motions in general dynamics 



185 



we have iv^ = 0, and hence 



Q (w) = 1 + 





whose inverse is 



1 



1 

 -tH^ 



Thus we require, to first power of ju,, 



/ 1, OX 

 \-tH., l) 



Q. 



'){- 



or 



1 , 



1 , 



t {y, y) Ho 



{y, x), 



{x, x}. 



(y^y) 

 {x,y) 



1 



a 



H- 



a.. 



where A = {y, x) 



B = {y, y), 



C=-{x,x} + t [Ho {y, X) + {X, y) Ho\ - tm^ {y, y) H^, 

 D^ - {x,y) + tHo iy, y) . 

 Here the matrizant Q, to first power of ju., is given by the definition 

 D {/x fA, 



\C, 

 and the product is 



+ /X 



B\}=1 + fi (QA, • QB\ 

 Dj \QC, QDJ 



^-tHo, l) 



QA , QB \ 



tHo . QA + QC, - tHo . QB + QDj 

 that is / 1 + ixQA , [jlQB 



[- tHo (1 + l^Q^) + QC, 1 - /^«^o • QB + l^QDy 

 In the paper referred to the rule reached for determining the 

 characteristic exponents was (p. 163) to pick out the coefficient 

 of t in the matrix such as this, as it occurs outside trigonometrical 

 signs, then put t = 0, and from this form a determinantal equation. 

 That rule was founded on the assumption (p. 162) that a certain 



matrix, Q]^ (u), had, in case of equal roots, only linear invariant 

 factors. It will be proved below that for the application of the rule 

 the assumption is unnecessary; this is important in the present 

 case since two of the characteristic exponents are necessarily zero, 

 as we shall see. Of the quantities A, B, C, D, which are given 

 exphcitly above, only parts are material for the result. Denote 

 then provisionally the coefficient of t in the matrix last written by 



where under the trigonometrical signs t is put zero; the deter- 

 minantal equation is then 



jxa — p , /LtjS = 0. 



— Ho+ {xy, /xS — p 



