the Force of fired Gunpowder . 
269 
is variable ; and moreover it is a curve convex towards the line 
A B, on which x is taken ; and this circumstance proves that 
the ratio of y to x is continually- increasing. 
Though these experiments all tend to show that the ratio of 
y to x increases as x is increased, yet when we consider the 
subject with attention, we shall, I think, find reason to con- 
clude that the exponent of that ratio can never be less than 
unity ; and farther, that it must of necessity have that value 
precisely, when, the density being taken infinitely small, or 
= o, x and y vanish together. 
Supposing this to be the case, namely, that the exponent of 
the ultimate ratio of y to x is = 1, let the densities or successive 
values of x be expressed by a series of natural numbers, 
the last term = 1000 answering to the greatest density; or 
when the powder completely fills the space in which it is con- 
fined ; then, by putting % = the variable part of the exponent 
of the ratio of y to x, 
To each of the successive 
The corresponding value! 
of y will be accurately ex- >o I+2 , i 1+z , 2 l + z , ^ l+z , 4 I+Z , &c< 
pressed by the equationsj 
For, as the variable part ( z ), of this exponent may be taken 
of any dimensions, it may be so taken at each given term of 
the series, (or for each particular value of a:), that the equa- 
tion x l+z =y, may always correspond with the result of the 
experiments ; and when this is done, the value of z, and the 
law of its increase as x increases, will be known; and this 
will show the relation of x to y, or of the elasticities of the ge- 
N n 2 
o, 1, 2, 3, 4, &c. to 1000, 
values of x = 
