334 
DRS. J. AND E. HOPKINSON ON DYNAMO-ELECTRIC MACHINERY. 
plane making an angle a. We want a rough idea of the extension of the area between 
the plates by the spreading of the lines of induction beyond the boundary. We know 
that the actual extension of the area will be greater than we shall calculate it to be if 
we prescribe an arbitrary distribution of lines of force other than that which is 
consistent with Laplace’s equation. 
Assume, then, the lines of force to be segments of circles centre O, and straight 
y 
lines perpendicular to OA. The induction along a line PQPt will be ■ _^ ^ ^ "V 
being difference of potential between the planes, and the added induction from OPB 
[X X 
will be ~ 
I m nr — 
Vdx 
0(7 T—U)X- 
■t 
V , ( 7 T — cd)X-\-1 
- log - —7-. 
Thus if «=.; we have for x=t, 2 1, &c. 
(-IT — 
X 
t 
2 1 
3 1 
4 1 
5 1 
10 1 
IT — « 
log 
•599 
•904 
1T09 
1-263 
1-387 
1-792 
a)x + t 
showing that the extension of the area of the field is likely to be considerable. 
(2) The failure of the actual curve to reach the maximum indicated by approximate 
theory is because the theory assumes that all tubes of induction passing through the 
magnets pass also through the armature. Familiar observations round the pole pieces 
of the magnets show that this is not the case. If v be the ratio of the total 
induction through the magnets to the induction in the armature we must, in our 
expression for the line integral of magnetising force, replace the term fl by f ^ 
^3 ! J V-W 
v is not strictly a constant, as we shall see later ; it is somewhat increased as I 
increases owing to magnetisation of the core of the armature, and it is also affected 
by the current in the armature. For our present purpose we treat it as constant. 
There is yet another source of error which it is necessary to examine. Some part 
of the induction in the armature may pass through the shaft instead of through the 
iron plates. An idea of the amount of this disturbance may be readily obtained. 
Consider the closed curve ABCDEF, AB and FEDC are drawn along lines of force, 
AF and BC are orthogonal to lines of force. Since this closed curve has no currents 
passing through it, the line integral of force around it is nil; therefore, neglecting 
force along ED, we have force along AB is equal to force along FE and DC. In the 
machine presently described we may safely neglect the induction through the shaft; 
the error is comparable with the uncertainty as to the value of l } ; but in another 
machine, with magnets of much greater section, the effect of the shaft would become 
very sensible when the core is practically saturated. 
