312 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



By the first of equations (11a) we obtain also the value of fli; conse- 

 quently that of 



0:, = </i fli 



«5=</i <u (U, etc. 



If, in the computation of 51 we take a sufficient number of fractious, 

 as, for instance, up to A'19, we have thereby also performed the greater 

 l)art of the numerical computation for q^, 55, and q-. 



This remarkable method of determining the constants was by La- 

 place applied to the theory of the tides. Its true importance was tirst 

 recognized again by Sir William Thomson, who defended it against 

 Airy.* Without Thomson's commentary Laplace would not be easy 

 to understand. In our case the matter presents itself very similarly. 

 The differential equation (11), when we replace q) by E, is of the second 

 order, and should have an integral with two arbitrary constants. These 

 can be determined v/hen on two arbitrary circles of latitude, certain con- 

 ditions are to be fulfilled, such for instance as f =0, or t=0. One con- 

 stant drops out when we let one of the parallel circles coincide with the 

 pole; the other is in this case to be determined as if the second i)ar- 

 allel was the equator itself. At the equator, on account of the sym- 

 metry, we must have 6=0. The equatorial plane is to be considered as 



a fixed partition. 



a, 

 The computation assumes that - — converges towards as i increases. 



"'1—2 



If we assume for a^ not the value that results from the computation of 

 the continued fraction but some other arbitrary one, and therewith 

 compute ^3, «5, etc., by equation (11a), we obtain a series that diverges 

 for the equator, where sin &? = 1. 



I have computed the constants with two values of A-. First, 



A; = 2.5 ^ 4x10" 12 = 287.0 



o = — rr— — 



W = : 



To = 298.70 



24 X 00 X 60 



And second, for 



/.• = 2.7352 To = 2730 



If we also write 



AVe find- 



■axG instead of ^i, 



a-iC instead of 03, 



l3iG instead of 61, 



>/?3C instead of 63, 



r = C sin a? {nt + A), \ 



<;? = C cos &? (n-i sin cj + 0-3 siu^ Gj-f . . .) > (12) 



e = C sin {nt + A) [/ii sin co + /is sin^ 00 -\- (i^ sin^ cy + . . .J ) 



* Airy ; " On an Alleged Error in Laplace's Theory of Tides." PMl, Mag., 1875 (4), 

 Tol. L., p. 227. 



