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LXXII. Another treatment of the Equations, which conduct to the 

 Skew Surface of Mr. Cayley in the Supplementary Number to 

 Vol. 24 of this Journal. By Dr. Adolph Steen, Professor at 

 the University of Copenhagen*. 



IN searching for the equation of the skew surface generated 

 by a line which passes through three directors,, viz. the 

 plane cubic 



(^ + ^)xy-(x^ + y 3 y^ = } 

 and the two lines 



(# — mz = 0, y — nz = 0), 



(*-* =0, */-/3=0), 



Mr. Cayley arrived at the following system of equations : — 



(a 3 + /3 3 )^-(tf 3 + Z/ 3 )«/3 = 0, 



X=# + AZ, 

 Y=y + BZ, 



(n-B)x-{m-A)y = 0, 



B(x- a )-A(!/-/3)=0. 



Eliminating A, B, x, y, he obtains thence the equation of the 

 surface in a most beautiful and ingenious manner, in which, how- 

 ever, I missed some of that sublime simplicity so frequently met 

 with even in transcendental researches. After some trials I suc- 

 ceeded in finding a very plain and elementary solution of the 

 problem. 



We first eliminate x and y in the three first equations, whence 



(a 3 + /S 3 )(X-AZ)(Y-BZ)-«/5((X-AZ) 3 + (Y-BZ) 3 ) = 0. 

 The same values of x and y put in the last two equations give 



(>i-B)(X-AZ)-(m-A)(Y-BZ) = 0, 



B(X-AZ-a)-A(Y-BZ-/3) =0; 

 or, with a slight transformation, 



(Y-7?Z)A-(X-™Z)B = mY-rcX, 



(Y-/3)A-(X-«)B =0; 



that is, two equations of the first degree to determine A and B. 

 Thence we find 



A B (Y-wZ)A (X-7rcZ)B 



X-« " Y-/3 ~ (X-a)(Y-nZ) ~ (Y-/3)(X-mZ) 

 rnY — nX. 



= (X-«)(Y-tzZ)-(Y-/3)(X-™Z) ; 



* Communicated by the Author. 



