PAPER RV PROF. HELMHOLTZ. 97 



moment of rotation, the ratio ^^^. is then negative, and the tangent to 



the boundary surface cuts the celestial vault below the pole. The 

 colder, more slowly rotating mass, which we will designate by tlie sub- 

 script (2), lies in the acute angle between the boundary surface and that 

 part of the terrestrial surface which is on the polar side of the given 

 poiut. 



When now at the boundary surface of the two strata, a mixture 

 takes place of the component masses nii and Wj, then will the moment 

 of rotation (fl) of the mixed masses be given by the equation 



{mi-\-m2)fl=mi£li + m2fl2, 



since the sum of the moments of rotation does not vary when no exterior 

 rotatory forces are at work. Equally will the potential temperature 

 of the mixture be given by 



(mi+W2) 0=mi 6i+m2 02- 



If now in equation (1), we at first substitute the mixture in place 

 of the cooler mass (2). in order to find the direction of the 

 boundary line between the mass (1) and the mixture, and indicate by 

 dpi and dvi the corresponding values of dp and <?r, then our equation (1), 

 after an easy transformation, gives 



3^r^ fir-^ wi Hi {ni-n2? 



^r^tdp~dp_\-mi-{-m2 (Jz-Si • • • • l^»; 



Since in stable equilibrium ^><^i, therefore this equation shows that 



dvi dr dpi dp 

 (fpi dp ^^ dri dr 



that is to say, that the boundary surfaces between mass (1) and the 

 mixture must ascend more steeply with reference to the horizon than 

 the boundary surface between (1) and (2). 



Similarly it follows that the ratio ^^ between the cooler mass (2) 



dp2 



and the mixture will be given by the equation — 



^G rdr2 dr 

 r^ Jlp2 dp_ 



7M2^2 (i^i —0.2 



nii+mz 61—62 



Therefore ^— >^!-; that is to say, the boundary surface between the 

 dp2 dp 

 cooler mass (2) and the mixture must make a more acute angle with 

 the horizon towards the pole than does the boundary surface between 

 the mixture and the warmer mass (1). 



It is to be noted that the ratios — are positive when the tangent to 



the boundary line is more inclined than the line to the pole — in the 

 other cases they are negative — and furthermore that the increase of a 

 negative quantity means the diminution of its absolute value. 

 80 A 7 



