96 REPORT—1846. 
the two other, being only approximately required for this purpose, have been 
merely calculated by theory, viz. by those values of constants which we are 
about to correct. 
The correctness of the numbers in those 283 primary equations for the 
twenty-four unknown, has then been controlled by determining the theoretic 
values of the X, the Y, the Z, or of the two or three of them that were required 
for the composition of w, of tang @ or tang 2, a second time in a somewhat 
different way. It consisted in calculating according to 
X=F(u)+F' (u) cos A+F" (wu) sin A+ F™ (w) cos2 A 
2 eee +F™(u) cos 4A+F"™"(u) sin 4A, 
or to a quite analogous expression for Y and Z, for which the nwmerical values 
contained in F(w), F\(u)..... E(w) are given as resulting from the pre- 
liminary values of the twenty-four constants in M. Gauss’s theory of terres- 
trial magnetism, § 27. 
This part of the task being completed, the second, and by far the most 
laborious one, consisted in forming out of the coefficients in each primary 
» 25 x 26.5 ws 
equation 9 = $325, and therefore altogether 325 x 283=91975 products 
of two factors, each according to this form,— 
ae A CAGE Ye Ieee Ne sisnis tna tae ole vosce'ocedsncnnbes ans n.(c.Ag'), n(c.Ah!"), 
(c.Ag*°).(c.Ag*°), (c.Ag*°).(c.Ag*!),...(c.Ag?).(e.Ag''), (e.Agt?).(c.Ah!1), 
(c.Ag*!).(c.Ag*!), ... (e.Ag*!).(e.Ag'!), (e.Agt!).(c.Ah!"), 
SOR e eee reese see sesessseseee 
See mem eee eee eeeeeeeeeeeeeseses 
(c.Ag?').(e.Ag'"), (c.Ag'!).(e.Ah''), 
(c.Ah!)!).(¢.Ah!"). 
The 283 products, assembled under each of these 325 titles, were then 
separately summed up, and by this means (marking by [ ] a sum of analogous 
terms) the twenty-four final equations of the following form were obtained: — 
— [n.(e.Ag*?) ]=[(c.Ag*?).(e.Ag*?) ].Agt?+ [(eAgt)(c.Agh!) ].Agt!+.. 
+ [(c.Ag*?)(e.Ag'!) Ag+ [(c.Agt)(c.Ahe!)]. Ahh, 
_ inane ee [(c.Ag*!).(c.Agt) ].Agt° + [(e.Ag*!)(ce.Ag*!) ].Agt!+.. 
eee et + L(¢.Ag*"')(e.Ag'!) ].Ag!! + [(e.Ag*!)(e.Ahb!)]. AA, 
— [n.(c.Ah*!)] = [(e.Ah*!).(c.Agt?) ].Agt? + [(e.dht!)(c.Agt!)].Agt! +. 
Boch EA + [(e.Ah*!)(e.Ag'!)].Ag'! + [(e.Aht!)(c.Ahb!)] Ak'!, 
—[n.(e.Ah'})] = [(e.Ah!) (e.dg4) ].Ag49+ [(cAh!!)(c.Agt!)].Agtt +. 
Srseestsane + [(¢.Ah).(e.Ag'!)].Ag!! + [(c.Ah!)(c.Ak}!) ].ARL 
The numerical expressions of these equations will be found in the table 
marked VI. Hitherto they have been controlled by the calculation leading 
to them from the primary equations, being repeated a second time in the 
same manner as the first, but with the suppression of one decimal figure in 
the products and in their sums. To obviate the danger, arising from the ad- 
dition of such extensive rows of numbers, lest the compensation of opposite — 
errors might produce an illusory agreement, M. Petersen, besides the forma- 
tion of new primary equations, has proceeded to subject these final equations 
to another kind of control,—I mean the process usually employed in similar — 
re eee 
ii 
1, 
qu 
