100 Prof. W. C. Rontgen on the Change in the Double 
marked. On examining the quartz sphere marked with ink- 
spots, the result obtained was that the position of these points 
is simply determined by three planes which cut each other at 
an angle of 120°, in the diameter parallel to the principal 
axis, and contain the three axes of no piezoelectricity. The 
direction of the principal axis determined in this way agreed 
very well with that subsequently found by optical means. 
It hence follows that all straight lines lying in any of the 
three planes specified are directions of no piezoelectricity ; a 
pressure exerted upon the crystal in one of these directions 
produces no electricity at the points of pressure. These planes 
are therefore called planes of no piezoelectricity. 
The following values were obtained with the sphere de- 
scribed, by measurement of the six angles included between 
the three planes : — 
58°, 61°, 60°, 60°, 59°, 62°. 
These angles ought to be exactly 60°. The deviations from 
this number may result in part from experimental errors; 
they are, however, probably also a consequence of small 
deformations and irregularities of the quartz, the presence 
of which could not be determined by optical methods in con- 
sequence of the spherical form of the crystal. Such defor- 
mations have considerable influence on the distribution of 
piezoelectricity, as Hankel has shown with crystals having 
natural faces. Thus, for example, with another sphere which 
plainly showed visible irregularities, I obtained the following 
angles : — 
51°, 54°, 69°, 57°, 64°, 65°. 
The experiments with the first crystal were several times 
repeated — thus, for example, once at a temperature of about 
10° C, a second time at about 31°C. I found always the same 
position of the three planes. 
(2) After the position of the planes of piezoelectricity on 
the sphere had been determined and marked, I examined in 
what way the fields lying between these meridians were piezo- 
electric. The result was that the electricity excited at all 
points of pressure lying in any one of the six fields was of the 
same kind, but that it changed on passing from one field into 
the next. The whole sphere may therefore be divided into 
six fields alternately positively and negatively piezoelectric. 
For the clearer understanding of what follows, it will be well 
to designate them 1, 2, 3, 4, 5, 6, and to assume that at points 
of pressure lying in the first field positive electricity was pro- 
duced : consequently the fields must be marked in order + , 
-, + , -, +, -. 
