DR. T. R. ROBINSON ON THE REDUCTION OF ANEMO GRAMS. 
419 
cerned are periodic functions of the time-angle is that given by Bessel, in which, calling 
the quantity u and the angle $, we have 
w=K+A cos $+B cos 2$+C cos 3$+D cos 4$ + &c. 
+ 0 sin 3 + P sin 2$ + R sin 3$ + S sin 4$+&c. 
But as the monthly variations must be represented as well as the horary, a formula of 
this nature including two variables would be very complicated ; and it seems best to 
obtain, as proposed by Bessel, the horary formula for each month, and to regard the 
constants of this formula as themselves periodic functions of the monthly time, and 
develop them in similar formulas of the month-angle, <p. Stopping at terms of the 
fourth order, we should have nine of these for each component ; and for a given day of 
the year and hour of the day we must compute the constants for the <p of the day, and 
multiply each of the last eight by the cosine or sign of the corresponding multiple of $. 
The calculation of the horary constants is shortened by observing that for the angles 3, 
18O + 0> 180 — 0, and 360 — 3 the sines and cosines have the same numerical value ; and 
hence the calculation need only be made for the first quadrant. 
Supposing the circle divided into 2 n equal parts, and that 3 contains to of these, the 
u corresponding to any $ may be characterized as u, that corresponding to $ + 180 as 
u , and the sum or difference of these two as 5, d . 
As the cosines and sines of odd multiples of $ and 180 + $ differ in sign, but those of 
even multiples agree, the expressions of A, O, C, and R will contain only d, those of the 
others only s. The signs of s and d are easily determined in each case. Thus for the 
first multiples of 3 the cosine and sine are + for to through the entire quadrant ; they 
are — and + for n — to. For the second multiples the sine is + through the quadrant, 
the cosine is + up to 45°, — through the rest ; for n — to the cosine is the same as for 
to, the sine different. I take, as in the first instance, the horary division in which 
n= 12, and Bessel’s formulae become 
K=+4\fs+s+s .... +s j, 
U 1 2 11 J 
A= 1 +j(Z+ (d—d'j cos 15°+ ^cos 30°+ (d—d'j cos 45°+-J^Z— d^-\- (d— ^cos 75°j, 
Bzr-J^js— s+-^s+s — s— s'j + ^s+s — s— s^cos 30° j, 
C — d'j + p d — d — | d — d-\-d — d^j ~ ] sin 45 |, 
D= *{+? +i (?+?,) -i (+?J ~ (f +f) ■ -i( s +i) +++)}’ 
0 = ti‘! ^Z+o^sin 15 +-£ (d~\~ddj + ^Z+(Zjsin 45°+ ^fZ+ ^Zjsin 60°+ ^Z+<Zjsin 75°+ fZ j, 
P s+s— s\+s— s+ (s— s+s— s\cos 30 o l, 
l \1 11 s 71 3 9 \2 10 4 8/ J 
3 k 2 
