8 Transactions of the Royal Microscopical Society. 
section. Doubly refracting crystals give two images, one or both 
bifocal, and have one or three principal focal axes, according as they 
have one or two optic axes. These focal axes, which correspond to 
Fresnel’s so-called axes of elasticity, are invariably perpendicular to 
the plane of polarization of the images, and their direction may 
thus be determined with accuracy, if the section can be examined 
sideways with polarized light. In the case of any bifocal image, 
one of the apparent indices is true only when the corresponding 
rincipal focal axis is parallel to the plane of the section. Ib 
therefore, a natural or artificial plate of a crystal be parallel to 
two principal axes, each image gives one true index, and the third 
can be calculated. If, however, parallel to only one axis, only one 
index can be determined, whilst if not parallel to any axis, none of 
the true indices can be directly measured. 
When the section is parallel to two of the focal axes, the three 
true indices being fi, jju, fji!', the four observed indices are 
From lines perpendicular to the plane of polarization {x fx 
tx"^ u"2 
From lines parallel to the plane of polarization . . . . — ^ 
^X fX 
Eepresenting these observed quantities by a, h, c, and d, we 
have u!’=\/ac 01 \/hd and ^ : but since the measured 
values may be affected by errors of observation and also by those 
depending on the section not being cut accurately, these relations 
may not be found to be strictly true, and there may be a consider- 
able difference between the two calculated values of In this 
case their mean may be adopted as the most probable approxima- 
tion ; but it could not be correct if the section does not lie more or 
less closely in the plane of two of the three focal axes. This may, 
however, be at once known by observing whether the two images 
are superimposed or separated ; provided that the two polished 
surfaces are truly parallel to one another. If with or without 
alteration in focal adjustment the two images are seen to be 
directly superimposed, the section must be almost or quite in the 
plane of two axes, whereas if there be a lateral displacement the 
section must be inclined to that plane. 
Some anomalous results are seen in minerals which have such a 
strong dichroism that one image is completely invisible, or which 
are so made up of alternating plates of different refractive powers 
that the lines parallel to them are obscured by reflexion ; and various 
other special peculiarities might be named if time would permit. 
On the whole we may draw the following conclusions : crystals 
having no double refraction give only one image, which is strictly 
unifocal ; crystals having only one optic axis give two images, one 
