4-30 Mr. T. Weddle on Asymptotic Straight Lines, Planes, 



simultaneous values in (2.) ; the resulting equations will be 

 those of the asymptotes to the surface that pass through the 

 point («i3y). 



Since (4.) is of thepth degree and (6.) of the (/?— l)th, the 

 equation resulting from the elimination of/ (suppose) from (4.) 

 and (6.) cannot exceed thep (jo— l)th degree, and consequently 

 there cannot be more than p (p—l) values of the ratio m-^n. 

 From this we learn, that through any point in space there 

 cannot be drawn more than /^(p— 1) asymptotes to a surface 

 of the pih degree. 



This theorem suffers an exception, however, which I pro- 

 ceed to consider. 



It may happen that the point (a/3y) through which the 

 asymptotes are to be drawn may be so taken as to cause (4.) 

 and (6.) to have a common factor ^^ (which I shall suppose 

 to be their greatest common measure). In this case (4.) and 

 (6.) will be satisfied if ;i^, = 0; and eliminating l,'m,n from 

 this equation by means of (2.), we have 



%,(•«•-«> J/-/3, ^-7) = 



for the equation to the asymptotic cone, which is the locus of 

 the innumerable asymptotes that pass through the point (a/3y). 

 (The factor ;)^,^ may sometimes be resolvable into other factors, 

 and then the preceding asymptotic cone of the qth degree will 

 in fact consist of several cones of inferior degrees.) 



The division of (4.) and (6.) by X'l will give two equations, 

 ^'p_qZ=0, andx"p-q-i = 0) which admit of no common measure. 

 Now (4.) and (6.) will be satisfied by these two equations; but the 

 equations ^'p_q = 0, x"p-q-i =^j will determine not more than 

 iP~^){P~9~^) ^^^^ of values of the ratios l-^-m-^n, hence 

 (excluding the generators of the cone corresponding to x<j) 

 not more than (/> — §')(/'— 2'—!) asymptotes can pass through 

 the point (a, /3, y). 



In order to find those points (if any) which are the vertices 

 of asymptotic cones, eliminate one of the quantities /, »?, n from 

 (4.) and (6.), and find thosevaluesof a,/3,y thatwill renderallthe 

 coefficients of the resulting equation equal to zero. If no such 

 values be possible, the surface (1.) does not admit of an asymp- 

 totic cone ; but if values a^, /3,, y^ of a,/3,y can be found, then 

 the point (aj /Sj y,) will be the vertex of an asymptotic cone. 

 To find the equation of this cone, we must substitute «i, /3j, y^ 

 for 01, ^,y in (6.), and ascertain Qg the common measure of (4.) 

 and (6.) thus modified ; then wilH,^(A'— «j, y— jSj, ^ — yj)=:0 

 be the equation to the asymptotic cone, having its vertex at 

 the point (aj /Sj yj). If the equation resulting from the elimi- 

 nation of/, m, or n from (4.) and (6.) can be rendered identically 



