578 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where k = w't/4. ttN^'D, Z> = ^ [4 + tt (/x - l)]/4. 

 For this problem, (55) yields 



H^ — H^ 



= e-6-316<^ (56) 



and a comparison of (54) and (56) shows that the eddy currents in a 

 core of this wire, one millimeter in diameter, have practically no effect 

 in slowing the changes in magnetism of the iron. 



If the core of the given solenoid were made up of rods one centimeter 

 in diameter, 7nb or a would be given as the roots of the equation 



Jo (ir) ■ (1 - 0.01366 x'') = \OxJx {x), (57) 



and it is not very difiicult to show by a process of trial and error from 

 Meissel's Tafel der BesseV schen Functionen, that the first three of these 

 roots are approximately equal to 0.4411, 3.8525, 7.0204, and that the 

 corresponding values of Jq{x) and Ji(.r) are 0.951946, — 0.402672, 

 0.300112, and 0.215229, —0.008352, 0.001444. 



If, with these roots, we wish to determine such a set of coefficients, 

 (Zi, X2, X3) as shall make the mean square of the difference between 

 unity and 22/ • Jf^{m7-) as small as possible, for the range from r = to 

 /• = &, we have to solve the equations 



Ai ' L\ + B11 • L2. + Biz • Lz = Ci, B12 • Li + ^2 • -^2 "I" -^23 • Lz=- C^ 

 Biz • Li + ^23^2 ■{• AzLz= Cz, where 



Ci = 2^b'- Ji{xi)/xi, Ai = 7rb'\ [J,(xi)f + [Ji(xi)f}, 



B12. = V" \Xi • JJ^X^ • Ji{Xi) — X2 • Jo{Xl) ■ Jl(.?'2)]/(^/ — ^2^), 



as Professor Byerly's theorems show. The computation here indicated 

 shows that fl = 6.096 or 0.320, approximately, according as ^ = or 

 ^ = 0.1, whereas, if eddy currents were wholly cut out, the correspond- 

 ing values would be 6.316 and 0.336. These figures illustrate the 

 comparatively slight effect of subdividing the core in the particular 

 case here considered. The results would, of course, be somewhat 

 different numerically, with different assumed values for the constants 

 of the circuit. 



It is clear that the inversion of sign in the magnetic moment of a 

 straight iron bar, when the magnetic excitation is suddenly removed, 

 accompanies, at least, a large demagnetizing factor due to the ends of 

 the bar, and no one seems to have observed the phenomenon in the 

 case of closed cores. In rings, however, as in straight bars, the ulti- 



