TERRESTRIAL MAGNETISM. 2SS 



mixed causes, we attach the same meaning as before to the signs 

 V, P°, P', P", &c., v,p°,p',p", applying the former to the causes 

 acting from within, and the latter to the causes acting from 

 without ; and if we further put V + v = W, P° + p° = 11°, 

 P' +p' = W, P" + f = U", &c., then on the surface of the 

 earth, 



~ = U° + U' + U", &c. 



where 11^ satisfies the same partial differential equation as P^"^ 

 (Art. 18.) ; and the two components of the horizontal magnetic 

 force there existing are expressed by 



__dW_ _ dW ^ 

 R du ' R sin u d\ 

 The propositions mentioned in Articles 15. and 16. retain there- 

 fore their validity in this case, and we can determine the magni- 

 tudes n', 11", W', &c. simply from the knowledge of the hori- 

 zontal forces, but without being able in any degree to conclude 

 from hence only as to the existence of mixed causes. But if we 

 consider the vertical force by itself, and bring it into the fonn 



Q" + Q' + Q" + Q" +, &c. 

 so that Q^**-* satisfies the above-mentioned partial differences, 

 then Q° = P° 



Q =2P -p' 

 Q" = 3P" - 2p" 

 Q" = 4P"' - 3/", &c.; 

 and, consequently, 



3P' = n' + Q', 3/ = 2n' - Q' 

 5 F' = n" + Q", 5p" = 3 n" - Q" 

 7 F" = W" + Q", 'Jp'" = 4 n"' - Q'". 

 Thus, by the combination of the horizontal force with the verti- 

 cal, we obtain the means of dividing W into its constituent parts 

 V and V, and thus of learning whether a sensible value must be 

 assigned to the latter. Only the constant part of v, namely, j9°, 

 is left wholly undetermined by the observations, the reason of 

 which is plain from Art. 38. 



Hence it appears important, in this interesting point of view 

 likewise, to consider the horizontal magnetic force by itself, and 

 we see in this an additional reason for the recommendations in 

 Art. 21. 



