| ‘MATHEMATICAL SECTION. | 107 
But we have, by (20), 
sin (¢’/ + 4m’ _|y +24’ _|p’ %)-/ 
=sin{ Sey t4m_y+2u_dby (8) 
therefore, m |= — m _ly (24) 
# ig aig (25) 
Oe ae, Se OC, 
and S=5a" = 
Anticipating here the definition of the highly important con- 
stant, the nome q, which is such a prominent feature in the brilliant 
researches of Jacobi and Abel, we have 
ls 
q=e * hy ie 
and the nome q’ of the transformed integral is 
_|6 m’ 1 _|p mip 
ee ee Oe ae | = = bee GD) 
Thus it appears that the nomes of the given and transformed 
integrals are in a relation 
q” — d n! 
where n and n’ are integers, and, if n = 1 and n’ = 2, we have the 
quadric transformation. 
Landen’s transformation consists in assuming 
sin (2y’ — ¢) =—sing (28) 
which is Legendre’s convenient form for computing the amplitude 
gy’. Differentiating, we have 
(2d¢’ — dg) cos (2¢’ — ¢) = —cos gy dg 
dy dg’ 
or” ahge  t(as¢-+ccos¢) (29) 
