ON THE RECENT PROGRESS OF ANALYSIS. 297 



mentary. In the lemniscate, which is a case of Cassini's ellipse, 

 one of these arcs disappears, and the moduli of the two integrals 

 are equal, each being the sine of half a right angle. So that 

 M. Serret's theorem is an extension of the known property of 

 the lemniscate. 



M. Serret has since considered the subject of the representa- 

 tion of elliptic and hyper-elliptic arcs in a very general manner. 

 His memoir, which was presented to the Institute and ordered to 

 be published in the Savans Etr angers, appears in Liouville's 

 Journal, X. 257. He had remarked that the rectangular co- 

 ordinates of the lemniscate are rationally expressible in terms of 

 the argument of the elliptic integral which represents the arc, 

 for if we assume 



/- z + z s , /- z 



, 



-r z i -f- z 



we shall have 



=, 



and if between the first two of these equations we eliminate z, we 

 arrive at the known equation of the lemniscate *. So that if we 

 state the indeterminate equation 



(x, y and Z being real and rational functions of 0), the lemniscate 

 will afford us one solution of it ; and every other solution will 

 correspond to some curve whose arc is expressible by an elliptic 

 or hyper-elliptic integral. Of this indeterminate equation M. 

 Serret discusses a particular case. He succeeds in solving it 

 by a most ingenious method, which is applicable to the general 

 equation, and shows from hence that there are an infinity of 

 curves, the arcs of which represent elliptic integrals of the first 

 kind. M. Serret's researches however have not led him to a 

 geometrical representation by means of an algebraical curve of 

 any integral of the first kind, though his results are generalized 

 in a note appended to his memoir by M. Liouville. In order 

 that the curve may be algebraical, it is necessary and sufficient, 



* On reducing the integral I . to the standard form of elliptic integrals, 



J vi+2 4 



we find that it is an elliptic integral of the first kind, of which the modulus is the 

 sine of 45. 



