308

THEORY AND PRACTICE

TRACT 37.

of this is a X hyp. log. —— a = 0; hence the hyp. log. of

— = 1 , and ~ == the number whose hyp. log. is 1 , that is

— = 2-71828, and 4“ = or the length of the charge

a is rather more than f of the length of the gun bore. Sothat a : b : : 1 : 2-71828, or as 4 to 11 nearly, or nearer as 7to 19 ; that is, the length of the charge, to communicate thegreatest velocity, by the above formula, is the ^ 0 f the lengthof the bore, or nearer ^ of it, or about f. By our experi-ments it has been found (art. 124 Tract 34) that the chargefor the greatest velocity, is but little less than that which ishere computed from theory, the correspondent parts therefound, for the four guns, being nearly T %, T 3 ? , - 3 %; the

parts here varying as the gun is longer, which allows moretime for the greater quantity of powder to be fired, and forthe flame to act on the ball, before it is out of the bore.

184. But we shall be likely to come nearer to the experi-mented quantity, if, instead of the, first or incorrect formula,Ave employ the more exact one, which includes the weight ofa portion of the powder, as well as that of the ball; viz, the

theorem X hyp- qp, or, omitting the con-

stant factors, X hyp. log. must be a maximum. Nowit will be convenient here to express p and w in terms of aand d. And first, the content of the ball being fed 3 in cubicinches; and the cubic inch of cast iron weighing 4-3 ounces;therefore 4 - 3 X fed 3 = 2-8 fed 3 is = zv the weight of the ballin ounces nearly. Also the length of the cylinder of the pow-der and bag being a, its content will be acd 2 inches; and theweight of the cubic inch of the same being half an ouncenearly, or rather -54, therefore 'Si'acd 2 is = 2 p the weightnearly, or p =: -27 acd 2 . Substituting now these values of pand w in the denominator of the maximum above mentioned,

it becomes ■ ■ - X hyp. log.—, or ,;-y ■— x hyp.

log. ~ the maximum. Then putting the fluxion of this