538 Br. C. Chree : Applications of 



require to be taken into account, and no very great increase 

 in the accuracy aimed at by Marek and Benoit will be 

 possible without making such an allowance. The fact that 

 compressibility might come in was not overlooked by Marek, 

 who gives (I. c. tome iii. p. D. 80) a formula equivalent to 

 &v/v=—p/k, where p is the pressure at the level of the C.Gr. 

 of the kilogram in a liquid in which it is being weighed. 

 He gives, however, no explanation of how the result was 

 reached, and it had probably no strict mathematical basis, 

 but simply represented an assumption that the change of 

 volume was the same as if the pressure over the surface of 

 the kilogram had everywhere the mean value for the volume 

 of liquid displaced. When the variation of pressure over 

 the surface of the solid is relatively small, it is of course 

 obvious a priori that an assumption of the kind cannot be 

 much in error. 



Solid surrounded by Varying Medium. 



§ 5. The results (8) and (9) treat the density of the medium 

 surrounding the solid as uniform. Strictly speaking, this 

 can never be true. Ordinary liquids are much more com- 

 pressible than solids ; e. g. the values of k for water and for 

 alcohol are of the orders 20 x 10 6 and 12 x 10 6 in grammes 

 wt. per sq. cm. Thus increase in the depth to which a solid 

 is immersed in a specific gravity experiment may be of most 

 importance from its enhancing the density of the liquid. 

 If p' and p' represent pressure and density in the liquid at 

 depth z, II' the atmospheric pressure on 2=0 ; p ' the density 

 of the liquid if under zero-pressure, we have- — neglecting 



(p'lk'y—. 



dp'=gp'dz, 



p'= P(t '(l+p'jk'), 



and when z=0, p' = U'. 



Proceeding by successive approximations, treating p \ti as 

 very small, we get 



p'=p '{l + (Il+gp ' S )lk'},. ..... (10) 



1 ,r=iL'+gpj*\i+Q2'/*')+m'*IPV)l ■ (U) 



Also for the weight of the liquid displaced by the solid we 

 easily find 



W= gPo >v{l + (U'+ ff p 'Q^\, . . . (12) 



where v represents the volume of the solid as reduced by the 

 pressure to which it is subjected, its density, however, being 

 treated as Uniform. The formulae (8') and (9') assumed the 



