362 Fariaiions of the Terrestrial Magnetism 



a little If^ss than that which M. Huniboldt observed, and 

 the difference is equal to i*69 ; but it must be considered 

 also that the formula from which we derive these values 

 supposes the position of the magnetic equator to be per- 

 fectlv determined ; but it may not be so with the utmost 

 exactness, according to the only two observaticms of Lapey- 

 rouse and Humboldt, which we have employed. It is there- 

 fore bv siudving the progress of the formula, and comparing 

 it with the observations, that we are able to appreciate it 

 justly ; after which we may think of remedying the small 

 errors with w hich it may be accompanied. 



To obtain the result I have here mentioned, and which 

 is, as it were, the limit of all those which may be obtained 

 by Efiving to K different values, it is lo be remarked that 

 the quantity 



a + 2 K COS. ?/ + K-)"3' + (I -r- 2 K C OS. V ^-K^)^ 



^ ^ — — — -r 



(1 + 2Kcos. M + K-)'^ — (l-2Kcos.a + K^)- 



becomes when K is null, but by applying to it the 



methods of known quantities it will be found that its value 



in this supposition is really determinate and equal to - — ^ — . 



By substitutmg this in the formula wc shall have 



sin. // 

 tang. jS = 



3 COS. u 



ail equation which may be reduced lo this form : 



sin. 2 n 

 tans:, p = ■ — — — ; — r ; 



^ '^ COS. 2 M + i * 



which will easib; oive the vaku- of j3 : and when this value 

 is known wc shall have the inclination I, by the following 

 foi nulla : 



I = 100 + 7/ — ;3, 



which will serve throughout the whole extent of the two 

 hemispheres. 



From the progress T ha\-e traced out ii is seen that the 

 preceding formula is not merely an einpyric construction ol 

 observations ; on the contrary, it is totally independent, and 

 '>n!v supposes the inclination of the magnetic needle to be 



produced 



