1877.] Presents. 363 



i 



whence 1 -f- cos -j = 0, which gives 

 P* 



_1_ 



97r 2 ' ~ 25:r a 



,p= — or — x-t, or — 



that is to say, the series will be convergent if — , — , ■— — . . . are all 



less than unity — that is, if x is less than which accords with the 

 result given by Cauchy. 



Let us next consider the series 



€ x*+x Mq _j_ u ^ x U j* m # # ^ 



we easily obtain by differentiation the following equation for deter- 

 mining u n , 



O + l) u n+ i = w Jt -f 2u n _ h 



which may be written 



.— =— + 2. 



As w. increases without limit, ^dl — J^L. ^ } an d the equation becomes 



u n u n—l 



p 2 = 0, or p =x 0, whence the series is convergent however great a; may 

 become. 



Presents received, June 14, 1877. 



Transactions. 



Batavia : — Bataviaasch Grenootschap van Kunsten en Wetenschappen. 

 Tijdschrift voor Indische Taal-, Land-, en Yolkenkunde. Deel 

 XXIII. AA.5&6; Deel XXIV. Afl. 1-3. 8vo. 1876. Notulen 

 van de Algemeene en Bestuurs-Vergaderingen. Deel XIV. No. 

 2-4. 8vo. 1876-77. Catalogus de Ethnologische Afdeeling van 

 het Museum. Tweede Druk. 8vo. 1877. Verslag van eene Ver- 

 zameliiDg Maleische, Arabische, Javaansche en andere Hand- 

 schriften. 8vo. 1877. Het Maleisch der Molukken. 4to. 1876. 



The Society. 



Boston [U.S.] : — Bussey Institution. Bulletin. Vol.11. Parti. 8vo. 



1877. Harvard University. 



Cherbourg : — Societe Rationale des Sciences Naturelles. Compte- 



Eendu, 30 Dec. 1876. 8vo. 1877. The Society. 



Dublin : — University Biological Association. Proceedings. Vol. L 



No. 2. 8vo. 1876. The Association. 



