Cones and Cylinders to Algebraical Surfaces. 433 



so that it becomes (Py,_<=0, there will be no asymptotic cone; 

 but if this be not the case, then the reduced equation must be 

 combined with fl,y = 0, in the same way as directed for (8.) and 

 d =0, and we may get asymptotic cones though not more than 

 s of them. I proceed to establish the last assertion. 



It has been shown above that if {5^}* be the highest power 

 of ^q that is a factor of <P;;, then tlie equation to be combined 

 with 6^ = will be of the form 



rI/.{D5j' + vI;'.{D5j'-i + =0, . . (13.) 



where 4/, \p' ... do not involve «, /3, y, and t may be equal to, 

 but cannot be greater than s. Now if there be a correspond- 

 ing asymptotic cone, let (ai/3,yi) denote its vertex j then if 



(which I shall denote by Difl^) be substituted for DS, in (13.), 

 the resulting equation will be satisfied by aid (if necessary) of 

 d, = 0; hence (13.) must be divisible by D^, — D^^q, so that it 

 may be written 



(D9^-D,y(vI/.{DfiJ'-» + ....) = 0. . . (U.) 



Also,if a2j/32»72 beanothersetofvaluesofa,|3y, satisfying (1 3.), 

 they must reduce the second factor of (l-t.) to zero, for the first 

 is of a lower degree than 5 . Hence vJ/.jDfl^^}^"^ + . . . . must 

 be divisible by 'D^q — 'DcP)\ and so on. In this way we shall, 

 after a certain number {v) of divisions, get an equation, 



4/.{D9j '-«+.... =0, 



which either does not contain Dflg (and hence «,/3, y) at all, or 



which cannot be satisfied by any values of a,/3,7. Rejecting 



this factor then as affording no solution, (13.) is equivalent to 



(D9^-D,g(D9^-D,y (D9^-D„y = 0, 



and each of these factors will give but one set of values of«,|3,y; 

 hence there will be but v asymptotic cones, 



^.(■^-"i. y-^v 2-yi)=0 .... e,(a:-«„, y-/3„, ^r-yj = 0; 



and since v cannot exceed t, nor t exceed s, it follows that 

 there cannot be more than 5 asymptotic cones resulting from 

 a factor of (p^ of the form {9 }*• 



When ^q is of the first degree, it is clear that instead of an 

 asymptotic cone we shall have a plane ; and since any point 

 in it may be regarded as the vertex, every straight line drawn 

 in it will be an asymptote ; hence the asymptotic cone will in 

 this case become a conical asymptotic plane : also since fl, is 

 here of the form A/+Bw + Cw, 



Dfi^ = A« + B/34-Cy, 



Phil. Mag. S. 3. Vol. 31. No. 210. Dec. 184-7. 2 F 



