510 XII Elliptic Functions
Matematicheof Count Fagnano, a copy of which had been sent them by the author.
The papers which aroused Euler’s interest had in fact already appeared in an obscure
Italian journal between 1715 and 1720. Fagnano had shown first how a quadrant of a
lemniscate could be halved, then how it could be divided algebraically into 2m, 3 · 2 m
or 5· 2 mequal parts. He had also established an algebraic relation between the length
of an elliptic arc, the length of another suitably chosen arc and the length of a quadrant.
By analysing and extending his arguments, Euler was led ultimately (1761) to a general
addition theorem for elliptic integrals. An elegant proof of Euler’s theorem was given
by Lagrange (1768/9), using differential equations. We follow this approach here.
Let
gλ(x)= 4 x( 1 −x)( 1 −λx)= 4 λx^3 − 4 ( 1 +λ)x^2 + 4 xbe Riemann’s normal form and let 2fλ(x)be its derivative:
fλ(x)= 6 λx^2 − 4 ( 1 +λ)x+ 2.By the fundamental existence and uniqueness theorem for ordinary differential equa-
tions, the second order differential equation
x′′=fλ(x) (7)has a unique solutionS(t)=S(t,λ), defined (and holomorphic) for|t|sufficiently
small, which satisfies the initial conditions
S( 0 )=S′( 0 )= 0. (8)The solutionS(t,λ)is an elementary function ifλ=0or1:S(t, 0 )=sin^2 t, S(t, 1 )=tanh^2 t.(For other values ofλ,S(t)coincides with the Jacobian elliptic function sn^2 t.)
EvidentlyS(t)is an even function oft,sinceS(−t)is also a solution of (7) and
satisfies the same initial conditions (8).
For any solutionx(t)of (7), the functionx′(t)^2 −gλ[x(t)] is a constant, since its
derivative is zero. In particular,
S′(t)^2 =gλ[S(t)], (9)since both sides vanish fort=0.
If|τ|is sufficiently small, thenx 1 (t)=S(t+τ)andx 2 (t)=S(t−τ)are solutions
of (7) neart=0. Moreover,
x′j(t)^2 =gλ[xj(t)] (j= 1 , 2 ),since these relations hold fort=0. From
(x 1 x 2 ′+x′ 1 x 2 )′=x 1 fλ(x 2 )+x 2 fλ(x 1 )+ 2 x 1 ′x 2 ′and
(x 1 x′ 2 +x 1 ′x 2 )^2 =x 12 gλ(x 2 )+x^22 gλ(x 1 )+ 2 x 1 x 2 x′ 1 x 2 ′