j ON THE RECENT PROGRESS OF ANALYSIS. 81 
rithmic functions. Either may, under certain conditions as to the form of 
@ 2, disappear, but while gz is merely known as the polynomial of the Ath 
degree, we cannot decide whether the integral is to be referred to the second 
or third kind. 
I may mention here a very elegant result due to M. Jacobi. It appears in 
the thirtieth volume of Crelle’s Journal, p. 121, and is a generalisation of the 
fundamental formula for the addition of elliptic arcs. With a slight modifi- 
cation it may be thus stated. If gz involve only even powers of z, the 
: 2 pr2m 
highest being 2", then the sum of the integrals To is equal to 
0 
the product of their arguments, that is of the different quantities denoted by 
the symbol z. In this case then the logarithmic function disappears, and 
the integral belongs to the second kind. 
In the twenty-ninth volume of Crelle’s Journal there is a paper by M. 
Richelot on a question connected with hyper-elliptic integrals. The reader 
will find in it a good many fully-developed results, which may be considered 
as particular cases of Abel’s theorem. They illustrate the learned author's 
criticism of Legendre’s classification of hyper-elliptic integrals, though they 
_ are not adduced for that purpose. 
The function M (vide ante, p. 38) is a function of the arbitrary quantities 
a, 6, ...c, which, as has been remarked, may themselves be considered func- 
tions of the arguments 2,, z,,...2,. To determine M as a function of the 
last-written quantities is a necessary ulterior step in almost any special appli- 
cation of Abel’s theorem, and this M. Richelot has done in several interesting 
cases, establishing at the same time the relations which exist among the 
quantities in question. His investigations, however, have an ulterior purpose, 
| and are not to be considered merely as corollaries from Abel’s theorem. 
Another paper of M. Richelot, on the subject of the Abelian integrals, is 
found in the sixteenth volume of Crelle’s Journal, p. 221. The aim of it is 
to furnish the means of actually calculating the value of the Abelian integral 
_ of the first class by a method of successive transformation, that is, by a method 
analogous to that used for elliptic integrals. M. Richelot’s process depends 
essentially on an irrational substitution, by means of which we can replace 
the proposed integrals by two others which differ only with respect to their 
limits. In the development of this idea the author confines himself to the 
first kind of the Abelian integrals of the first class, though the same method 
“may m.m. be more generally applied*. From the formula which expresses 
the proposed integral as the aggregate of two others is deduced another, in 
which it is expressed by means of four integrals, the inferior limits of all being 
zero. The first and second of these integrals differ only in their amplitude, 
| and the same is true of the third and fourth. There are two principal trans- 
‘formations, either of which may be repeated as often as we please ; and though 
it might seem that the number of integrals would in the successive trans- 
formations increase in a geometrical progression, yet by the application of 
Abel’s theorem we can always reduce them to the same number. But the 
‘development of this part of the subject M. Richelot has reserved for another 
“occasion t. 
* 
{M +Nz}dz 
WV {2(1~z) (1—#2z) (1-222) (1—p2z)} Mh at 
7 The integral to be transformed is 
q 
N, &c. being constant. ; 
__ t His transformations ultimately reduce ‘the hyper-elliptic integral to elliptic integrals ; 
the latter may be considered known quantities, ‘vel per paucas adjectas transformationes 
‘directe computentur.”’ 
