PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 181 



It is possible to show by a rather long application of the method of 

 trial and error, using third differences in Meissel's table, that the value 

 of the first root is 0.044654+ and this corresponds to m = 0.89308, 

 /3- = 6.31351, Jo(mb) = 0.9994891+. 



If, then, we consider the single term 



Q = ff, e-«-3i35« . To (0.89308 r), (103) 



Q will satisfy (70) and will vanish when t is infinite. "When t is zero, 

 Q will be equal to Hq for r = 0, and wiU differ from Hq by about one 

 twentieth of one per cent when r = b. The second root of (102) is 

 roughly equal to 3.8 and the corresponding value of /3^ is about 45,000, 

 so that the exponential factor would soon be very small. An inspection 

 of the graph of ./(, {x) shows that if we were to use several terms of the 

 form L • e~^'' ■ Jo (^'^^)) "^6 could easily form a function which should 

 differ very little from H^ for any value of r up to b, when t was zero ; 

 but it is clear that after the lapse of about l/5000th of a second, all 

 the terms beyond the first would be negligible, and there is no practi- 

 cal advantage in using more than one term. 



We may assume then that the value of H in any one of the iron rods 

 is given fairly accurately, except at the very beginning, by (103). Since 

 AttNC = JIs the current in the solenoid falls in the first second to 

 0.001808 of its original value, or to 0.001812 times that value accord- 

 ing as eddy currents are absent or present. These fractions differ 

 from each other by about one two hundred and fifty thousandth part of 

 the original current strength. Another close approximation to the 

 value of H may be made by dividing (103) by J^ (mb) and another by 

 multiplying the second member of (103) by 



l + J^, (104) 



These changes would not affect the relative rate of decay of the 

 current. 



The nearness of the approximation to the value of the field attain- 

 able by a single term is evidently much increased as the diameter of 

 the iron wire of which the core is built up is decreased. If as before 

 (1 = 5, but \ib = 1/200, n = 1000, the value of the first root of the 

 equation for mb will be 0.00446616, nearly, and the value of Joimr) 

 will not change by so much as 1/I0(i000th part of itself as r changes 

 from to b. A single term, therefore, will represent H with great 

 accuracy. In this case the effect of eddy currents is wholly inappre- 



