50 THE VOYAGE OF H.M.S. CHALLENGER. 



XL Equilibrium of a Vertical Column of Water. 



In Canton's second paper we have the following interesting statement : — 



"The weight of 32g feet of sea-water is equal to the mean weight of the atmo- 

 sphere : and, as far as trial has yet been made, every additional weight equal to that 

 of the atmosphere, compresses a quantity of sea-water 40 millionth parts ; now if this 

 constantly holds, the sea, where it is two miles deep, is compressed by its own weight 

 69 feet 2 inches ; and the water at the bottom is compressed 13 parts in 1000." 



Either Canton overestimated the density of sea-water or he underestimated the 

 amount of an atmosphere, for undoubtedly 33 feet is a much closer approximation to 

 the column of sea-water which produces 1 atmosphere of pressure. He does not give 

 his process of calculation, but it was probably something like this : — The pressure 

 increases uniformly from the top to the bottom (neglecting the small effect due to 

 change of density produced by compression), and everywhere produces a contraction 

 proportional to its own value. Hence the whole contraction is equal to that which 

 would have been produced if the pressure had had, at all depths, its mean value, i.e. 

 that due to half the whole depth. This process, with Canton's numbers, gives nearly 

 his numerical results. 



If, then, a be the depth, and p the original density, gp cc/2 is the mean pressure. 

 If e be the compressibility, the whole contraction of a column, originally of length a, is 

 egptfx 2 \1. Now, a mile of sea-water gives nearly 160 atmospheres of pressure, so that 

 the loss of depth of a mile of sea (supposed at 10° C. throughout) is 



160 x 0'000045 x 5280/2 = 19 feet, nearly. 



For other depths it varies as the square of the depth ; so that for two miles it is 76 

 feet, and for six miles 684 feet nearly. 



This, however, is an overestimate, because we have not taken account of Perkins' 

 discovery of the diminution of compressibility as the pressure increases. The investiga- 

 tion for this case is given in Appendix G, where the change of depth is shown to be 



/ 2*r *r2 s 



Wo« 2 /2( 1 -3ff + 25*--) 



rs being the pressure at the bottom in tons weight per square inch, and II (by Section 

 VIII.) being 38 in the same units. 



For six miles of sea this is, in feet — 



684 (l - A + ^ - &c.) = 620 nearly. 



In the Appendix referred to I have given a specimen of the hydrostatic problems 

 to which this investigation leads. Any assigned temperature distribution, if not 



