224 
[33 
33. 
ON THE REDUCTION OF WHEN U IS A FUNCTION OF 
THE FOURTH ORDER. 
[From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 70—73.] 
It is well known that the transformation of this differential expression into a similar 
one, in which the function in the denominator contains only even powers of the corre- 
r Sajl 
sponding variable, is the first step in the process of reducing j /TT to elliptic integrals. 
JJU 
And, accordingly, the different modes of effecting this have been examined, more or 
less, by most of those who have written on the subject. The simplest supposition, 
that adopted by Legendre, and likewise discussed in some detail by Gudermann, is that 
u is a fraction, the numerator and denominator of which are linear functions of the 
new variable. But the theory of this transformation admits of being developed further 
than it has yet been done, as regards the equation which determines the modulus of 
the elliptic function. This may be effected most easily as follows. 
Suppose 
U = a + 4 bu + 6cu 2 4- 4 du 3 + eu 4 , 
P — ax 4 + 4sbx?y + 6cx 2 y 2 + \dxy z + ey 4 . 
Also let 
P' = aV 4 + 4 b'x' 2 y' + 6c'x' 2 y' 2 + 4 d'x'y' 3 + e'y' 4 
be what P becomes after writing 
x = Xx' + fiy' , 
y = X/ + yyf : 
and let 
U' — a’ + 46V + Qc'u' 2 + 4 d'u' 3 + eu 4 .