( 107 ) 



for tills fiiiK'tloJi, l)C('ausc It satislics the differential eqiuitioii (jr-\-z G=0, 

 the same expansion holds as for F{t), bnt while In the series for 

 LPZP^ the derivatives are taken with regard to increasing- time, 

 those in that for LP^ZP nnist be considered with regard to decreas- 

 ing time. If we make nse of the symbols i, 5, ^"^ etc. to denote deri- 

 vatives of z with regard to increasing time, the signs of the odd 

 derivatives of z in the expansion for LP^ZP must be reversed. 

 Hence we obtain for LP^ZP : 



LP,Z P 



-3^.) 



2 ) 



b! 



{^z,é, - 2^,^") -^+ 



' rn'' 

 __!_ I _öviii 



(t — n) du 



and bj summation of the two series, for t = r, 

 ^LP,ZP, ^ r, z, + 

 1 



Vp 



2 



-, + (- 



"^^17 + 



-h 



1 



-(z ^ — oz \-^~{z '■ 



3^.)i7-(^- 



■2c,"^)-(3--2^,"^)l-^ + 



^ ^- 13^,:f3 + 10V-5^,i V -^/ ^ 13^,~, + 10i,' 



[' 



pi [i^^VIII (^_„) _|_ ^VlII (^_,,)| j,,^ 



IV 



— + 



It appears that in this formula the tei-ms with even powers of 

 T^ can be transformed into series of terms with the higher odd powers 

 of T^. In order to do this I derive an expansion in series by which 

 this aim is reached in a general manner for the difference ƒ (y) — /(''')>/ 

 being an arbitrary function which between x and y does not show 



singularities. Let here x be put for // — x, and m for '- then 



f{y)—f{x)z=z I /'(»4 + u)du and after integration by parts: 



/(;y) - f{^) = Y [/ {y) + / (•^0] -ƒ ^'/" (^^ + ^0 du. 



-V. 



Proceedings Royal Acad. Amsterdam. Vol. VIII. 



