(us46ar) 
XIX.—On the Nature of the Curves whose Intersections give the Imaginary 
Roots of an Algebraic Equation. By THomas Bonn SpraGuE, M.A., F.R.S.E. 
(Plate XXIV.) 
(Read 20th February 1882.) 
“Tf the roots of the equation 
al OE tla ta ta CUP 0. ce Dew: va ly cena vey 0 
“are all real, then all the roots of the equations 
cos (A*) f(a) =s(a) Ea) +P @)— . . . . =0 
“and 
= sin (45) f (0) = J '(#)- = aif (a) + ae ee |) 
“are also real, and one of the »—1 roots of the latter lies between each adjacent 
“two roots of the former.” 
In DE Morean’s Trigonometry and Double Algebra, and in ToDHUNTER’S 
Theory of Equations, there is given a theorem of CAaucuy’s, which lays down a 
rule for determining the number of imaginary roots of the equation f(a) = 
that lie within any assigned limits. ToDHUNTER says of it that it proposes to 
effect with respect to the roots in general what Sturm’s theorem effects with 
respect to the real roots; but it seems to me that this is scarcely correct, and 
that the corresponding theorem as to the real roots, is that the number of such 
roots between a and 0 is the number of changes of sign of f(x) when all values 
of « from a to b are substituted in f(z). 
Tf h+7k is a root of f(x) =0 (where 7 is put for /—1), we have 
f(h+ik) =6"* f(b) = jeos (7, i) +isin (eG)! Ne 
whence 
cos (45, J (hk) =0, and sin (47) f (h) =0 
It follows that 2 and & must satisfy these two equations ; or, in other words, 
they are the co-ordinates of a point of intersection of the two curves repre- 
sented by the equations 
cos (uz) f (y=, “sin (vz. 7) May= 
The number of points of intersection will be the number of roots (real or 
imaginary) of the equation f(z)=0, & being of course zero for the real roots. 
Writing p=0, g=0, for the last two equations, so that f(x+2y)=p+zq, and 
VOL. XXX. PART II. 4¢ 
