432 Mr. T. Weddle on Asymptotic Straight Lines, Planes, 



+ (Pp_i = 0, reduces to (Pp_i = 0, and consequently there will be 

 no asymptotic cone unless fl^ be a factor of fp_i ; if so, let <Pp-i 

 =4''.^^, then 



becomes 4/'.D5^ + (pj5_2 = 0, which is of the first degree in a.,^,y, 

 and this (instead of (8.)) being combined with 9,y = 0, may give 

 an asymptotic cone. If {6 }2 however be a factor of <py,_i, then 



becomes (pp-2 = 0, and there will be no asymptotic cone unless 

 $^ be a factor of (Pj,_2. If this be the case, assume f^-i =4/'. {5, }% 

 and <pp_2=\J/".fl^, then 



reduces to 



and this equation, which replaces (8.)} combined with fi,=0, 

 may give one or two asymptotic cones (but not more, as will 

 be shown below), unless 6^ should enter both fJ,_■^ and ip^.a in 

 a higher power than has been supposed ; we shall then have 

 (Pp_3 = 0; and hence fl,^ must (if there be an asymptotic cone) 

 be a factor of f^-s. Suppose therefore 



<p^_j = vl/'.{aj3, ^^_^=4,//.|5j2^ and ^p_3 = vl;"'.0„ • 



then 1 _. , 



becomes 



and this equation (which cannot be satisfied independently of 

 a, /3, y, for 9^ is not a factor of rj/), combined with 5^ = 0, may 

 give four asymptotic cones. 



Similarly, if {fl^}* be the highest power of 9^ that is a factor 

 of <Pp, it may be shown that «, /3, y enter the equation to be com- 

 bined with 3j = 0, only through D9^, and that this equation 

 may rise to any degree in D^^ (except the (5— l)th) not ex- 

 ceeding 5. 



Hence when a power {s) of 5^ is a factor of <Pp, we must 

 ascertain the highest powers of fl^ that are factors of (p^,_„ <Py;_2, 

 ...<Pp_<+i, also (pp^t the first term of (1.) that has not 9,^ for 

 a factor ; we must then equate to zero the coefficient (reduced 

 as above) of the highest power of r in (5.) that does not vanish 

 independently of a,^,y. If a, /3, y disappear from this equation 



