566 EEPORT — 1891. 



one position, until it is compresse.d to such an extent that its form becomes lenti- 

 cular, tlie edges of the lens being just iu contact with the sides of the tube when 

 it commences to ascend. There appears, therefore, to be a special temperature for 

 each liquid, at which the angle of contact of its meniscus with the walls of the 

 containing vessel is zero. 



Depaetmext II. — Mathematics. 



1. Interim Eeport of the Committee on Mathematical Functions. 

 See Reports, p. 129. 



2. Interim Report of the Committee on the PelJian Equation Talks. 

 See Reports, p. 160. 



3. 0?!. Periodic Motion of a Finite Conservative System.^ 

 By Sir William Thomson, Pres.B.S. 



4, On a Geometrical Ilhi.itration of a Dynamical Theorem. 

 By Sir Robert Ball, F.E.S. 



It was observed in this paper that a dynamical system Avheu moving in any 

 way could be constrained to adhere to the same mo'tion, so that every element 

 fihould continue to twist about the same screw as it was twisting: about at the 

 moment. The forces to be applied for this purpose could be simply expressed, and a 

 geometrical construction was given in the particular case of a rigid body, which 

 was possessed of three degrees of freedom. It was shown that the screws about 

 which a body so restricted could twist might be represented by points in a plane 

 made on two ellipses, one representing the screws about which the body could 

 twist with zero kinetic energy, the other representing the screws of zero pit'ch. It 

 was then shown that two homographic systems of points could be constructed such 

 that if any point P be joined to its correspondent Q, tlien the hole of the ray with 

 regard to the pitch ellipse represents the screw on which a wrench could be placed 

 ■which should just steady the motion. The pole of the same ray with regard to the 

 knaetic ellipse gives the acceleration of the body if permitted to pursue its move- 

 ment without interference. A complete account of the investigation will shortly 

 appear in the publications of the Royal Irish Academy. 



6. 071 the Transformation of a Differential Resolvent. 

 By the Rev. Robert Harley, M.A., F.B.S. 



If there be two algebraic equations such that they can be changed, the one into 

 the other, by assuming, without loss of generality, certain relations among their 

 variables; and if the differential resolvent of one of these equations is known, how can 

 we pass directly to the differential resolvent of the other, without having recourse 

 to a separate and independent calculation ? That is the question I propose to 

 consider in the present paper. Nearly thirty years ago, when seeking to determine 

 the form of the differential resolvents of two trinomial algebraic equations connected 

 in the manner above described, I endeavoured to effect a passage from one differ- 

 ential resolvent to another by a simple transformation, but was stopped by what 

 seemed to me at the time to be an anomalous result. Fortunately the result was 

 placed upon record for future discussion ; it will be found in Art. 13 of a paper read 

 before the Literary and Philosophical Society of Manchester (November 4, 1862), 



' Printed in extenso in Phil. Mag. October 1891. 



