752 REPORT — 1898. 



SATURDAY, SEPTEMBER 10. 

 The Conference did not meet. 



MONDAY, SEPTEMBER 12. 



The following Report and Papers were read :— 



An Account of the late Professor John Couch Adams's Determination of 

 the Gaussian Magnetic Constants. By Professor W. Grtlls Adams, 

 F.E.S. See Reports, p. 109. 



2. On a Simple Method of obtaining the Exjrression of the Magnetic 

 Potential of the Earth in a Series of Spherical Harmonics. By Arthur 

 Schuster, F.B.S. 



The methods which have been employed so fiir to represent the earth's magnetic 

 potential in a series of spherical harmonics sutFer from the serious defect that the 

 different coefficients are not determined independently of each other. Thus, in 

 the latest and most accurate computation of Adolph Schmidt, the value of the first 

 and largest coefficient was found to he 1,872, 1,913, or 1,021, according as the 

 expansion is supposed to end with terms of the third, fifth, or seventh order 

 respectively. If the value of the potential were known at all points of the earth's 

 surface it is well known how by a direct integration over tlie surface of the 

 sphere each coefficient may be separately determined. But there are large tracts 

 of the earth over which the magnetic forces have not been directly observed, and 

 hence some form of interpolation is always implied in whatever method is employed. 

 A certain amount of uncertainty results from this interpolation ; but perhaps less 

 than is commonly supposed, owing to the fact that neglecting magnetic masses 

 ■actually situated in the surface, the potential, and all its different coefficients must 

 be continuous. A complete knowledge of the potential over any finite part of the 

 earth's surface is therefore theoretically sufficient to fix it all over the globe, and, 

 at any rate, the continuity of the potential and of its derivatives facilitates and 

 justifies the process of interpolation. 



The whole interest of the expansion in a series of spherical harmonics centres 

 round the first few coefficients. It would be waste of labour to obtain a complete 

 representation of the potential, as we know that a very large number of terms 

 would be necessary for the purpose. The sole object of the expansion can only 

 lie in the separation of the outside and inside forces, and for the present, at any 

 rate our interest in the outside forces must be confined to the first few terms. 

 Hence I consider it a matter of great importance to obtain these terms separately 

 in such a way that their value does not depend on the number of terms which are 

 taken into account. I believe that the following method solves the difficulty. 



I write Pn for the zonal harmonic, T ° for the tesseral harmonic defined by 



<f P„ 



where B is the colatitude and /x = cos ^. 



If it be required to represent a function, V, of 6 and the longitude X, in a series 

 of spherical harmonics, it is known that the coefficients will depend on integrals of 

 ihe form 



JVT/ cos o-X d<o, 

 « being a surface element. The method I propose depends on a transformation 



