1890.] Images, and on Amici's Prism Telescope. 87 



as above ; and 



tan 7] _ cW 

 tan e dcf> ' 



where t] is the inclination of the image to the vertical, and <f> is the 

 angle of incidence of the axial ray on the first face. 



(3) On some theorems connected with Bicircular Quartics. 

 By R. Lachlan, M.A., Trinity College. 



The object of this paper is to extend to Bicircular Quartics, 

 and twisted quartics, Sylvester's theory of residuation in connection 

 with the plane cubic. 



1. A curve of the 2nih order, having multiple points of the 

 nth order at each of the circular points, may be called a circular 

 curve of the 2?ith order. Such a curve is determined by n (n + 2) 

 points, as is seen at once by writing down its equation ; and two 

 circular curves of the 2nth and 2mth orders intersect in 2mn points, 

 other than the circular points. 



Let U, V be any two circular curves of the 2nth order passing 

 through n(n + 2) — l given points, then U+kV will represent 

 any circular curve of the 2?ith order passing through these points, 

 bat such a curve must obviously pass through all the points in 

 which U and V intersect. Hence it follows that any circular 

 curve of the 2nih order which passes through n (?z + 2) - 1 fixed 

 points must pass through (n — l) 2 other fixed points. 



2. Further we may show that every circular curve of the 2nth 

 order which passes through 2np — (p — l) 2 points on a circular 

 curve of the 2pth order (p being < n) meets this curve in (p — l) 2 

 other fixed points. For if we draw any circular curve of the 

 2(n— p)th order through (n — p) (>i—p + 2) assumed points, then 

 since 



2np -(p- l) 2 + (n -p) (n -p + 2) = n (n + 2) - 1, 



any circular curve of the 2nth order which passes through the 

 given 2np — (p — If points on the curve of the 2pth. order and 

 also through the assumed points on curve of the 2 (ii—p)th. order, 

 must pass through {n — l) 2 other fixed point. But the given curve 

 of the 2pth order and the curve of the 2 (n — p)th order make up 

 one such circular system ; hence these (n — Yf points must lie on 

 one or other of these curves of lower order. The most that can 

 lie on the curve of the 2 (?i — jj)th order is 



2« (n - p) - (n - p) (n-p + 2), i.e. (n - 1/ - (p - l) 2 ; 



