322 report— 1865. 



Among the new genera are — 



1. Palinurina .- 2 species 



2. Aeger 3 „ 



3. Glyphsea 2 „ 



4. Pseudoglyphaea 2 „ 



5. Scapheus 1 „ 



besides the genus Eryon, to which two or three new species have to be added. 



I have likewise detected a minute species of Squilla. All these seven 

 genera (save one) characterize also the Solenhofen limestone of Bavaria* 

 (Upper White Jura). 



I have now to notice a remarkable burrowing Crustacean of the family 

 Thalassince (a genus of which (Callianassa) occurs in the uppermost bed of 

 the Cretaceous series at Maestricht), as occurring in our Hempstead series 

 (Upper Eocene) in the Isle of "Wight, and another species in the Greensand 

 formation of Colin Glen, Belfast. 



The death of my brother and colleague Dr. Samuel P. Woodward (my 

 best scientific adviser during the past eight years), has materially retarded 

 my accustomed work ; I beg therefore to be allowed to speak of this as my 

 first report only, and that I may be permitted next year to oifer a more com- 

 plete and detailed statement of my researches in this interesting group. 



Report on the Theory of Numbers. — Part VI. By H. J. Stephen 

 Smith, M.A., F.R.S., Savilian Professor of Geometry in the Uni- 

 versity of Oxford. 



124. Application of the Theory of Elliptic Functions to Quadratic Forms. 

 — The Theta Functions of Jacobi. — It will be for the convenience of the reader 

 to give in this place a brief statement of a few principles and results which 

 belong to the theory of elliptic functions, and to which we shall have occasion 

 to refer in the following articles. 



The Theta functions of Jacobi are defined by the equation 



TO=-f0Q r * 1 "1 



m = — oo 

 or if g«™=2, by the equation 



e*v («,«)= 2 (-i)»»v 



m= — oo 

 In these equations, p and v are given integral numbers ; w is an imaginary 

 constant, having for the coefficient of i in its imaginary part a quantity dif- 

 ferent from zero and positive ; so that the analytical modulus of q is inferior 

 to unity, and the series defining the Theta functions is convergent for all va- 

 lues of x real or imaginary ; lastly, a is a constant at present undetermined, 

 but to which we shall hereafter assign a particular value depending on that 

 of hj. When it is not necessary to specify the value of to, we shall write 

 e i*, v (#)> instead of M> „ {x, ui). The following equations are immediate con- 

 sequences of the definition of the Theta functions : 



* See Oppel's Palaeontolog. Mittheilung. Munich, 1863. 



