434' Mr. T. Weddle on Asymptotic Straight Lines, Sfc. 



which does not involve /, m or w. Hence to determine the 

 conical asymptotic planes (if any) to the surface (1.), we must 

 take those factors of spp that are of the first degree, and proceed 

 as directed above for asymptotic cones ; with this modification, 

 however, that Dd, not containing /, m or n must be regarded 

 as a single constant, and consequently the process will be much 

 simplified. If Vj, Vg^.V^ {t not -Ji s) be the values of T>Qq 

 corresponding to the factor 



{S^Y={Al+Bm + Cn}% 

 we shall have 



A.r+By + C^ = Vi, Ax+Bi/+Cz=y^... Ax+By+Cz=Yt 

 as the equations to the conical asymptotic planes relative to 

 this factor. 



It appears from the preceding reasoning, that if the equation 

 (4'.), or, which is the same thing, the highest homogeneous 

 function in the equation to the surface (1.) can be resolved 

 into a factors of the first degree, b factors of the second de- 

 gree, c factors of the third degree, &c. (here a factor of the 

 form {fl }* is to be accounted s factors), then the surface may 

 admit of, but cannot have more than a asymptotic cones of the 

 first degree, that is, a conical asymptotic planes, b asymptotic 

 cones of the second degree, c asymptotic cones of the third 

 degree, &c. Some of these cones may have the same vertex ; 

 and since a + 2b + 3c . , . . =p, the degree of the aggregate of 

 all the asymptotic cones to a surface can never exceed that of 

 the surface itself. 



It will be seen that unless equal factors enter the highest 

 homogeneous function, the asymptotic cones to a surface de- 

 pend only on the two highest homogeneous functions in its 

 equation ; and hence (the above case excepted) all surfaces 

 having the two highest homogeneous functions in their equa- 

 tions identical, will have the same asymptotic cones. Also 

 conversely, it is plain that those surfaces that have the same 

 asymptotic cones must have the two highest homogeneous 

 functions in their equations identical, providing the degree of 

 the equations to the surfaces be exactly equal to that of the 

 aggregate of the cones. Now this aggregate may be consi- 

 dered one of these surfaces ; hence if 



Wl = 0, Uq=:0, .... Ui = 



be the equations to cones, the aggregate of which is of the 

 j9th degree, the equation to all the surfaces of the j5th degree 

 having these for asymptotic cones may be denoted by 



UiU^...Uf-i-Xp-2{^^z) +%p-3(a?2/2) +Xi('^^2;)+xo=0- (15.) 



Wimbledon, Surrey, Nov. 10, 1847. 



