538 REPORT— 1881. 



As a special case of (4), if is the middle point of AB and describes a circle 

 whose radius is a, while AB revolves uniformly, the areas A and B will be equal 

 epitrochoids, and (4) becomes 



whence 



A = 7r(«* + A*«), 



in which a is the sum of the radii of the fixed and rolling circles, h is the distance 

 of the generating point fi'om the centre of the rolling circle, and A-1 is the 

 number of the branches of the epitrochoid. 



14. On the Equatimi of the Multiplier in the Theory of Elliptic Transforma- 

 tion. By Professor H. J. S. Smith, M.A., F.B.8. 



15. On a Linear relation between two Quadratic Surds. By Professor 

 H. J. S. Smith, M.A., F.B.8. 



16, On a Class of Binodal Quartics. By Professor R. W. Genese, M.A. 



If A B be third diagonal of a quadrilateral Q, then any quartic with nodes at 

 A, B, and circumscribing Q, has the following properties : — 



1°. If P be any point on the curve, and A P, B P meet the cui-ve again at Q, R, 

 then B Q, A E intersect on the curve ; or, we may state it thus, if A P Q be any 



secant, and B P, B Q meet the curve again in R, S, then R S passes through A, 

 2°. If A P Q tm-n round A, then B P, B Q belong to a fixed involution. 

 The equation to the curve in biangular co-ordinates is 



{a^x- + b^x + Cj)i/ + {OnX^ + bnX + Co)?/ + (a^x"^ + b^x + Cj) = 

 "With the condition 





= 0. 



SATURDAY, SEPTEMBER 3. 

 The following Papers were read : — 

 1. On a Class of Differential Equations. By Professor Halphen. 



2. On the Aspects of Faints in a Plane. By Professor Halphen. 



3. On a Connection between Homoqraphies in a Straight Line and Points 

 in a Space. By Cypaeissos Stephanos. 



I 



