50 
Sill (.4. (IREEXHTLL UN ELECTIIOMAGNETIC INTEGRALS. 
Thence any fonnnla of the Landen qnadiic transformation, first and second, can 
be interpreted geometrically on tig. 1 and 2, and we reconcile the baftiing and 
conflicting notation of previous writers on the subject. 
Interpreted dynamically, witli e proportional to the time, t = for period T, Q 
circulates round the circle on iVB in tig. 1 with velocity due to the level of F, or 
proportional to EQ or DQ, and beats the elliptic function to modulus y, while T 
circulates round the circle on (JE witli velocity due to the level of D, or S at the 
same level will oscillate, beating elliptic functions to modulus k. 
So also for the motion of P round the circle on ED, with f proportional to the 
time t, and velocity due to the level of O, or proportional to BP or AP, gravity being 
reversed. 
11 . Oombined into one quadric transformation, the first and second of Landen, 
from modulus y to k, ami then k to y. 
( 1 ) 
( 2 ) 
v/zc sn(2eK+/KV) 
y sn (eG+./’Gb) cn{eG + fG'i) 
di;(eG+/GT) 
= y sn (cG+y Gti) sn (G — cG—yGT), 
dn (2eG+ 2/’GT) 
l-^'SiP(2eK+/Kti) 
l+^sid(2cK+/KT) ’ 
and then f or e is made zero. And 
(3) 
(4) 
cn (2eK+,/KT) 
did(eG+/Gb:)-y' 
(l-y')dn(eG+/GT) 
dn (cG+./-GT)-dn (G-cG-./TTb) 
dn (2e'K 4 f 1^'?’) 
did(eG+./Tlb:) + y 
(1 + y') dn (cG-f-,/ G^'l) 
dn (cG+./'Gb:) + dn (G-eG-./'Gb:) 
1 + y 
(5) 
^ cn (2cK +./‘Kb:) = T _ 1 (lu(G-eG-/Gti) ^ 
^ “ V y v y 
( 3 ) 
dn (dcK+_/Kiti) ^ dn (cGa-/ G^'i) , j dn (G — cG — / Gti) 
/ - 2 /ti t2 ^7 - 
^ G y \/ y 
