TRACT 37.

OF GUNNERY.

229

Then z = '0115 — 6Q0x — yV 600 =— '005007,

or z = '029396- 1200.T — yV\ 200 =z - '005007,and z ='0475205 - 1800.Z —yV 1300 = - '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

(■00003146V - '0000968^/ v - '005) v = r,or ('00003 l 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 ,/v 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 and 3rd terms, '0001 \/v and '005 are al-ways both very small in respect of the first term '00003146v,but also that the 2nd term, -0001 Vr, may be generallytaken at about | of the 3rd term '005, or nearly =: '003 ona medium; therefore these two small terms together arenearly = '008; consequently the foregoing formula becomes('00003146v — '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 ('00002665v — '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 defect. Thiscircumstance then naturally suggests the idea of taking themedium between the two, for a still nearer formula, that is,half the sum

of ('00002665v — '004)v = r,and (-00003146v — 'C08)v = r,which is ('00002906v — '006)v = r,or nearly ('0000291v — -006)v = r,the diameter of the ball being 2 inches; and therefore, forany other diameter d, the resistance will be

('0000073 v — -0015)vt/ l = r.