262

Hydrostatics.

B G is half that upon the Bottom C D;and consequently, the total Pressure against

the

6. Let the Sum of all the Pressures, or the total Pres-sure against the Side of a Vessel, be represented by S,the Number of Terms in the Scries by N, and thegreatest Term by G ; then since S — £ NG, and sinceN and G are ever proportional to each other, we shallhave S a$ J N N, or 2 S as N 1 ; and since halvesare in the fame Ratio with their Wholes, we have S asN" ; that is, the Sum or total PreJJure will be as the Squareof the Number of ' Tersns , or Altitude of the Fluid,

7. For Example, in the above Series, if we take theSum of 4 Terms, it will be ost-r st-2st-3 —6 ; and thenof 8 Terms, the Sum will be o-p 1+2+3-J-4-}-5st-6+ 7= 28, which is 4 Times 6, and 4 over. Nowfhele Sums ought to be as 1 to 4, because the Altitudeswere as 1 to 2 ; and though the Sum of any Number ofTerms will always be a little more, than 4 'l imes theSum of half that Number of Terms in this Way ofcomputing, yet when the Number comes to be exceed-ing great, the Excess will become indefinitely small,and therefore may be neglected in the Cafe of Fluids.

8. Or yet more clearly and accurately in Symbolsthus ; since S is always as { GN or as GN ; and inthe Series above adapted to Fluids, G —N—1, thereforeGN—NN—N ; and so S will always be as NN—N ;but when N is indefinitely great, N 1 will be infinitelygreater thanN, which therefore will vanish in that Cafein the Expression N N—N ; and so S will ever be as N%that is, the Sum or total Pressure will be as the Squareof the Altitude of the Fluid.

9. This Way of considering the Quantity of lateralPressure by the Arithmetical Series is universal, whereasthe common Method restrains it to the Property of a:iequicrural right-angled Triangle, and to a Vessel of acubical Form; which I shall here give for the fake ofsuch as would fee the Demonstration of a Thing in se-veral Ways. ABCD is a Vessel of a cubical Form, thatIs, whose Side BC is equal to the Length of the Bot-tom C D ; if then the Diagonal-B D be drawn, woshall have the Lines is—Hi, 2S—B2, 3^—83, ^4