398 PROCEEDINGS OF THE AMERICAN ACADEMY 



False 2i = — false z, = TjAl—K-r^ i e-^ i e 2 "^Tl — 7" )^ — 1- 



We thus see that, by neglecting the resistance, we get for the value of 

 Zj a quantity which requires only a minute correction in order to give 

 the imaginary part of the true z^. The same thing is not true for z^ 

 and z^. Now, //j is O di\ided by the principal period of oscillation of 

 the pendulum upon the flexible stand. This is the quantity which we 

 wish fo determine ; the others have only to be known approximately 

 for the purpose of calculating the small correction to this. The loga- 

 rithmic decrement of the amplitude of oscillation of the pendulum in 

 the unit of time, so far as it is due to internal friction, is the quantity 

 ^j. After these two quantities have been approximately ascertained, 

 we may approximate to the quantity (J/ -[~ V>^) by means of the 

 equation 



Then, by eliminating a between the two equations 

 2(5, + a = f^. 



2[(5.' + 1,')^.. + (-V + O-'J = T^.. 



we obtain S.„ and consequently jJj- The values so obtained must sat- 

 isfy the equation 



4^,5, + .V + 1/ + ^,^ + ^,^ = ?^'. 



Before proceeding to the consideration of the elastic after-effect, I pro- 

 pose to apply the equations thus obtained to the calculation of the cor- 

 rection of the seconds' pendulum for the flexure of the stand, supposing 

 the internal friction to be proportional to the velocity. 



For the pendulum used by me we have the ajjproximate values : — 



/ = 1.00 ; h (heavy end up) = 0.30; h (heavy end down) == 0.70 ; 

 g (New York) = 0.993 X Q- = 9-89 ; y = ^.tjtj^jtstj = 4706; 

 j/j = 1.00. 



The accompanying table shows that J^ = 0.000008. From this, we 

 calculate tliat with heavy end up £^ = 0.08, r^.^ ■= 257 ; with heavy 

 end down J^ = 0-17> % = ^92. From this, it appears that the cor- 



