124 REPORT— 1898. 



equations for all the different belts of latitude we have the final equation 

 from the series (X), which may be represented by the form 



2[x;:' x;;;. ^r] < + 2[X"_!„ X". w\fi,, + 2[(x;;;)2 w] <?- + ctc.=2 [x-. lo. P] ( i) 



Equations similar to the above will be derived from the series (Y) and 

 from the series (Z). 



These equations may be solved separately, and the values of the 

 magnetic constants determined from each series, taking series (X), 

 series (Y), and series (Z) separately. 



The series (X) and the series (Y) may also be conveniently combined 

 into one equation in the same way as the above equations for different 

 latitudes in X have been combined, in which case the coefficient of y" in 

 the final equation for p-,™ will be 



2[X-X™.«'] + 2[Y;;'Y;r,H 



and the coefficient of £/™ in the final equation for £?;;' will be 



2(Xr.)2M; + 2(Y;r,)V. 



We have seen above that in the case of a sphere the coefficients of each of 

 the magnetic constants in this equation (4), except the coefficient of </™, 

 will vanish. 



The corresponding coefficients on the spheroid will be small, depending 

 on the value of the square of the eccentricity ; but this will only be the 

 case when the summation is taken all over the surface. 



The right-hand side of the equation becomes under these conditions 

 2[X;;',. P.w] + 2[Y;;'_.Q.tr] for the equation of (/;i in turn for all values 

 of ?^^. Hence when the successive belts are sufficiently near together the 

 coefficient of {[/''a+g'-,) in the final equation for g™ is approximately 



«(n-f-l)r (H;;')"rf/t 



w(?z + l) (« — ?u) ! (u + «i) 



■ 9 



2n + l ^ "[1.3.5... (2«-l)p' 

 and, as before, the right-hand side of the equation becomes 



rx',r.p.c?^t+r Y;r.Q.f^/,. 



In the present state of our knowledge as to the values of P, Q, R, (fee, 

 which are derived from the observations of the magnetic elements, the 

 charts giving the values of those elements are exceedingly defective for 

 our purpose, and the observations taken in high latitudes are not sufficiently 

 numerous and appear to be doubtful — no great reliance can be placed 

 upon them. Under these circumstances we propose to solve these equa- 

 tions, taking into account the data as derived from magnetic observations 

 over the portion of the surface of the Earth between latitudes 67|° N. and 

 671° S., taking only the equations of condition for belts between these 

 latitudes, and taking only those terms in these equations for values of 

 m from to 6 and for values of n from 1 to C inclusive. These equations 

 will give values for 48 constants, and no equation will contain more than 

 three unknown quantities. 



