478 THOMAS BOND SPRAGUE ON THE CURVES WHOSE INTERSECTIONS 
second asymptote, the sum is 27; and so on. Lastly, if we suppose y to have 
the fixed value 4, and assign to o the values, 7, 27, 37... ., we get (n—1) 
values of x ; and since the values of o lie singly between two adjacent values of 
o for the curve P, we see that the »—1 points in which the straight line y=A, 
cuts the curve Q, lie singly between the » points in which it cuts P. Thus the 
proposition enunciated at the outset is completely established. 
When the roots of f(«)=0 are not all real, the equations (P) and (Q) still 
admit of a simple geometrical interpretation. Suppose there is a pair of 
imaginary roots f+ig, f—7g; then the corresponding factors in f(x + 7) become 
Bath —ig +1y)(x—f+ ig +iy)=R,R,(cos w +7 sin p)(cos vy +7 sin v) 
=R,R,{cos (u+v)+7 sin (u+v)} 
if Ri=(«@—f)+(y—g), Ri=(x@—fy+(y +g)’, tan pait tan polit, 
It is easy to see from this that cos (us;) J (a) =0 leads to 
a ot Be a esl -1Yt9 aye 
tan "pg t ban aoe : . +tan oe . pray Da Sy tm. 
In fig. 19, let R. be the point. (2, y), OA=a, OB=5,.... OF=f, 
RS=RT=g; then the equation expresses that the sum of the angles RAX,, 
BB Xtras st Siw LER Se is is equal to = , or on, or 3m, &c.; or the 
curve P is the locus of a point for which this is the case. Similarly the curve 
Q is the locus of a point for which the sum of the angles is z, 27, 37,.... 
Whether the roots of f(~)=0 are real or imaginary, the  asymptotes of P and 
n—1 asymptotes of Q, are all real; but when some or all of the roots are 
imaginary, our demonstration that the infinite branches lie below the asymp- 
totes, no longer applies ; for p, may then be either positive or negative. 
In conclusion, it may be useful to give a few examples of the actual forms — 
of the curves. . 
. First, we will take a case where all the roots are real 
at — 25a? + 60a—36 =(«—1)(x—2)(x2—3)(a +6) =0. 
Then the equations to the two curves are 
oy —6a'y? +a*+25(y’—a’)+60x—36=0 . . . . (P) 
2y°x — 2a? + 25%—30=0 pele ei ers Te (2) 
and the curves assume the shape shown in fig. 20, where the thin branches 
belong to P and'the thick belong to Q. Consistently with what has gone before, 
P and Q do not intersect, but a branch of Q lies between each adjacent two. 
branches of P; and, the equation being of the fourth degree, P has four asymp- 
