Action of a Galvanic Coil on an external small Magnet. 197 



verse plane, and a the distance of the same point from the axis 

 of z. Also in the same plane let r and 6 be the polar coordi- 

 nates of the given point P referred to the point of intersection 

 as pole, and to the straight line through the pole parallel to the 

 plane xy. Then, the rectangular coordinates of P being x, y, z, 

 if we put R for (a? 2 + z/ 2 )" 2 , and suppose the velocity parallel to 

 the plane xy to be F(R, z), and that in the transverse plane to 

 be f(r, 0), we have 



«= /(r^)^.|+F(R,r)|, 



w=-f(r,6) 



z—h y J?/T> x x 

 R 



/M) — -t-^»E- 



together with the equalities r 2 = (z — h)* + (R — a) 2 , and 



tan 0= p . By analytical operations, the details of which, as 



being somewhat long but presenting no difficulties, are not in- 

 serted here, it may be shown (1) that -7- -|- — + -7- = : 



J dx dy dz 



(2) that udx + vdy + wdz is not an exact differential; (3) that by 

 substitution in the equation (a) there results the following equa- 

 tion of condition connecting the functions /and F : 



r df a d¥ , 7N /T) N d¥ , 



37. Respecting this equation we may, first, remark that since 



/if 

 it does not contain -j4, it shows that the assumed motion requires 



that / should be a function of r only, and consequently the mo- 

 tion in planes transverse to the axis of the wire is proved to be 

 circular. This result is in accordance with the original assump- 

 tion, that the transverse section of the wire is circular, as should 

 plainly be the case, since the surface of the wire bounds the cir- 

 cular motion. 



38. The proof that / is a function of r only having taken no 

 account of the magnitude of a, and being clearly independent of 

 that of h, we may infer that the function has the same form 

 whatever be the radius of the axis of the wire, and therefore the 

 same as if the radius were infinite, in which case a finite portion 

 of the wire might be considered to be a straight cylinder. But 



we have shown (art. 35) that for the straight cylinder /(/) = -\. 



Consequently, substituting this value of /(?•) in the equation (b), 



