dol 
electrostatic problems showing the same geometrical configuration, 
on account of the well-known general analogies. The details and 
proofs are to be given elsewhere. 
Round armatures. Considering in the first place surfaces of 
revolution, more especially cones, the coincident vertices of which 
both le in A, the field in this point is known to be 
B 
D= H, + H, = Aa § sin vers B + An § sin? a cos a log Fu (1) 
The notation sufficiently appears from Fig. 1. Both terms are 
generally of the same order practically; the first corresponds to the 
truncated frontal planes, the second to the conic surfaces; the latter 
shows a maximum for a = tan—V2 = 54°44’, 
In order to judge of the field’s uniformity we now consider the 
second derivatives, which are related to one another by LAPLACE’s 
equation and the symmetry of the case. The x-component, $,, of 
the field is everywhere meant, though the index 2 is mostly omitted 
for simplification. For the centre A, where the first derivatives 
evidently vanish, the following values are found 
07H, arene 0D, ie 07), ay Sain? Boost 8 5 3 sin‘ B cos 8 (1) 
Ou? Oy? dz? ae b? 
Now the term , always shows a minimum in the centre A, 
when passing along the longitudinal z-axis, corresponding to a 
maximum along the equatorial transverse axes, because the numerator . 
suv? B cos °B remains positive for 0< @< ar/2; in particular this is 
a maximum, and accordingly the non-uniformity is greatest, for 
= fan—V */, = 39°14’. 
The term 4, behaves exactly in the opposite way, its second 
derivative vanishing for that same angle. This well-known result also 
follows from the general formula, which I now find, viz: 
07.9, dd, en) den Rd 
DE =d De —=-3 eo ae An. ae at cos a (5 cos? (5 — =e 
As B > b this expression evidently is + for a = cos! //*/, == 39°14"; 
accordingly £, shows a longitudinal minimum and transverse maximum 
for smaller semi-angles, whereas for larger ones the reverse holds, 
so as to make the field weaker on the axis than in its lateral 
surroundings. Finally for the total field 
0? (D,E 3 b? 
Es =e +) y aR 2 sin‘ B cos B +- sint a cos a(5 cos? a-3) (: -z) |© 
22* 
