and considers that more detailed investigations can “ hardly 
upset the conclusion that the greater part of the diurnal 
variation is due to disturbing causes outside the earth’s 
surface.” 
Now, following Gauss, the full expression of the potential 
on the surface of the earth, which is equivalent to the 
periodical part of a system of forces, is 
"V = [Qi] + [Q 2 ] + + [Qi] + (1) 
where Y is the potential at a point which has u for its co- 
latitude (measured from the north pole from 0° to 180'’) 
and X for its longitude (east); and [QJ consists of (2i-j-l) 
terms, being Legendre’s coefficients in order, each multiplied 
by a periodical function of the form — = 
^icos0 + ^isin0 + P 2 COS 20 + g2shi20 + 
The different periodical functions, which have no neces- 
sary relation to each other, may be symbolised by the 
letters A, B, C , or (say)™ 
[Qi] = Acobu + Bsiii^<cos\ + Csin-MsinX 
[Q 2 ] = ~ I) E^^sinwcostiCosX) + F(3sin?iCoswsinX) 
+ G(3sinVcos2X) + H(3siiF'Wsin2X) 
&c., &c. 
For the values of X^, Y^, the deviations, at the time 9,^^ 
from tlie mean values, for the full period, of X, Y, the north 
and west components of the horizontal force, we obtain— 
X = 
^ adu adu 
A . B , C . . , 
= — sin-M - —cosifcosX — cos^^sinX 
a a a 
-f ? ?sin2z^ - 35cos2tecosX - 3-cos2iisinX 
Z a a a 
- 3?sin2?«cos2X - 35sin2^isiii2X 
a a 
( 2 ) 
where a is the radius of the earth. 
* The time being expressed by the hour angle ^ of the mean sun from the initial 
meridian. 
