296 ON THE RECENT PROGRESS OF ANALYSIS. 



assumed, in consequence of the discoveries of Abel and M. 

 Jacobi, is that which it will probably always retain, however 

 our knowledge of particular parts of it may increase* What has 

 since been effected relates for the most part to matters of detail, 

 of which, however important they may be, it is difficult or im- 

 possible to give an intelligible account. 



26. It does not fall within the design of this report to con- 

 sider the various applications which have been made of the 

 theory of elliptic functions ; but I shall briefly mention some of 

 the geometrical interpretations, if the expression may so be used, 

 which mathematicians have given to the analytical results of the 

 theory. 



The lemniscate has, as is well known, the property that its 

 arcs may be represented by an elliptic integral of the first kind, 



the modulus of which is = . The problem of the division of its 



V2 



perimeter is accordingly a geometrical interpretation of that of 

 the division of the complete integral, and was considered by 

 mathematicians at a time when the theory of elliptic functions 

 was almost wholly undeveloped. Besides Fagnani, whose re- 

 searches with respect to the lemniscate have been already noticed, 

 we may mention those of Euler, who however did not succeed in 

 obtaining a solution of the problem. Legendre, who seems to 

 have attached considerable importance to geometrical illustra- 

 tions of his analytical results, assigned the equation of a curve 

 of the sixth order, whose arcs measured from a fixed point 

 represent the sum of any elliptic integral of the first kind and an 

 algebraical expression. He showed also that an arc of the curve 

 might be assigned equal to the elliptic integral, but in order to 

 this both extremities of the arc must be considered variable, so 

 that in effect the integral is represented by the difference of 

 two arcs measured from a fixed point (Traite des Fonctions 

 Elliptiques, I. p. 36). 



M. Serret, in a note presented to the Institute in 1843 

 (Liouville's Journal, viil. 145), has proved a beautiful theorem, 

 viz. that the sum and difference of the two unequal arcs, inter- 

 cepted by lines drawn from the centre of Cassini's ellipse to cut 

 the curve, are each equal to an elliptic integral of the first 

 kind, and that the moduli of the two integrals are comple- 



