NORMAL SULPHATES OF POTASSIUM, RUBIDIUM, AND CESIUM. 
475 
For the screws : I 
8600 + 4-56 --^^' 
{U — h) for the first interval, and 
similarly for the second interval, substituting for U. 
For the compensator; 
10 ®(2204 -h 2’12 (O ” h) for the first interval, 
and a like expression for the second interval with A substituted for U. 
In each case the actual expansion of the metal is thus calculated by multiplying 
the length (thickness) of the metal by the mean coefficient of the linear expansion 
between the two temperatures, that is by a + for that metal where t is the mean 
of the limiting temperatures of the interval, namely -g (h -fi A) or ^ (d + t^); and also 
by the amount of the temperature interval, that is, — h or — L- Actually, of 
course, one uses h instead of 2fi. (^i + ^ 2 )- 
The diflferences between the amounts of expansion of the screws and the compen¬ 
sator are given in the next two columns headed “ correction for non-compensation,” 
The correction is obviously positive, given an expanding crystal, when the screws 
expand most, and negative when the compensator expands to the greater extent. 
For in the former case the effect is to increase the thickness of tiie air-film, and 
consequently the amount of diminution of the thickness of the air-film due to the 
expansion of the crystal is not fully evident, the actually observed amount being less 
than that really effected by the expanding crystal by the amount of this excess of 
expansion on the part of the screws. This latter amount should, therefore, be added. 
The inverse is the case when the excess is on the part of the compensator; causing, 
as it does, additional diminution of the thickness of the air-film, it should be sub¬ 
tracted. The values given in the last two columns, representing the actual expansions 
of the crystal during the two intervals of temperature, and were 
obtained by applying the correction for non-compensation, in the sense just indicated, 
to the apparent expansions'^A/2 andji^'X/2. 
In the last portion of the table are given the calculated values of 6, (f), and Lq, and 
of the two required constants of the coefficient of linear expansion, the coefficient 
at 0° and h, half the increment of the coefficient per degree of temperature. In the 
last column are given the values of the coefficient of linear expansion, a, for 50°, 
calculated by means of the formula rx — a 2ht. Fizeau invariably gave the 
coefficient at 40°, a specific temperature in the neighbourhood of the mean of the 
extreme limits employed by him, in addition to a and h. As 50° is nearer the mean 
of the author’s limiting temperatures, this specific temperature has l^een chosen in 
})reference, for which to record a particular calculated value of a. 
The Results. 
In the following tables are presented the results of the determinations and calcu¬ 
lations, 
3 P 2 
