for a Convertible Pendulum, 37 



sented by P must be a minimum, or since 2 w is given, that 



2 , u z^ + 2 . w a;^ 

 must be the least possible. 



In order that the first condition of optimism may be at- 

 tained, the solid* may be regarded as one of revolution : put 

 then r for the radius of the solid at the position x, then ir r' 

 8.Z is the element of the mass, and 



S ^ 7-2 a^; = M 

 is one of the conditions ; another condition is, that z be con- 

 tained between two limits zi and — z.,, and that z^ -{' z^ ^ I ; 

 while a third condition is, that 



S T /•' 5^ + S = min. 



Whatever may be the form which results from the above 

 conditions, there is still another consideration that may 

 completely overturn the result : that is, the compressibility 

 of the material of which the pendulum is to be made. Had 

 we perfect command of the density and strength of matter, 

 absolutely the best form would be to compress the whole 

 weight into a minute point suspended by a fine wire ; but 

 then this minute wire would be distended so as to vitiate the 

 admeasurement. 



In the general use of the convertible pendulum, some of 

 the parts which are distended while it is in the one position, 

 are compressed when it is in the other, so that the compres- 

 sibility of the pendulum rod comes to interfere with the results. 

 The error arising from this source will be the less the stronger 

 the rod is made in comparison with the weight which it has to 

 sustain : and the error of distension will consequently be the 

 least when the whole weight is concentrated in the rod. 



From this consideration, it would almost result that a con- 

 vertible pendulum may be best constructed of a simple cylin- 

 dric rod of a rigid and heavy metal ; the one-knife edge being 

 placed as near as possible to the one end, and the conjugate 

 knife-edge at about two-thirds of the length of the rod 

 distant. 



When this form of the pendulum is used, the projection of 

 the knife-edges in one plane causes an inequality in the times 

 of oscillation in different directions, which must be removed 

 by making gratuitous projections at right angles to the edges, 



