396 Royal Irish Academy. 



where H = — — The ratio H is constant for all bodies of pris- 

 ds r 



matic form ; and for these, therefore, the moment of resistance is 



_MK 



H U '' 



M denoting the moment of inertia / r 2 dm. 



The differential equation of motion is, therefore, 



dta Xu . a K 



— — = — -f- sin 9 to. 



dt M H 



rf 9 



But w = — — - ; and, being small, we may substitute for sin 0. 

 dt 



The equation thus becomes 



d^ + Kdj_ Xj, 

 dt Hd* x M 



Making, for abridgement, — = 2 A, — |- = B-, the integral is 



= (c cos V B 2 — A 3 . t + c' sin V B* — A 2 . *) e~ A '. 

 But, A being small, we have approximately 

 e- A ' = l-A*; 

 and, if T denote the time of vibration, 



VB a -A a .T = tf. 

 Hence the preceding equation may be put under the form 



0=(1 — A/) (ccosifL 4. c'sin*4Y 



Now, let 0, and 0' denote the values of 0, when t becomes / — T 

 and t + T. It will be seen at once, on substitution, that 



0, + 2 + 0' = 0. 



Hence by combining the three readings according to the preceding 

 formula, the deviation of the magnet from its mean position, arising 

 from the vibratory movement, is completely eliminated ; and it will 

 readily appear that the same result may be attained by any greater 

 number of readings, taken and combined according to the same 

 law. 



Now, let the value of contain an additional term, + p t, propor- 

 tional to the time : or, in other words, let us suppose that there is a 

 progressive change of the declination, which may be regarded as 

 uniform during the whole interval of observation, "it is then mani- 

 fest that / -J-20 + 0' = 4^f; and accordingly that the quantity 



£(0 / + 2 + 0') 

 will give the mean place of the magnet corresponding to the epoch t. 



