TERRESTRIAL MAGNETISM. 207 



much too soon to do so : the scantiness of the data which we 

 now possess does not allow of our dispensing with the assistance 

 of the vertical part. It is a confirmation of the theory, if we can 

 show the agreement of the different elements when reduced to 

 one principle. 



22. 



Although we are a priori certain that the series for V, X, Y, Z, 

 converge, nothing can be determined beforehand as to the de- 

 gree of convergence. If the seats of the magnetic forces be 

 limited to a moderate space around the centre of the earth, or if 

 there were such a distribution of the magnetic fluids in the earth 

 as to be equivalent thereto, the series would converge very ra- 

 pidly ; on the other hand, the further the seats of the magnetic 

 forces extend towards the surface, and the more irregular the 

 distribution, the slower we must be prepared to find the con- 

 vergence. 



In the practical application, absolute exactness is unattainable ; 

 we have to desire only a degree of approximation commensurate 

 with the circumstances. The slower the convergence, the greater 

 will be the number of members which must be taken into account 

 to attain a certain degree of accuracy. 



Now, P' contains three members, and requires, therefore, the 

 knowledge of three co-efiicients ^'■°, g^'^, A'' ; P" requires five 

 co-efficients; P'^' seven; P'^nine, &c. As we consider P', P", P'", 

 Sec. as magnitudes of the first, second, and third order, and so 

 on, it is clear that if the calculation is to be pushed to magni- 

 tudes of the order n inclusive, the values ofn^+ 2 n co-efficients 

 must be determined ; therefore, for example, 24 coefficients, if we 

 would go as far as the fourth order. 



Every given value of X, Y, or Z, for given values of u and X, 

 furnishes an equation between the co-efficients, whilst for each 

 place where the complete elements of the terrestrial magnetic 

 force are known, three equations are given. If we could venture 

 to assume that the members have a sensible influence only as far 

 as the fourth order, complete observations from eight points 

 would be sufficient, theoretically considered, for the determina- 

 tion of all the co-efficients. But such a supposition can hardly be 

 ventured upon, and the accidental en'ors which beset all obser- 

 vations, together \\'ith the neglected members of higher orders, 



