56 SIR G. GEEENHILL ON ELECTED MAGNETIC INTEGRALS. 



Expanded in a series 



(2) th-' b - 2 



PQ- ^ 



where, as before, in (3), 3, 



(3) 6 = 2w, r~ = r, 2 cos 2 a + r* sin 2 = r/ A 2 (, y), y' = - 2 , 



* 



f 2 "" 7i 1 /7i\ 2n + 1 f 9/- /-7 1 / 7> 2 \+i 



/ i \ j.1 _i " 7/1 -^ * O \ f>Oi Oidi 1-^. 1 I \ t\ / \ 



(4) j/h'pQ-^^^^-^g J^^pl^^^^^j P.( M ), 



where P,, (?/.) is the toroidal function, introduced by W. M. HICKS, ' Phil. Trans.,' 



1881-4, defined by 



(5) P (%)=["- d ~ f'Y EA - EB V" 1 '* 



Jo (ch w + sh it, cos $)" + * Jo \ EQ 2 



2 v/r' r ('^y 



Jo \ y / 



y 

 given by the substitution 



(6) 



and P n satisfies the differential equation 



with C = ch u, and the sequence difference equation 



( 8 ) (2n + ])P n+1 -4nCP n +(2n-l)P n _ 1 = 0. 

 Expressed otherwise, with u = eG, v = G + 2/G'i, 



(9) , , 



= 2m' 



Jo 



am w. 



