272 ON THE RECENT PROGRESS OF ANALYSIS. 



Crelle's Journal (in. pp. 192, 303, 403, IV. p. 185) notices, mostly 

 without demonstrations, of the progress of his researches. Al- 

 most everything in the first and second of these notices is 

 found in the Fundamenta. In the third we find a remark- 

 able algebraical formula for the multiplication of the elliptic 

 integral of the first kind. The fourth and last relates to ul- 

 terior investigations, which it was the intention of the author 

 to develope in a second part of his work. It contains an in- 

 dication of a method of transformation depending on a partial 

 differential equation*; values of the elliptic functions of multiple 

 arguments ; a method of transforming integrals of the second 

 and third kinds ; a most important simplification of the method 

 of Abel for the division of any integral of the first kind, &c. 

 Of this simplification he had already given some idea in a note 

 in the preceding volume of the same Journal, p. 86. 



17. It may not be improper in this place to observe, that 

 in 1818, and thus in the interval between Legendre's first and 

 second systematic works on the theory of elliptic functions, 

 M. Gauss published the tract entitled Determinatio Attrac- 

 tionis, &c. The illustrious author begins by remarking that 

 the secular inequalities due to the action of one planet on 

 another are the same as if the mass of the disturbing planet 

 were diffused according to a certain law along its orbit, so 

 that the latter becomes an elliptic ring of variable but infini- 

 tesimal thickness. The problem then presents itself of deter- 

 mining the attraction exerted by such a ring on any external 

 point. In the solution of this problem M. Gauss arrives at two 

 definite integrals; they can readily "be reduced to elliptic in- 

 tegrals of the first and second kinds. For the evaluation of 

 the integrals to which he reduces those of his problem, M. Gauss 

 gives a method of successive transformation, analogous in some 

 measure to that of Lagrange. But the transformation of which 

 he makes use is a rational one, and is in fact the rational 

 transformation of the second order. The discovery of this 

 transformation appears therefore to be due to M. Gauss. He 

 has remarked, though merely in passing, that his method is 

 applicable to the indefinite as well as to the definite integral. 



* Mr Cayley, to whose kindness I have been, while engaged on the present 

 report, greatly indebted, has communicated to me a demonstration of the truth 

 of this equation. 



