MAGNETIC DECLINATION AND HORIZONTAL FORCE. 
17 
The fact of harmonious relations of this kind being found to subsist between results 
derived from long series of observations of two independent instruments, we cannot 
but regard as strong testimony to the reality of the phenomena now brought to light; 
neither can we refrain from claiming for such results a modest place amongst the 
phenomenal laws of terrestrial magnetism that must ultimately stand in the same 
relation to a physical theory of terrestrial magnetism that Kepler’s laws stand in 
towards the theory of gravitation. 
11. At this stage the question was put—With what approach to completeness does 
the typical variation in each case represent the four actual variations from which it 
was derived, or how much of a mean lunar variation is there in these over and above 
the typical variation ? Subtracting (say) f e .Jh) from the observed variation at new 
moon and full moon, and adding it to the observed variation at first quarter and last 
quarter in such a way that the Oth, 6th, 12th, and 18th hours of f c .z(h) are compared 
with the Oth hour of new moon, first quarter, full moon, and last quarter respectively ; 
the 1st, 7th, 13th, and 19th hours of with the 1st hour of new moon, first 
quarter, &c., respectively, and so on, we obtain four sets of residual variations, each 
commencing with the Oth hour of the lunar day; and, taking the mean of these four, 
we obtain the residual lunar diurnal variation that is left after appropriately elimi¬ 
nating the typical variation for the four quarters. A similar procedure, using the 
observed variations at the eighths phases, gives corresponding residual variations for 
the eighths phases. 
12. On curving these residuals for each quarter of the year—eight in all for each 
magnetic element—and comparing each curve with the corresponding typical variation 
curve, they were all found to be of small relative range, but most of them had a 
definite character, in which the principal harmonic element was that which has the 
lunar day for its period. In the latter fact we found a suggestion that, although our 
formula disposes of the hulk of the phenomena for which an expression is to be found, 
the addition to it of two more independent terms would not only make it mathe¬ 
matically more complete, but would render it further expressive of an otherwise 
neglected, but significant, element in the luni-solar variation. Making this addition, 
the formula becomes 
Mh) cos 
(^) +/•.#) cos 2 +fs,(h) sin 2 
* If 0 be written for the angle -p t, and 6 for the angle the formula may easily be transformed 
into 
K.i (0) cos + l\i(0) sin + F„. 2 (0) cos 2 (|q^) + F,. 2 (0) Sin 2 
in which F C1 (0), F s ] (0) are variations, of constant types, having the period of a lunation, and F C2 (0), 
f- 5 . 3 ( 0 ) are variations, of constant types, having a period of half a lunation; and all these swell and 
contract with a wave-like oscillation—F c . 1 (0), F 51 (0) in the period of a day, and F<.. 2 (0), F,- 2 (0) in the 
MDCCCLXXXVIT.—X. D 
