Field of an Electrical Current. 213 



and it will be found that 



|jR 2 cm. X sm^ x =iJ x (R"-ff)sin X d X = A^, . (48/) 

 WwdR. XS m2 X d X = iJ x (R«_E 4 ) wx2 X d X 



in which % has been taken from to co only, so that in the 

 correction (48 e) these must be doubled. 



Hence the whole of the correction (48 e) amounts to 



sin cf> c 2 



iM 8 ^-M 2 ^ + £> in2 *}' 



2a in 

 so that (43) is replaced by 



-[(6L-5)m 2 + C 2 +£ r2 ] S in2</> J, (48 i) 



the right-hand side being the value of the resultant conical 

 angle subtended by the circuit at the point P. The magnetic 

 potential at P is therefore this right-hand side multiplied by 

 t, the total current traversing the cross-section of the wire. 



It will be found that this value of © satisfies, both for the 

 terms of the first order and for those of the second, the con- 

 dition (44). 



If the density of the current at any point Q in the cross- 

 section be assumed to vary as any power of the distance of Q 

 from the axis OV, the conical angle subtended at a point near 

 the wire is found just as easily as in the case in which the 

 density is supposed to vary inversely as the distance. Thus, 



if it is proportional to — , (48 c) will be replaced by 



ti n (n -J- 1 ) 



1 (RcosO — racos<£) + 2 J (Rcos0 — ^cos(/>)' 2 , 



and we have merely the same terms (48/), &c, as before. 



We see then that when small quantities of the first order, 



rn 

 indicated by the fraction — , are taken into account, the maff- 



J a & 



netic potential, and therefore the magnetic force, at any point 

 are not the same as if the whole current were condensed into 

 an infinitely thin filament traversing the centre of the wire, 



