ON THE GAUSSIAN CONSTANTS FOR 1829. ' 95 
a 
O=AZ+2 cos uw. Ag'°+2 sinu. (Ag'! . cos A+ Ah!) . sin A) 
+.B eos? w—1) . Ag? +3 cos u . sin w (Ag! .cos A+ Ah! . sin A) 
} 
. + 3sin2u(Ag*?.cos 2A + Ah®*.sin 2 A)+ (4 cos$ u -2 cos u) Ags? 
+ (4 costa —= )sin & (Ag>:!cos A+ Ah! sin A) +4 cos u sin? wv. 
(Ag32.cos2.A + Ah32.sin 2A) +4sinw.(Ags.cos3 A+ Ah®.sin3a) ¢(3-) 
+ (5 cos wa cos? + =).ag'" + (5 cos? u-2 cos u). 
sin u (Ag*! cos A+ Ah*! sin A)+ (5 cos? u—2) sin? u(Ag*®.cos2A 
+ Ah*!2 sin 2 A)+5 cos 4. sin’ w (Ag? .cos 3 A+ Ah*3 .sin 3 A) 
+5 sint uw. (Ag**.cos 44+ Ah**. sin 4A); 
and denoting by AX, AY, AZ their just-mentioned developments according 
to the corrections of constants, 
O=Aw —cosd. AX Se) Se er C2) 
O=w.Ad.+sin 3. AX 008 GA Yo,s (ope. oe ijouihen Ga) 
O=w. Ai +sin i(cos .AX+sin d.AY)—cosi.AZ . . (6.) 
and then 283 numerical primary equations, relating to magnetic elements, 
observed on a line from Berlin to the east coast of North Asia, at the port 
of St. Peter and St. Paul, have been formed, by alternately recurring to one 
or the other of these six formule. For the sake of uniformity, their first term, 
which always meant the value of the magnetic element calculated with M. 
Gauss’s numbers—the observed value of the same element,—has always been 
marked by the letter x, independently of its having been derived by the Ist, 
the 2nd, ...... the 6th of these formulz ; and also the whole numerical primary 
equation has been represented by 
O=n + coeff.Ag*°.(Ag*°) + coeff.Ag*!.(Ag*!) + coeff-Ah*}. 
4q 4 0 meter + coeff.Ah's!.(Ah"), 
_ independently of their origin from formule (1) (2) ............ or (6). 
In five of the accompanying tables you will find, according to these de- 
- nominations,—1. the numerical values for 
log. n, log. coeff. Ag+°, log. coeff. Ag*', log. coeff. Ah*...... log. coeff.Ah!>! 4, 
furnished by each single element; 2. the name, the latitude and the lon- 
gitude of a station to mark the place of observation; and 3. one of the letters 
_X, Y, Z, w, ¢ or i, which respectively indicate that the observed element has 
been the northern or the western component of horizontal force, the vertical 
force, the whole horizontal force, a declination or an inclination*. In the 
three last cases in which therefore ~ denotes the value of Aw, of wAd, or of 
WAitt, it must still be noticed that the values of the declination (¢) in the 
first, and respectively of the horizontal force (w), or of the total force (f) in 
‘ 
: 
; 
| 
n* of is understood that, according to M. Gauss’s memoir, the meaning of the letters em- 
Dloy is— 
X Northern horizontal force. 3 Declination. 
Y Western horizontal force. i Inclination. 
w Total horizontal force. wu North polar distance of the station. 
Z Vertical force. a Longitude east from Greenwich of the 
_ 4p Total force. station. 
Instead of log. coeff. the further abbreviation |.c. being usually employed. 
The arcs Ad and Ai being previously changed to the ratio of their sines to unity. 
