Elliptic Integral of the Second Kind. 505 
Of course for the actual calculation of the numerical value 
of K corresponding to a given value of 0, the second formula 
is much to be preferred ; for the trigonometrical factors dimi- 
nish rapidly, whereas in ‘the first series they have no tendency 
to diminish, and the gradual convergence of the series is due 
solely to the numerical coefficients, Thus in (1) the nth 
term is of the order z but the corresponding term in (2) is 
of the order 1 inom20 When @=0, the series in (1) is of 
the form 0x, the true value being 4; when 0=47, the 
series is infinite in value, as it should be. 
§ 2. Gudermann’s process is in effect as follows. He shows 
that the transformation which converts k into e-? converts 
K into 3e°(K—iK’), where & denotes $7—0@. Starting with 
(2), he thus finds 
K—7K/ ike ee lego? 
mee ot oa 
from which (1) is immediately deducible by equating the real 
arts. 
: At the end of the investigation Gudermann, after referring 
to the slow convergence of (1), which renders it of no prac- 
tical value, adds: —“Aus diesem Grunde iibergehen wir auch 
die Herleitung einer abhnlichen Reihe fiir den Quadranten HE.” 
The main object of this note is to give the series for 
which corresponds to (1). It does not admit of quite such 
simple derivation as the series for K. 
§ 3. Gudermann’s transformation corresponds to the change 
of qinto ig*; and this change may be supposed to be produced 
by the change of q into g?, followed by the change of g into —g. 
By the change of g into 1g?, k’ is changed into e~**, and K’ 
is changed into 4e(K’—iK). Starting with the series 
910" +. &e., 
2K’ - 12. 3% be ea-e 
H1t et orp + op ga ge bt he., 
we thus find 
Me as tk WAS an Vad ty 
as Ht gpet + on gee + on ge ge tee 
§ 4. In order to apply the same process to the series for 
Ki’, viz. 
20’ 1 ere ea 
oe ee ome Bag) a 
it is necessary to determine the quantity into which H’ is 
converted by the change of gq into 2°. 
