[ 217 1 



IX. The Third Elliptic futff/ml. and the KUipsotomic Problem. 

 By A. G. GREENHILL, F.R.S. 



Received December 22, 1903, Read January 21, 1904. 



THE ABEL Centennial Ceremony, held in Christiania, September, 1902, has directed 

 the attention of mathematicians to the great influence of AHKL on modern analysis, 

 and to the history of elliptic functions, and of the foundation by CRELLE of the 

 "Journal fur die reine und angewan Ite Mathematik." 



ABEL'S article in the first volume of ' OKELLE'S .Journal,' 1826. 

 " Ueber die Integration der Differential- Formel 



P </f 



wenn R und p gauze Functionen sind," is of great importance as indicating the 

 existence of what is now called the pseudo-elliptic integral; the present memoir is 

 intended to show the utility of this integral in its application to mechanical theorv. 



Provided with a list of these integrals, proceeding from the simplest case and 

 extending as far as possible, the student of applied mathematics will be able to effect 

 the complete solution of many interesting mechanical problems now abandoned in an 

 unfinished state; at the same time, the exploration along the simplest lines of 

 progress is effected of the general analytical field, and mathematical research is 

 guided in a path likely to arrive at useful development in the theory of elliptic 

 functions. 



The problem of the Division, and thence also of the Transformation, of elliptic 

 functions is solved incidentally ; as also the difficulties raised by LEUKNDHE in his 

 letter to ABEL, of January 16, 1829 : 



" Mais de la nait une difficulte sur la division generale des fonctions elliptiques. 



" Admettrons-nous cette difference enorme entre les fonctions de 3" 1C espece a 

 parametre logarithmique et les fonctions a parametre circulaire, savoir que les 

 premieres peuvent s'exprimer par une fonction de deux variables, facilement reductible 

 en table, et que les autres ne le peuvent pas ? II y aurait done par le fait quatre 

 especes de fonctions elliptiques au lieu de trois, et la quatrieme sera it bien plus 

 composee que la troisieme. C'est un point qui merite d'etre examine et mis au clair. 

 Je le recommande a votre investigation et a celle de M. JACOBI." 



VOL. ccm. A 367. 2 F 3- 8M 



