r 



OF THE ROOTS OF ALGEBRAIC EQUATIONS. 345 



or wp may write these, 



x = p,+ p„_, p cos + p, _.. p cos 9.0 + + p" cos nO | 



= p„_, sin 6 + p„_:. p sin 20 + + p""' sinwOJ 



5. I now proceed to discuss these equations, and shall consider first the equation of the 

 projection of the curve on the plane of a^y. 



This equation is in polar co-ordinates, 



p"-^ &m lid + p, |o"'- sin (h - \)Q + + p„_i sin 9 = (9). 



To find the asymptotes, I observe that p will be infinite when sinw0 = 0, except for = 



and = TT ; hence there will be infinite values of p for = — , — , 



11 n n 



Again, 



sin wd + — sin {n - 1)0 + + ^~ sin = 0; 



P P"' 



.'■ n cos n0 — sin {n - 1)0. ^ 73 = f when ^ = Co, 



p~ do 



dO p, sin (» - 1)0 



or, p' -r- = a ' 



dp 11 cos 710 



And if we put = k —, k having any value from I up to n — 1, 



de . k-ir pi 



p^ — = - sin — . i-. 

 'dp n n 



Hence there will be an asymptote corresponding to each infinite value of p, and these will 

 lie on the left of the corresponding infinite radius vectors looking from the pole. If however 

 we suppose the given equation deprived of its second term, that is, if />, = 0, then the polar 

 subtangent vanisiies and the asymptotes pass through the origin and coincide with the radius 

 vectors ; and since this condition may always be fulfilled, I shall generally suppose that pi = 0, 

 and then it may be stated that the projection of the imaginary branches of the curve on the 

 plane of ay has w - 1 asymptotes, which pass through the origin, are equidistant from each other, 

 and make the same angle with each other as the first of them makes with the axis of x. 



The symmetry of tiiese infinite branches with respect to the origin when p, = seems to me to 



point out a kind of geometrical explanation of the great simplicity introduced in the solution of 



equations l)y first depriving them of their second terms. 



dp 

 (i. To determine where p is a minimum, we have by differentiating (9) and putting — — = 0, 



«p""'cos 710 + (n - i)p,p"''' cos (ti - 1)0 + = (10), 



which equation together with (<)) will give the required values of p and 0. Now if we make 

 = o, which satisfies (9), (10) becomes 



np"-' + (n - i)p,p"~' + = 0, 



<"•, /(p) = 0, 



wliich (if ,r; be written for p) is tl)e equation for determining the maxima and minima of the real 

 branch of the curve; hence p is a minimum for the j)rojection of such ])oints. Hesides these 

 there may be other minimum values of p lying between the different pairs of asymptotes. 

 Vol. VIII. Paut III. Yy 



