Cones and Cylinders to Algebraical Surfaces, 427 



and since («|3y) may be any point in each asymptote, (6.) de- 

 notes the locus (a, /3, y being the variable coordinates) of the 

 asymptotes parallel to the same generator of (7.) ; this locus is 

 therefore a cylindrical asymptotic plane, and it is parallel to 

 that tangent plane of the cone (7.) which touches along the 

 generator. Hence, to find the equation of a cylindrical asymp- 

 totic plane, we have only to take such values of /, m, n as 

 satisfy (4.) and substitute them in (6.). It thus appears that 

 when the cone (7.) is not imaginary, there is an indefinite 

 number of cylindrical asymptotic planes ; one indeed parallel 

 to every tangent plane of the cone (7.)> with a few excep- 

 tions, which I shall consider presently. 



Should (4.), or, which is the same thing, (7.) be resolvable 

 into factors, then (7.) will in reality denote as many conical 

 surfaces ; and if any of these factors be of the first degree, 

 the corresponding conical surface will degenerate into a plane. 



Let ^q be any factor of ip^, and put 



hence (6.) becomes 



when dj = 0, this reduces to 



and this equation, together with fl,^=0, will supply the place 

 of (4.) and (6.) for those cylindrical asymptotic planes that 

 are parallel to the tangent planes of the cone $q{x}/z)=:0. Also 

 similar equations may be found for every factor of <pp. 

 If the equations 



^^-^' ll -^' dm -^' dn -^* 



can be satisfied by simultaneous values (/j m^ n^) of/, m, u, (6.) 

 cannot be satisfied unless (pp-i also =0; if <P;,-i should not 

 = 0, there will be no cylindrical asymptotic plane correspond- 

 ing to these values of I, m, n ; but if <p^j_i=0, so that we have 



^^-^' if-^' f^' ■^-^' '^p-'-^^r V9.) 



then (6.) will be satisfied independently of a, /3, y. We have 



only to recur however to (5.), and equate to zero the coefficient 



* Since ^p 'n a homogeneous function of /,j?j,n of the/>th degree, we have 



dl dm d7i 



hence the equations (9.) amount only to four independent equations — the 

 last four. 



