400 PROCEEDINGS OF THE AMERICAN ACADEMY 



same for the staud and for the pendulum, a certain change in the total 

 relative amplitude might occur in this way, but only an excessively 

 minute one, nothing like what M. Plautamour thinks he has observed. 

 But it is so improbable that the motions of the stand and pendulum 

 depart much from the forms (1) that it would be wrong to accept M. 

 Plantamour's results, until they are confirmed by a purely optical 

 observation free from any possible influence from the machinery 

 attached to the stand. Such an observation has been made by me ; 

 and, though I admit it was rather rough, it is entirely opposed to M. 

 Plantamour's conclusions. Should the latter be confirmed, they would 

 totally nullify the attempt to correct for the effect of flexure, as they 

 would show the inapplicability of the analysis which has been proposed 

 for the solution of that problem, without affording us much hope of 

 being able to replace it; and it would seem to be necessary in that case 

 to reject all the work which has been done with the reversible pen- 

 dulum. 



If the pendulum were started in the manner proposed, and if for any 

 cause the amplitudes of pendulum and stand were altered in different 

 ratios, there would be a perpetual force at work tending to restore the 

 old ratio, so long as the phases of the motion were the same in the 

 pendulum and stand. But, if the phases differed, a part of this force 

 would go to diminishing the amplitudes, and would act so strongly in 

 this way that there would be a rapid decrement on account of this cir- 

 cumstance. Suppose, for instance, that in the differential equations 

 we were to put instead of D/'s, l),'Sp where Sj is the value of s at a 

 time later than t hy a constant. The result of this would be (neglect- 

 ing terms involving a) that instead of the square of the exponent of 

 the Neperian base being the sum of two negative quantities, one of 

 them very small compared with the other, the smaller of these quan- 

 tities would be multiplied by an imaginary root of unity. This would 

 have but little effect on the imaginary part of the exponent of base, 

 which determines the period ; but it would add a considerable real 

 part, which would represent a corresponding decrement of arc. 



It seems difficult to conceive of a force which should greatly change 

 the relative amplitudes of oscillation of the pendulum and stand, with- 

 out at the same time producing an enormous decrement of the ampli- 

 tude of oscillation, such as certainly does not exist. It is for those 

 who believe that the existence of such a force has been experimentally 

 proved to show how great an effect it would have upon the period of 

 oscillation. M. Plantamour supposes that the formula given by me in 

 my paper, " De I'influence de la flexibilite du trepied sur roscillation 



