118 GRAVITATIONAL METHODS [Chap. 7 



With the last expression, the arc correction is 



2.3 X 32 logio ao — logio «/ 



If the effect of this reduction is to be less than 0.1 milligal, the amplitude 

 must not exceed lj°. With a simplification permissible for such ampli- 

 tudes, the reduced coincidence interval is 



, 2 w(2n — 1) ,„ -._ s 



Wred. „=o = n + a . -^-— - — - . (7-23c) 



The correction may be further reduced if referred to (constant) average 

 amplitude am and a mean coincidence interval w^:^^ 



Wred.a^ = 71+ -^-^ [2(a„ - a!„w„) — (a — a„)]. (7-23d) 



3. In the temperature correction it is sufl&cient to assume a linear change 

 of period with temperature. The change in length of invar pendulums 

 ranges from 1.2 to 1.6 ju per meter and degree Centigrade. Quartz pen- 

 dulums expand much less, while brass or bronze pendulums increase in 

 length as much as 20 n per meter and degree Centigrade. The reduced 

 period is 



IVed.temp. = T - CeO - GJ, (7-24a) 



where Ce is the "temperature coeflBcient," determined by experiment. 

 The correction on the coincidence interval is 



nred.terap. = W + Co(© " Op), (7-246) 



where the relation between Ce and Ce is 



<^e = 7^^ iT2-<^e' ^^ ^o = (^Wred. - D^-Ce- (7-24c) 



(2i red. — 1)'' 



4. The air-pressure correction arises from the fact that the period of the 

 pendulum is lengthened by the buoyancy of the air, by its hydrodjTiamic 

 effect, and by its viscositj'" (interior friction). The buoyancy effect is the 

 most important and depends not only upon the pressure of the air but also 

 upon the amount of water vapor in it. That is, the buoyancy effect is 

 less for saturated air than for dry air at the same pressure. For this 

 reason a "hygrometer correction" has to be applied to the observed air 

 pressure and the reduced air density is computed from the relation 



F, = P6 - 0.377p^.h (7-25a) 



760[1 + 0.003665(9 -e„)]' 



" H. Schmehl, loc. cil. 



2« H. Schmehl and W. Jennie, Zeit. Instrumentcnkunde, 49(8), 396-406 (1929). 



