318 PHILOSOPHICAL TRANSACTIONS. [aNNO i75g. 



view of the problem, the load of an engine, when the effect is a maximum, and 

 the force a current, is determined by computation to be -f* of the weight which 

 would cause the engine to rest. This then being suited only to a uniform ve- 

 locity both in the lever and obstacle, Mr. B. considers the case of a uniformly 

 accelerated one in repeated vibrations. The maximum which corresponds to it 

 is adapted to the steam-engine, and of no less importance to be determined than 

 the other. 



A general expression for the time of a stroke in such vibratory engines, will 

 easily lead us to a computation of their effects. 



A a c b D 



»» A rt 



Let AD be a lever, whose brachia are a and /', and supposed without weight. 



Let 7n be a power, and n a weight. Then a : b ::n:—, the balance for n at A, 



and m is the effective force at a, which multiplied by the lever a gives ma — 



nb for the efficaciousness of that force in the angular velocity of the power and 

 weight. Now by the principles of mechanics, the inertia of any bodies revolv- 

 ing about a centre, is as the quantities of matter into the squares of the brachia; 

 in the present case therefore, the whole inertia of m and n is as wa* -|- nb'^. 

 Hence then, and because the velocity generated in a given particle of time is as 

 the force directly and inertia inversely, we have ^^^^— as the accelerating force 

 or the measure of the angular velocity of the power and weight at the end of the 

 said given particle of time. But again, the times of descent by means of uni- 

 form forces, through a given space, are inversely as the square roots of the ac- 

 celerating forces, or measures of the velocities generated in a given particle of 

 time ; therefore V ^^—^ is a general expression for the time of a stroke. 

 This being had, the solution is easy ; for, supposing n only to be variable, say 

 as s/ — 3—- : n: : 1, a constant or given tmie: n\/ — ; the effect m 



ma — nb '^ ma^A-nb^ 



time 1, ex hypoth. the greatest effect which can possibly be produced in the said 

 given time. Taking then, as usual, the fluxion equal O, we have, after a pro- 

 per reduction, 2a^ m^ — Sa'mnb + amnb^—2n^¥ = O, and n = ^' ^/^-{-(iiiZ^)* 



— " '"^ AM~ • Therefore, in these sorts of engines, when the brachia are 



given, the weight: power ::;Vf+(^?V-"-%-^ = 1: and if tl,e 

 brachia are equal, i. e. if a = b, the weight : power : : V *. — ^ : i, viz. O.618 : 



* The theory however is erroneous which brings out ^ of the weight j it' should be J instead of 

 ^, as may be seen fully explained in vol. 3, p. 1 14, &c. of the Transactions of the American Phi- 

 losophical Society j or in my Dictionary, v. 2, art. mill. C. H. 



