36 REPORT—1846. 
rectification of elliptic arcs, or, as we should now express it, is reduced to 
elliptic integrals. But of these instances Legendre has remarked that they 
are isolated results, and form no connected theory. MacLaurin is charged, — 
in a letter appended to the works of Fagnani, with taking from the latter, — 
without acknowledgement, a portion of his discoveries with respeet to the — 
lemniscate and the elastic curve. 
4. In 1761, Euler, in the ‘ Novi Commentarii Petropolitani’ for 1758 and 
1759, published his memorable discovery of the algebraical integral of the — 
equation ; 
m dx n dy 
(A+ Ba4+ Ca? + Das + Eat}? (A+By+Cy+ Dy + Ey)? 
m and m being any rational numbers. 
He says he had been led to this result by no regular method, “sed id 
potius tentando, vel divinando elicui,” and recommends the discovery of a — 
direct method to the attention of analysts. In effect his investigations re- 
semble those of Fagnani: he begins by assuming a symmetrical algebraical 
relation between the variables, and hence finds a differential equation which 
it satisfies. In this differential equation the variables are separated, so that 
each term may be considered as the differential of some function. With one 
form of assumed relation we are led to the differentials of circular, and with 
another to those of elliptic integrals, and so on. It is in this manner that 
Dr. Gudermann, in the elaborate researches which he has published in Crelle’s 
Journal, has commenced the discussion of the theory of elliptic functions. 
5. In the fourth volume of the Turin Memoirs, Lagrange accomplished 
the solution of the problem suggested by Euler. He integrated the general 
differential equation already mentioned by a most ingenious method, which, 
with certain modifications, has remained ever since an essential element of 
the theory of elliptic functions. He proceeded to consider the more general 
equation da dy 
where X and Y are any similar functions of a and y respectively, and came 
to the conclusion, that if they are rational and integral functions, the equa- 
tion cannot, except in particular cases, be integrated, if they contain higher 
powers than the fourth. He also integrated this equation in a case in which 
X and Y involve circular functions of the variables. It had been already 
pointed out in the summary of Euler’s researches, given in the ‘ Nov. Com. 
Pet.’ t. vi., that if X and Y are polynomials of the sixth degree, the last- 
written equation does not in general admit of an algebraical integral, since, 
if so, it would follow that the solution of the equation wisi = 4 , which 
[+23 1+ 
(as the square of 1 + 23 is a polynomial of the sixth degree) is a particular 
case of that which we are considering, could be reduced to an algebraical 
form. Now this solution involves both circular functions and logarithms, and 
therefore the required reduction is impossible. This acute remark* showed 
that Euler’s result did not admit of generalisation in the manner in which it 
was natural to attempt to generalise it. It was rese#ved for Abel to discover 
the direction in which generalisation is possible. 
6. The discovery of Euler, of which we have been speaking, is in effect 
the foundation of the theory of elliptic functions, as the generalisation of it 
by Abel, or more properly speaking, the theory of which Euler’s result is an 
* M. Richelot, in one of his memoirs on Abelian or hyper-elliptic integrals, quotes it, in 
a slightly modified form, from Euler’s ‘ Opuscula.’ 
