Mutual Induction of a Circle and a Coaxial Helix, $-c. 195 



)Q,,, (2m l)Q_, ......... (iv), 



)P,,,-( 2m - 8 ) ( a " t + 1 ) p. ( T ), 



2m 1 



'HJ* = (&,_! + cQ* ........................ (vi), 



' 2 Q /K = (2m-c' 2 )Q wt -(2m-l)Q / _ 1 . . . ..... ( v ii), 



(viii), 



where c' 2 = 1 c 2 , and the dotting of a function denotes differentia- 

 tion with regard to c. 



It will be observed that Q and Q_i are respectively the complete 

 elliptic integrals (F and E) of the first and second kinds with regard 

 to modulus c. 



4. In equation (3) put 



l) 



~ 9 "" 



so that M@ = (A + a)c 2 2 ( l)"K w . 



Then we can find by a double application of (v) a relation between 

 K Mi +!, Km, and K,,^!, viz. : 



2m(2m + l) / (2m- 1) (2m-3) 



m "" 2m. 2m K '"- 1 



where d 2 = -- f 2 and e 2 = - 



This formula renders the calculation of successive terms of the 

 series sufficiently easy. 



5. Hence to find M , given A, a, and #, we have to calculate the 

 following quantities in order : 



