584 
LANDEN. 
[791 
Euler in their researches on the same subject. He reproduces and further develops 
and defends his own views in his Mathematical Memoirs, and in his paper in the 
Philosophical Transactions for 1785. But Landen’s capital discovery is that of the 
theorem known by his name (obtained in its complete form in the memoir of 1775, 
and reproduced in the first volume of the Mathematical Memoirs) for the expression 
of the arc of an hyperbola in terms of two elliptic arcs. To find this, he integrates 
a differential equation derived from the equation 
interpreting geometrically in an ingenious and elegant manner three integrals which 
present themselves. If in the foregoing equation we write m = 1, g — k 2 , and instead 
of t consider the new variable y = t-r-( 1 —k'), then 
which is the form known as Landen’s transformation in the theory of elliptic functions; 
but his investigation does not lead him to obtain the equivalent of the resulting 
differential equation 
äy = (1 
(1 + k') dx 
V1 — y 1 .1 — \ 2 y 2 Vl — 
— x 2 .1 — k 2 x/ 
due it would appear to Legendre, and which (over and above Landen’s own beautiful 
result) gives importance to the theorem as leading directly to the quadric transformation 
of an elliptic integral in regard to the modulus. 
The list of his writings is as follows:—Ladies’ Diary, various communications, 
1744—1760; papers in the Phil. Trans., 1754, 1760, 1768, 1771, 1775, 1777, 1785 ; 
Mathematical Lucubrations, 1755; A Discourse concerning the Residual Analysis, 1758; 
The Residual Analysis, book I., 1764; Animadversions on Dr Stewart’s Method of com 
puting the Sun’s Distance from the Earth, 1771 ; Mathematical Memoirs, 1780, 1789.