416 
MESSES. T. E. THOEPE AND A. W. ETTCKEE ON 
between the specific gravities of the solutions employed, which are 
•004355, -003748, and -004160 
respectively, differ in the fourth place of decimals, we are led to the conclusion thaty(s) 
is a linear function of s, and that differences in the fourth place of decimals in the value 
of the specific gravities do not involve important changes in the law of thermal 
expansion. 
From this it follows that equation (1), when written at length, is of the form 
v=l J r bt-\-ctf-\-df-\-(b't-\-c'f-{-d l f)( 2 )-\-qs), (2) 
or 
As this expression is of no theoretical value, it is unnecessary to describe, step by step, 
the arithmetical operations by which the constants were determined ; it is sufficient to 
say that a larger number of equations than that required for the purpose may be obtained 
by equating the coefficients of t in (2) to their values as determined for the solutions 
submitted to experiment and given in Table XII., and that the numbers given by the 
equations of condition so obtained have, been in some cases slightly modified. The 
values taken were : — 
6= -00008097, &' = --5509xl0- 5 , 
-c= -0000049036, c'= -020198 x lO” 5 , 
d= - -000000012289, d'= - -00033276 x 10~ 5 . 
The expression^) 4- ^ ma y he conveniently written in the form 
ll-95-940(s-l-02), 
in which it is unnecessary to express s to more than 4 places of decimals. Table XIV . 
shows the agreement between the volumes calculated by the mean formulae and the 
general expression thus determined. 
Table XIV. 
Sample A. 
Sample B. 
Sample O. 
Sample D. 
t° c. 
Mean 
General 
Diff. 
Mean 
General 
Diff. 
Mean 
General 
Diff. 
Mean 
General 
Diff. 
formula. 
formula. 
formula. 
formula. 
formula. 
formula. 
formula. 
formula. 
0 
1-00000 
1-00000 
0 
1-00000 
1-00000 
o 
1-00000 
1-00000 
0 
1-00000 
1-00000 
0 
2 
18 
18 
0 
14 
14 
0 
11 
10 
-1 
7 
7 
0 
4 
41 
40 
-1 
32 
33 
+ 1 
26 
26 
0 
20 
19 
-1 
6 
66 
66 
0 
55 
56 
+ 1 1 
46 
47 
+ 1 
38 
36 
-2 
8 
96 
95 
-1 
82 
83 
+ 1 1 
71 
72 
+ 1 
61 
58 
-3 
10 
131 
130 
-1 
114 
115 
+ 1 
100 
101 
+ 1 1 
88 
86 
-2 
12 
167 
167 
0 
149 
150 
+ 1 
133 
135 
+ 2 
119 
118 
-1 
15 
232 
229 
—3 
210 
209 
-1 
191 
191 
0 
174 
173 
-1 
18 
301 
298 
-3 
278 
277 
- 1 
257 
258 
+ i ! 
238 
237 
-1 
21 
379 
376 
-3 
355 
353 
-2 
331 
333 
+ 2 
310 
310 
0 
24 
463 
461 
-2 
439 
437 
-2 
412 
415 
+ 3 
390 
390 
0 
27 
555 
553 
-2 
529 
527 
-2 
501 
503 
+ 2 
477 
477 
0 
30 
653 
652 
- 1 
626 
623 
-3 
596 
597 
+ 1 
570 
569 
-1 
33 
758 
758 
0 
728 
72 6 
-2 
697 
697 
0 
668 
665 
-3 
36 
869 , 
870 
+ 1 
836 
834 
-2 
803 
802 
— ! 
772 
766 
-6 
* The numerical constants involved in the above formula are given in the forms in which they were, for 
facility of calculation, determined. The expression can, of course, be easily transformed to the simpler form, 
Y=F l (t)+sF i (t). 
