470 THOMAS BOND SPRAGUE ON THE CURVES WHOSE INTERSECTIONS 
either of P or Q, the proper sequence of signs is obtained and the rule applies. 
We thus see that between any two infinite branches of P on which there is an 
intersection, there must lie an odd number of infinite branches, either of P or Q ; 
and the same is true of any two such branches of Q. 
Applying the principles we have established, it is easy to show how to draw 
very complicated curves to which the rule will apply ; but it seems unnecessary 
to pursue this part of the subject further. 
Caucuy does not seem to have investigated the properties of the curves 
P and Q, but it will be found that an examination of them leads to a variety 
of interesting results, one of which I have stated at the outset of this paper. 
As a first step in this direction we expand the equations 
d : ‘A 
cos (v2 )\f (x) =0, and sin (v2) r(@) 00 
and thus get 
Bos y* iv sical —0 P 
Sit) MOTT =O 
and 
uf'(e) 4 7"@)+ GF @—.... =0. 
The latter is satisfied by y=0, or the z-axis forms part of the locus of the 
equation: and putting y=0 in the former equation, we get /(z)=0. It is 
obvious that the points thus determined correspond to the real roots of the 
original equation, and that the imaginary roots correspond to the intersections 
of the curves represented by (P) and by 
fay Lp EF = aidn, 20.0) ile ee 
The only case I propose to examine in any detail on the present occasion is 
when /(z)=0 has no imaginary roots. The two curves must then have no 
intersections corresponding to finite values of y, and we shall presently prove 
that this is the case. For this purpose we must trace the two curves, which 
for brevity we will call P and Q. 
We first observe that since y enters the equations (P) and (Q) only in even 
powers, both curves are symmetrical with regard to the a-axis. Putting y=0, 
the points in which the curves cut that axis are determined by / (2) =0, /’(x) =0, 
respectively. By the theory of equations, since all the roots of the former are 
real, all the roots of the latter (7—1 in number) are also real, and one of them 
lies between each adjacent two roots of the former. We next observe that, if 
J (z)=0 has no equal roots, then /’(7)=0 has none, and each of the curves cuts 
the z-axis at right angles; that is to say, P has m branches and Q has n—1 
branches, all of which cut the z-axis at right angles. In fact, if / is a value of 
