22 Prof. R. Clausius on the Relations between the 



further, for abbreviation, //, with the signification 

 _ (n + 2)(n-l) 



fi = 



n + 3 



(47) 



is introduced : — 



«=i, 



a x — + 



\/n + 3 

 n— 1 1 



rc + 3 3 



* 3=± ^^T3 2. 

 ?z + 5 1 



« 4 =-^ 



«*= + 



rc + 3 3 3 . 5 



^ 2 3 .3 + //, 



' 5 ~aA+3 2 4 .3 3 .5' 



rc + 5 2_._3 2 -^ 

 fl e--/ A n + 3 ' 3\5.7* 



2 4 .3 3 .5 2 -2 6 .3V-13V 



a'= + 



^ 



--s/w + 3 



2 6 .3 5 .5 2 .7 



(48) 



(49 



By the employment of the new variable, z, equation (44) is 

 changed into the following, in which the limits of the integra- 

 tion are definitely given : — 



j _ _1_ P*= VT^ d(a + a x z + a^ + a 3 z s + &c.) 



* Z-- VI— ?2 V a 



The integral herein signified falls, according to the terms of 

 the series, into an infinite number of integrals, the values of 

 which can easily be given. Every term that contains an even 

 power of z gives as an integral the value zero. For the terms 

 with odd powers the following general equation (in which v is to 

 be an uneven integer) holds good : — 



J 



V1-92 



d{?) 



=- V1-9 2 * 



1.3.5.7...V 

 1.2.4.6...V-1 



(W 



Vr-.l 



) 2 7T. 



If we apply this formula to the odd terms of the preceding equa- 

 tion, we obtain first 



Ji=|[«,+ |«3(l-? 2 )+|^« S (l-ffT+&c.]. . (50) 



In this equation, for the coefficients a , a 3 , &c. we have to put 



