TERRESTRIAL MAGNETISM. 205 



sinuK d\ '^ d\ '^ dX +'^^-J 



Z=2P' + 3P" + 4P"' +,&c. 



20. 



If we combine^ then, with these propositions, the known theo- 

 rem, that every function of \ and u, which, for all values of A,, 

 from to 360°, and of u, from to 180°, has a determinate finite 

 value, may be developed into a series of the form 

 po + pi ^pn + pm +^&c. 



the general member of which, P^' satisfies the above pai-tial dif- 

 ferential equation, — that such a developement is only possible in 

 one determinate manner, — and that this series always converges, — 

 we obtain the following remarkable propositions. 



I. The knowledge of the value of Fat all points of the earth's 

 surface is sufficient to enable us to deduce the general expres- 

 sion of V for aU external space, and thus to determine the forces 

 X, Y, Z, not only on the smface of the earth, but also for all ex- 

 ternal space. 



It is clearly only necessary for this purpose to develope -^ 



into a series according to the above-mentioned theorem. 



In the sequel, therefore, unless it is expressly stated otherwise, 

 the symbol V is always to be taken as Umited to the surface of 

 the earth, or as that function of A, and m which follows from the 

 general expression, when r is made = R : thus 

 V= R{P' + P'< + P'" +,&c.) 



II. The knowledge of the value of Xat all points of the earth's 

 surface is sufficient to obtain all that has been referred to in 



Prop. I. In fact, according to Art. 15, the integral f ''^ Xdu = 



t/ 



^ , V ° signifying the value of F at the north pole, and 



the developement of / X du into a series of the form referred 



to must necessarily be identical with 



yo - pi- pi' _ piii^ &c. 



III. In like manner, and under the considerations in Art. 

 16, it is clear that the knowledge of Y on the whole earth, 

 combined with the knowledge of X at all points of a line run- 



VOI.. II. PART VI, O 



