1877.] 



On certain Definite Integrals. 



507 



division of Tortoises is confined to Aldabra and never extended to the 

 Mascarenes proper, and that the Tortoises from the latter islands can 

 be externally, though not osteologically, distinguished as a whole from the 

 Galapagos Tortoises, as will be seen from the following synopsis : — 



I. Nuchal plate absent. Frontal portion of the skull flat. Fourth 



cervical vertebra biconvex. Pelvis with broad symphysial bridge. 



A. Grular plate double ; sternum of moderate 



extent Galapagos Toetoises. 



B. Gular plate single ; sternum short Mascaeene Toetoises. 



a. Carapace thin, thickened towards the margins ; centre of 

 the last vertebral plate raised into a hump, which is 

 separated from the penultimate vertebral by a trans- 

 verse depression : Tortoises of Mauritius (T. triserratay 

 T. ine^ta, T. indica, T. leptocnemis). 



h. The entire carapace extremely thin and fragile, all the 

 bones very slender: Tortoise of Rodriguez {T . Vosmceri). 



II. Nuchal plate present. Frontal portion of the skull convex. Third 



cervical vertebra biconvex. Pelvis with narrow symphysial bridge. 

 Gular plate double. Carapace thick. Aldabea Toetoises (T. 

 elephantina^ T. Daudinii, T. jponderosa, T. hololissa). 



II. On certain Definite Integrals .^^ By W. H. L. Russell, 

 r.R.S. Received December 5, 1876. 



The following paper is a continuation of one recently inserted in the 

 * Proceedings of the Eoyal Society.' 



a + 4 



(13.) O c70log,(l + 2acos'0cos20 + a'cos'0)=7rlog, 

 Jo 



(U) f °° ^^^\/ l + acos0+ V'l + 2a cos 4- = 

 Similarly we may find 



J. - na, -1 



dd\/ cos 20 + ^ cos 30 + x/l + 2fji cos + yu log, ^^x-a- 



ne ^ loge COS COS 20 _ TT l-f^T 



^ •'^ J ^ 1^2^c^"40+^^ ~ 8<l-a7'0 T^x 



/-.^r^^''.7n siu + .1?' siu 30 /l+V^l* 1^ -l/'l 



(17-) l-2..-cos40 + .- = Vlt^"g-Vw^) +2^^^ ^"/' 



