the Force of Fh^cd Gztnpowder, 145 



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

 jv 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 j to A" is continually in- 

 creasing. 



Though these experiments all tend to show that the 

 ratio of jy to x increases as x is increased, yet when we 

 consider the subject vv/ith 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, ;c andj vanish together. 



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

 ponent of the ultimate ratio of y to a* 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 2; = the variable part of 

 the exponent of the ratio of y to x, 

 To each of the successive values of 



^ = o, I, 2, 3, 4, &c. 



The corresponding value of y will be accurately ex- 

 pressed by the equations 



0'+% V + % 2'+% f+% 4'+% &C. 



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 x'+^ = y iurj always correspond 

 with the result of the experiments ; and when this 



VOL. I. 10 



