280 Mr. L. Fletcher on the Dilatation of Crystals 
The mineral we select for this illustration is rock-crystal; 
and for the following reasons : — 
1st. Any section through the morphological axis contains 
two lines which are at once thermic axes and atropic lines for 
all temperatures; 
2ndly. The expansions along these atropic lines have been 
determined by Fizeau with great accuracy; and 
3rdly. These expansions are very different, one being almost 
double the other. 
The precision attained by the method of Fizeau* will be 
better appreciated if we mention that two determinations 
made with especial care at times separated by an interval of 
about twelve months, during which the apparatus had been 
heated and cooled some hundreds of times, gave for the co- 
efficient of expansion in the direction of the morphological 
axis 0-0000078118 and 0-0000078117 respectively. 
We may here conveniently call attention to the fact that 
Fizeau's method when employed alone gives the variation in 
the thickness of the plate and not the variation in length of a 
crystal-line initially normal to its faces, the two quantities 
only being identical when the same line of the crystal is per- 
manently normal to the plate. If h be the initial thickness of 
the plate, and h! the length at the second temperature of a 
line which, though at the first temperature normal to the 
plate, is at the second temperature inclined to the normal at 
an angle 6, the quantity measured by Fizeau is h! cos 6— h, 
and differs from the true expansion of the crystal-line, namely 
h'—h, by a quantity 7/(1 — cos 6), which, though always 
small, is in many cases quite appreciable ; if, for instance, for 
an interval of temperature of 100° C. the normal be rotated 
through an angle of 10', as is sometimes the case, the correc- 
tion of the coefficient of expansion for that interval is "0000042 
and for an interval of 1° C. is "000000042, quantities scarcely 
to be neglected. 
The present illustration is so chosen that this difficulty does 
not present itself, the expansions having been determined by 
Fizeau from plates normal to permanently atropic lines. 
Let X O Z (PI. III. fig. 1) be any section of a crystal of 
quartz of which O Z is the morphological axis and O X any line 
perpendicular to it, these lines being permanent in direction in 
space for all changes of temperature ; let the unit lengths 
along O X, O Z at the initial temperature t x become a, /3 at the 
temperature t 2 ; let L' W be the position at the second tempe- 
rature of the line which initially has the position L N, and let 
* Comptes Rendus, 1868, vol. lxvi. p. 1006. 
