[ 384 ] 
LIX. Proceedings of Learned Societies, 
ROYAL SOCIETY. 
(Continued from p. 233.] 
May 24, 1860.—Sir Benjamin C. Brodie, Bart., Pres., in the Chair. 
> me following communications were read :— 
“Qn a new Method of Approximation applicable to Elliptic 
and Ultra-elliptic Functions.’ By C. W. Merrifield, Esq. 
On the Lunar-diurnal Variation of Magnetic Declination at the 
Magnetic Equator.’’ By John Allan Broun, F.R.S., Director of the 
Trevandrum Observatory. 
This variation, first obtained by M. Kreil, next by myself, and 
afterwards by General Sabine, presents several anomalies which re- 
quire careful consideration, and especially a careful examination of 
the methods employed to obtain the results. The law obtained seems 
to vary from place to place even in the same hemisphere and in the 
same latitude, and this to such an extent, that, for example, when the 
moon is on the inferior meridian at Toronto it produces a minimum 
of westerly declination, while for the moon on the inferior meridian 
of Prague and Makerstoun in Scotland it produces a maximum of 
westerly declination. No two places have as yet given exactly the 
same result ; though the result for each place has been confirmed by 
the discussion of different periods. 
In order to obtain the lunar diurnal action, it has been usual to 
consider the magnetic declination at any time as depending on the 
sun’s and moon’s hour-angles and on irregular causes. Thus, if 
at conjunction, H, be the variation due to the sun on the meridian, 
and h, be that due to the moon on the meridian, H, the variation 
for the sun at 1, 4, for the moon on the meridian of 1", and so on; 
it is supposed that we may represent the variations for a series of 
days by the following expressions, where the nearest values of / to 
the whole hour-angles are given :— 
Ist day. H’, +A’, +2), H+’, +2, ... H’,, +2, +2’. 
2nd day. H",+h",, +2" H" +h", +2", ics HA", "5 
nth day. H) +4, +e, Hy +h) +a. ... H+A +, 
where 2 is qos to imaeatalit causes, and n is 7” aed of days ina 
lunation nearly, 
Summing these quantities we have nae 
2H, +3) ji+3e, 3H, +2, Atde, , ++. SH, +23, Ate, (A) 
saa the means are, 
H+ ¢ +=", H+ ¢+=,... ul eee 
Here the aa means are affected by the spine due to the total 
action of the moon on all the meridians, and by variables depending 
on disturbing causes. If, on the other hand, we arrange the series as 
