MAGNETIC INCLINATION AT GOTTINGEN. 649 
tan(2e—3(9 +9'—f—f')) = 
pees ey ne os te), 
fami Ftoth' +0) 
For the case when ¢ is not required, but only 2 is to be deter- 
mined, an elegant rule for the calculation is given in Erman’s 
Reise, part 2. vol. ii. p. 22. 
20. 
The relations of the observations to the inclination and to 
the other elements which have been hitherto developed apply 
generally, whether the deviation of the centre of gravity from 
the axis of rotation be great or small. When the latter is 
the case, which it always will be in needles executed by a 
really good artist, as long as they remain unaltered by any 
external causes (for example, by rust, wear, taking out the 
axles, or by weights intentionally added) the formule admits 
of a very material simplification. As long as 4 or < do not 
exceed the value 0°03, the difference between the sines of 
S+e-—i,g—c-—i, f'+¢ —i, g¢ —c— i, and the arcs them- 
selves, cannot amount to a second ; and considering the mode- 
rate degree of exactness of which observations with an inclina- 
torium are susceptible, the arc and the sine may be interchanged 
without scruple, even with considerably higher values of - 5 ane 
m° m! 
With the four needles of Robinson’s Inclinatorium the values 
are within even much smaller limits, and I shall therefore treat 
the observations to be here given according to such an abbre- 
viated method, first subjecting to it the example which has been 
hitherto under our consideration. 
21. 
_ If, for the sake of brevity, we write 
206265" qcosQ _ 
a 
206265" 9 sinQ _ 
Smee ee 
then under the supposition that f+c—i, g —ce—i, f' + ¢ —i, 
g' — ¢ —i are so small that they may be used for their sines, 
the equations (1.), (2.), (4.), (5.) of Article 15 assume the follow- 
ing form :— 
3 
