Among the most noticeable papers in the Collected Mathematical Papers of H. J. S. Smith we have the hitherto unpublished “Memoir on the Theta and Omega Functions,” xliii. (vol. ii. pp. 415–623), written in connexion with Dr Glaisher's Tables of the Theta-Functions and originally intended as an Introduction thereto. It appears that in 1873 or 1874 Dr Glaisher asked him, as a member of the British Association committee for the calculation of the Tables, whether he would contribute an Introduction. His reply was that he did not see his way to writing anything appropriate to the tables themselves, but that he “could say something with respect to the constants at the head of the pages.” These constants were K, K′, E, J, J′, &c., the numerical values whereof were given for every minute of the modular angle. The memoir grew in extent, and it was finally decided that it should follow these yet unpublished tables with the before-mentioned title, “Memoir on the Theta and Omega Functions,” but fortunately it has at length appeared in the Collected Papers as above.

In explanation of the title and scope of the memoir, it will be remembered that the Theta-Functions are functions of two arguments, x and q; so that giving to x the value zero or any numerical value, or any value depending on that of q, we obtain a series of functions containing the single argument q, or writing as usual q = eiπω, say the single argument ω; and in the memoir, the attention is directed chiefly but not exclusively to these functions of a single argument which are termed Omega Functions.