84 Dr. J. W. Nicholson on the Effective 



It is convenient to take a new variable &/> = #, so that 



(d 2 Id 1 \ . cos 2 a dA /gn v 



a? + ; ; s +1 "~iM ^"is ""■«&• [m 



\c?A' 2 xdx Ik 2 a 1 dx/ u 2kax dx y v y 



Now (20) may be written 



. did. cos 2 « <#A n 



Aj+ -, T #A 1= = —j r J? , 



a# a? ax ka dx 



or 



/l d d \1 d A _ cos 2 a fZ 



\# rfa? rf# I no dx 2 &aa? da? 



dA 

 da?' 



and the elimination with (21) may be effected at once, 

 yielding 



(? 1 *L 4- ^ f— 4- X £. -4- _ cos2 «(2-3cosV) _^\ A 

 Wa? 2 + a? ^ + J \_dx 2 + x dx + War X dxj ° 



cos 4 a 1 d dA 

 2Ic 2 a 2 xdx dx 



'=£•> ■ ( 2 2) 



and it is at once evident that A is only altered to the second 

 order by the bending of the wire. It is henceforth supposed 

 that not only a, but ka is large, as this case includes nearly 

 all of practical interest. When ka is small, a special investi- 

 gation is necessary. 



Ignoring the effect of bending, the differential equation 

 becomes 



/ d* Id , \ 2 . n 



(d^ + xdx +1 ) A «=°'> 



so that, for a solution finite on the central curve, 



/ d 2 Id , \ A T , . 

 {dx* + xdx +1 ) A °« J ^)'> 



and finally 



A o =AJ o 0r) + B.a i (^) ? 



where A and B are arbitrary. In determining the self- 

 induction and resistance, we only require the modification, 

 inside the metal, of the form J (x), and in the outer medium, 

 of the form ~K (ix). The solution xj x (x) may be discarded, 

 as it corresponds to nothing in the case of the straight wire, 

 to which the formulae must lead when a is infinite, or a = 0. 



