704 



cular surfaces so narrow vertically, in proportion to their breadth, 

 as they are in the cervical vertebrae of the Pterosauria : in the 

 dorsal series the cup and ball present more ordinary Saurian pro- 

 portions. 



Besides these principal and more general characters, those also which 

 distinguish the vertebrae of the several regions of the spine, together 

 with the specialities of the atlas and axis, and of other individual 

 vertebrae, are pointed out and described. 



The Paper is illustrated by numerous figures, which (excepting 

 two from the Aptenodytes) belong to the Pterodactyle. 



March 31, 1859. 

 Sir BENJAMIN C. BRODIE, Bart., President, in the Chair. 



The following communications were read : 



I. "The Higher Theory of Elliptic Integrals, treated from 

 Jacobi's Functions as its basis." By F. W. NEWMAN, Esq., 

 M.A., Professor of Latin in University College, London. 

 Communicated by the Rev. Dr. BOOTH. Received March 



3, 1859. 



(Abstract.) 



The peculiarly beautiful properties of these integrals, as treated by 

 Jacobi and (in his two supplements) by Legendre, are obtained 

 through so very elaborate and difficult a process, that few students 

 can afford the time to study them. Professor De Morgan, in his 

 * Integral Calculus/ declines to enter even the Lower Theory, on the 

 ground that the subject requires a detailed treatise. That in some 

 sense it is analogous to trigonometry, which no one would desire to 

 be treated fully in the differential and integral calculus, has been re- 

 cognized by several writers. Legendre, in his second supplement, 

 sixth section, took the first steps toward treating Jacobi' s functions 

 (A and 0) on a wholly independent basis, by investigating their pro- 

 perties from the series which they represent: but after only two 

 pages of this sort, he aids his research by assuming their relations 



