52 Prof. Sylvester's Astronomical Prolusions. 



occurred, at which " moment the mercury was precisely at its 

 minimum height"*. 



Having in 1843, at the request of Sir Charles Lemon, the 

 then President, written an account of the extraordinary oscilla- 

 tions of the sea of the 5th of July in that year, I have consi- 

 dered it due to this Society, as well as to myself, to state thus 

 fully my reasons for still regarding the hypothesis I then ad- 

 vanced as the only one capable of reconciling all the known facts 

 connected with these disturbances. 



VII. Astronomical Prolusions: commencing with an instanta- 

 neous proof of Lambert* s and Eider's Theorems, and modulating 

 through a construction of the orbit of a heavenly body from two 

 heliocentric distances, the subtended chord, and the periodic time, 

 and the focal theory of Cartesian Ovals, into a discussion of 

 motion in a circle and its relation to planetary motion. By J. J. 

 Sylvester, F.R.S.f 



THE original demonstration by Lambert of the celebrated 

 theorem which bears his name was a geometrical one. See 

 Monthly Notices of the Astronomical Society, vol. xxii. p. 238, 

 where this demonstration is reproduced, or rather recapitulated 

 by Mr. Cayley. See also Lambert's own Insigniores Orbitce 

 cometarumproprietates, Augusta Vindelicorum [Augsburg], 1761. 

 It occupies seven or eight pages of this celebrated tract, and, 

 elegant as may be considered the chain of geometrical enuncia- 

 tions from which it is deduced, is, as a specimen of geometrical 

 style, little worthy of the inconsiderate commendations which 

 have been heaped upon it, containing, as it does, a hybrid mix- 

 ture of algebraical, geometrical, and trigonometrical ratiocina- 

 tion. The late Professor MacCullagh, as I am informed by my 

 ingenious coadjutor Mr. Crofton, one of his hearers at Trinity 

 College, Dublin, greatly improved upon Lambert's method, and 

 succeeded in reducing it to a purely geometrical form. Lagrange 

 has given no less than four distinct demonstrations of the same, — 



. * Personal Narrative, vol. iii. pp. 316-318. 



t Communicated by the Author. A portion of this paper has appeared 

 in the Monthly Notices of the Astronomical Society of London for 

 December last, viz. so much of it as relates to Lambert's theorem proper. 

 The portion concerning circular motion formed the subject of a communi- 

 cation to the London Mathematical Society at the Meeting of December 

 18, 1865. The part which presented itself last to the author's mind, and 

 is consequently the least developed, is that which relates to the determi- 

 nation of the forces in any orbit to any two (or more) centres of force. 

 The general expression for such forces will be found stated further on in a 

 footnote, where the equation of radial work is defined and employed to 

 obtain the solution in a form of unexpected simplicity. 



