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



kh-s''' 



fp-k- 



2.3. ..(s-l) 



+Df;,_,+, + fp-s)rP-'> =0*, 



where D denotes the operation 



^:^D— ^^_,+ !► (3.) 



d ^ d d 



dl 



dm 



dn 



This equation will determine the values of r at the points 

 in which the straight line (2.) cuts the surface (1.); now for 

 all lines parallel to an asymptote, one of these points is evi- 

 dently at an infinite distance ; hence a root of (3.) being infi- 

 nite, we must have 



f;> = 0; (4.) 



and this equation determines the directions of the asymptotes. 

 The equation (3.) hence becomes 



+ 



2. ..(5-1) 



= 0; 



(5.) 



in which values of/, m, n satisfying (4.) must be substituted. 

 Now an asymptote being a tangent at an infinite distance, it 

 follows that the asymptote will be distinguished from all lines 

 having the same direction by a root of (5.) being infinite ; we 

 must therefore have 



that is, 



Dfj, + fp-i = 0; 



^a+$^/3 + 



d'Pn 



dl- dm'-^-d^'^-^^^'-'^^- 



(6.) 



The equation (4.) shows that every asymptote is parallel to 

 some generator or other of the cone 



^/.(•^i/2r) = 0; (7.) 



* In this paper I restrict 6,^,^,x (either with or without a letter or figure 

 subscribed) to denote homogeneous functions only ; and when these sym- 

 bols stand alone, they are to be understood as functions of /, m, n ; in otiier 

 cases the symbols of quantity must be written ; thu? X(i(^I/^) ^^ homogeneous 

 function of x,i/,z of the qth degree) means the same function oix,y,z that x„ 

 does of /, m, n. 



