ON THE RECENT PROGRESS OF ANALYSIS. 301 



number of them can be reduced. As we know, this number is 

 unity in the case of elliptic integrals, and by Abel's theorem we 

 find that it is two in the first class of the higher transcendents, 

 three in the next, and so on. 



Following the analogy of elliptic integrals, Legendre pro- 

 posed to recognise three canonical forms in each class of hyper- 

 elliptic integrals, and thus to divide it into three orders. The 

 sum of any number of functions of the first kind will, when the 

 required conditions are satisfied, be equal to a constant ; that of 

 any number of the secondhand third kinds respectively will, 

 under similar conditions, be equal to an algebraical or logarith- 

 mic function. 



Much the greater part of the remainder of the supplement 

 consists of a discussion of the particular transcendents 



It contains a multitude of numerical calculations, and if the 

 writer's age be considered (he was then almost eighty), is a very 

 remarkable production. By means of the numerical calculations 

 he recognised, as it were empirically, the values to be assigned 

 in different cases to the above-mentioned constant : what these 

 values ought to be, he did not attempt to determine a priori. 



At the close of the supplement we find a remarkable reduc- 

 tion of an integral, apparently of a higher order to elliptic in- 

 tegrals. The method employed has been generalised by M. 

 Jacobi, in a notice of Legendre's Supplements, inserted in the 

 eighth volume of Crelle's Journal (p. 413). 



29. In the ninth volume of Crelle's Journal (p. 394), we 

 find a most important paper by M. Jacobi ( Considerationes Ge- 

 nerates, &c.} which may be said to have determined the direction 

 in which the researches of analysts in the theory of algebraical 

 integrals were to proceed. 



The writer proposes two questions, both suggested by the 

 cases of trigonometrical and elliptic functions. First, as in these 

 cases we consider certain functions to which circular and elliptic 

 integrals are respectively inverse, and which are such that func- 

 tions of the sum of two arguments are algebraically expressible 

 in terms of functions of the simple arguments, what are the coi> 



