Astronomical and Nautical Collecliom. 353 



X somewhat smaller, we shall reduce the expression to a per- 

 fect square, and we shall find that (.16 + .25 A)^ = .0256 + 

 .080 A + .0625 A' will represent the diminution with great 

 accuracy, giving .187, .168, and .152, for the respective arcs of 

 1.09, 1.00, and .92: and this expression has the advantage of 

 affording a very easy integration for the arc. 



For, if t be the number of vibrations divided by 1920, we have 



-dA-(.16 + .25A)=d^o,nd^-^^^^, == dr. but 



d ^ = -^^^^ ,and 1 = ^ + c or 



.16 + 25A (.16 + 25AV .16 + .25A 



4 16 



.16 + .2.5 A = , and .64 + A = : whence, put- 



t+ c t + c 



. . • . , 7 I 16 J 16 



tin? .64 + A = B, and its initial value h, b = — , and c := -— ; 

 ' c t> 



consequently B z; , and _- = -— + -— . 



^ J 16 B b 16 



In many of the series of experiments, it is necessary to make 

 some variation in the constant coefficients, on account of the 

 state of the atmosphere, and we may take in general B = A + C, 



and -L = J- + _L,thefactor9,in thecasealreadycomputed, 

 B b q 



being made either 16, or 16 x 1920, accordingly as we wish to 

 take the interval of the coincidences for the unit of time, or to 

 express it in seconds ; and C, in some of the series of experi- 

 ments, appearing to be about 1° or even 2°, instead of 0°.64. 

 The supposition of C = 1° is equivalent to that of D = .04 + 

 .04 A 4- .01 A', y becoming in this case 25.4 instead of 16. 

 The constant part of D, expressed by x, causes in half a vibra^ 



tion a retardation of J- ^ = 0°.000067 = 0'.004 = 0".24, 

 3840 



which happens to agree singularly well with the 0."20 deduced 



by Euler from Newton's experiments. ■-'^' 



We may easily compute, from the value of A thus determined, 



the total retardation depending on the vibration in a circular 



2 A2 



