32*

NEW EXPERIMENTS

TRACT 34.

oscillation, as deduced from the chord of the arc which isactually described.

Having thus obtained two different expressions for thevelocity of this centre, independent of each other, let anequation be made of them, and it will express the relation

of the several quantities in the question : thus then we havebiv 5*6727 c ,pso+ lii A , - , . . i , •

-p =-v--rr. And from this equation we obtain

/£ + ot r f>g + bi 1

5*6727 c

v = — - [(pgo + bit) x (pg + 6i)]» ^ ie true expression

for the original velocity of the ball the moment before itstrikes the pendulum. And this theorem agrees with thoseof Messrs. Euler and Antoni, and also with that of Mr. Robinsnearly, for the same purpose, when his rule is corrected bythe parapraph which was by mistake omitted in his bookwhen first published ; which correction he himself gave in apaper in the Philosophical Transactions for April 1743, andwhere he informs us that all the velocities of the balls,mentioned in his book, except the first only, were computedby the corrected rule. Though the editor of his works,published in 1761, has inadvertently neglected this correc-tion, and printed his book without taking any notice of it.And that remark, had M. Euler observed it, might havesaved him the trouble of many of his animadversions on Mr.Robins’s work.

22. But this theorem may be reduced to a form muchmore simple and fit for use, and yet be sufficiently near thetruth. Thus, let the root of the compound factor (pgo -f bii)X {pg + bi) be extracted, and it will be equal to {pg + bi.~Y~) X Vo, within the 100000th part of the true value,in such cases as commonly occur in practice. But, sincein our experiments, is usually but about the 500th,or 600th, or 800th part of pg', and since bi differs fromonly by about the 100th part of itself; therefore

b g + bi is within the 50000th part of pg + bi.?-j-^. Con-sequently v — S' 6727 c . ver y nearly. Or, further,

if g be written for i in the last term b i, then finally v —