204 Prof. Baker, The theory of stability of rotating masses of liquid 



§ 5. The conclusion so reached is in accordance with a note ap- 

 pearing in the Cwnpt. Rendus (clxx, 5 Jan. 1920, 38; "Calculs de 

 G. H. Darwin sur la stabilite de la figure piriforme," Note de 

 M. Pierre Humbert) long after the above was written, to which 

 my attention was called by Mr F. P. White of St John's College. 

 The author has calculated Darwin's series to a further approxima- 

 tion, and finds a result not strengthening Darwin's prevision. 



But the conclusion is made almost certain by a comparison 

 with Liapounoff's paper, "Sur un probleme de Tchebychef," St 

 Petersbourg Memoires, xvii. For the statements there given refer- 

 ence is made to another memoir, of which the first, the theoretical, 

 part, appears not to be obtainable in England; indeed, were it 

 otherwise, there might be little justification for the preceding sum- 

 mary reniarks, save perhaps on account of the total difference of 

 method, in view of the rigour with which Liapounoff's results in 

 these problems are developed. As was pointed out to me by Mr 

 S. E. U. Savoor, of Trinity College, Liapounoff makes the remark 

 {loc. cit., p. 27) that the terms of various orders in his development 

 of what is here called A, though presenting themselves in the first 

 place as infinite series, can be summed, and then take the form 

 above remarked (§ 4) as belonging to the expansion which can be 

 deduced from Mr Jeans' work. 



Liapounoff however also remarks {loc. cit., p. 30) that the in- 

 stability of the pear follows, when the ellipsoids have been ex- 

 amined, from the sign of one term only, which, in a footnote, he 

 identifies with that above denoted by ii\^— which he states he has 

 expressed in finite terms as an algebraic function of the axes of the 

 ellipsoid ; and gives further, also without proof, a general expression 

 for the increase of angular momentum in passing from the ellipsoid 

 to the pear, with the remark that the (positive) sign of this also 

 follows from the sign of N*. (See Peters. Mem. xxii, 1908, 126-131.) 



n\*i "^^^ following references to Liapounoff's papers may be useful to the reader: 

 (1) 1884, 'Sur la stabilite des figures ellipsoidales," Toulouse Annates, vi, 1904 

 (translated from the Russian) ; (2) 1903, "Recherches dans la theorie de la &^\ire " 

 fA '^nn^^^o'*^ Memoires, xiv; (3) 1904, "Sur I'equation de Clairaut," ibid, xv; 

 (fl ^^y?' ^"'' un probleme de Tchebychef," ibid, xvn; (5) 1906, Sur les figures 

 deguthbre peu differeyiies des ellipsoides, Part I— published separately, unobtain- 

 able m England; (6) 1908, "Probl6me de minimum...," St Petersbourg Memoires, 

 fr^^k^n Toulouse Annates, ix, "Probleme general de la Stabilite du mouvement"; 

 (7) 1909, second part of the memoir (5) ; (8) 1912, third part of the same. The second 

 and third parts are in the British Museum (as was first discovered for me by Mr 

 t. P. White); (9) 1916, "Sur les equations qui appartiennent aux surfaces des 

 ^!JT.It 'I'JJ^'''''® derivees des eUipsoides d'un liquide homogene en rotation"; 

 and Nouvelles considerations relatives a la theorie des figures d'equilibre derivees 

 des ellipsoides dans le cas d'un liquide homogene," St Petersbourg Bulletin, 1916. 

 Ihese last two references I take from an article by L. Lichtenstein, "Gleichge- 

 wichtsfiguren rotierender Fliissigkeiten," Math. Zeitschrift, vn, Berlin, 1920 132 

 The same writer also gives {ibid, i, 1918, 232) reference to a fourth part of the memoir 

 (8), under date 1914, and to a memoir, Ann. de r^c. Norm, xxvi, 1909, 473. 



Mr fe. K. y Savoor has made a detaUed application of the method of Liapounoff 

 to the case of the rotating cylinders, of which it is hoped that a summary may be 

 published in the Trans. Camb. Phil. Soc. ' ^ "^ 



1 



i.irJ, ■ 



