248 REPORT— 1897. 



this branch of mathematica confer Dirichlet's ' Gedachtnissrede auf Jacobi ' 

 (Jacobi's ' Werke,' vol. i. p. 9). 



(7) The suggestion made by Euler that one should find a direct method 

 of integrating the diflferential equation proposed by him (art. 5) was 

 carried out by Lagrange, who by direct methods integrated this equation, 

 and in a manner which elicited the great admiration of Euler. (See 

 ' Miscell. Taurin.,' iv. 1768 ; or Serret's ' CEuvres de Lagrange,' vol. ii. 

 p. 533.) 



(8) The consideration of relations between integrals that have different 

 moduli gave rise to a theorem due to Landen (and proved somewhat 

 differently by Lagrange), in accordance with which an elliptic integral 

 may be transformed into another integral of the same kind by means of 

 algebraic transformations. Landen (' Phil. Trans.,' 1775, p. 285 ; or 

 'Mathematical Memoirs,' by John Landen, London, vol. i. 1780, p. 33) 

 proves that in general the hyperbola may be rectified by means of two 

 ellipses, with the addition of an algebraic quantity.' 



The germ of the general theory of transformation is contained in this 

 theorem, as has been observed by Legendre.'-^ 



By means of algebraic tranformations Landen was able to reduce 

 elliptic integrals of the first kind into forms that had the same modulus, 

 and showed that an elliptic integral of the first kind could be transformed 

 into an elliptic integral of the first kind with smaller modulus, or into an 

 integral of the first kind with smaller amplitude and greater modulus. 



Lagrange^ showed that the integration of any irrational function 



which contains the square root of a function <^ may be made to depend 



P (•«) 

 upon the integration of a function of the form — -^ where P is rational ; 



and that if <^ is not higher than the fourth degree in x, the integration 

 may be reduced to that of 



Ncfcc 



V(l±;j2x2)(l-!-92a;2)' 



N denoting a rational function in cc^, and ^; and q constants. If the 

 elliptic integral be reduced to this form, Lagrange showed by the intro- 

 duction of a new variable that this integral may be transformed into 

 another of similar form, but in which v and q become two new quantities 

 jp' and q' , and that if ^^ is greater than q, p' becomes greater than f and 

 ^' less than q. By the repetition of this process the factor corresponding 

 to \A2_(fx^ maybe made as near unity as we desire, and consequently the 

 integral may be expressed by a circular arc or logarithm ; if, however, 

 the transformations are made in the other direction, the functions corre- 

 sponding to l+^j^a;^ and l+^'-a;- become as near equal as we wish, and 

 thus the elliptical integral reduces to a lower transcendent.'' 

 Legendre investigated the general integral given above, 



f '^dx 



J >/ a + ySa; -f- yai^ 4- Sa;3 + ex* ' 



' An interesting geometric construction of this transformation is found in a letter 

 of Jacobi to Hermite (Jacobi's Werke, bd. ii. p. 118). See also a geometric jKoof by 

 MacCullagh {Tra,ns. of the Royal Irish Academy, vol. xvi. p. 76). 



■ See Ellis, p. .S7. 



' Memoire de VAcad. de So., 1784-85 ; (Euvres ii. p. 253. 



* Ellis, p. 44. Casorati, Teorica delle funcioni, kc, p. 6. 



