DIURNAL VARIATIONS OF TERRESTRIAL MAGNETISM. 
21 
of weight. The fifteen Northern observatories were thus given a total weight six, 
and the three, four, or five Southern observatories (according to the number available 
in the different cases) received a total weight one or one and a-half. 
Where two harmonic functions were involved in the representation of one set of 
data, as in Tables III. (y) and III. ($), the same general method of weighting and 
combining the data was used, except that two equations had to be formed, to give 
the two coefficients. Usually the middle four or five values of a n or b n were used in 
one equation, and the remainder in the other. 
§ 9. The Harmonic Representation of the Magnetic Variation Field. 
A potential function which varies with the local time, but is otherwise the same at 
all stations along any parallel of latitude, can always be expanded in a series of 
spherical harmonic functions of the form 
(3) </'»'= E" 
T>m + 2’ 
^i+IUw^rlcos^+fE" 
r 
m (b) 
T>m + 2\ 
+ I B W« sin nt 1 Q m " (0). 
»(*) 'm+l 
Here E" m(a) , E" m(w , I" m(a) , I” m (b) are numerical coefficients ; t is the local time reckoned 
in angle at the rate of 15 degrees per hour (t = \ + t r , where X is the longitude 
measured towards the East from some standard meridian, and t is the time corre¬ 
sponding to that meridian); r is the distance from the earth’s centre to the point 
considered, at colatitude 6 (in this paper measured from the North pole as origin), 
and It is the earth’s mean radius. The function Q m " (0), or Q„ t ” as it will generally be 
written, is the ordinary tesseral harmonic of degree m and order n ; it can readily be 
calculated from the formula 
(4) Q„” = 
(2m)! 
2™ . m! (m — n) 
sin” 6 ■[ cos™ " 6 - 
(m —%) 
2 2 .1! (m— jX 
cos™ - ” -2 6 
+ 
(m — n) 
2 4 .2!(m—i) 
4 cos™ - "- 4 
in which the factors of the form p s , where s is a positive integer, denote 
P {P- 1 ){P~2) ••• ( p-s+l ). 
The part of (3) which depends on r™ is continuous and satisfies Laplace’s equation 
within the sphere r = It. In the case of the magnetic variation potential, 
consequently, it arises from an electric current system outside the earth. The 
remaining portion of (3), which depends on r~ m ~ x , similarly results from a current 
system within the earth. The letters E and I are chosen to indicate the respectively 
external and internal origins of the corresponding parts of the potential. 
A term f m n in the magnetic variation potential would lead to the following 
