122 REPORT— 1898. 



Y,7 and Z" only odd powers. Hence, if the coefficient of cos m\ in either 

 of the quantities X, Y, Z be denoted by a„ and the coefficient of sin mX 

 by 6,„ for a given north hititude, and if «'„„ i'„i denote the similar quantities 

 for the corresponding south latitude, then we have, when m— m is even, 



iK-«'m)=2(X;r?;r + X"_'„^'!„), andi(6,„-6',„)=2(X™/C + X!:!„A':!„), 

 \{K + h',^ =2(y;;y„''+Y™„^'!!„), and -iK + «'™)=2(Y™7C+Y-„7t!l„), 

 l(a„ + a'„)=2(Z;X + Z-„ <7™„ ), and \(Jy,„ + 6'„) =2(Z,7A- + Z™„/r„ ), 



and when n—m is odd 



^K + a'„,)=2(X;X + X™„y'::„), and i(6„ + 6',„)=5(X-/C + X™„7^-„), 



l(a„-a',„)=2(Z;;'5r;r + Z-,9»„), and i(6„-6'„)=2(Z:7C + Z-„7i-.). 



Hence the equations for the quantities 7C and 7i!;„ will be found from the 

 equations for g'^ and </'!'„, when 7i — ??i is even, by substituting 



\(J^m — i'm) for ^(a„, — «',„) in the equations for X, 



— 5(«„j + «'„,) for ^(&m + &'m) ii'^ tl^e equations for Y, 



and i(6„, + £',„) for 5(a„, + «'m) in the equations for Z. 



And similarly the equations for 7t™ and 7i'"„ will be found from the equations 

 for jr^ and rfl,„ when ?i — 7)i is odd, by substituting 



\ibm + b'm) for i(«m + «'m) ii^ tlic cquatious for X, 

 — ^(«m — '^'m) for ^(6„— 6'„) in the equations for Y, 

 and \{bm — b'm) for ^(«„, — «'„,) in the equations for Z. 



In the first solution of the equations, the absolute terms {i.e. the terms 

 derived from the observed values of the magnetic elements) are taken from 

 Sabine's magnetic charts for the period about 1845, as published in the 

 ' Philosophical Transactions of the Royal Society.' In the second solution, 

 the observed values of the magnetic elements are taken from the Admiralty 

 charts for 1880 prepared by Captain Creak, kindly lent by the Lords of 

 the Admiralty. 



The values of X, Y and Z are calculated for every 10° of longitude and 

 every 5° of latitude from the declination (S), the dip (t), and the horizontal 

 force (w) as given in the charts. Then the values of X, Y and Z are 

 analysed for belts of latitude 5° in breadth around the earth by the 

 formula 



f^o + rti cos X + 6i sin X+o^ cos 2A+62 sin 2 A + kc. 



The values of these coefficients for the different belts of latitude were 

 obtained and tabulated. Then if a,„ and 6,„ denote the values of two of 

 these coefficients for a given northern latitude, and «'„, h'^ the correspond- 

 ing values for an equal southern latitude, then the values of ^ia^ + a'^), 

 \{<^m — «'m), M^m + ^'m)) ^^^ M^"'— ^'m ) ^ud of their logarithms are deter- 

 mined. The values of these quantities are determined for each of the 

 periods for which the magnetic constants are required. 



Each system of equations of condition will involve a single value of 

 m combined with all even values of n, or with all odd values of n. 



There will be one system for the coefficients X™, X"',„ another for the 

 coefficients Y'^, Y™„ and a third for the coefficients Z™, Z"l„. 



