APPENDIX G. 



EQUILIBRIUM OF A COLUMN OF WATER. 



First, suppose the temperature to be the same throughout. Let a be the whole 

 depth, p the density, on the supposition that gravity does not act. Then, if p be the 

 density at the distance f from the bottom, when gravity acts, we have by the 



hydrostatic equation 



dp 1 



- gp = - ffPo— 



Tl+p 



if we adopt the rough formula of Section VII. for the compressibility. The integral is 



p(l - A) + An log.(n +p) = C - g Po t 

 Now the conditions are — 



(1) £ = f (the altered depth), p = ; 



(2) £ = 0, p = gp a = * suppose. 

 So that 



£ = a(l-A)+^I]og. I l±|^ 



9Po n 



= «(l-A) + ^log.(l +g ) 



Since, even in the deepest sea, ex/TI is not greater than 1/6, we may expand the 

 logarithm in ascending powers of this fraction. We thus obtain 



*— -a{i-5(5-^+^-...)} 



The second term is the diminution of depth required. We may write it, with 

 change of sign, as 



m gp ° a ' 2 ( l -'m + m ~ &c -) 



As the factor A/n stands for what is called e in the text, the first term is the 



