4(j SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 



In MINCHIN'S dissection of the circle on AB by lines radiating from M, ' Phil. 

 Mag.,' February, 1894, the solid angle cut out by a complete revolution of PQ about 



PM at a constant angle is (l - ^) 27r, so that for an elementary angle d,,, 



and P1Q 2 d n = MY . a d9, 



if MY = A cos + a is the perpendicular from M on the tangent at Q ; so that 



7 Aa cos + a 2 bd9 , ^o/MO^ 



(8) dti = d n - -^r- pQ = 4r- 1Q). 



In a complete circuit of the circle on AB, ,, grows from to 2-r, if M is inside the 

 circle on AB (a > A, / < |), 



(9) Q = 2,r-S2(MQ). 

 Replacing Kn cos by J(MQ 8 -a 8 -A 2 ), 



r-'"r( 2 -A 2 bde i fbde 



(10) v iuv *;/ - 2 I *rr)2 Of) ' -' 1 PC) 



J 1 1 IT J_ \fJ _L \^ J -L v^J 



= 2B + ^G 



'3 



= 27T/+ 2G zn 2/G' + 2G 7 ' sn 2/G', 

 (11) Q = 2*- (1 -/)- 2G zn 2/G' - 2Gy' sn 2/G'. 



This agrees in making Q = 2?r when P is at E and AB is viewed close up, and 

 12 = when /= 1 and P is at D, where the circle AB is seen edgeways ; and then, 

 with this value of B in (12), 4, 



(12) L = 7rPab + %W>-7r(a 2 -A. 2 ) (2* Q). 



In making the circuit of the circle EPD, and starting from E, where /= 0, it = 2,-n-, 

 then / grows from to 1, and Q diminishes from 2-n- to at D. After passing D, / 

 grows from 1 to 2, and 12 is taken negative for the reverse aspect of the circle AB, 

 and on arrival at E again with /= 2, Q = 2-jr. 



Thus 4-7T must be added in crossing AB if P circulates counter-clockwise. But 

 with the clock, the other way round, 4ir must be subtracted in crossing AB, just as 

 twelve hours is deducted on the clock in passing through XII o'clock. 



