the Force of Fired Gunpowder. 145 



But A D is a curve, and this shows that the ratio of 

 x tojy is variable ; and moreover it is a curve convex to- 

 wards the line A B, on which x is taken, and this circum- 

 stance proves that the ratio of y to x is continually in- 

 creasing. 



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 conclude 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 andjy vanish together. 



Supposing this to be the case, namely, that the ex- 

 ponent of the ultimate ratio of y to x is I, let the 

 densities or successive values of x be expressed by a 

 series of natural numbers, 



o, i, 2, 3, 4, &c. to 1000, 



the last term = 1000 answering to the greatest density ; 

 or when the powder completely fills the space in which it 

 is confined ; then, by putting z = the variable part of 

 the exponent of the ratio of y to x, 

 To each of the successive values of 



x = o, i, 2, 3, 4, &c. 



The corresponding value of y will be accurately ex- 

 pressed by the equations 



n i+z T i + z n l + z oi + * A l + z &C 



u , i , .i , j ,4 , occ. 



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 

 x\ that the equation # 1+z ;= jy may always correspond 

 with the result of the experiments ; and when this 



VOL. I. 10 



