292 



8 



generally equal to the number s of degrees of freedom. In special cases, however, 

 this number may be less than s, viz. in such cases where, for all values of the I's, 

 Ihere exist one or more relations of the type 



^ muwk = 0, (13) 



where the m s are a set of integers possessing no common divisor. In fact it is 

 easily seen that by means of n relations of this kind it is possible to eliminate n 

 of the quantities wu in the expressions rj^^ + ... Zstos, so that these expressions 

 assume the form TjCMj -|- . . . ts-n^fs-n- Conditionally periodic systems for which 

 relations of the type (13) hold are called "degenerate" and play an important part 

 in the quantum theory. In § 2 we shall meet with a typical example of a degene- 

 rate system. 



We shall now proceed to derive expressions for the values of the coefficients 

 C, which occur in the expansion in a trigonometric series 



where /"(Çj, ■ • qs) is a one-valued function of the 7's. According to Fourier's theo- 

 rem we have') 



Cr„ .. r, ji/C?!, • • • g,)e-2^'<^'"''+ - +'v«','rfWi • • dws, (14) 



where the g's are regarded as functions of the ty's and the /'s. We shall transform 

 this expression into a multiple integral taken over the g's, instead of over the w's, 

 by means of the transformation formulae (8), which by means of (7) may be written 

 in the form 



The functional determinant of this transformation is given by 



5(9,, .:. 9,) - \ dhdqi \ ~\ dh\~ ' 



and consists of the sum of a finite number of products of functions which each 

 contain only one of the q's. Transforming (14) we now get 



dSi 



Cr„ ..Ts=).... yiq„ ...qs) e-2"'f P'- JT,^ àdq, . . . dqs, (15) 



') See C. V. L. Charlier, Die Mechanik des Himmels. I, p. 1Ü6. It will be noted that the method 

 followed in the present paper is a simple generalisation of the well known method by which the coor- 

 dinates of a planet performing a Keplerian motion are expressed, by means of a simple Fourier series, 

 as functions of the time. 



