254 ON THE RECENT PROGRESS OF ANALYSIS. 



replaced by unity, and the elliptic integral is therefore reduced 

 to a circular or logarithmic form. Or by successive transforma- 

 tions in the opposite direction we come to an integral in which 

 p l and <f are sensibly equal, in which case also the elliptic 

 integral is reduced to a lower transcendent. This most in- 

 genious 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 Legendre's first essay on the subject to which he after- 

 wards 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 problems. 



He begins, not with a general form as Lagrange had done, 



but with the integral IVl tf sin 2 (f)d(f>, which as we know re- 

 presents an elliptic arc, and shows how other functions, for 

 instance the value of the hyperbolic arc, may be expressed by 

 means of it, and of its differential coefficient with respect 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 demonstration of Landen's principal theorem. 

 This demonstration is founded on Legendre's own methods, 

 and he deduces from it the remarkable conclusion, that if of 

 a series of ellipses, whose eccentricities are connected by a 

 certain law, we could rectify any two, we could deduce from 

 hence the rectification of all the rest. The law connecting the 

 eccentricities of the ellipses is that which would be obtained 

 by making use of Lagrange's method of transformation, with 

 which accordingly this result is closely allied. 



Legendre's next work was an essay on transcendents*, pre- 

 sented to the Academy in 1792 and published separately the 

 year after. It contains the same general view as that which is 



* A translation of it appeared in Leybourne's Mathematical Repository, Vols. II. 

 and in. The original I have not seen it has long been scarce. 



