68 REPORT—1846. 
ees. 
The question presents itself, what is the connection between the irrational 
transformation (that of which Lagrange’s is a particular case) and the rational _ 
transformation of even orders? Perhaps the simplest answer to it (though 
every question of the kind is included in the general investigations contained 
in Abel’s ‘ Précis’) is found in a paper by M. Sanio in the fourteenth volume 
of Crelle’s Journal, p.1. The aim of this paper is to develope more fully 
than Mr. Ivory has done the theory of transformations of even orders, and 
particularly of the irrational transformations, which M. Sanio considers more 
truly analogous to the rational transformations of odd orders than the rational 
transformations of even orders; and also to discuss the multiplication of 
elliptic integrals by even numbers, a subject intimately connected with the 
other. We have already mentioned the existence of what are called com- 
plementary transformations, each of which may be derived from the other 
by an irrational substitution, by which two new variables are introduced. In 
the case of transformations of odd orders, the original transformation and the 
complementary one are both rational, and are both included in the general 
formula given by M. Jacobi’s theorem; but to the rational transformation of 
any even order corresponds as its complement the irrational transformation 
of the same order. This remark, which, as far as I am aware, had not before 
been made, sets the subject in a clear light *. 
22. In the twelfth volume of Crelle’s Journal (p. 173), Dr. Guetzlaff has 
investigated the modular equation of transformations of the seventh order: it 
is, as we know from the general theory, of the eighth degree, and presents 
itself in a very remarkable form, which closely resembles that in which 
M. Jacobi, at p.68 of the ‘Fundamenta Nova,’ has put the modular equa- 
tion for the third order. Dr. Sohncke has given, at p. 178 of the same vo- 
lume, modular equations of the eleventh, thirteenth and seventeenth orders, 
none of which apparently can be reduced to so elegant a form as those of 
the third and seventh. Possibly the transformation of the thirty-first order 
might admit of a corresponding reduction. The whole subject of modular 
equations is full of interest. Dr. Sohncke has demonstrated his results in a — 
subsequent volume of the Journal (xvi. 97). 
In the fourteenth volume of Crelle’s Journal there is a paper by Dr. Gu- 
dermann on methods of calculating and reducing integrals of the third kind. 
I have already quoted from this paper the expression of the opinion of its 
learned author, that it is impossible to express the value of integrals of the 
circular species in terms of functions of two arguments. If this be so, it is 
which is M. Gauss’s, and is termed in M. Jacobi’s nomenclature the rational transformation 
of the second order. It satisfies the equation 
dy Ss fe ha il hal 
Jaap dara OT aaa iaeaiy 
where, as before, _2Vve 
1+e 
* Lagrange’s transformation being 
ay Lagan ae V1 a! 
= 1 t = a d = = 
” atnen |e ite V1—y? = en 
then we find that _ (+2) 2’ 
while the differential equation becomes 
dy’ Ness A 
Vay amy ~ OF) aaa) Pa) 
where =1—F?*. 
