Explosion of Coal Gas. 281 



fined channel, let the length occupied initially by the gas be 



a : then, when it becomes ax, the elasticity -will be diminished 



— - 

 m the ratio of 1 to x * ; and the force will be expressed by 



nx * — ] , and the fluxion of half the square of the velocity by 



nax '^J:—aJ:; and the fluent will be— 47iaj;~* — ax, which is 



initially — — 4raa — a, and finally, when 7ix~* — 1 = 0, and 



X = M'*, and n = X*, = — 4ax — ax; and the difference 

 showing half the square of the velocity generated, is (4» + 1 



— 5«^) a. When n = 20, the expression becomes 26a ; 

 when n = 20 000, 66 204a ; and in order to make these values 



equal, the latter value of a must be of the former; and 



2540 



15000 



= 5.9. 



2540 



In a similar manner, when a is the radius of a sphere, or of 

 a hemisphere, which expands in every direction, the elasticity 



_i 5 . . ,_ 1 5 



will vary as x 4 , and the fluxion will be nax i x —ax, and 



4 -li- 



the fluent — — nax ^ — ax; which, being corrected, gives 



for the half square ( — n+1 — — n^ 1 a. 

 ^ V 11 11 / 



The fluent, thus found, may be compared with the feet in 

 which the force of gravity would produce an equal velocity, by 

 increasing it in the ratio of the pressure of an atmosphere to the 

 weight to be moved : that is, for a brick wall 18 inches thick, 

 multiplying it by 11 : so that, when n =: 20, and a = 15, 

 .5242a = 865 feet, the height of ascent : or, supposing the space 

 doubled, and n r=: 10, and a about 18^, the height would be 

 430 feet. 



Where the explosion of the gas takes place without an ob- 

 stacle, the mean force being about — ! atmospheres, the 



velocity of expansion will be about 2000 feet in a second ; or, 

 perhaps, a little greater, on account of the lightness of the gas ; 



