FORMATION OF THE FINAL EQUATIONS. 597 



Formation of the Final Equations. 



5. Each equation of condition, as above found, will give rise to as 

 many final equations as there are magnetic constants in the equation to 

 be determined. The equations of condition are multiplied by the weights 

 w a , w b &c. of the observations for their respective belts of latitude. In 

 the following solutions of the equations the weight of the observations 

 for any belt of latitude has been taken to be proportional to the area 

 of that belt. The weight of each equation from the equatorial belt (s) is 



taken as ->,, since this belt extends only 1\ on each side of the equator. 



Then the final equation for each magnetic constant g is formed by 

 multiplying each equation so formed by the coefficient of g in the 

 corresponding equation of condition, and adding together the resulting co- 

 efficients of g from the different belts of latitude (a), (b), (c) &c. 



Thus the type of the final equation for g" is 



x < + & c . = 



with similar equations for Y and Z. The absolute term 2 \_X' wx'^\ is 

 different, according as n m is even or odd, the value of x' m being arrived 

 at as indicated in the last article. 



For the convenience of the ready calculation of the coefficients in the 

 final equations, a series of numerical equations has been formed from the 

 equations of condition by multiplying the equation of condition for each 

 latitude by the square root of the weight of the observations in that 

 latitude. These equations so formed are given in the previous Section 

 (see pp. 554 587) and are of the types 



g^) = w^x' m , and 



with similar equations for Y and Z. 



6. From the above equations (2) the final equations for any magnetic 

 constant g or A for a given latitude are formed by multiplying by 

 (X*w*), (Y*w*), and (Z' m n w h ), i.e., by the coefficient of gl or h* in the 

 above equations for X, Y and Z respectively. 



