16 REPORT — 1872. 



logrt, — go; log &, 9-9070 h; log c, 9-77JL0 ji; logcZ, — qo; log e, 9-8753 n; 

 log/, 9-7393 n ; log g, 9-8467 n ; log h, 9-3498, in ^Y, according 

 to schedule (4). 

 And for 



lY, «ai=0-00, 5a = + 3-13, ca3= + 0-81, cZa^=0-00, ea,= -\-1-lG, 

 fa,= -2-72,}a,= +2-U, ha,= -0-13; 

 therefore 



2y=[rtaj + 5a„+ .... +/'. «s]= + 10-39; Y + aY=182-31=the re- 

 duced Y. 



log a, 0-2693 ; log b, 9-G387 ; log c, 0-7747 n ; log d, 0-1932 ; 

 log e, 9-7830 ; log/, 9-9190 n • log g, 8-9926 n ; log /;, 9-5895 », 

 in ^Z, according to schedule (5). 



And for SZ, 



«a,=— 0-59, 5a,= -l-02, on3= + 8-15, (?a^=-S-32, ec,.= -5-79, 

 /a„=-4-12, gu,= + 0-30, 7ia,= + 0-22; 



therefore 

 SZ=[aa^ + ha^+ .... +/(f,J= -11-83 ; Z + ?Z=1597-91=the reduced Z. 



.II. Compviatlon of the fwenti/-fonr Gaussian Constants from values 

 ohserved in or reduced to 1829. 



The numerous applications which we made of these means of reduction, 

 not only have added considerably to the number of empirical data for our 

 intended research, but they have also increased the intrinsic value of the 

 ■whole stock of such data. Indeed many observed elements Avhich by their 

 reduction to the epoch 1829 became api)licable to our purpose, related to 

 points of extensive regions where all knowledge of magnetic i^henomena had 

 been hitherto wanting. Such were, for instance, the beautiful series of 

 magnetic measurements which English navigators have executed in the 

 antarctic and North-American glacial oceans, and also many magnetic 

 determinations in the interior of the United States. Therefore the materials 

 now collected must amply suffice for our purpose ; but it seemed at first sight 

 as if for its attainment two cntirelj' different ways were left to our option. 

 Further consideration, however, has convinced us that of these ways or modes 

 of operating only the one which we have adopted was admissible ; but this 

 consideration, together with the doubt which it settled, merits to be shortly 

 explained here. 



According to a first plan of operation, we had to begin by calculating for 

 everj' newly added magnetic element its excess n over the theoretic value 

 assigned to it by the old approximations for the Gaussian constants — then, 

 having formed for each of these results (with a^, rf„. . . . w.,^ for given functions 

 of u and \, and Ar/''' '*, A/t"' ", every v and /i respectively varying from 1 to 4 

 and from to 4, for the corrections of constants) the expression 



to derive from each of these primary equations its correspondhig contributions 

 to the twenty-four final equations for A(/'", Ac/'' A/i'*''*; and lastly, 



