u 



( 660 ) 

 [?/] so that,, if in the equation (II) we replace Fs{ii) by <j2k{^^), 

 (_l)/.-i^eU = ,,2Z,-i ■''!•%, Y'^^l .... (Ill) 



the thus (Icliiied function c is algebraically expi-essed in ./• and y. 

 Hut as well as in the e(|nations (1) and (II) " is still disconlinnoiis 

 for integer values of ./• in e(iualion (111). By a slight alteration it 

 is possible to make these discontinuilies disap{)ear. Without altering 

 the value of z for ijiteger values of .r. and // ^ve may \vrite instead 

 of (III) 



(_l)/.-i ^ c2^• ^ ,^k-\ j ■' V{,2/. 0^ 4. ,/,/. ^0) _!_ ,,,^. (,^) p ^.,) j ..(IV) 



2/-! ( ;/=l V •'■ / -^ ) 



and the function : has become continuous ever^'where. The same 

 however is not true for the partial derivatives of : with res])ect to 

 ,1; or //; besides there is as in e<piations (1) and (II) a lack of symmetry. 

 By interchanging x and y \\\r value of z alters. To sonie extent 

 these disadvantages may be eliminated. The process of integration is 

 apt to level finite discontinuities, moreover symmeti-y may be intro- 

 duced by it. And indeed a suitable e.\i)ression of : in the form of 

 a definite ijitegral can i>e given. 



Wc consider the function j delined bv 





K^ 11'' \ <lk{'Vn)<iic{;ii')<JH {V) 



Now z depends symmetrically on ./, and y and is continuous 

 througiiout. The function has continmms derivatives; we may different- 

 iate z a mnnber of /: — 1 limes with i-espect to ,7 and also /—I 

 times with resj)ect to y, either sejtarately or sidisecpiently, before 

 the derivatives lose their continuity, so thai by making /• larger 

 and larger the behaviour of z tends more and more to that of an 

 analytical function of two real variables. 



We now again substitute in (Vj .i'=z((I). 1/ =1 ^^1) and as the 

 trigonometrical series (jkU'ii) and </k{;/") are absolutely convergent 

 (under the supj)Osition /■ ^ J ) we may multij)ly termwise and 

 integrate the partial [)roducts. 



Hut after integration a nojivanishing amount is furnished oidy by 



those [)artial products 



!<iH t<ln 



2nliaBu ^.tl^Du, 



cos COS 



in Avhich we have 



