Page:Scientific Memoirs, Vol. 2 (1841).djvu/219

 Rh 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.

Although we are a priori certain that the series for $$V$$, $$X$$, $$Y$$, $$Z$$, converge, nothing can be determined beforehand as to the degree 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 rapidly; 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 convergence.

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-efficients $$g^{1\centerdot0}$$, $$g^{1\centerdot1}$$, $$h^{1\centerdot1}$$; $$P$$ requires five co-efficients; $$P'$$ seven; $$P^{IV}$$ nine, &c. As we consider $$P'$$, $$P$$, $$P'$$, &c. as magnitudes of the first, second, and third order, and so on, it is clear that if the calculation is to be pushed to magnitudes of the order $$n$$ inclusive, the values of $$n^2 + 2 n$$ 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 $$\lambda$$, 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 determination of all the co-efficients. But such a supposition can hardly be ventured upon, and the accidental errors which beset all observations, together with the neglected members of higher orders,