476 THOMAS BOND SPRAGUE ON THE CURVES WHOSE INTERSECTIONS 
J (x) =(e—a)(x—b)(a—c).... (aA). 
Hence f(atty)=(x—at ty)(a—b + ty)... . (@—l+ ty) 
Now let —/_=tan a, —*, =tan B, rt ET 1 =tan d; 
and put =R=,/{(«—a)?+y} J {(a—by ty... . S{(a—-PPt+y}. 
Then, by an obvious transformation, 
f(x+ty)=R (cos a+7 sin a) (cos +7 sin B).... 
=R{cos (a+B+....+d)+ésin (a+6+ ....+))} . (B). 
But we have seen that : | 
jf (e+ ty)=cos (v2 )s@) +7 sin (ut) Fle) te 
Hence cos (uz:) £(@)=0 leads to cos (a+@6+....+A)=0, and a+@+.... 
+h=3 +m, where m is an integer ; or to 
tan**_+tan7 4, 4+... +tan, =F + mr. 
This equation admits of an obvious geometrical interpretation. In fig. 17 let 
R be the point (x, y,)and A, B,C... be points on the «x-axis at distances 
from the origin a, b,c... . ; then the equation expresses that the sum of the 
. 2 . 
angles RAX, RBX, ROX,.... is equal to 7, or “a, or =a, or.... 
Mies s a. When y is positive, each of the angles is <7 and their sum is there- 
fore <n; and the curve has n branches, corresponding to these n values 
anne) 
x» 97,.... , and passing through the points A, B,C .. . . respectively. - 
Considering first the branch passing through A, we see that when R moves up 
to A (see fig. 18), the angle RAX is equal to 7:2 and each of the angles RBX, 
RCX .. ... vanishes, so that the sum is equal to 7:2. Next considering the 
branch passing through B, when R approaches B, RAX becomes equal to z, 
RBX to 7:2, and RCX,... . all vanish; so that the sum of the angles is 
37:2. Similarly we see that for the branch passing through C, the sum of the 
angles when R approaches C, is 57:2; and so on, for all the » branches. ~ 
Consider next the infinite branches. We have seen that the asymptotes 
eet 5 aw 3m Sar 2n—1 
are inclined to the w-axis at angles 55, 5,.,5,>-+ ++ 9,73 and when the 
