(11.) 



Cones and Cylinders to Algebraical Surfaces. 4-29 

 Hence (10.) becomes 



that is, 



which evidently denotes two parallel cylindrical asymptotic 

 planes ; also since /,?w, n are here only connected by the equa- 

 tion Q,j = Oy it appears that there are in general two cylindrical 

 asymptotic planes parallel to every tangent plane of the cone 



fl,(^7/2) = 0. 



Generally, let{0^}% {^J^'S {d^}'-^- 5, be factors of 



<Ppy fp_i, <Pp-2 <Pp-s+i, and put 



(here the subscribed letters relative to ^^ ^', &c. are omitted 

 for simplicity), then it may easily be shown that when 5^ = 0, 

 we have 



Df^=0, B\=0 .... D-'fp=0, D*<p^=2.3...s.4/.{D9J*,&c. 



Moreover, the equation to the asymptotic cylinder parallel to 

 a generator of the cone fl^(a:j/2;)=0, will, by equating to zero 

 the first coefficient of (5.) that does not vanish independently 

 of a, j3, y, be found to be 



■2jZs^''^'-^2j:^{^=T)^'^^^^^ 

 and this, by what precedes, reduces to 



it is evident that the asymptotic cylinder degenerates into s 

 cylindrical asymptotic planes, all parallel to a tangent plane 

 of the cone (fg{xj/z) = 0; and there is in general the same num- 

 ber parallel to every tangent plane of this cone. 



The asymptotes to the surface (1.) passing through a given 

 point (a/3y) will be found by determining the ratios l-^m-r-n 

 by (4-.) and (6.), and substituting, in succession, each set of 



'*"•••} . (12.) 



