130 Mr Sang on the Manner in zchich 



linear measures of different nations may be referred. As might 

 have been expected, a subject of such vast importance has re- 

 ceived the attention of the most distinguished analysts. La- 

 place has given the general equation of tlie curve which will 

 produce isochronism in a medium whose resistance is a function 

 of the first and second powers of the velocity, while Poisson 

 has attempted the investigation when the resistance is supposed 

 minute ; but neither philosopher gives to the result a form 

 suited to practical purposes. 



For the purpose of illustrating the succeeding investigations, 

 1 shall first consider the motion of the cycloidal pendulum when 

 affected by a constant friction. 



Let V be the vertex of a cycloid, in Avhich the motion of a 

 body is retarded by a constant friction, acting in a direction 

 tangent to the curve. The most convenient way for repi'esent- 

 ing the friction is to assume a distance VF or Vf along the 



curve, such that the tendency of the body to descend from F 

 ory may be just equal to the friction. When the body is at A, 

 its tendency to descend is, from the nature of the cycloid, pro- 

 portional to VA, while the friction is proportional to VF, where- 

 fore the surplus tendency to descend must be proportional to 

 FA. When the body has descended and reached the other side 

 of F between F and \', Va the tendency down is less than the 

 fi-iction VF, and therefore the motion is retarded as if by a 

 pressure Fa. And again, when the mobile reaches the other 

 side of V also, both the friction and gravitation tend to retard 

 its motion, so that the -etarding pressure is still proportional to 

 its distance from F. It thus appears that the body oscillates 

 about F just as it would have oscillated, at equal distances, 

 about V, had tliere been no friction. 



The time of oscillation in the cycloid is, then, not altered hij 

 (I constant friction. 



