DE. T. E. EOBINSON ON THE SEDUCTION OE ANEMOGEAMS. 
425 
Let the true constants of the formula be denoted by small italic letters, so that 
u=K-\-a cos d-\-o sin 6-\-b cos 24+&c., 
1 
then, as the mean of u through the space D'—8=^'ud0, we have 
mean u= {JK^+Jacos 6d&- j-J o sin 6d0 &c. 
Let 8=$— [A ; and as all the pairs of terms are of the same form, 
a cos j()8-\-o sin p8, 
p p 
integrating this will do for all. The integral is 
which within the limits 
K0-f +« 
p 
sinjofl 
P 
=k>. . . . + J™M±M_ sin M-M 
A p p 
2a cosp^ sinj3«’ + 2 o sin/3\J/ 
=2K^+ p . 
/3j«.-i-sinjo/x 
f cos (pb+pfi) 
T p 
and dividing by 2p=d— 8, we obtain, calling 
cos(jp4— pit) ] 
P 1 
, cos 4 o sin 4 , 7 cos 24 _ 
mean w=K + a — — -f- — +6 — - — + &c. 
Now we might form the n equations for u and treat them by minimum squares; but 
as in this case none of the terms would vanish on summing, though all (except the one, 
say «, whose square appears) are small, the labour of eliminating 12 quantities 13 times 
p 
over would be truly formidable. This might be evaded by substituting in each sum for 
the true constants those given by the series of ®, which differ little from them, and all, 
except A, are multiplied by small coefficients. This will give A a with close approxi- 
p p 
mation. The process may be repeated with the corrected values, but A a alone will 
p 
have any notable effect. Yet even with this simplification the labour is very great. 
But it may be superseded thus. We have the above equation for u\ but we have also 
u'= K-j-A cos pfi-0 sin p+B cos 2<p+P sin 2<p &c. ; and equating the two values, 
„ , a cos 4 . o sin 4 b cos 24 n sin 24 „ , . . _ . , „ 
K+— ^+-y^+— 7 ^+^- r -^=K+A cos <p+ O sm <p+&c. 
It is evident that if we put a=Ar C ° S f , o=0 and s0 on, the equation would 
1 cos 4 sin 4 
3 L 
MDCCCLXXV. 
