Surface reciprocal to a given one. 467 



(2^ + 3c) (2n^-2?z-26-3c-l)-3(3i + 4c) (re-2) + 18^+ I27+ 9< 



-(46 + 6c) (n-2) (n-3) + 8^ + 18/t-36/3-247-12t'+126c. 

 that is to say, 



= n (76 + 12c) - W- - ^c" - 86- 15c + 8i-+ 18/« - 18/3- I27- 12f+ 9^ 



As a first verification of the preceding formulae, we take the case where the 

 surface is a complex one made up of several others. In this case the complex 

 surface must be considered as having for a double line (in addition to whatever 

 double lines the surfaces may have, considered separately) the aggregate of the 

 curves of intersection of each pair of surfaces, on which every point of intersec- 

 tion of three surfaces will be a triple point. The effect of this double line must 

 be to reduce the degree of the reciprocal of the complex surface to the sum of 

 the degrees of the reciprocals of the simple surfaces. 



We shall then have 



71= 2^1 ; 6 = 261 + 271,712; c=2cr, 



A = 2/11 + 2c,C2 ; /3 = 2|3,; 7 = 27, + 20,71^ ; 



< = 2^1 + 271,712723 + 26i?!2 ; i - 2/, ; 



p = 2/), + 2a,722 j "■ = 2<7, ; 



>fc = 2A,+2 ^'^''^^'7^)^"--^) + 26.62 



4- 2?i, 2tt,ri2723 - 327J,W27i3 + 26i2?j,?i3 — 226i7Z2 ; 



and on substituting these values the equations (A) and (D) are satisfied iden- 

 tically, and the reduction in the degi-ee of the reciprocal surface, caused by the 

 curves of intersection of the simple surfaces, is 



3271,^712 + 62n,n2«3 - 2271,712 ; 



but this is just the difierence between 



(2?z,)'-2(2?2,)' + (27i,) and 2 (?«,'- 2?z,= -|-?z,). 



II. — Developable Surfaces. 



I come now to the application of the preceding theory to the case of deve- 

 lopable surfaces. I use with respect to these surfaces the notation employed 

 by Mr. Cayley ("Liouville," vol. x. p. 245 ; " Cambridge and Dublin Mathe- 

 matical Journal," vol. v. p 18). The degree of the developable surface is r. 



