248 NEWTON S PKINCIPIA. 



Assume these to be the true equations of motion when /is 

 not zero, then a and b are functions of t. 



And since the second equation is the differential of the 

 first, 



sin. (n t + b) -= + a cos. (n t + b) -7- = 0; 

 d t d t 



and since they satisfy the original equation of motion, 



, TX da . . N e?& 



TZ cos. (nt -{- b) -7 w sin. (nt + b)-j- = /. 



Hence, solving these 



-T- = sin. (n t + b) 



d t na 



These equations, when solved, will give the changes in the 

 arc and time produced by the causey. 



Suppose /to be a small force. Then the variations of 

 a and b are very small, and being multiplied on the right 

 hand side by the small quantity f we may neglect them. 

 Hence if , and b / be the altered values of a and b, 



1 /&quot;&quot; 



aj a = - If cos. (n t + b) d t 

 n^J 



b,-b= A/sin, (n t + b) d t. 



Hence we learn that if/ consist of two disturbing causes, 

 the total disturbance will be nearly equal to the sum of 

 the separate disturbances. 



Suppose / to be the resistance of a medium varying as 

 the m th power of the velocity /= x v m . The velocity in 

 moving from the lowest point is 



