18 
MR. 0. CHAMBERS ON LUNI-SOLAR VARIATIONS OF 
each term of which is symbolical of a definite physical conception, viz., that an other¬ 
wise constant variation swells and contracts with a wave-like motion, as the age of the 
moon increases, between the limits —f(h) and 
13. If the initial new moon occur at li hours of the initial solar day, from the 
beginning of which time is reckoned in solar astronomical hours, the age of the moon 
• • ~-7T 
will become h — N, the period of a lunation 24P; and for _ p t may be substituted 
2tt 
{li—li ) : and if, further, /(h) be expressed in the form 
, 2tt 
a-, cos—A -j- b 1 sin h + a 3 cos 2 
^4: 
2-7T, 
24 
ii 
+ h s ' m 2 ) + &c., 
our extended formula may easily be transformed into 
A 1 C0S {yJ 1 ( 1_ fi) + a }+ B i C0S {24 2/ { 1 -fiI>) + / 3 j 
-fiCi cos {^ 4 2 ^^1 + — ) + y} + Di cos j— h ^1+ p ) + ^| 
+ A 3 cos ^ — h (l— ~ j-fe j-fB 3 cos p) + C| 
-f C 3 cos p j+T? j +D 3 cos j— li ^1+ p )+0j, 
where the numbers A l5 C 1} D x , &c., and the angles a, /S, y, 8, &c., are constants ; 
that is to say, it may—inclusive of the first four terms of f(h )—be transformed into 
eight simple waves whose periods, in solar hours, are 
24 
P-1 ’ 
2P 2P P P 
12 2P-1 ; 12 2P +1 ’ 24 P + 1 ’ 24 P-2 ’ 
12 
12 
P-1’ P+1’ 
and 24 
P+ 
9' 
Of these periods the first and sixth are the lunar day and half-day respectively ; from 
which it follows that, even if our extended formula be a substantially correct expres- 
period of half-a-day. It thus appears that 9 and 0 are reciprocally related, so that the period of either 
may he regarded as that of the variation of constant type, and the period of the other is then that in 
which the variation of constant type oscillates, whilst the complex variation of the formula remains 
identically the same; and this result is general so long as the number of terms in the formula is the 
same as the number of terms in each of the variations of constant type. The name—the luni-solar 
variation—has been chosen to distinguish the variation expressed by the formula as one involving the 
periods of a lunation and the solar day, and of sub-multiples of these periods; and the Dame—typical 
variation—has been given to the variations of constant type. 
