24 PKOFESSOR J. D. EVERETT'S INVESTIGATION OF AN EXPRESSION 



a series which converges with extraordinary rapidity, and may be used even for 

 large values of z. Extracting the square root, we have 



45 56/0 



8. For the value of the acceleration a we have 



P (c+e J. sin 2 

 tan ' '^ • ^ - - ^ 



cos z 



Now, the first factor of the numerator is obviously greater than the first factor of 

 the denominator, and it may be shown" that the second factor of the numerator 

 exceeds the second factor of the denominator by the quantity, — 



2^ 2* 2® 2* „ 



which is positive for all values of z not greater than unity. Hence, for all such 



values of ^, tan (t + «) is greater than unity, and therefore a is positive ; that 



is to say, the phases of temperature are earlier for the mean than for the centre 

 of the stratum. 



e^+ e 



9. In the expression — — -^^ tan z^ the first factor is always finite and posi- 



e — e 



tive for finite and positive values of z. Hence, when tan z is zero or infinity, 

 tan (I + a j is also zero or infinity. And we know from general considerations that 



when 2;=0, a=0. Hence it may be shown, that when z is of the form ^, where 

 ?■ is any integer (not including zero), ~ + a=z, or a=z — '^. For all other values of ^;, 



z — z 



J + « will be unequal to z, since the factor ^^"^^_^ is never equal to unity. 



e — e 



10. We have hitherto been supposing the expression for the temperature at 

 the centre to be, — 



?; = sin t. 



To render our results applicable to the general term 



A sin (nt + E), 



in the expression for the temperature at the centre (§ 3), and the corresponding 



term 



m A sin (nt + E + a) 



