Action of a Galvanic Coil on an external small Magnet. 199 



lengths; and as any portion of a uniform conductor of any form 

 may be supposed to be similarly composed, the expressions for 

 f{r) and F(r) for a circular wire would appear to apply generally, 

 if the radius a be taken to represent the varying radius of curva- 

 ture of the axis of the wire. This question, however, requires 

 more consideration than I can now give to it. 



41. Returning now to the circular conductor, if in the expres- 

 sions for u, v, w in art. 36 the values found for f(r, 6) and 

 F(R, z) be substituted, we shall have 



c^x{z — h) C<£f 

 U ~ *~~R? + Wr 

 c x y[z-h) ___ CjX 

 Rr d EV 



w= — 



r° 



Hence, since R 2 = # 2 + ?/ 2 and r 2 = (s — A) 2 + (R~«) 2 , it will be 

 found that 



, 7 c, (z — h)dR — (Ti — a)dz c ydx—xdy 

 udx + vdy + wdz = -i • ±- ' v 4~ + ~ ' 2 , 2 - 



Consequently the right-hand side of this equation becomes an 

 exact differential when multiplied by the factor r. Before pro- 

 ceeding to the next step, it is necessary to take into account that 

 in the foregoing investigation the arbitrary constants c x and c 2 

 have been introduced in such manner as to show that they 

 are wholly independent of each other. Hence, on equating 

 r{udx-\-vdy-\-wdz) to zero, we must have separately 



(z— h)dH — (R — a)dz _„ ydx—xdy_~ 



c i (*_A) 2 + (R-fl)* ' ° 2 ' x Ci + if ' 



which means that both the motion transverse to the axis of the 

 wire and that parallel to the same are such as require a factor 

 for making udx + vdy + iudz integrable. Both are steady motions 

 and therefore coexist. Instead of the above two equations we 

 may, by introducing an arbitrary constant factor A, employ the 

 single equation 



(z — h)dB>—(R — a)dz ydx—xdy^ 



C] (*_A)S+(R- fl )« + A ^ 2 ' f- + a* -°' 



Hence, by integration, 



z — ~ h v 



C = c, tan" 1 -p h Xc 2 tan - 1 *- = c x 6 + \c 2 <£, 



xii — a x 



y z ~~~ h 



supposing that - = tan </>, and, as before, that — -==tan0. 



x }x — a 



