78 REPORT—1846. 
attach a definite sense to the equation « = 9 x + @y, or tosee how the value 
of w is determined by it *. 
31. Two divisions of the theory of the higher transcendents here suggest 
themselves, which are apparently less intimately connected than the corre- 
sponding divisions in the theory of elliptic functions, viz. the reduction and 
transformation of the integrals themselves, and the theory of the inyerse 
functions. 
But before considering these I shall give some account of what has been 
done in fulfilment of the suggestion made by M. Jacobi at the close of 
the ‘ Considerationes Generales.’ Mathematicians have succeeded in effect- 
ing the integration of the system of differential equations to the consideration 
of which we are led by Abel’s theorem, and which is commonly designated 
by German mathematicians as the ‘“‘ Jacobische system ;” its existence and 
its integrability having been first pointed out by M. Jacobi. 
In Crelle’s Journal (xxiii. 354), M. Richelot, after modifying the form in 
which Lagrange’s celebrated integration of the differential equation of 
elliptic integrals is generally presented, extended a similar method to the 
system of two differential equations which occurs when we consider the 
Abelian transcendents of the first class. He thus obtains one algebraical 
integral of the system. In the case of Lagrange’s equation one integral is 
all we want; but in that which M. Richelot here discusses we require 
two. Nowif in the former case we replace each of the variables by its reci- 
procal, we obtain a new differential equation of the same form as the original 
one, and integrable therefore in the same manner; and if in its integral we 
again replace each new variable by its reciprocal, that is by the original va- 
riable, we thus, as it is not difficult to see, get the integral of the original 
equation in a different form. That the two forms are in effect coincident 
may be verified @ posteriori. But the same substitutions being made in M, 
Richelot’s equations, which are of course those we haye already mentioned 
at p. 76, the first of them becomes similar in form to the second, and vice 
versé the second to the first. Thus the system remains similar to itself; and 
if in the algebraical integral we obtain of it we again replace the new vari- 
ables by their reciprocals, we fall on a new algebraical integral of the original 
system ; this integral being, which is remarkable, independent of that pre- 
viously got. Thus the system of two equations is completely integrated. 
Extending his method to the general system of any number of equations, 
M. Richelot obtains for each two integrals, but of course these are not all 
that we want. At the conclusion of his memoir M. Richelot derives from 
Abel’s theorem the algebraical integrals of the “ Jacobische system.” 
Though in this memoir M. Richelot only obtained by direct integration 
two of the m—1 algebraical integrals of the “ Jacobische system,” yet he put 
the problem of its complete integration into a convenient and symmetrical 
form. As there are m variables and m — 1 relations among them, we may 
suppose each to be a function of an independent variable ¢ Lagrange, as 
we know, in integrating the equation 
ae dy _ 0 
A Beri AX i000 
* The difficulty here mentioned may perhaps be met by saying that the value of ¢ de- 
wddx 
termined by the integral ergs is necessarily determinate, and so likewise is that of w. 
0 
That considerations connected with the conception of a function inyerse to ¢# make the 
latter quantity appear indeterminate is undoubtedly a difficulty ; but it is, so to speak, a 
difficulty collateral to M. Jacobi’s theory, and therefore need not prevent our accepting it. 
