448 Dr. W. F. Gr. SWamTorc the Motion oj the 



Case where ~F(t) is of the form P(l — e~ M ) *.— The solution 

 of (1) corresponding to this case is 



(2) 



The conditions 6 = when t = and # = when t = give 



*r(4\-7tT) 



tail6 ~27r 2 -2A, 2 + /aT* 



It is unnecessary for our present purpose to write down the 

 expression for c. The ideal motion of the needle is given by 



6=0^1- e~ h % (3) 



where 1 is written for the maximum deflexion P/a. 



Owing to the presence of the term involving e~ h \ it is 

 impossible to take the observations in such a way as to give 

 directly a result entirely free from errors due to inertia; we 

 can, however, easily arrange the observations so as to be 

 readily susceptible of correction. If we write 1 — k for 



KL 2 /a-bhla + l, 



we can write (2) in the form 



= (l-£)-i?|l-^-£-c(l^^ 



Jc may easily be expressed in terms of X and T in the form 



4/aT-PT 2 

 k ~~ 4(tt 2 + X 2 ) ' 



If the first and successive observations are taken when the 



periodic term is zero, the difference of any pair of these 



observations will give (1 — A-) -1 times the difference for the 



corresponding pair in the ideal motion. In order to obtain 



exactly the same values as for the ideal motion, the first 



T 

 observation should be taken when t = — 8, and the succeeding 



observations at intervals of a period, the observations should 

 all be multiplied by (1 — A:), and the fraction h of the maximum 

 deflexion P/a should be added. 



h may be obtained sufficiently accurately for substitution 

 in these correcting terms, by assuming the observed values 

 of d to obey the ideal relation (3). 



* All the expressions corresponding to the preceding problem may o( 

 course be obtained from those of the present problem by putting h equal 

 to infinity, and P equal to zero, subject to the relation 17/ = X. 



