ON THE MEASUREMENT OF MAGNETIC HYSTERESIS. 
41 
where B is the average value of the induction over the section of the specimen at 
any time. Here L and L' are constant, while M varies slightly as the suspended coil 
turns round. 
By the principle of the conservation of energy, the work done by the voltage E in 
any time is equal to the energy dissipated in the specimen, together with the heat 
produced in the resistances B and S, and the kinetic energy acquired by the moving 
coil of the dynamometer and the increase in the magnetic energy of the system. 
Now let Wj, Wo be the energy dissipated by hysteresis j)er cub. centim. for two 
semi-cycles, so that the loss per cycle is W = W;^ + W 3 . Further, let X^, Xo 
be the space-averages^^ of the energy dissipated by eddy currents, so that the loss 
per cycle is X = Xj -b Xo. Then the total energy dissipated in the specimen in a 
semi-cycle is A/(Wj -j- X^^). 
If xjj be the deflection of the suspended coil, then the couple tending to increase \jj 
Is CcclM/cl^ dyne-centim. The rate of working of this couple at any instant is thus 
CctlM/clxfj.dxfj/dt or CcdM/dt, and hence the] total work done is | (dcdd^ijdt.dt. This 
is therefore the kinetic energy acquired by the coil. 
Then, if T, T' be the magnetic energy at the beginning and end of a semi-cycle, we 
have 
I mdt = (Wi + Xi) M -f I (RC3 -f + GcdKIdt) dt -f T' - T. 
But from ( 10 ) 
J'ECd^ = \W^dt + |C I (N/AB -f L'C Me) dt . 
Comparing these expressions, we And, since U\GdOldt.dt = 0 for a cycle or a semi¬ 
cycle, 
(Wi -f Xi) M = |C (N/AdB/c/^ + micjdt) dt-^\chU - T' fl- T. 
The integrations are to be effected between the limits i = 0 and ^ = co , where 
^ = 0 denotes any instant before the primary current begins to change, and t = <x> 
denotes some instant towards the end of the change when the primary current has 
with sufficient accuracy reached its final value C^J. It follows that c = 0 at both 
limits. 
From ( 11 ) we find 
- S I chit = I (wAB + MC) dt , 
since Ijcdcjdt vanishes on integration. Adding this expression to the last one, we 
obtain 
(w, + x.)a; = I 
cKf+ m|) + c^OAB+MC) 
dt - T -b T 
* tVe take the space-average because the rate of heat janduetion is not uniform over the section; when 
the section is circular and dBjeU is very small the rate at any point is proportional to the square of the 
distance of the point from the centre of the section. (See Appendix I.) 
VOL. CXCVIII. — A. 
G 
