40 SIR G. GEEENHILL ON ELECTROMAGNETIC INTEGRALS. 



3. In the reduction of these integrals to a standard form, for the purpose of a 

 numerical calculation by use of the tables of LEGENDRE, it is convenient to put 

 6 = 2w, w = ABQ, in fig. 1, and to introduce a new variable t, and constants T, ,, t s , t s , 

 in accordance with the notation of WEIERSTRASS, such that, m denoting a homogeneity 

 factor, 



(1) 



PA 2 = r 2 = m a (t 1 - 0, PB 2 = r 2 = m 2 (*, - ,), 



MA 3 - r 8 a -6 a = ( + A) 2 = m s (T-* 3 ), MB 2 = r/-6 2 = (a- A) 2 = m 2 (r- 2 ), 



PA 2 -PQ 2 = r-i* = 2Ao(l- costf) = 4A sin 2 , = m 2 (t-t 3 ), 



PQ'-PB 8 = r 2 -?-/ = 2Aa (1 + cos 0) = 4Aa cos 2 eo = m 2 (t 2 -t), 



PA--PB 2 = r 3 2 -r/ = 4Aa = m 2 (t 2 -t,), 



(2) > 1 >T> s >>* g >oo. 

 Tn the notation of LEGENDRE 



(3) PQ 2 = r 2 = // cos 2 + r/ sin 2 = -, 3 A 2 (, y), y' - 



/ 4 \ 



4oG 



PQ ?- 3 A' o PQ r 



and P is the rim potential of the circle on AB, with 



T-y 



Jo A (, y) 



(6) Q = p -gede f (2 Sin2 _ 



" 



= [(2-y 2 )G-2H], H = E(y) = 



and Q cos <p is the magnetic potential of the circle on AB, with uniform magnetisation 

 parallel to AB. 



In the Third Elliptic Integral keep to the Weierstrassian form, with the variable t, 

 ( 8 ) ^ dt = 2Aa sin B dQ = m 2 </(t 2 -t . t-t,) dd, 



(9) 



PQ mv/T ' PQ 



