NORMAL SULPHATES OF POTASSIUM, RUBIDIUM, AND CESIUM. 
473 
quantities has been determined at least six independent times on three different 
crystals, and with respect to five of the quantities eight determinations have been 
made on four separate crystals. In all sixty-four independent determinations have 
been carried out, on different days, and using the twenty-nine different crystals of 
which the details have already been given (p. 465). 
Each determination afforded, as already fully explained in connection with the 
determinations of the expansion of platinum-iridium and aluminium in the dilato- 
meter memoir {loc. cit. p. 352), the two constants required for a complete statement 
% 
of the thermal behaviour, namely, the constant a, the coefficient of expansion at 0°, 
and 6, half the increment of the coefficient per degree of temperature, the coefficient 
not being a fixed quantity for all temperatui'es but varying regularly with the 
temperature. The coefficient of thermal expansion is signified by a, and the expres¬ 
sion for the actual coefficient at any temperature t, as also for the mean coefficient 
between any two temperatures whose mean is is : 
a = « fi- 2ht. 
The mean coefficient of expansion between 0° and f is, however 
a -f- &L 
The data afforded by observations of the positions of the interference bands at 
three adequately separated temperatures, and of the number of bands passing the 
reference point during the intervals between these temperatures, together with a 
knowledge of the original thicknesses of the block of crystal and of the aluminium 
compensator, and the length of the platinum-iridium screws projecting above the 
tripod table or its raised points,-are ample to enable the two constants a and h to be 
calculated. For it is only necessary to insert respectively in three equations of the 
form 
= Eo (l cit ht") 
the known values of the three temperatures and the lengths (thicknesses) of the 
crystal block at those temperatures, and to solve the three equations thus provided, 
for the three unknown quantities Lq, a, and 6. 
The solution of these equations furnishes expressions for the three required 
quantities of the forms 
0 <h 
a = — , h =^, and Lq = — Bti — 
-L'O -1^0 ^ 
in which 0 and (f) are terms involving the differences of the lengths, at the 
three temperatures ti, U, and and the sums and differences of those temperatures. 
The actual expressions for 0 and (j) employed throughout the observations were : 
0 _ (fi + 4) (L<3 ~ Lf,) (fi + i-i) (L 4 ~ ^ 
(4 ^1) (4 ^2) (4 ^1) (4 ^2) 
/ _ — fiti __ 1^4 ~ , 
(4 4) (4 4) (4 4) (4 “ 4) 
3 p 
VOL. CXCII.-A. 
