DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 123 



Each belt of latitude will contribute an equation to each system. The 

 belts, 5° in breadth, are distinguished by the letters (a), (b), (c), <fec., 

 starting from 87^° N. latitude. 



Then if P, Q, R be quantities given by observation we shall have 

 equations of the form 



X^" g-?^ + X^'^ g]!l + Xf </™ + X ™ gl\ + &c. = P, 

 Yr g"^ + Y_"^ g^l + Yf gr + Y ™ g^, + &c. = Q, 

 Z^ gf + Zi'a <7 ™ + Zf g'l + Z_™ ^ J'| + &c. = R, 



for even values of n, and similar equations with other quantities P', Q', R', 

 given by observation, for odd values of n. 



Thus for latitude 60"', the set (f) will furnish the three following 

 equations to the respective systems X, Y, Z, corresponding to m^i, 



' -[9-6479108] i/l- [9-6397698] ^1,- [9-3739435] ^,1-[9-3627519]^i6=P, 

 [9-7120302] i/l + [9-7022392] (714 + [9-5314878] ^^ [9-5173452] ^io=Q, 

 [9-5118188] ^t-[9-4012092](7i, + [9-4766723]^;!-[9-3934121]^io=R; 



and the three following to the similar systems corresponding to odd values 

 of n, 



-[9-5471920] (7|-[9-5374280]^l5-[9-1267947](7^-[9-1145742]^ij=P', 

 [9-6499180]^H[9-6379512]^lg + [9-3653414]^^ + [9-3490233]^ij=Q', 

 [9-5284778]i7^-[9-4344144]i/i5 + [9-3682450]^^-[9-2923736](/i,=R'. 



These equations of condition are multiplied by the weights ?o„, w^^, &c., of 

 the observations for their respective belts of latitude, the weight of each 

 equation from the set (s) corresponding to 2^° on each side of the equator 

 being \ w^. Then the final equation for each magnetic constant ^™ is 

 formed by multiplying each equation so formed by the coefficient of ^,"' in 

 the corresponding equation of condition, and adding together the resulting 

 coefficients of g™ from the different sets (s), (r), (q), &c. 



To form this final equation for each constant multiply each equation of 

 condition by \/ weight and then multiply the resulting equation by the co- 

 efficient of that constant g'™ in it which has to be determined. Then 

 integrating or adding up the coefficients of the several magnetic constants, 

 we get the equation in the form 



2[(X:')-M^] 5r™+2[X™ X-.U.] ^!!„4-&c.=S[X;;,'. to. P], 



with similar equations for Y and Z for even values of n, and with other 

 similar equations with P', Q', R', for odd values of n. 



We shall have a separate final equation for each value of n ; thus the 

 final equation for f/™ is 



2[X-X:,«;^™ + X"4 X- t«^!!„ + (X„";)2«;5-™ + .fec.]=X- wV, (3) 



for even values of «i, and a similar expression with P' instead of P for 

 odd values oi n^. 



Then adding up, for any constant ^", the coefficients in the final 



' Where [9-6479108] is employed to express the number of which 9-647910S is 

 the logarithm. 



