CHAP, xi] LIOUVILLE'S SERIES OF EIGHTEEN ARTICLES. 339 



Pepin 26 proved all the theorems in Liouville's first five and last two 

 articles, and (L) of the .sixth. 



E. Meissner 27 proved all the theorems in Liouville's articles VII-XVL 

 Thus there remain unproved essentially only (N) and (Q) of the sixth 

 article. [Piuma, 20 pp. 197-201, proved (N).] ' 



P. Bachmann 28 gave an exposition of selected formulas from Liouville's 

 series. 



A. Deltour 29 proved (a) and recalled how it implies that, if m is odd, 

 the number of decompositions of 4m (or 8m) into a sum of 4 (or 8) odd 

 squares equals the sum (or sum of cubes) of the divisors of m. 



P. S. Nasimoff 30 proved formulas (a) and (c) of Liouville, 1 (F) of Liou- 

 ville, 2 (P) of Liouville, 4 one of Liouville, 18 and related results. 



26 Jour, de Math., (4), 4, 1888, 83-127. 



27 Zurich Vierteljahr Naturf. Ges., 52, 1907, 156-216 (Diss., Zurich). 



28 Niedere Zahlentheorie, 2, 1910, 365-433. 



29 Nouv. Ann. Math., (4), 11, 1911, 123-9. 



30 Application of Elliptic Functions to Number Theory, Moscow, 1885. French re'sume' in 



Annales sc. de 1'ecole norm. supe"r., (3), 5, 1888, 147-64. 





