554. 



REPORT — 1901. 



considered as covered by Routh's ^ and Weierstrass's = investigations as to the 

 stability of a state of steady motion. 



Amongst other results, Charlier finds a method for reducing the disturbing 

 function to a canonical form. As I have recently indicated '' a process for the 

 reduction in the more general case of any steady motion, it may be worth while to 

 show how my method is simplified in Charlier's case. Using the notation of my 

 own paper, Charlier's disturbing function is given by writing b,s = 0, c =a'„ 

 so that 



H2 = A2a,, ,r, X, + i2.2,,, f ■ ^, {r, 5 = 1,2,. . ., w) 



where 



<'ra 



and the equations of motion are 



dt di,' dt dx^' 

 Then the determinant which I employ is 



0, 0, . . „ ;., 



which is readily reduced to the form 



/u+M". fvv 



•) Jm 



Jinf 



fn.., . ■ ; fun + M' 



where As = f?,.iri',,, + «,.,«,,+ . . . +«,„«„„ so that f,, = f . It follows by a 

 theorem due to Frobenius * that the values of /i- are equal to those of - X-, where 

 X is a root of the equation 



., a.n i =0 



«,i-X, 



«1o, 



a.,., — X, 



«..n 



By another theorem due to Frobenius, the invariant factors of the equation 

 in fx are linear, as a consequence of the linearity of those of the equation in X . 

 That the latter are linear was proved by Weierstrass {I.e.) (' Berliner Monats- 

 berichte,' 1858, p. 21-5; 1868, p. 336). It follows that if X = a„ a„, . . ., a„ are 

 the (real) roots of the equation in X ; then fi.= ±ia^, ±ia„, . . ., '±?a„ are the 

 roots of the equation in ^ (of which any number may be* equal) ; and so the 

 method of § 3 of my paper already quoted can be applied to bring Hj to the form 



i2a,x,'i/ (r = l,2,. . .,«) 



the canonical equations of motion being unaltered. In this form the reality of 



' Adams Prize Essay, 1877, Stahility of Motion ; cf. Thomson and Tait, Natural 

 Philosophy, art ,34.3 m. 



^ Berliner Monatsherichte, 1879, p. 430. 



' Proc. Land. Math. Soc, vol. xxxii. 1900, p. 197 (see also p. 325). 



■* Crelle's Journal f. d. Math., Bd. Ixxxiv. 1878, p. 1 (see p. 11, Satz iii. and p. 25, 

 Satz v.) i- \ r ! i- ) 



