ON THE MEASUEEMENT OF MAGNETIC HYSTERESIS. 
43 
d (wAB + MC)/c/^, viz., Sc = — c/(wAB + We may also treat M as constant. 
When this is done we obtain 
n dt) 
(„Af+ M 
dt - T' + 
T. 
If we add together the results for a pair of semi-cycles, the quantities T and T' 
disappear when a cyclic state has been established, for then the magnetic energy is 
the same at the end as at the beginning of a cycle. We may then replace W^ + 
by W -b X and 6-^ by 6^ + do, but we must remember that the integral is to be taken 
completely round a cycle. 
The expression can be put into a more convenient form if we notice that by the 
rules of approximation c is to be taken as zero in the terms under the integral sign. 
We thus replace IttNC by H, and then writing 47rNdB/dH.c?C/c/^ for cZB/dq we 
have, on division by al. 
W 
Nf'P 
N r /47r?dA 
SW J \ l rfH 
+ L)(4™ANb| 
( 12 ). 
the final formula. 
Now, unless the specimen be so thick, and the variation of H so rapid, that during 
part of a complete cycle the average induction for the whole section increases vdiile 
the applied magnetic force diminishes, c/B/c/H is always positive. Hence, since dGjdt 
always has the same sign as r/C, the correcting integral is always positive. Thus the 
true value of W -f- X is less than that calculated from the throw of the dynamometer 
on the assumj)tion that S is infinite. 
§ 9. Before we can apply equation (12) in finding W, we must either know that X 
* We can easily obtain with rough approximation the condition which must be satisfied in order that 
ludc/dt may be negligible in comparison with dfiiAB + MC)/dt From (11) we have 
(s + L c = - ^ («AB + MC), 
Avhence 
^ + “C) = (s A® ■ ■ 1- 
Thus the approximation -Sc = d(nAB + MC)/df is valid if L/S.cPfnAB + MC)/dP is small in com¬ 
parison with d{nAB -f MC)/(/^. A sufficient idea of the magnitudes involved is obtained by treating B as 
proportional to C. In this case L/S.d-C/d/^ must be small in comparison with dCjdt. But, if E 
denote the voltage of the battery, the characteristic of C is 
KdC/dt + EC = E, 
where E is now the resistance of the whole of the primary circuit, and K depends mainly upon the 
choking coil (§ 33) in the circuit. Hence, treating K as constant, 
Kd'^C/df + 'RdCjdt = 0. 
Thus the condition for the validity of the approximation can be expressed in simple form by saying that 
L/S must be small in comparison with K/R. 
G 2 
