812 THE MECHANICS OE THE EAETIi's' ATMOSPHERE. 



By the first of equations (11a) we obtain also the value of a x ) conse 

 quently that of 



(h=q\ fli 

 «5=</i <h «ij etc. 



If. in the computation of q { we take a sufficient number of fractious, 

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

 part of the numerical computation for q 3 , q 5 , and </ 7 . 



This remarkable method of determining the constants was by La- 

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

 recognized again by Sir William Thomson, who defended it against 

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

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

 The differential equation (11), when we replace cp by E, is of the secoud 

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

 can be determined when on two arbitrary circles of latitude, certain con- 

 ditions are to be fulfilled, such for instance as e =0, or &=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 par- 

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

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

 a fixed partition. 



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



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

 the continued fraction but some other arbitrary one, and therewith 

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

 for the equator, where sin go = 1. 



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



ft = 2.5 4 x 10 7 R = 287.0 



2tt b ~~ 2n T = 298.7° 



71 ~ 24 X GO X 00 



And second, for 



ft = 2.7352 T = 273° 



If we also write 



We find- 



■a x C instead of«i, 

 a 3 G instead of a 3 , 

 fiiC instead of b u 



-/3 3 C instead of b 3 , 



r = G sin go (nt + A), \ 



(p — C cos &) (a x sin go -f a 3 sin 3 go -f . . .) i (12) 



e = G sin (nt + A) [fi x sin gj 4- (3 3 sin 3 &? + /3 S sin 5 &? + . ". .J ) 



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

 vol. L., p. 227. 



