657]
NOTE ON THE THEORY OF ELLIPTfC INTEGRALS.
141
that is,
and consequently
j = ^(m + s)±|V , i(to — s) 2 + nr,
where, by what precedes, the integer under the radical sign is negative: and we have
thus the above mentioned theorem.
As a very general example, consider the two rational transformations
z = (x, u, v); mod. eq. Q (u, v) = 0;
dx
Vl — z 2 . 1 — V 8 Z 2 Vl — X 8 . 1 — u 8 x 2 ’
y = (z, v, w); mod. eq. P (v, w) = 0 ;
Mdy
dz
Vl — y 2 . 1 — w 8 y s V1 —
Z 2 .1 —
viz. 0 is taken to be a rational function of x, and of the modular fourth roots
u, v; and y to be a rational function of z, and of the modular fourth roots v, w;
the transformations being (to fix the ideas) of different orders. We have y a rational
function of x, corresponding to the differential relation
MNdy
dx
Vl — y 2 .1 — w 8 y 2 Vl — X 2 . 1 — U 8 X 2 *
Suppose here w 8 = u 8 , or say w = 6u, 6 being an eighth root of unity: we then have
Q (v) = 0, P (-v, 6u) = 0, equations which determine u. The differential equation is then
MNdy
dx
V1 — y 2 .1 — u 8 y 2 V1 —
X 2 .1 — U 8 X 2 ’
an equation the algebraical integral of which is y = a rational function of x as above:
hence, by what precedes, we have
a half-rational numerical value, as above.
To explain what the algebraical theorem implied herein is, observe that the
equations Q (u, v) = 0, P (v, 6u) = 0, give for u an algebraical equation. Admitting 6 as
an adjoint radical, suppose that an irreducible factor is </> (u), and take u to be
determined by the equation <f>u = 0; then v, and consequently also any rational function
°f u > v > can be expressed as a rational integral function of u, of a degree which
is at most equal to the degree of the function <pu less unity. The theorem is that,
in virtue of the equation <f>u = 0, this rational function of u becomes equal to a half-
rational numerical value as above. Thus in a simple case, which actually presented
itself, the equation <pu = 0 was u 2 — 4^ +1 = 0; and had the value u — 2, which
in virtue of this equation becomes = + V — 3.
