50 SIR G. GREEXHILL ON ELECTROMAGNETIC INTEGRALS. 



Thence any formula of the Landen quadric transformation, first and second, can 

 be interpreted geometrically on fig. 1 and 2, and we reconcile the baffling and 

 conflicting notation of previous writers on the subject. 



Interpreted dynamically, with proportional to the time, t = %eT for period T, Q 

 circulates round the circle on A.B in fig. 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 OE with 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. Combined into one quadric transformation, the first and second of Landen, 

 from modulus y to K, and then K to y, 



(1) 



du (eG 

 = y sn (eG+fG'i) sn (G-eG-fG'i), 



(2) . dn (J 



and then,/ or e is made zero. And 



(3) on (2dU/K'0 - 



/ne+ 



dn(cG+/GV)-dn(G - t '.G - f G'O 

 ' ' 



(4) dn(2eK+/K'i) = 



(l+y')dn(eG+fG'i) 



dn (eG+fG'i) + dn (G-eG-fG'i) 

 1+y' 



(5) ^ en (2eK +/K'.') = i ne+/ _ , 



\/y' 



(6) dn(2eK+/K'Q __ t du (eG+fG'i) ^ dn (G-eG-fG'i) 



'' ' 



