DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 125 



The coefficients S[X^,'X;',\.t6-], ikc, on the left-hand side of the above 

 equations of condition will be the same for 17"' and for A"', but the right- 

 hand sides of the equations 2:;[X;"((; P], <fec., the absolute terms derived 

 from the observations, will be diflerent. Hence the equations for solu- 

 tion may be conveniently arranged as follows. 



From (X) taken separately 



From the series for (X) combined with the series for (Y) we may also solve 

 the equations, of which the type will be as follows : 



{2[X- X-.t(;] + S[Y;r Yl.io\]g^ 

 + {2[(x;;' )-^.r] + --[{Y';ifw]) gl + etc. = {^^'"f^^'^f'''') 

 and {S[(X: l^r^w] + 5:[Y™ Y-?o]} A™ 



+ {2[(X';;j..] + ^~[{Yifw\] h:\ + &c. = (''''' fo).^;^™'™) 



the absolute terms being derived in this case from the series for (X) and 

 for (Y) combined. In general the values of the same constants derived 

 from these equations will differ somewhat from one another, and the ques- 

 tion arises which solution will give the truer value. 



Probably in the present state of our knowledge of the magnetic 

 elements over the surface of the Earth the equations derived from the 

 series for (X), (Y) and (Z) combined, for all latitudes from 67|°S. to 67^° N., 

 will give the most trustworthy values of the constants of terrestrial 

 magnetism, that we may hope to attain from any magnetic charts derived 

 from observations previous to the Admiralty Charts of 1880. 



Let us illustrate the mode of solving these final equations by taking 

 the case given above, in which «i=4 and n odd, taking the equations of 

 condition up to latitude 77^° inclusive, and combining the equations for X, 

 Y and Z, supposing the constant corresponding to negative values of n to 

 be non-existent. We will include the terms involving n=7. 



The coefficients for gl and h* being the same, we may take ni to stand 

 for either (1) ^5 or (2) hi, taking the absolute term for g* in the first case 

 and 14 in the second, and the final equations for gl and hi for the period 

 1845 may be written thus ; 



From 



(for gl) (for A1) 



(X) 3-4034960 o^--3898572 c4=-2416593 or --0159063. 

 (Y) 9-4158541 a^-}- -4092903 c.^=-0589245 or 4- -3418323. 

 (Z) 15-3871472 a^ + -0223528 c.^,=-4657356 or -f -1824818. 



Adding these together, we have 



28-2064973 c,^ + -0417859 <.^=-7663194 or -5084078 («). 

 Similarly the final equations for a^ and h* may be written thus : 



(X) --3898572 a.^4--2637326o^ = -0204205 or -0140404. 

 (Y) -4092903 «^-l- -3081774 a^=-0454171 or -0373065. 

 (Z) -0223528 aH'6536612c.^=-0056358 or -0882180. 



