ISO Mjp Sang m the Manner in whkh 



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 the 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. 

 I shall first consider the motion of the cvcloidal pendulum when 

 affected by a constant friction. 



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

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

 tangent to the curve. The most convenient way for represent- 

 ing the friction is to assume a distance YF or \f along the 



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a: 



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

 or/* may be just equal to the friction. When the body is at A, 

 its tendencv 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 V. Vfl the tendency down is less than the 

 friction 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 retarding pressure is still proportional to 

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

 about F just as it would ha\e oscillated, at equal distances, 

 about V, had there been no friction. 



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

 1 1 constant friction. 



