302 ON THE REGENT PROGRESS OF ANALYSIS. 



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 inte- 

 grals of differential 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 ques- 

 tions are, it is obvious, intimately connected. 



M. Jacobi first takes the particular case in which the poly- 

 nomial under the radical is of the fifth or sixth degree. If we 

 call this polynomial Jf, it follows from Abel's theorem, that if 



and fax = 



we shall have the equations 



(pa -+ (f>b = <px - 

 faa + fab = fax + fay + fax 1 + fay\ 



where a and b are given as algebraical functions of the indepen- 

 dent quantities x, y, x 1 , y 1 . 



fax + fay = v, fax 1 + fay 1 = v 1 . 



Then x and y are both given as functions of u and v. We 

 may therefore put 



and similarly, 



and as (pa + <pb = u + u 1 



we shall have a = X (u + u 1 , v + v') 



c\ v + v 1 ). 



o, of which the algebraical integral is 



x xi-y 3 +y *i-x* = C. 



Each term of this differential equation is a differential of a transcendent function 

 sin- 1 ^ or Bi*r l y. 



