■Tract 38.

PROBLEMS.

321

ing : so that it may be a question whether, on the whole, thebusiness maybe effected in the less time by a greater heightof the machine, or whether there be any limit to the height,so as to produce the greatest effect in a given time.

To answer this question, let x denote the indeterminateheight from which any weight w is to fall, z the time ofraising it after a fall, which time is supposed to be as theheight x to which it is raised, also m the given time of pro-ducing a proposed effect; then = the time of the weight

failing ; thereforeyV v + z = the whole time of one stroke;conseq. j—~— or ——— is the number of strokes made in

1 i ^/1 + z ^/x + 4s

the given time m, and hence - miw - — the whole force or ef-y . v'. r + 4:

feet in the time m. Now this effect or fraction increases con-tinually as x increases, because the numerator increasesfaster than the denominator, since the former increases as x,while, in the latter, though the one term z increases as x,yet the other term >/x only increases as the root of x. Sothat, on the whole, it appears that the effect, in any giventime, increases more and more as the height is increased.

problem nr.

To determine how far a man, who pushes with the force of100 lb, can thrust a sponge into a piece of ordnance, whose di-ameter is 5 inches, and length 10 feet, when the barometerstands at 30 inches : the vent, or touch-hole, being stopped, andthe sponge having no windage, that is, fitting the bore quiteclose ?

A column of quicksilver 30 inches high, and 5 in diameter,is 5 2 X 30 X “7SS4 = S89“05 inches; which, at 8'102 ozeach inch, weighs 4-772“48 oz, or 298 - 28lb, which is thepressure of the atmosphere alone, being equal to the elas-ticity of the air in its natural state ; to this adding the lOolb,gives 398“28lb, the w'ho’e external pressure. Then, as thespaces which a quantity of air possesses, under differentpressures, are in the reciprocal ratio of those pressures, it