226

THEORY AND PRACTICE

TRACT 31.

formula, though serving very well for some particular resist-ance, or even for constructing a complete scries or table ofresi-tunces, is not proper for the use of problems in whichfluxions and fluents are concerned, on account of the mixednumber 2-^,, in the index of the velocity v.

We must therefore have recourse to an expression in twoterms, or a formula containing two integral powers of thevelocity, as k 1 and v, the first and second powers, affectedwith general coefficients m and n, as mv 1 + nv = r, the re-sistance. Now, to determine the general numerical valuesof the coefficients m and n, we must adapt this general ex-pression mi? 4- nv — r, to two particular cases of velocity,at a convenient distance from each other, in one of the fore-going tables of resistances, as the last for instance. Now,after making several trials in this way, it is found that thetwo velocities of 500 and 1000 answer the general purposebetter than any other that has been tried. Thus then, em-ploying these two cases, we must fust make v = 500, andr— 74'4 oz = 4‘65lb, its correspondent resistance, and thenagain v = 1000, and r = 362 oz = 224-lb, the resistancebelonging to it: this will give two equations, by which thegeneral value of m and of n will be determined. Thus thenthe two equations being

500*'m + 500n = 4-65,and 1000b» ■+• 1000 n — 22'625 ;dividing the 1st by 500, and C 500m + n = ‘0093,the 2d by 1000, they are t 1000m + n = ’022625;the dif. of these is . . . 500wi = *013325,

and therefore div. by 500, gives m = *00002665;hence n=*0093 — 500>m = *0093 — *013325 = _ *00665=m.Hence then the general formula will be •00002665k 1 —•004025k = r, the resistance nearly in avoirdupois pounds,in all cases or all velocities whatever.