Surface reciprocal to a given one. 469 



the results derived from both are identical, appears on eliminating 7 between 

 these equations, when we have 



m (r^ -r) = 6h+ mn + 2mx + 10/3 - 2a + 5n + 2m, 



an equation which that just given reduces to 



6/t - Zni' + 10/3 - 2a + 5w + 2m = 0. 



And this equation is immediately verified by adding the two given in Mr. Cay- 

 ley's memoir just cited, 



6/j + 8i3 + ?i = 3w(m-2), 2(j3-a) = 4 {m-n). 



It appears then, that of the six equations (E), three may be verified by the 

 theory of developable surfaces already known, while the remaining three deter- 

 mine the three quantities, t, the number of " points on three lines" of the system, 

 7, the number of points of the system through which pass another non-conse- 

 cutive line of the system, and k the number of apparent double points on the 

 nodal line of the developable. It is obvious that the corresponding reciprocal 

 singularities may be determined in like manner, that is to say, the number of 

 " planes through three lines," &c. 



It is possible to verify the value just found for 7 by investigating this quan- 

 tity directl}'* in the ca^e where the edge of regression of the developable is the 

 intersection of two surfaces U and V, the former being supposed to be of the 

 degree k, and the latter of the degree I. The points where the line joining two 

 given points meets each of the surfaces is determined, by the method already 

 given, from the equations, 



X'U^ V'-V AZ7-f '^p^^'U^ &c. = 0, 



V V + \'-V A V+ ^^ A^ F-f &c. = ; 



but if the line joining the two given points be a line of the system, and xyzw 

 Its point of contact, we have U= 0, V= 0, AU=0, A V= 0. Introducing these 

 values, and eliminating \, /i, between the equations, we shall have the condition 



* The method of investigation employed is the same as that ty which Mr. Catlet has deter- 

 mined the number of points of infle,xion and double tangents of plane curves, of which I have else- 

 where given an account. — (" Higher Plane Curves," pp. 77, 86.) 



VOL. xxm. 3 Q 



