ON THE MEASUREMENT OF MAGNETIC HYSTERESIS. 
97 
Now (2) is satisfied if we write 
u — d^jdij, V — — d^jdx .( 5 ), 
while (1) now becomes d^^jdx"^ + d^(f>ldif = q .(6). 
The solution of (6), approj^riate to the problem in hand,"^" is 
, _ _ / _! V'+'‘_ cos(2?>t + l)7rxla . cos(2» + l)7r,# _ 
^ ~ TT' ^ {2m + 1)(271 + 1) {(2m + l)VU(d + {2n + 1)VV&2} ’ 
This value of </> satisfies vV — because, m and n both ranging from 0 to oo , 
cos (2 m + 1) TTi'la _ 4 ^ , 
2m+1 — 
1) 
, cos (2?-<, + 1) Tryih 
2n + 1 
within the limits x — V = 
Now by (5) and (7) 
IQq ^ ^ \m+n 4" 1) 7r.r/(X . sin (2?i + 1) Tryjh 
u 
— VV/ m 
dy 
(2?i + l)7r 
TT" 
(2m + 1) {2n + 1) {(2m + 1)- TT-jcd + {2n + l)^ ttUS-} 
with a similar expression for v. These expressions satisfy (3) and (4), and thus all 
the conditions are fulfilled. 
The rate at which heat is generated is given by 
ahd^jdt = cr 11 {ir + v~) dxdy .(8). 
The necessary integrations are easily effected, for the integral 
COS (2A + 1) TTxIa . cos (2k +1) TTxja . dx 
-ia 
is zero unless h and k are equal. When h = k, its value is ^a. Similar results hold 
when two sines are substituted for the two cosines. 
We thus obtain 
dX. 64:q"acd 1 
~ TT® ' (2m + l)'^(2;i + ly {(2m + 1)' + {2n + If ay¥} ‘ 
It is not convenient to calculate dJL/dt from this double series, on account of the slow 
convergence. We therefore transform it into a single series. Now 
cosh 7TZ/2 = (1 + (1 _|_ 2^3=) (1 + zy5~) ... * 
Differentiating the logarithm of both sides, we obtain 
itani' -2 + !/ + .■ • 
{m from 0 to oo ). 
* The method of solution was suggested l)y notes taken by one of us at a course of lectures on 
hydrodynamics, given by Mr. R. A. Herman, Fellow of Trinity College, in November, 1889. 
VOL. CXCVIII. —A. 
O 
