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768] NOTE ON landen’s theorem. 
corresponding to an equation given by Landen. And we thence have 
1 _ tea? = \ {2 — te — i 2 + V7 7 }, = \ {1 + k' 2 -t* 4- *JT\, 
which is the required expression for 1 — tex-. 
The trigonometrical form sin (2<p' — <£>) = c sin <£ of the relation between y and x 
does not occur in Landen; it is employed by Legendre, I believe, in an early paper, 
Mém. de l’Acad. de Paris, 1786, and in the Exercices, 1811, and also in the Traité 
des Fonctions Elliptiques, 1825, and by means of it he obtains an expression for the 
arc of a hyperbola in terms of two elliptic functions, E (c, (f)), E(c', <£')> showing that 
the arc of the hyperbola is expressible by means of two elliptic arcs,—this, he observes, 
“ est le beau théorème dont Landen a enrichi la géométrie.” We have, then (1828), 
Jacobi’s proof, by two fixed circles, of the addition-theorem (see my Elliptic Functions, 
p. 28), and the application of this (p. 30) to Landen’s theorem is also due to Jacobi, 
see the “ Extrait d’une lettre adressée à M. Hermite,” Crelle, t. xxxii. (1846), 
pp. 176—181 ; the connection of the demonstrations, by regarding the point, which 
is alone necessary for Landen’s theorem as the limit of the smaller circle in the 
figure for the addition-theorem is due to Durège (see his Theorie der elliptischen 
Functionen, Leipzig, 1861, pp. 168, et seq.). 
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