28 



of the pupil, do not converge to a point at any position within the 

 eye, but converge so as to pass through two lines at right angles to 

 each other, and, in the ordinary position of the head, inclined to the 

 vertical, as formerly described (Transactions of the Society, vol. ii.). 

 As the luminous point is moved further from or nearer to the eye, 

 the image of the point becomes a straight line in one or other of the 

 positions above-mentioned. Since 1825 the inclinations of the two 

 focal lines to the vertical, their length, and their sharpness do not 

 appear to have undergone any sensible change, but the distances at 

 which the luminous point must be placed to bring the focal lines re- 

 spectively exactly upon the retina are increased, having been formerly 

 3"5 and 6 inches, and being now 4*7 and 8*9 inches. Thus while 

 the shortsightedness of the eye is diminished the astigmatism remains 

 the same. 



On the Geometrical Representation of the Roots of Algebraic 

 Equations. By the Rev. H. Goodwin, late Fellow of Caius Col- 

 lege, and Fellow of the Cambridge Philosophical Society. 



The changes of value of any function of x, f (x), may be very 

 clearly, and for some purposes very usefully represented, by tracing 

 the curve defined by the equation z=f (a?) ; and the positive and 

 negative roots of the equation f (x) = will be the distances from 

 the origin at which the curve cuts the axis of x. 



In this memoir a similar method is applied to the representation 

 of the changes of value of a function of (x), corresponding not only 

 to real values of x, but also to values of the form x +y \J — 1 . If we 

 make z=-f (x + y V —1). and restrict ourselves to real values of z, 

 the equation separates itself into two, which, it is shown, may be 

 represented symbolically by 



* =cos (/£) /0c) 

 ando=:in (^) /Gr) ' 



and these will correspond to a curve of double curvature, the inter- 

 sections of which with the plane of xy will determine by the distance 

 of those points from the origin the imaginary roots of the equation 



/O)=o. 



The properties of this curve are fully discussed for the case of/ (a?) 

 being equivalent to x n +p x x n -i + p 2 %n-2+ .... +pn, where jo, p% 

 . . . . p n are real ; and the following results are obtained. 



1 . The ordinate of the curve admits of no maximum or minimum 

 value. 



2. The curve goes off into infinite branches, which lie in asymp- 

 totic planes equally inclined to each other, and which tend alter- 

 nately to positive and negative infinity. 



3. Any plane parallel to the plane of xy cuts the curve in n points 

 and no more. 



