THE RECENT PROGRESS OF ANALYSIS. 299 



Besides one or two other papers I may mention a tract by 

 the Abbe Tortolini, on the geometrical representation of elliptic 

 integrals of the second and third kinds. This tract, however, I 

 have not seen. 



Lag-range long since proved (vide Theorie des Fonctions Ana- 

 lytiques, p. 85), that by means of a spherical triangle a geome- 

 trical representation of the addition of elliptic integrals of the 

 first kind may easily be obtained, and that hence by a series of 

 such triangles we are enabled to represent the multiplication as 

 well as the addition of thesejuitegrals. 



M. Jacobi ha-s given (Crelle's Journal, ill. p. 376, or vide 

 Liouville's Journal, X. p. 435) a geometrical construction for the 

 addition and multiplication of elliptic integrals of the first kind. 

 It is founded on the properties of an irregular polygon inscribed 

 in a circle, and the sides of which touch one or more other 

 circles. It is to be remarked that Legendre, in giving an ac- 

 count 4 of this construction in one of the supplements to his last 

 work, has only considered its application to multiplication and 

 not to addition, and has been followed in this respect by M. Ver- 

 hulst, whose treatise on elliptic functions has been already men- 

 tioned. In consequence of this, M. Chasles was led to believe 

 that until the publication of his own researches, no construction 

 for addition excepting that of Lagrange was known. But he has 

 recently (Comptes Rendus, January 1846) pointed out the error 

 into which he had fallen. 



27. In the Transactions of the Eoyal Irish Academy (ix. 

 p. 151), Dr Brinkley gave a geometrical demonstration of 

 Fagnani's theorem with respect to elliptic arcs, and in the 

 sixteenth volume of the same Transactions (p. 76), we find Lan- 

 den's theorem proved geometrically by Professor MacCullagh. 



M. Chasles has considered the subject of the comparison of 

 elliptic arcs by geometrical methods, and with great success. 

 His fundamental proposition may be said to be, that if from any 

 two points of an ellipse we draw two pairs of tangents to any 

 confocal ellipse, the difference of the two arcs of the latter respec- 

 tively intercepted by each pair of tangents is rectifiable. Or, 

 what in effect is the same thing, if we fasten a string at two 

 points in the circumference of an ellipse, arid suppose a ring 

 to move along the string, keeping it stretched, and winding it 



