80 REPORT—1846. 
sum, and the independent variable being ¢. From the first integral of this 
equation it is easy to eliminate the differentials, and we have thus an alge- 
braical relation in a, y, ...z, from which, in the manner already mentioned, 
M. Richelot deduces another. We now see that if we could find any other 
symmetrical function which would lead to an integrable equation we should 
get a finite relation among the variables. : 
In the next volume of Crelle’s Journal M. Jacobi took the following func- 
tion as his principal variable, 
V{(e—2) (uy). (e- 2} 
p being a root of X =O or fe =0 if we suppose X=fz. Calling this 
function v, we get a simple differential equation in v and ¢, and a correspond- 
ing integral of the system. Now fx =0 has 2m or 2m — 1 roots, and we 
only want m — 1 integrals. The integrals therefore which we get by making 
p the first, second, &c. root of fx = 0 are not all independent. 
In the twenty-fifth volume of Crelle’s Journal M. Richelot resumed the 
subject of his former paper, and discussed it in a very interesting memoir. 
This fundamental or principal result may be said to be a generalisation of 
M. Jacobi’s. It is that in the function 
V {(u — 2) (uy) ++ (n— 2)} 
f» may have any value whatever. The resulting differential equation, 
though rather more complicated than when, with M. Jacobi, we suppose 
a root of fz = 0, is still very readily integrable. We have thus an indefi- 
nite number of algebraical integrals, since the quantity ~ is arbitrary, but 
of course not more than m — 1 of them are independent. 
In the same volume of Crelle’s Journal, p. 178, there is a curious paper by 
Dr. Heedenkamp, in which the algebraical integrals of Jacobi’s system are 
for the case of a polynomial of the fifth degree under the radical deduced 
from geometrical considerations. It is shown that in a system of curvilinear 
coordinates (those of which MM. Lamé and Liouville have made so much 
use), the equations of the system are the differential equations given by the 
Calculus of Variations for the shortest line between two points. Conse- 
quently the finite equations of a straight line are the integrals sought. This 
very ingenious consideration is afterwards generalised. 
32. In the twelfth volume of Crelle’s Journal, p. 181, M. Richelot has 
considered the Abelian integral of the first class. The principal result at 
which he arrives is, that the only rational transformation by means of which 
such an integral may be changed into one of similar form is linear in both 
the variables which it involves. By means of this substitution, he transforms, 
under certain conditions, the integral in question into a form analogous to 
the standard forms of elliptic integrals. The subject of the division of hyper- 
elliptic integrals of each class into three genera is also considered, and the 
same principle of classification as Legendre’s is made use of. The paper 
concludes by pointing out an error which Legendre committed in the appli- 
cation of his principle. Legendre had thought that the formula of summa- 
tion given by Abel’s theorem for integrals of the form of Vaz could not 
involve a logarithmic function. Thus these integrals would belong to the 
first or second kind, according to the value of the index e, and of A the degree 
of gz. But in reality, though the integrals in question are of the first kind 
(that is, they admit of summation without introducing either an algebraical 
or logarithmic function) if e be less than a certain limit, yet if it be not so 
their formula of summation will in general involve both algebraical and loga- — 
