208 Prof. G. M. Minchin on the Magnetic 



OV (&g. 2) be £ Then we should have 



dz* + d? + ? d? ' 

 or if %=u— 7), where ?? = QN, 



dm am L_^_ 



dz 2 drf- ol — 97 di) 

 Now the term in brackets in (31) is 



Lz 



Z7] 



and, observing that J& = z'*-\-rf, 



dz - P 2 ' drj ~ P 2 ' « -77 " a + a 2 C ° 5 ' 



we find v 2 ^ = 0. 



Before obtaining the effect at P due to the whole current 

 flowing through the cross-section, pk'Bk, it is necessary to 

 express a., the radius of the circular filament through Q, in 

 terms of a the radius of the filament through D. If PD = m, 

 we have « = « + Peos 6 — m cos <j), where <£>= Z.PDO; and to 

 the same order of approximation as before, (31) becomes 



= 2t7— 2 J0+ ^sin<9 



+ ^^[(3-2L)Rcos^+8(L-1)twcos^] }, . (32) 

 where L = \og e ^. 



Resultant Conical Angle. — The element of area of the cross- 

 section at Q being rfS, the resultant conical angle subtended 

 by the circuit at P is (YlrfS, if the density of the current 

 in the cross-section is assumed to be constant. This we shall 

 assume for the present. Let L QPD = %, and take 

 d$ = HdHdx. Then the first two terms in (32) will give 

 2(7r— 6)d$, or 2(7r — <£ + %)<iS, the integral of which is 

 2(7r — (f)) .A, where A is the area of the cross-section, pWBk. 

 The term mZS obviously vanishes, since % is negative for 

 points Q on the lower side of the line PD. 



If the tangent from P to the circle pJc'BJc makes the angle 

 co with PD, the values of % run from -co to co ; and it is 

 obvious any integral of the form 



V(cos%)sin % ^ 



J— w 



