138 
THE REV. W. WHEWELL ON THE SOLAR INEQUALITY 
and have their maxima aud minima at the same seasons. This latter condition may 
perhaps be slightly modified, so that a may be somewhat different for different values 
of A h. 
The curves drawn with full lines in Plate XVII., are drawn under these conditions, 
the ordinates being those contained in Table (C. H.). Their agreement with the ori- 
ginal curves is such as to entitle us to consider them as correct interpolations, for the 
coincidence is almost complete in the cases where the corrections are the largest ; as 
the hours 0 h 30 m , l h 30 m , 2 h 30 m , 6 h 30 m , 7 h 30 m , 8 h 30 m ; and there are no material dis- 
crepancies except in the lines 4 h 30 m and 10 h 30 m ; and even in these the difference is 
only a displacement of a maximum of four inches by about a month. The obser- 
vations, therefore, prove that there is a solar correction of the heights which follows, 
nearly, the law suggested by the equilibrium-theory. The greatest amount of the 
periodical part of this correction is about half a foot plus and minus ; and to this 
must be added the non-periodical part, which at some seasons amounts to one fifth 
of a foot. 
1 /. We may expand the formula above given for Ay into a non-periodical and a 
periodical part. We have 
Ay = 
h + h' cos 2 (<p — «) 
Ah, 
i / {A 2 + A' 2 + 2 / 1 / 1 ' cos 2 ( 1 p— «)} 
h' c + COS 2 (<p — a) 
+ h") 
VI 1 + 7777 cos 2 (<p- a)} 
.Ah, 
Making c = y. Expanding and omitting c 2 , &c. 
Ay = 
+ { V + C0S 2 (P ~ a )) ( J ~ c cos 2 (<p — a)) A h 
= | c + cos 2 {<p — a) — c cos 2 2 (<p — a) j A h 
]j} C c c 1 
= I -Q + cos 2 (p - a) - 7 - cos 4 (cp - a) j A h 
Hence the periodical part is 
(Jp + A' 2 ) ^ cos 2 (<p ' K ) ~2 cos 4 a) ^ • 
This vanishes for two values of <p, which are a little less than 90° or 6 1 ‘ from each 
other when c is small. In Table (B. H.) it appears that the periodical solar cor- 
rection vanishes for two values of the hour-angle which are nearly 6 h from each 
other in each month. Hence we may suppose that the periodical part involving 
cos 2 (?) — a) is alone sensible. This vanishes when cos 2 (<p — a) = 6 h or 18 h , <p = 
3 h + a or 9 h + a. Hence we find that in January and September a is 2 h 30 m ; in 
March, April, June, July, October a, is about 2 h ; in August and November it is 
l h 30 m ; in December it is 3 h . 
