640 MISCELLANEOUS PAPERS. 



The effect of the screws aud the rollers serving as guides for the 

 cylinders was next investigated and allowance therefor duly made, 

 likewise for such other parts as were not of an eliminating character. 

 After making all corrections that presented themselves the result was 

 the desired moment for the normal position of the pendulum. The 

 next thing was to find what change was produced upon this moment 

 by giving to the pendulum a certain position ; that is, when the pendu- 

 lum is at a known point, as shown by the reading of the reflection on 

 the scale. By combining both elongations certain terms in the com- 

 putation disappeared, thus simplifying the labor. l(2cp(a) represents 

 the moment in the normal position of the pendulum, regarding a as 

 the distance from the origin, h the linear deviation of the middle ol the 

 ball, e the deviation, likewise expressed in linear measure, caused by 

 attraction, the moment will be expressed by q){a-\-b—e) for the first 

 position of the weights and (p{a—b—e) for the second. The sum of 

 both is 2(p(a—e)-^b'^q}"{a—e), but since (p"{a—e)=0, the resulting 

 moment can be expressed by 2(p{a)—2e(p'{a), a value which can be ac- 

 cepted as sufficiently accurate when it is considered that e never ex- 

 ceeds two millimeters. 



Although the attraction of the cylinders upon the instrument is the- 

 oretically determined with suflicient precision, yet the accuracy of the 

 final result is limited by the closeness with which the constants are 

 known. Therefore it is preferable to eliminate from the final result 

 those constants which can not be ascertained with the desired accu- 

 racy. This can be done in the following simple manner: 



Suppose a series of observations with the complete apparatus gave 

 for the mean density J, and a second series after diminishing the 

 weights on the upper end of the pendulum gave /},,', the difference be- 

 tween these two values rests upon a false value for the specific gravity of 

 the pendulum, not regarding errors of observation for the present. If X 

 is the error in the moment arising from this cause, the true value can 



be derived from z^ = J.-4- -' , z/= J,,4- "^ ,, in which m, and w,, are ob- 



m/ m,/ 



tained from the observations. In this manner the attraction of the 

 weights upon the pendulum-rod appears to be perfectly eliminated 

 from the final result. It is conditioned upon the value accej)ted as the 

 attraction of the balls alone, in which no uncertainty is involved, while 

 the agreement z/, and /},, furnishes a thorough control over the experi- 

 ments. 



Correction for the«resistance of the air was applied, and a law for the 

 decrease in the amplitude sought. This was found to be proportional 

 to the first power of the velocity, a result which agrees with the expres- 

 sion found by Cornu and Bailie for the effect of the air upon the torsion 

 balance. 



The constants were determined with great care, and in each case the 

 units of weight and measure were compared with the standards, while 



