4-28 Mr. T. Weddle on Asi/mptotic StraigJit Lines, Planes, 



of the first power of r that does not vanish independently of 

 any relation among a, /3, 7. If this coefficient be that of r^-^, 

 we have 



(10.) 



This equation denotes a surface which is evidently the locus 

 of the asymptotes which are parallel to that generator of (7.) 



whose equations are — = ^ = — . Hence (10.) must denote 



1 ^^ 1 ^^ I 



a cylindrical surface; and as its generators are all asymptotes, 

 it is an asymptotic cylinder of the second degree (which may 

 in certain cases degenerate into one or two cylindrical asymp- 

 totic planes). Should the values of /,?«,» satisfying (9.) also 

 cause a, /3, y to vanish from (10.), there will be no correspond- 

 ing asymptotic cylinder, unless <Pp_2 = 0; and in this case we 

 must equate the coefficient of rP~^ in (5.) to zero, and we shall 

 have an asymptotic cone of the third degree ; and so on. 



Hence, to determine the equations of the asymptotic cylin- 

 ders to the surface (1.), we must find such values (if any) of 

 l,m,7i as satisfy (9.), and substitute them in (10.); if all the 

 terms of (10.) also vanish, we must recur to the coefficient of 

 yrp-s in (5.) ; and so on. There will be as many asymptotic 

 cylinders as there are sets of values of /, m, ?i satisfying (9.), 

 unless, after substituting any set in (10.), &c., the only term 

 that does not vanish is that independent of a, /3, 7, in which 

 case there will be no asymptotic cylinder for this set of values. 



If (p contain a factor of the form {5^}% the first four equa- 

 tions of (9.) will be satisfied by Qq = 0; and this, combined with 

 c j = 0, will give determinate values for the ratios l-i-m-^n, 

 and the corresponding asymptotic cylinders will be determined 

 in the way just mentioned. It may happen however that 9^ is 

 also a factor of (p^-i; and if so, all the equations (9.) will be 

 satisfied by fl^ = 0, and (10.) now admits of simplification as 

 follows. Let 



9p={^y-^P-2g, and <Pp-, = Q^.^'p_^_,, 

 then it may easily be shown that when 9^=0, 



