110 REPORT— 1898. 



my brother, it is important for the accuracy and trustworthiness of the 

 resulting values of the magnetic constants that the observations shall be 

 taken from stations distributed as uniformly as possible over the Earth's 

 surface ; whereas we see that in the northern hemisphere the Observatories 

 which exist are very unequally distributed, and that in the southern 

 hemisphere there are only three first-class magnetic Observatories where 

 continuous records are taken, viz. those of Batavia, Mauritius, and 

 Melbourne. 



For the more ready development of the theory of terrestrial magnetism, 

 Professor J. C. Adams established simple and convenient relations between 

 successive Legendre's coefficients and their derived differential coefficients 

 regarded as functions of the colatitude = cos"'//.. 



Taking P„ to represent Legendre's «,"' coefficient, he employed the 

 notation Ql" to denote the value of 



and found certain simple and useful relations between successive values of 

 Q for different values of n and 7n. 



He also employed the symbol G,"' to represent the Gaussian function 



,,»-m_ (w-»^ ) ( n-m-1) „_„^, , 



m 



and found it convenient to employ the symbol H™ as = G™ (1 — /i^)''' • 



He worked out very simple relations between successive values of G for 

 different values of n and m, and proceeded to determine the numerical 

 values of these functions (1) for every degree of latitude on a sphere, and 

 (2) for every degree of the geographical colatitude on a spheroid of 

 eccentricity equal to that of the Earth itself. He also obtained very simple 

 relations between successive values of H and its differential coefficients for 

 different values of n and m, and expressed the magnetic potential V and 

 the magnetic forces X, Y and Z in terms of these symbols H™. He also 

 determined the values of these functions H™ for belts of latitude 5° apart 

 (1) on a sphere, and (2) on a spheroid whose eccentricity equals that of 

 the Earth's surface. Two distinct schemes of calculation were employed 

 to determine the numerical values of G™ and also of H;;' for different 

 values of n and m, including all values of n and m from to 10, and these 

 calculations were made by different people and the results of the calcula- 

 tions compared to ensure the accuracy of the results. 



In the case of the spheroid, the functions G;;' and Hj^' are regarded as 

 functions of the geoyraphical colatitude Q, and /x = cos 0, and the symbols 

 G'^ and H'™ are the same functions of the geocentric colatitude & of the 

 same point, where ;«,' = cos & . 



A new theorem giving the values of G' — G' for different values of 

 n and in is established, by means of which the accuracy of the calculated 

 values of G and G' may readily be tested. 



Taking V to represent the magnetic potential at a point of the Earth's 

 spheroidal surface where A is the longitude, Q the colatitude, and r the 

 distance from the Earth's centre, X, Y and Z the magnetic forces in three 

 directions at right angles to one another, X being the force towards the 

 north perpendicular to the Earth's radius, Y the force perpendicular to the 



