TRACT 36.

THE WHIRLING-MACHINE.

203

In the 6th column are the several resistances to a velocityof 12 feet per second, as deduced from those in the 4th co-lumn, to which number these are nearly equal, by statingthe resistances to be proportional to the 2*04 power of thevelocity, agreeably to the rate we have before determinedfor slow motions, that are so nearly equal to each other.

In the 7th column, the resistances of the 6th column arereduced, in the proportion of the distance of the plane tothe distance of the actuating weight; and these are the truemeasures of the resistances, in thousandth parts of an ounce,to the velocity 12 feet in a second of time, for e,ach positionof the plane.

In the 8th column are set the ratios of all these resistances,to the last or greatest of them ; which are obtained by divid-ing each of them by that greatest; being done in order tocompare them with the sines of the same angles of inclina-tion, as placed in the 9th or last column Whence it ap-pears, that the resistance is in no wise proportioned to thosesines: for, from the direct vertical position, or 90 degrees,the sine of the angle of inclination decreases faster than theresistance, till at the angle of 60° inclination, where they are.again equal. After which, the resistance decreases fasterthan the sines; but no where however so fast as the squaresof the sines, and much less of their cubes. So that the re-sistance, it appears, is in practice not as any single power ofthe sines, neither 1st, nor 2nd, nor 3rd power of them.

53. But, if we wotdd search out some powers or functionof the sines, that should be proportional to the resistances,it is manifest that the exponent must be some variable quan-tity, which at 60° must be = 1, or nearly so; below 60°, itmust be greater than 1, but less than 2, and above 60° itmust be between 1 and 0. Now the cosines of the anglesincrease from 90° down to. 0, in such sort, that their doubleshave the above requisite property, viz, that at 60°, 2c is = f;above 60°, 2c is between f and 0; and below 60°, 2c is be-tween 1 and 2; where c denotes the cosine of the angle ofinclination. Let us compute then the proportions of fop