Cones and Cylinders to Algebraical Surfaces. 431 



zero by other simultaneous valuesof a,/3,y, there will be as many 

 asymptotic cones as there are sets of values. When the eli- 

 mination referred to above is effected by the process for the 

 common measure, the factor fl,^ will be the last of the remain- 

 <lers that do not vanish when «„|3i,yjare substituted fora,/3,y. 

 It will sometimes be found, however, that (4.) and (6.) have 

 a common measure independently of «,/3,y, arising from{6^^}^ 

 and 6 being factors of <p^ and (p/,_i ; and in this case we must 

 proceed with this common measure in the way to be noticed 

 presently. 



When we know that (4.) cannot be resolved into factors, 

 the determination of the asymptotic cone is very easy ; for 

 since (4.) admits of no measure but itself, and (6.) is of an in- 

 ferior degree^ it is evident that if there be an asymptotic cone, 

 (6.) must be identically zero ; hence if such values «i, /Sp yj can 

 be given to oL,^,y as to cause the coefficients of (6.) to vanish, 

 there will be an asymptotic cone of the pth degree, namely, 



but if the coefficients cannot be rendered zero simultaneously, 

 there will be no asymptotic cone. Since «,i3,y enter (6.) in the 

 first degree only, there will evidently be at most only one set 

 of values of a,^,y that will render (6.) identically zero; and 

 hence a surface of the pih. degree may have one asymptotic 

 cone of the pih degree, but not more, and it is plain that there 

 cannot be an asymptotic cone of a higher degree. 



If (4.) admits of being resolved into factors, and these fac- 

 tors can be found, the asymptotic cones may be determined 

 as follows. Let 9^ be one of the factors of ip^, and let fl^ itself 

 be irresolvable into factors. Arrange (6.), or rather (8.), and 

 6,/ according to the powers of either /, m or n {I suppose), and 

 divide the former by the latter until the remainder is of lower 

 dimensions in / than d, ; then since fl^ is irresolvable into fac- 

 tors, it is clear that this remainder must be identically zero : 

 find therefore a.^, /3j, y^ the values of a,|3,y, that make the coeffi- 

 cients of the remainder vanish, then ^^{x—Uy, y—^^ z-—'/^) = 

 will be the asymptotic cone. As ci,^,y enter (8.) in the first 

 degree and do not enter 9,^, there cannot be more than one 

 set of values of «,/3,y, it indeed there be any. The same pro- 

 cess being repeated with each of the other prime factors into 

 which (4.) is resolvable, we shall have all the asymptotic cones 

 which the surface admits of. 



The preceding process requires modification when the 

 second or any higher power of 6,^ is a factor of (j&p. As an ex- 

 ample, suppose that {d }"* enters as a factor into <p , and put 

 ^^=^|;.{fl^^)'* (d^ not being a factor of ^), When d^=0, Df^ 



