768] 
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768. 
NOTE ON LANDEN’S THEOEEM. 
[From the Proceedings of the London Mathematical Society, vol. xiii. (1882), pp. 47, 48. 
Read November 10, 1881.] 
Landen’s theorem, as given in the paper “An Investigation of a General Theorem 
for finding the length of any Arc of any Conic Hyperbola by means of two Elliptic 
Arcs, with some other new and useful Theorems deduced therefrom,” Phil. Trans., 
t. lxv. (1775), pp. 283—289, is, as appears by the title, a theorem for finding the 
length of a hyperbolic arc in terms of the length of two elliptic arcs; this theorem 
being obtained by means of the following differential identity, viz., if 
where 
to- — rv 
then 
(to — n) 2 — t 2 ± 
(m + n) 2 — t 2 4 
(to + n) 2 — t 2 ' 
(to — n) 2 — ~t 2 
(this is exactly Landen’s form, except that he of course writes x, t in place of dx, 
dt respectively): viz., integrating each side, and interpreting geometrically in a very 
ingenious and elegant manner the three integrals which present themselves, he arrives 
at his theorem for the hyperbolic arc; but with this I am not now concerned. 
Writing for greater convenience to=1, n = Je', and therefore g = fc 2 , if as usual 
k 2 +k' 2 = 1, the transformation is 
C. XI. 
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