290 MATHEMATICAL NOTES. 
MATHEMATICAL NOTES. 
1. On Linear Asymptotes in Algebraic Curves : 
A method of finding asymptotes, given by D. EF. Gregory in 
Vol. IV., p. 42, of the Cambridge Muthematical Journal (to which 
my attention was called by Prof. Irving), is so elegant and simple 
that it is surprising it has not yet found its way into the text-books. 
Let the equation to the curve, expressed in rational and integral 
form, be of 2 dimensions, and be arranged in homogeneous functions 
of # and y in descending order, as follows: 
i ny) ae ae Ey tees ee NO 
Then the equations to the asymptotes, (2’, y’ being current coordi- 
nates), are given by 
Jn (GY) =9 
a! Ze (2, y) + y’ a5 fo y) SP ea (x, y) =0 
The expression is left by Gregory in this form, but a little further 
reduction will give it us in a shape in which the equation to an 
asymptote can at once be written down by inspection merely. Thus 
let ; — 2 be a factor of f,(a, y), and let } (#, y) be the quantity 
containing the remaining factors, so that the equation to the curve 
may be aritten 
wv 
GF —2) oN +h Gy) + ne = 0 
then the equation to an asymptote is 
ip > 
G— 2) ohm + fri m) =0. 
The case of an asymptote parallel to one of the axes (e.g., that 
of y) is included in this by making 7 = 1, m — « and evaluating 
( : fa— 1) in the usual way. 
The method fails when the above equation becomes indeterminate 
by the simultaneous vanishing of ¢ and f,_,. which can only hap- 
pen when ¢ (a, y) contains the same factor F — 2 ); that is, when 
there are parallel asymptotes. Perhaps the easiest way of treating 
this case is to substitute in the equation to the curve f (2, y) = 0, 
for x and y the quantities 77 + 2, m7 + y, and to arrange in de- | 
scending powers of r. Then, as before, 7, (/, m) = 0 will give the 
