TRACT 37.

OP GUNNERY.

229

Then z ~ ‘Oil'S— 600.r — yV 600 = — '005007,or 2 = -029396-l200.r -^Vl200 = - -005007,and 2 = ’0475205 — 1800.T — t/\/1.300 = — "005007,all the three values the same, showing so far the truth ofthese deductions. Hence the general formula for the resist-ance will be nearly

(•00003 146'v - "0000968^/ v - -005) v — r,or (-00003 1 Sv - -0001 </v — -005) v — r,

which is quite correct for the three velocities, 600, 1200,1800, and very nearly so for all the others.

29. But, further, the middle turn -0001^^ may be ex-punged, and so the formula rendered much more simple, inthe following manner. By adapting this formula to a fewof the velocities, in different parts of the series, it is found,not only that the 2nd anil 3rd terms, -0001 and -005 are al-ways both very small in respect of the first term -00003146®,but also that the 2nd term, -0001 Vr, may be generallytaken at about | of the 3rd term -005, or nearly 2 = -003-ona medium; therefore these two small terms together arenearly =’008; consequently the foregoing formula becomes(-00003146® — -008) v = r nearly.

30. Computing now, by this formula, all the series of theresistances, or values of r, for all the several velocities, it isfound that they come out a little nearer than those by theformer theorem (-00002665® — "004) v = r, but having dif-ferences from the true resistances the reverse way, most com-monly, the one from the other, that is, the one being in ex-cess, for the most part, when the other is in detect. Thiscircumstance then naturally suggests the idea of taking themedium between the two, for a still nearer formula, that is,half the sum

of (-00002665® — -004)® = r,and (-00003146® — -008)® = r,which is (-00002906® - -006 )v — r,or nearly (-0000291® - -Q06)® = r,the diameter of the ball being 2 indies; and therefore, forany other diameter d, the resistance will be

(-0000073 ® — -0015)®^ = r.