NEWTON S PBINCIPIA. 241 



material particle suspended from a fixed point by an in 

 flexible inextensible string without weight. The length 

 of this string is called the length of the simple pendulum. 



A perfect simple pendulum is only a mathematical idea ; 

 we may approximate to such an instrument, but we can 

 not accurately construct it. It will, therefore, in many 

 cases be necessary to have the means of determining, when 

 any compound pendulum is given, the length of the equi 

 valent simple pendulum. The first person who solved this 

 generally was Huygens in his &quot; Horologiurn Oscillato- 

 rium,&quot; 1673. The principle on which he proceeded was not 

 so simple as that to which this and such like problems are 

 now referred. But the result is that if h be the distance 

 of the centre of gravity from the axis of suspension, and 

 mi 2 the moment of inertia about an axis through the centre 

 of gravity parallel to the axis of suspension, the length of 

 the equivalent simple pendulum will be 



y _ 



This is on the supposition that the body moves in vacuo 

 under the action of gravity. If it move in a medium 

 resisting according to any function of the velocity, the 

 above will still be the length of the equivalent simple pen 

 dulum. For the resistance on a unit of area being supposed 

 to be x v n 9 according to the usual theory of resistance, 

 the moment of the whole resistance on the surface will be 



A . v n , 



where A depends on the form of the surface, and not on the 

 nature of the motion. Hence to make the simple pendu 

 lum move in the same manner, we have merely to suppose 

 that the weight of the particle is equal to the weight of 

 the pendulum, and that it experiences a resistance which 



B 



