80 Proceedings of the Royal Society 
of a given quantity of water, the line thus produced would be a 
parabola. Having traced the 
curve, let ordinates A and 
t 2 B be drawn corresponding to 
the observed temperatures t v t 2 , 
and let the points A and B at 
which these cut the curve, be 
joined by a straight line A B. 
This line is not parallel to 
the line of abscissae, but is in- 
clined so as to show the rate of 
expansion of the glass ball, and 
a line drawn parallel to it to 
touch the curve will do so, not 
on the ordinate which shows 
the maximum density of the 
water, but on the ordinate 
mid-way between t x and £ 2 ; 
which ordinate D C would in- 
dicate the temperature at which 
the water would seem to have 
the greatest density when in- 
closed in a thermometer tube 
made of the same kind of glass. 
The distance E A, or F B be- 
tween the tangent and the chord is, according to the well-known 
properties of the parabola, proportional to the square of the inter- 
cepted absciss D£ 2 , or to that of its double # x t 2 . 
Hence the square of the observed interval t x t 2 between the two 
temperatures at which the ball passes the thermometer bulb, when 
multiplied by some constant number to be determined by means of 
the known formula for the expansion of water, will give the dif- 
ference between the glass and the water, when each is of the tem- 
perature half-way between t x and t 2 . 
If now we mix in the water some soluble salt, so as to change 
slightly its specific gravity, without perceptibly altering the law of 
its expansion, we shall thereby augment the interval between the 
temperatures at which the ball passes the thermometer ; and the 
