450 REPOET— 1882. 



that found by Regnault, and lead-glass and common German glass have given 

 nearly identical results. 



A method of treatment to prevent considerable changes of the zero points of 

 mercury thermometers is proposed to be applied to all thermometers before gi-adua- 

 tion, and a new method for graduating thermometers in short sections is recom- 

 mended, which is founded upon a series of determinations of the boiling points of 

 certain pure substances. 



SATURDAY, AUGUST 26. 

 The following Report and Papers were read : — 



1. Report on Recent Progress in Hydrodynamics. Part II. — Special 

 Problems. By W. M. Hicks, M.A.—See Reports, p. 39. 



2. On the Rotation of a Homogeneous Liquid Ellipsoid.^ 

 By A. G. Greenhill, M.A. 



3. On a Proof vf Wilson's Theorem.'^ By Professor Cayley, F.R.8. 



4. Integration of an Equation connected with the Elliptic Functions. 



By Captain P. A. MacMahon, R.A. 

 In ' Comptes Rendus,' tom. Lxvi. p. 1144, Allegret has considered the integral of 



where X = A + Bx + C.i- + D.r' 



Y = A + By + C/ + Dy=, 

 the result being 



where a is the arbitrary constant. 1 



He does not further rationalise it. To obtain the result in a rational form, ' 

 consider the equation 



^ , ^y =0 



{{x-a){x'-2px + q)}i {{y-a){f--2p^j + q)\i 

 and reducing to elliptic integrals hy the substitutions 



X Y 



we have 



= 0, 



where ^3_ q -p" . 



a'-22}a + q' 

 Euler's integral of which is 



^-!-^= — 1 ■ = .r, + W, + z., 

 x,-y, 1 1 ^1 1' 



where Xj = \/('-"i' — ^')j Yj = VC^i^ ~ V^)> and z^ is the arbitrary constant. 

 Now, s, is such that 



r ^^..,[- dy,r dz, 



' This has since appeared in the Proceedings of the Canibridge Philosophical 

 Society, vol. iv. 



' This has since appeared in the Cambridge Messenger of Mathematics, vol. xi. 



