ON THE RECENT PROGRESS OF ANALYSIS. 69 
a impossible to tabulate such integrals, and therefore our course is to devise 
series more or less convenient for determining their values when any pro- 
ple, e. g. that of the motion of a rigid body, to which Dr. Gudermann espe- 
 gially refers, requires us to do so. The formation of such series is accordingly 
the aim of this memoir, which contains some remarkably elegant formule ; 
one of which connects three integrals of the third kind with three of the 
second. : 
In the sixteenth and seventeenth volumes of the same Journal, Dr. Guder- 
_ mann has given some series for the development of elliptic integrals ; and he 
has since published in the same Journal a systematic treatise on the theory 
- of modular functions and modular integrals, these designations being used to 
denote the transcendents more generally called elliptic. The point of view 
from which he considers the subject has been already indicated (vide supra, 
p- 36). In asystematic treatise there is of course a great deal that does not 
profess to be original, and it is not always easy to discover the portions 
which are so. Dr. Gudermann’s earlier researches are embodied and deve- 
loped in his larger work ; and in some of the latter chapters (XXIII. 329, &c.) 
we find some interesting remarks on the forms assumed by the general trans- 
cendent when the biquadratic polynomial in the denominator has four real 
roots. Dr. Gudermann points out the existence of a species of correlation 
between pairs of values of the variable. 
23. The development of the elliptic function ¢ in the form of a continued 
product may be applied to establish formule of transformation. ‘This mode 
_ of investigating such formule was made use of by Abel in his second paper 
in Schumacher’s Journal, No. 148, which we have already noticed; and a 
corresponding method is mentioned by M. Jacobi in one of the cursory no- 
tices of his researches which he inserted in the early volumes of Crelle’s 
Journal. Mr. Cayley, in the Philosophical Magazine for 1843, has pur- 
_ sued a similar course. Another and very remarkable application of the same 
__ kind of development consists in taking it as the definition of the function ¢, 
and deducing from hence its other properties. It has been remarked that 
_ the continued products of Abel and M. Jacobi are derived from considera- 
tions which, although cognate, are yet distinct; those of the latter being 
_ singly infinite, while Abel’s fundamental developments consist of the product 
_ of an infinite number of factors, each of which in its turn consists of an in- 
| finite number of simple factors. Thus we can have two very dissimilar de- 
_ finitions of the function ¢ by means of continued products. M. Cauchy, 
who has investigated the theory of what he has termed reciprocal factorials, 
that is, of continued products of the form 
{i +2)(lféz)......}{(1+¢2-!)(14+ @271)......}, 
_ which is immediately connected with M. Jacobi’s developments, has accord- 
| ingly set out from the singly infinite system of products, and has deduced 
| from hence the fundamental properties of elliptic functions (Comptes Rendus, 
| kvii. p. 825). 
‘4 Mr. Cayley, on the other hand, has made use of Abel’s doubly infinite 
| products, and has shown that the functions defined by means of them satisfy 
_ the fundamental formule mentioned in the note at page 52, which, as these 
| equations furnish a sufficient definition of the elliptic functions, is equivalent 
__ to showing that the continued products are in reality elliptic functions. He 
| has therefore effected for Abel’s developments that which M. Cauchy had 
_ done for M. Jacobi’s. Mr. Cayley’s paper appeared in the fourth volume of 
| the Cambridge Mathematical Journal, but he has since published a trans- 
| lation of it with modifications in the tenth volume of Liouville’s. On the 
