^i~ {<^y\ - ^^1) (^Vi ~^S) > 



or, effecting the multiplications and replacing z^iv^ by Xyy.^, the 

 values of x^ y^, z^ lu^ contain the common factor ?/„ which may 

 be rejected. Also introducing on the left-hand sides the com- 

 mon factor MM', where U^=a^-r^B, M'^ = u'/3'-ry'8', the equa- 



