288 ON THE RECENT PROGRESS OF ANALYSIS. 



menta Nova. In Crelle's Journal, IV. p. 371, we find a paper 

 by M. Jacobi, entitled ' De Functionibus Ellipticis Commentatio.' 

 It contains, in the first place, a development of the method of 

 transforming elliptic integrals of the second and third kind, and 

 introduces a new transcendent H, which takes the place of , 

 with which it is closely connected. M. Jacobi proves that the 

 numerator and denominator of the value of y, mentioned above, 

 and which have been denoted by U and V, satisfy a single dif- 

 ferential equation of the third order. The remainder of the paper 

 relates to the properties of H (vide ante, note, p. 270). When 

 this function is multiplied by a certain exponential factor it be- 

 comes a singly periodic function, and, which is very remarkable, 

 its period is equal to one of the single or composite periods of 

 the elliptic function inverse to the integral of the first kind. 

 By composite period I mean the sum of multiples of the funda- 

 mental periods. The exponential factor being properly deter- 

 mined, its product by flt is equal to multiplied by a constant. 

 In considering this subject, M. Jacobi is led to introduce the 

 idea of conjugate periods. These are periods by the combi- 

 nation of which all the composite periods may be produced. It 

 is obvious that the fundamental periods are conjugate periods ; 

 and there are, as may easily be shown, an infinity of others. 



In the sixth volume of the same Journal we find a second 

 part of the ' Commentatio.' It contains a remarkable demon- 

 stration of the fundamental formulas of transformation of the odd 

 orders founded on elementary properties of elliptic functions. 



In a historical point of view a notice by M. Jacobi in the 

 eighth volume of Crelle (p. 413) of the third volume of Legen- 

 dre's Traite des Fonctions Elliptigues is interesting. It was 

 here, I believe, that M. Jacobi first proposed the name of 

 Abelian integrals for the higher transcendents, which we shall 

 shortly have occasion to consider. After some account of the 

 contents of Legendre's supplements, the first two of which con- 

 tain the greater part of M. Jacobi's earlier researches, he goes 

 on to generalise a remarkable reduction given by Legendre at 

 the close of his work. 



21. I turn to one of the very few contributions which 

 English mathematicians have made to the subject of this report, 

 namely, to a paper by Mr Ivory, which appeared in the Phil. 



