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ON THE RECENT PROGRESS OF ANALYSIS. 41 
course include Abel’s theorem, since the latter relates to cases in which the 
fractional power in question is the (4)th. Subsequently to the publication 
of this paper he presented to the Institute a memoir on the same subject, but 
gave to the functions to be integrated a different but not essentially more 
general form. This memoir, which was ordered to be printed among the 
‘Savans Etrangers,’ but which will be found in Crelle’s Journal (xxiii. 145), 
may be divided into two portions: the first contains results analogous to 
Abel’s theorem; the second relates to the discussion and reduction of the 
transcendents which they involve. In this part of his researches M. Broch 
has followed the method, and occasionally almost adopted the phraseology 
of a memoir of Abel, on the reduction and classification of Elliptic Inte- 
grals (Abel's Works, ii. p. 93). MM. Liouville and Cauchy, in reporting on 
the memoir, conclude by remarking that the author “ n’a pas trop présumé 
de ses forces en se proposant de marcher sur les traces d’ Abel.” 
M. Jiirgenson has contributed two papers to Crelle’s Journal on the sub- 
ject of which we are speaking. The first, which is very short, contains a 
general theorem for the summation of algebraical integrals* when the func- 
tion to be integrated is expressed in a particular form. This paper appears 
in the nineteenth volume, p. 113. In the second (vol. xxiii. p. 126) the au- 
thor reproduces the results he had already obtained, pointing out the equi- 
valence of one of them to the theorem established in M. Broch’s first essay. 
Besides this, he discusses a question connected with the reduction of alge- 
braical integrals. 
M. Ramus, in the twenty-fourth volume of Crelle’s Journal, p. 69, has 
established two general formule of summation ; from the second he deduces 
with great facility Abel’s theorem, and also another result, which Abel men- 
tions 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 Jour- 
nal, p. 89. It relates to the comparison of algebraical 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 Tal- 
bot contributed two essays, entitled ‘ Researches in the Integral Calculus.’ 
These researches may be said to contain a development and generalisation 
of the methods of Fagnani. 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 enlarged. 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. 
_Thave assuredly no wish to speak disparagingly of Mr. Talbot; his mathe- 
| matical writings bear manifest traces of the ability he has shown in so many 
_ * Ihave used the expression “ algebraical integrals,” though perhaps not correctly, to de- 
note the integrals of algebraical functions. 
