NEWTON S PRINCIPIA. 249 



v = an cos (nt + b) 



.\f=xn m a m cos (nt + b) 



.-. substituting and performing the integration between the 





limits nt + b= - and nt + b = we have 







m(m 2) (in 4) &c. 



b / - b = 0, 



Q &c. ] 

 -3)&c. I 



where a is ?r or 2 according as m is odd or even. 



Hence when small quantities of the second order are 

 neglected, the time of oscillation is unchanged and the arc 

 continually decreases, and the difference between the arc 

 described in the descent and that described in the sub 

 sequent ascent will be proportional to the same power of 

 the arc that expresses the law of resistance with the ve 

 locity. This will enable us to find the law. Also if the 

 difference of the arcs be represented by a series of terms 

 such as 



AV&quot; 



where V is the velocity at the point where it is greatest, 

 the resistance at this point will be represented by a series 

 of terms such as 



M O~ 2 ) ..... AV 

 / (m + 1) (m - 1) . . 



and the value of V, the maximum velocity, can be always 

 found by the formula 



Also putting R for the moving force of the resistance, and 

 W for the weight of the body, it is clear that 



R 



~ a 



