40 REPORT—1846. 
prochain. J’ose dire sans ostentation que c’est un traité dont on sera satis- 
fait. Je suis curieux d’entendre I’opinion de l'Institut la dessus. Je ne 
manquerai pas de t’en faire part.” Long before this memoir was published 
Abel had become “ chill to praise or blame.” He died at Christiania in the 
spring of 1829. 
M. Jacobi mentions in a note in Crelle’s Journal, that while at Paris he 
represented, and as he believed not ineffectually, to Fourier, who was then 
one of the secretaries of the Institute, that the publication of this memoir 
would be very acceptable to mathematicians. A long period however was 
still to elapse before the publication took place. It was possibly retarded by 
the death of Fourier. In 184] the memoir appeared in the seventh volume 
of the ‘ Mémoires des Savans Etrangers.’ It was prepared for publication 
by M. Libri. 
Thus for about fifteen years Abel’s general theory remained unpublished ; 
but in the meanwhile Crelle’s Journal was established, and to the third vo- 
lume of this he contributed a paper which contains a theorem much less ge- 
neral than the researches he had communicated to the Institute, but far more 
so than anything previously effected in the theory of the comparison of 
transcendents. ‘This is commonly known as Abel’s Theorem. Legendre, in 
a letter to Abel, speaks thus of the memoir in which it appeared :—* Mais le 
mémoire... ayant pour titre ‘ Remarques sur quelques propriétés géné- 
rales,’ &c., me parait surpasser tout ce que vous avez publié jusqu’a-présent 
par la profondeur de l’analyse qui y régne ainsi que par la beauté et la géné- 
ralité des résultats.” In a previous letter, with reference I believe to the 
same subject, he had remarked, “ Quelle téte que celle d'un jeune Norvé- 
gien !” 
Abel’s theorem gives a formula for the comparison of all transcendental 
functions whatever whose differentials are irrational from involving the square 
root of a rational function of z. 
In a very short paper in the fourth volume of Crelle’s Journal, which 
must have been the last written of Abel’s productions, the chief idea of his 
general theory is stated; and in the second volume of his collected works we 
find a somewhat fuller development of it, in a paper written before his visit 
to Paris, but not published during his lifetime. 
While Abel's great memoir remained unpublished at Paris, several mathe- 
maticians, developing the ideas which he had made known in his contribu- 
tions to Crelle’s Journal, succeeded in establishing results of a greater or 
less degree of generality. Researches of this kind may be presented in a 
variety of forms, because the algebraical function to be integrated, which 
we have called y, may be defined or expressed in different ways. For in- 
stance, if M and N are general symbols denoting any integral functions of x, 
VM a" ae : : 3 
N and y as precisely equivalent, since 
by an obvious reduction, and by changing the signification of M and N, the 
one may be transformed into the other; and so in more general cases. Thus 
the same function may assume a variety of aspects, and there will be a cor- 
responding variety in the form of our final results. 
In Crelle’s Journal we find a good many essays on this part of the sub- 
ject: of these I shall now mention several. 
M. Broch is the author of a paper in the twentieth volume of Crelle’s 
Journal, p. 178. It relates to the integration of certain functions irrational 
in consequence of involving a polynomial of any degree raised to a fractional 
power. For these functions he establishes formule of summation, which of 
the two suppositions 7 = 
