DETERMmATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 127 



The formation of the final equations for gl, g°s and gl will then be as 

 follows : 

 from equations for (X) 



53-575026 = 7-6331952 ^J--1138565 .9",- -0886747^9, 

 -2-456S63=--1138565r/? + 2-8880836(/!!- -1765112 (7?, 

 -•453S875=--0886747^?--1765112(7°i--3955108^»; 



from equations for (Z) 



85-065860=12-0G36234 ^?-2-1413469 r/«--7000106 gi 

 _lfl-292662=-2 1413469 ^; + 2-7856531 ^«--4744250 gl 

 -4-6678164=--7000106 £/?--4744250 r/« + -4394974 gt. 



Solving the equations for (X) we get 



ry°=7-01229, (/°= --56367, gl=-17302. 



These values agree almost exactly with those found from the whole of 

 the equations for (X) up to latitude 77^°. 

 Solving the equations for (Z) we get 



^"=6-951666, </°=--524544, and ^^=--11476. 



These values agree very closely with those found from the whole of the 

 equations for (Z) to the same latitude. 



The values of g° and gt agree fairly well with those found from the 

 equations for (X), but the values of gl have opposite signs. Probably the 

 neglected term in g'^^ may have some influence on this result. 



Takint^ the negative values of n into account, let us find approximately 

 what values of g\, gis, gU will bring the two sets of results into harmony. 



This may be done by substituting g°+g-„ for g°, in the (X) equations, 



and ff° \g~u for gl in the (Z) equations. 



n + l 



Hence we get 



,9;=6-971874, ,y%=-040416, ^«=- -541312, ^°_3= - -022358, 

 .79 = -01605, and (/%= -15697. 



Hence the constant g".- seems to be of great importance. 

 The values found for the two first of the above constants are, in 

 in British units, 



by Gauss by Erman 



^;=7-0155 ^"=6-9417 



,/0^_.1430. ^«=--4069. 



The values of these constants, derived from the above series of equa- 

 tions for (X), (Y) and (Z), combined for all latitudes from 671° s. to 67^° N., 

 are ^?=6-98081 and r/°=— -523986 for the period (1842-45) from Sabine's 

 charts. 



The values derived for the above constants from the above equations 

 of condition, taking m from to 4 and n from 1 to 4 only and neglecting 

 the other terms {i.e. taking those only which were determined by Gauss), 

 are (/?=6-9777 and gl= — -5310 for the same period. 



The values of the constants given in the two following tables are 

 derived from the combined equations for (X), (Y) and (Z) to equations 

 (e) inclusive (i.e. between latitudes 67^° S. and 67|° N.), supposing the 

 constants corresponding to negative values of n to be non-existent. 



