700 PROCEEDINGS OP THE AMERICAN ACADEMY. 



value for the fundamental ordinate A A x of that figure, and second, by 

 dividing this fundamental ordinate by the corresponding peripheral .vel- 

 ocity of the rubbing wheel. The resulting 71/is, therefore, the amplitude in 

 centimeters, of the motion which the middle point of the string would have 

 if the string were rubbed at that middle point with a speed of one centi- 

 meter per second. It should be a constant under all circumstances. The 

 smallness of the probable error in the second, fourth, sixth, and eighth 

 lines of the table is a verification of the speed law stated in the last para- 

 graph. Its smallness in the other four cases shows that the shape of 

 each individual envelope is that which Figure 1 predicts. This is shown 

 in another way in Figure 2, in which the observations of these groups 

 are plotted in the manner already mentioned. The envelope in each 

 figure is the best line of the given shape that could be drawn. And 

 finally the constancy of M throughout the table shows that the relative 

 magnitude of the envelopes is as described. This verification is not 

 quite so good as the individual probable errors would lead one to expect, 

 because of certain errors, largely in the adjustment of the brass bridge, 

 which are constant throughout any one set of observations, but vary from 

 set to set. The weighted mean of the results in the table is 



M= .000327 ± .000001 cms. 



In the case h to which this standard ordinate would correspond, the two 

 velocities under the bow would be equal, if Helmholtz's law is applicable 

 in longitudinal cases. If F be the frequency of the string, AMF would 

 be the distance travelled by its middle point in one second, and this should, 

 according to the velocity law, be equal to the corresponding velocity of 

 the bow, that is, to 1 cm. per second. Therefore F= 1 / 4 M. This 

 formula gives 



F (calculated) = 764 ± 2. 



A Valentine and Carr set of tonometer forks gave, by the method of 

 beats, 



F (observed) = 762, 



an excellent quantitative verification of Helmholtz's two laws for longi- 

 tudinal vibrations. 



§3. Aliquot Points — Theory. 



In the preceding section it has been shown that the amplitudes which 

 liave been measured in the various aliquot cases considered could all be 



