a Medium to small Oscillations. 117 



d^ cP^ dH ^ . . . . , 



T? Tfi* 773' ' ^' ^' resistance, in the case of very small 



oscillations, is a linear function of the velocity and its differ- 

 ential coefficients with respect to the time. 



28. If we retain the remaining terms of R, namely P^/, 

 P^'/, &c., we shall find 



vr 



J ^ R sin «/.£?/ = N « + N'«"»' + N"a'«" + &c., 



where N', N", &c. are quantities analogous to N. Hence, and 

 by (3.)) we may easily show that R does not sensibly differ 

 from P^ when the ball oscillates through any angle not ex- 

 ceeding 40', and consequently that, in such a case, R may be 

 assumed to be a linear function of the velocity and its differ- 

 ential coefficients. 



29. We shall now briefly show, from the second of the ex- 

 perimental facts above stated, that the resistance depends sen- 

 sibly upon the differential coefficients of the velocity, as well 

 as upon the velocity itself. 



Taking for granted what we have just proved, we may as- 

 sume that 



^-^'^^ + ^^^ + ^3^3 + &c-» 



where C^, C^, Cg are constants depending solely upon the con- 

 stitution of the medium and the shape of the ball. Now we 

 have for a first approximation, 



^^^ o. d^^ ^dQ dH ., „ 



^ ^^ =-^'^' di§ = - ''"re TT\ = ''^^ ^'- 



in virtue of which equations the expression for R becomes 



where 

 p=C,-Csn^+C,7i^ , q = C^-C,n^+Cen^ 



Hence the equation (1.) becomes, putting «^ = — , 

 d^Q d$ ^(1-g). 



If we examine the third term of this equation, we see that 



the coefficient of 9, instead of being — , is ^-^ ; in other 



words, the ball may be said to be deprived of the 5'th of its 

 weight by the resistance of the medium. Now if R depended 

 simply upon the velocity, and not upon its differential coeffi- 

 cients, q would manifestly be zero, and there would be no loss 

 of weight. Hence, since experiment clearly indicates a loss 



