320 Mr Glaisher, Notice of Prof. Smith. [Feb. 12, 



and its applications to Elliptic Functions he was without a rival, 

 and this was also true of his knowledge of Modern Geometry. 

 Besides his great report on the Theory of Numbers to the British 

 Association, which was well known, he was the author of two 

 memoirs of the highest importance on the subject, published in 

 the Philosophical Transactions, of which it might probably be said 

 with truth that nearly all the results on the subject that had since 

 been obtained on the Continent were included in them as cases of 

 the very general theorems they contain. The value of these 

 papers could not be overestimated ; the general principles enunciated 

 in them gave by a uniform process all the theorems discovered by 

 Jacobi, Eisenstein and Liouville relating to the representation of 

 numbers by four squares and other simple quadratic forms. 

 Besides these researches, Prof. Smith had given the complete 

 solution of the problem of the representation of numbers by sums 

 of squares. Eisenstein had shewn that the series of theorems 

 relating to the representation by squares ceased when the number 

 of them exceeded eight : the only cases that had not been con- 

 sidered were the extremely difficult ones of five squares and seven 

 squares. These yielded to ' Prof. Smith's powerful analytical 

 methods, and he gave the enunciation of the theorems for the case 

 of five and seven squares in the Proceedings of the Royal Society 

 for 1867. In ignorance that the problem had been solved fifteen 

 years before, the question of the resolution into sums of five 

 squares was proposed as the subject for the Mathematical Prize by 

 the French Academy last year. 



Much of the brilliant work performed by Prof. Smith in the 

 younger years of his life remains unfortunately still unpublished. 

 He looked forward to the termination of the labours of the Oxford 

 University Commission that he might devote more time to the 

 completion and publication of his mathematical work ; and in the 

 last year he revised more than 100 quarto pages of his great 

 memoir on the Theta and Omega functions, which was to appear 

 in the same volume as the Tables of the Theta Functions. In 

 this memoir he gives the arithmetical theory of the transformation 

 of Elliptic Functions and connects it with the transcendental 

 theory of Jacobi. He also published during the year one 'note' 

 and the half of a second one, occupying about 50 pages, in the 

 Messenger, on Elliptic Functions when the modulus is complex : a 

 subject of great difficulty and complexity and one well worthy of 

 his powers. There were to be altogether ten notes, most of which 

 are partially written. 



His work was of the very highest class : no mathematician 

 ever set before himself a higher standard of excellence or more 

 nearly reached perfection ; he attacked only important problems 

 that had real influence on the advance of the science, and the 



