GIVE THE IMAGINARY ROOTS OF AN ALGEBRAIC EQUATION. 477 
point R moves off to an infinite distance, all the angles RAX, RBX .... 
become equal, and their sum will be 7 : 2 for the first asymptote, 37:2 for the 
second, 57:2 for the third, and so on, being equal to the sums of the angles 
when R approaches A, B,C... . respectively. We have thus proved that.a 
branch proceeds from A to the first asymptote, another from B to the second 
asymptote, and so on; but it may be still more conclusively demonstrated as 
follows. 
We have seen that the x-axis cuts the curve (P) in ” real points, and that 
a line parallel to it at an infinite distance also cuts the curve in 7 real points, 
being coincident with the points in which it cuts the asymptotes; and I will 
now prove that any line parallel to the x-axis cuts the curve in 2 points. Sup- 
pose y to have the fixed value &, and x to receive all values from + to —o ; 
; k . . 7 
then as « decreases from + to a, tan~’ —— continually increases from 0 to = ; 
and as x continues to decrease to —o#, the angle similarly increases from 
a:2 to 7. <A similar proposition is true of each of the angles: hence, as x 
a as oo .... (which we may call 
o) continually increases from 0 to mz. In other words, as « decreases, each 
value of o occurs only once ; therefore, if we assign any value to o between 
0 and nz, there will be one value of «x corresponding to it, and only one. 
2 Re 2n—1 
Hence, assigning to o the values, > 3 = eet th: 5 a, we get for each, one 
decreases from +0 to—o, tan 
value of x; or in other words, whatever value & we assign to y, there are n 
real values of x which satisfy the equation (P). It is easily seen that if, instead 
: k : : : 
of supposing tan, to increase from 0 to 7, we suppose it to increase from 
a to 27, or from 27 to 3a, &c., and if we make the like suppositions with 
regard to the other angles, we shall always get the same values of x if we 
take in each case the proper value of oc. 
Similarly for the curve Q, the equations (B) and (C) (see p. 436) show that 
sin (vez) f(@)=0, leads to sin (a+ B+ ....+A)=0,0ra+B+.... =m7, 
where m is an integer ; or c=7, 27, 37. . . . (w—1)a7. Now we know that Q 
cuts the «-axis in (n—1) points, one of which lies between A and B, another 
between B and C, and so on; and reasoning as above we see that when R 
moves up to the first of these points, o becomes equal to 7; for the second, 27; 
and so on. Also the asymptotes are inclined to the z-axis at angles = F ; 
3 — é ! ; 
pape Si et a; and when R moves to infinity along the first asymptote, each 
nN N ; 
of the angles a, 8, y . . . . becomes equal to 7:n, and their sum is 7; for the 
