590 FORMATION OF THE EQUATIONS OF CONDITION. 



In these equations of condition, the theoretical expressions for the hori- 

 zontal and vertical magnetic forces in terms of the magnetic constants will be 

 of the same form for all periods of time. For a given period they are 

 equated to x' m , //, and z' m , the absolute terms derived from the magnetic 

 observations of the horizontal and vertical magnetic forces for that period, 

 and by solving the equations the values of the magnetic constants for that 

 period are determined. In the first solution, the absolute terms are derived 

 from the observations of the magnetic elements given in Sabine's charts 

 for the period about 1845 published in the Philosopliical Transactions of the 

 lioyal Society. In the second solution they are derived from the much 

 more complete and trustworthy observations recorded by Captain Creak in 

 the Admiralty Charts of 1880. By the kind permission of the Lords of 

 the Admiralty, reduced copies of these Charts are given at the end of 

 this Volume. 



The observations have been taken for every 10 of longitude and for 

 belts of latitude 5 in breadth around the surface of the Earth. 



The numerical values of the coefficients of the magnetic constants 

 for the spheroidal figure of the Earth have been calculated and recorded 

 in Section VII. for all values of n and m from to 10 both for internal 

 and external forces, and the equations of condition will contain two sets, 

 each of 120 magnetic constants: the values of these two sets of constants 

 may be found for any given period by substituting in these equations the 

 values of x' m , y' m , z' m derived from the observed values of the magnetic 

 elements for that period. 



Formation of the Absolute Terms of the Equations of Condition. 



3. The terms x' m , y' m and z' m as explained above are derived from 

 the observations of the horizontal force in the plane of the meridian, 

 of the horizontal force perpendicular to the meridian and of the vertical 

 force in the equations for X, Y and Z respectively. 



The values of the forces being given from the charts for A. = 0, 

 X=10, &c. to X = 350, they are analysed for each belt of latitude by a 

 formula of the type 



a + a a cos A. + b l sin X + a, cos 2A. + b. 2 sin 2\ + &c. 



The values of the coefficients a , a 1} 6 1} &c. for X, Y and Z for 

 the different belts of latitude for the epochs 1845 and 1880 were obtained 

 and tabulated. 



