ie ON THE RECENT PROGRESS OF ANALYSIS. 75 
_ kind will, when the required conditions are satisfied, be equal to a constant ; 
_ that of any number of the second and third kinds respectively will, under 
- similar conditions, be equal to an algebraical or logarithmic function. 
Much the greater part of the remainder of the supplement consists of a 
discussion of the particular transcendents 
dx dz 
V1 — x8 ane V1 +28 
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 @ priori. 
At the close of the supplement we find a remarkable reduction of an inte- 
gral, apparently of a higher order to elliptic integrals. The method em- 
ployed has been generalised by M. Jacobi, in a notice of Legendre’s ‘ Sup- 
plements,’ 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 im- 
portant paper by M. Jacobi (Considerationes Generales, &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 trigo- 
 nometrical 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 functions of the sum of two arguments are algebraically 
expressible in terms of functions of the simple arguments, what are the cor- 
responding functions to which the hyper-elliptic or Abelian integrals are 
inverse, and how by means of them can Abel’s theorem be stated ? 
Secondly, as in the same cases we obtain algebraical integrals of differen- 
tial equations, whose variables are separated, but which nevertheless can only 
_ be directly integrated by means of transcendents*, what are the differential 
equations of which Abel’s theorem gives us algebraical integrals? These 
two questions are, it is obvious, intimately connected. 
_ M. Jacobi first takes the particular case in which the polynomial under the 
_ radical is of the fifth or sixth degree, If we call this polynomial X, it follows 
_ from Abel’s theorem, that if 
- 
2 
i 
d x 
: ox= JG 
bi Lax 
iy and ¢,2 = Vx” 
we shall have the equations 
. Pat pb=gutoy+ out oy’, 
: Pitt PO=O, 2+ Py t+ G2 + Oy’, 
where a and 0 are given as algebraical functions of the independent quan- 
ae ities x, y, 2', y'. 
o> Let Or+oy=u oa'+ oy! =u! 
Pitt Pysv Put oy =v. 
#E ae dy ff Ces, +3 9 Dla Vix 
ba “Ge gt ic which the algebraical integralis 2 V1—y?-+-y V1—2?=C. 
ue ach term of this differential equation is a differential of a transcendent function sin wy or 
