56 REPORT—1846. 
however possibly arose from some misconception of 'M. Jacobi’s meaning. 
Dr. Gudermann, in the fourteenth volume of Crelle’s Journal, has given it 
as his opinion that the circular species of integrals of the third kind does not 
admit of the reduction in question; and remarks, that it occurs much more 
frequently than the other species in the applications of mathematics to na- 
tural philosophy. 
After having discussed at some length, and by new methods, the proper- 
ties of elliptic integrals of the third kind, M. Jacobi concludes his work by 
investigating the nature of two new transcendents which present themselves 
in immediate connexion with the numerator and denominator of the con- 
tinued product by which sin amw is expressed. One of them however 
M. Jacobi had already recognised by a distinctive symbol, in consequence of 
its intimate connexion with the theory of integrals of the third kind. 
Such is the outline of this remarkable work: before it appeared M. Jacobi 
gave in the third and fourth volumes of Crelle’s Journal (iii. pp. 192, 303, 
403, iv. p. 185) notices, mostly without demonstrations, of the progress of 
his researches. Almost everything in the first and second of these notices 
is found in the ‘Fundamenta.’ In the third we find a remarkable algebraical 
formula for the multiplication of the elliptic integral of the first kind. The 
fourth and last relates to ulterior investigations, which it was the intention 
of the author to develope in a second part of his work. It contains an indi- 
cation 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 ‘ De- 
terminatio Attractionis,’ &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 infinitesimal thickness. The problem then presents itself of 
determining 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 integrals 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. The rational 
transformation in question leads to a continually increasing series of moduli, 
or is, to use an expression of M. Jacobi a transformation “minoris in 
majorem.” The law connecting two consecutive moduli is the same as in 
Lagrange’s, which is, as we have seen, an irrational transformation ; so that 
M. Gauss’s method does not afford us a new scale of moduli. Nevertheless, 
as no rational transformation had I believe been noticed when his tract ap- 
* 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. 
