KECTILINEAR ASYMPTOTES. 297 



In equation (4) let x be supposed to increase indefinitely, 

 then we shall have different results depending on the value 

 of p. 



If p be greater than n 1 the value of /3 is infinite, and 

 there is no asymptote for the root /z x of the equation 



<(/*) = 0. 

 If p be equal to n 1 and (f> (/i t ) be not zero, the limit of 



/3 is T,/\ ; and the equation to an asymptote is 

 9 UV 



If p be fess than n 1 and 0' (/^) be not zero, the limit of 

 /3 is and the equation to an asymptote is 



y = w- 



In the last case the equations 



y = A*, </> OK) = o, 



give for determining the asymptotes 



hence when the equation to a curve can be exhibited in such 

 a form that the sum of a number of homogeneous functions is 

 zero, and the degree n of the highest of these functions ex- 

 ceeds by more than unity the degree of any of the others, 

 all the asymptotes in general pass through the origin and 

 may be found by equating to zero the homogeneous function 

 of the n th degree. We say in general because there is the 

 limitation that *j> (//.J is not to be zero ; that is, by the theory 

 of equations <f> (p,} = must not have equal roots. 



275. We will now consider the case in which <'(/ij) is 

 zero. 



First suppose p equal to n 1. 



If T/T (ftj is not zero /3 becomes infinite, and there is no 

 asymptote for the root /*, of the equation (/i) = 0. But if 

 = the value of ft becomes indeterminate. 



