PREFACE TO PART II. xi 



The symbol G'* is taken to represent the Gaussian function 



--- 



and the symbol H is taken to represent 6r (1 /n 2 ) 2 . 



Very simple relations are found between successive values of G for 

 different values of n and m, and the numerical values of these functions 

 are determined (l) 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. Very simple relations are also obtained 

 between successive values of H and its differential coefficients for different 

 values of n and m, and the magnetic potential V and the magnetic forces 

 X, Y and Z are expressed in terms of these symbols H. The values 

 of these functions H are determined for belts of latitude 5 apart (l) on 

 a sphere, and (2) on a spheroid whose eccentricity equals that of the Earth's 

 surface. The numerical values of G and also of H have been determined 

 for all values of n and m from to 10. Two distinct schemes of 

 calculation were employed, and the calculations were made by different 

 people and compared so as to ensure the accuracy of the results. 



In the case of the spheroid, the functions G and H are regarded 

 as functions of the geographical colatitude 6, and p. cos 6 ; and the symbols 

 G' and H' are the same functions of the geocentric colatitude 6' of the 

 same point, where ^ cos 6'. 



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

 n arid m is established, by means of which the accuracy of the calculated 

 values of G and G' may readily be tested. 



Section III. treats of the definite integrals of the product of two 

 Legendre's coefficients, which enter largely into the Theory of Terrestrial 

 Magnetism, and in Section IV. the product of any two Laplace's coefficients 

 is similarly dealt with. Section V. treats of the Theory of Terrestrial 

 Magnetism for the Earth regarded as a sphere, and contains new and useful 

 relations between the definite integrals of the products of the expressions 

 of the magnetic forces, which simplify the determination of the magnetic 

 constants. 



Taking V to represent the potential of the Earth's magnetic field, where 

 A is the longitude, 6 the colatitude of a point on its surface, and r the 



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