220

THEORY AND PRACTICE

TRACT 37.

the 100 feet velocity, with each of the following ones sepa-rately, first the 100 with the 200, then the 100 with the 300,then the first with the 400, ami so on to the end ; by whichmeans it appears that, commencing with the 2nd power orsquare of the velocity, at the very beginning or slowest mo-tion, the exponent of the power gradually increases, till, atthe velocity of 1500 or 1600 feet, it arrives at the 2*153power of the same, being nearly the 2-f pow r er, or 2 T 2 T morenear!}-, that is, the resistance to the 100 feet velocity, is tothat of the 1600 feet velocity, as 100 2 ' 153 to 1600* IS3 , or asl 1 * 53 to 16 2IS3 , or as ] to 16 2 ' 153 , since all the powers of 1 areequal to 1 only; so that, if r be put to denote 2*78 ounces,the resistance to the velocity of 100 feet velocity, then 2*78X 16 3 ' 53 will be the resistance to the 1600 feet velocity;and so on for the others. After the 1600 feet velocity,where the exponent (2*153) is greatest, it gradually de-creases again to the end. It may further be noticed, thatthe arithmetical medium among all these exponents of thepowers, is 2*104, or 2-^ nearly. The 6th or last column isadded, to show the resistances in avoirdupois pounds.

21. The method of determining these indices of thepowers, is the same as employed in art. 43 of the 36th Tract;viz, having given any two velocities v, v' , and their corre-spondent resistances r, r’; to find, according to what powersof v and H J, the resistances r, r' are proportional. Let x de-note the exponent of the power of the two velocities v, v ;then tf ; v*: : r : r'; hence, dividing the consequents by theantecedents, gives I : (^)*: : 1 : -£■; therefore (—f ~

taking the logarithms gives x x log. ~ = log. — ; hence x

— that is, the exponent of the power of the ve-

log. v— 1°S- v , 1

locities, to which the resistances are proportional, is equal tothe quotient of the differences of the logs, of the resistances,divided by the difference of the logs, of the velocities. Now,to apply this theorem to the calculation from the numbersin the preceding table of our experiments, beginning at thevelocity of 100, and thence to find the exponents for all the