10 
REPORT— 1847 * 
If this expression is differentiated with regard to;? and the differential is 
supposed equal to zero, the following equation is obtained:— 
{a - x) + 2 (b ~ ax^p + Sic - bx)p‘^ + 4 (rf — cx)f -Sdxf, 
which gives that value of;? for which the volume of the mixture is a mini¬ 
mum, or for which the greatest contraction of volume takes place. For the 
three first hydrates in Table III. the least value of and the corresponding 
value of;? is found by graphic constniction. 
In this manner the values of the maximum of contraction and of the con¬ 
tracted or true volume contained in the following table arc found, where the 
volume of the mixture is taken as unity. 
Table IV. 
The maximum of contraction corresponds to 
The true or 
Contraction 
for 
P “ 
1 or 
Goatr acted 
volume » 0 . 
» 1 - ®. 
SO, + laq 
SO, -f 2aq 
SO, + Saq 
SO, -f- 4aq 
SOj 4- 5aq 
SO, + 6aq 
SO,-f 7aq 
SO, -i- 9aq 
SO, + 12aq 
SO, -j- l5aq 
0*595 
0*370 
0*288 
0*254 
0*224 
0*198 
0*184 
0*158 
0*127 
0*109 
SO, -1- S-OOaq 
SO, 4* 7*S2aq 
SO, -f 10*92aq 
SO, + 12*97aq 
SO, + 15*30nq 
SO, 4- 17'9)aq 
.SO, 4- 10*604q 
SO, -f 23*57aq 
SO, 4- SO*a36aq 
SO, 4- 36'Uiui 
0*921486 
0*967734 
0*976805 
0*980735 
0*983692 
0*985972 
0*987754 
0*990421 
0*993011 
0-994668 
0*078514 
0*032266 
0*028195 
0019265 
0-016308 
0*014028 
0*012246 
0*009579 
0*006989 
0*005332 
As the volume of the mixture near its minimum only slowly changes its 
value. It is not to bo expected that the numbers in the second md third 
columns can have any protonsion to great accuracy ; they are stated only as 
a preluninary approxuuution, especially as 1 have not yet been able to dis- 
It of the per-centage of the acid employed, 
^xth numbers in the filth and 
n rcj)re8ents three successive terms of the 
So ZVn T ‘‘ + 4aq, SO, -f 5aq and 
P«Uto to oak lep;- ‘ ‘■‘>‘'‘’*‘"8 f™' 
(a + c):c^2aib. 
So for « = 0-019265, i = 0*016308, c = 0-01402S. is 
(a-l-c):c = 2*37, and 
2«;6 = 2-3C. 
• or 
0 
9 + rn ' a + n 
«hero « is the index for the different terms of the series, and a and fi con- 
