4 Transactions of the Royal Microscopical Society. 
compensate for small accidental errors, and care must of course be 
taken to avoid any that might he caused by imperfections in the 
instrument. If the section of the mineral be covered with thin 
glass, which in most cases is very desirable, its apparent thickness 
must be measured, and due allowance made for it in calculating out 
the results. It is also requisite to deduct from the indices given 
by the formula ^ a small quantity due to the effects of 
spherical aberration which varies with the aperture and correction 
of the object-glass, and also with the value of the index in each 
particular case. In order to obtain as accurate measurement as 
possible, a number of precautions must be taken, which are all 
simple enough, but it would occupy too much time to describe 
them in detail. With proper care the errors in the values of 
t and d ought to be certainly less than toW of an inch. The 
accuracy with which the indices of refraction can thus be determined 
depends much on the thickness of each specimen, but if it be from 
^ to of an inch, the errors ought to be limited to the third 
place of decimals. In practically employing this method it is of 
great importance to have some object which gives a very definite 
focus. In the first instance I made use of a glass plate having 
very fine parallel scratches, made with the finest emery paper ; 
but I soon found that it would be very convenient to have more 
definite and equidistant parallel lines, not in any way affected by 
moving the stage. This can be accomplished by having them 
ruled of an inch apart on a glass plate, fixed as far as 
possible below the lenses of an achromatic condenser, with a small 
central stop, which gives at the focus a much reduced image easily 
adjusted either a little below the lower or upper surface, or nearer 
the centre of the specimen, according as its shape may make it 
necessary, so that the light may pass to the object-glass as equally 
as possible from all sides. It is also extremely useful to have an 
iris diaphragm fixed just below the grating, so as to be able to 
obtain an image of a circular hole of any requisite diameter. I 
had two sets of lines ruled on the same surface at right angles to 
each other, in order that there might be less chance of mistaking 
any striae in the mineral for a single system of lines, and that 
either system might be used if the other were obscured. This 
arrangement has fortunately led to the discovery of an entirely 
new class of optical properties. 
Unifocal and Bifocal Images. 
On looking at the double system of lines without any inter- 
vening object, both sets of parallel lines are seen at the same focus. 
If a plate, with parallel flat surfaces of glass or of any transparent 
