ON THE RECENT PROGRESS OF ANALYSIS. 249 



nineteenth volume, p. 113. In the second (Vol. xxiii. p. 126) 

 the author reproduces the results he had already obtained, 

 pointing out the equivalence of one of them to the theorem 

 established in M. Broch's first essay. Besides this, he dis- 

 cusses a question connected with the reduction of algebraical 

 integrals. 



M. Ramus, in the twenty-fourth volume of Crelle's Journal, 

 p. 69, has established two general formulas of summation ; 

 from the second he deduces with great facility Abel's theorem, 

 and also another result. ..which Abel mentions in a letter to 

 Legendre, published in the sixth volume of Crelle's Journal, but 

 which he left undemonstrated. 



M. Rosenhain's researches (Crelle's Journal, XXVIII. p. 249, 

 and xxix. p. 1) embrace both the summation and reduction of 

 algebraical integrals. His analysis depends on giving the 

 function to be integrated a peculiar form, which he conceives 

 leads to a simpler mode of investigation than any other. 



A paper by Poisson will be found in the twelfth volume 

 of Crelle's Journal, p. 89. It relates to the comparison of alge- 

 braical integrals, but is not I think so valuable as that great 

 mathematician's writings generally are. 



Beside the memoirs thus briefly noticed, I may mention two 

 or three by M. Minding : that which appears in the twenty-third 

 volume of Crelle's Journal, p. 255, is the one which is most 

 completely developed. 



There is also a very brief note by M. Jacobi in the eighth 

 volume of Crelle's Journal. 



10. To the Philosophical Transactions for 1836 and 1837 

 Mr Fox Talbot contributed two essays, entitled Researches in 

 the Integral Calculus. These researches may be said to con- 

 tain a development and generalisation of the methods of Fag- 

 nani. They are however far more systematic than the writings 

 of the Italian mathematician, and if they had appeared in the 

 last century would have placed Mr Talbot among those by 

 whom the boundaries of mathematical science have been en- 

 larged. But it cannot be denied that they fall far short of what 

 had been effected at the time they were published, nor does it 

 appear that they contain anything of importance not known 

 before. I have assuredly no wish to speak disparagingly of 



