44 REPORT—1846. 
which are commonly called elliptic functions or elliptic integrals. Fora reason 
which will be mentioned hereafter the latter name seems preferable, and it is 
sanctioned by the authority of M. Jacobi, though the former was used by Le- 
gendre. Elliptic integrals then may be defined as those whose differentials are 
irrational in consequence of involving a radical of the form 4/{a@ + Baty x* 
+0a3+¢a*}. But it may perhaps be more correct to say that all such in- 
tegrals may be reduced to three standard integrals, to which the name of 
elliptic integrals has been given. 
In the Turin Memoirs for 1784 and 1785, p. 218, Lagrange considered, 
as has been already mentioned, the theory of these transcendents. He 
showed that the integration of every function irrational in consequence of 
containing a square root may be made to depend on that of a function of 
the form = P being rational, and R the radical in question ; and that if 
under the sign of the square root 2 does not rise above the fourth degree, 
dx 
Vl + pea) (1 + 9? a?) 
where N is rational in x. He thus laid the foundation of that part of the 
theory of elliptic transcendents in which a proposed integral is reduced 
to certain canonical or standard forms, or to the simplest combination of 
such forms of which the case admits. In Legendre’s earliest writings on 
elliptic functions there is nothing relating to this part of the subject. Having 
thus, in the simple manner which distinguishes his analysis, reduced the ge- 
neral case to that which admits of the application of his method, Lagrange 
proceeded to prove that if we introduce a new variable whose ratio to x is 
the subduplicate of the ratio of 1 + p*2® to 1 + ¢* x*, the last written inte- 
gral is made to depend on another of similar form, but in which p and g are 
replaced by new quantities p' and q'. If p is greater than q, p! will be greater 
than p, and q' less than q, and thus by successive similar transformations we 
ultimately come to an integral in which g isso small that the factor 1 + q' 2 
may be replaced by unity, and the elliptic integral is therefore reduced to a 
circular or logarithmic form. Or by successive transformations in the oppo- 
site direction we come to an integral in which p' and g' are sensibly equal, 
in which case also the elliptic integral is reduced to a lower transcendent. 
This most ingenious method is the foundation of all that has since been 
effected in the transformation of elliptic integrals, or at least whatever has 
been done has been suggested by it. Thus it is to Lagrange that we owe 
the origin of two great divisions of the theory of these functions. 
In the Memoirs of the French Academy for 1786, p. 616, we find Legen- 
dre’s first essay on the subject to which he afterwards gave so much attention. 
We recognise in it what may I think be considered the principal aim of his 
researches in elliptic functions, namely to facilitate, by the tabulation of 
these functions, the numerical solution of mathematical and physical pro- 
blems. 
He begins, not with a general form as Lagrange had done, but with the 
integral fs W1—e?sin? ¢d¢, which as we know represents an elliptic are, 
and shows how other functions, for instance the value of the hyperbolic are, 
may be expressed by means of it, and of its differential coefficient with re- 
spect to the eccentricity c. The memoir does not contain much that is now 
of interest. After writing it he became aware of the existence of Landen’s 
researches ; and in a second memoir appended to the first gave a demonstra- 
tion of Landen’s principal theorem. This demonstration is founded on 
it may ultimately be made to depend on that of 
