420 
DR. T. R. ROBINSON ON THE REDUCTION OF ANEMOGRAMS. 
[d-\- d — d~ j- f d-\-d-\- d -(- d — d — <^\cos 45 1, 
"1.2 10 6 \l 11 3 9 5 7/ J 
S = iVfs— s+s— s— (s — s-J-s — s\ cos 30°1, 
1 1 11 2 10 \4 8 5 7/ J 
and so on. 
These are all combinations of the groups s ± s , d^rd ; and by forming these 
groups the computation is evidently much simplified. 
This simplification is, however, only possible when n is an integer, and a the first arc 
of the series=^ or=0. 
Whatever be the value of a , Bessel’s formula fails generally to give G and U the 
cosine- and sine-coefficients of the nth order. The Q correspond to u=a-\-(m— 1)^, 
and this for the order becomes 1)^. Then cosine 0=cos(mz),sin 0=sine na ; 
both + for odd values of m, — for even ones. Thence the nth coefficient — 
u cos na= K cos w«+&c.+G cos 2 na + U sin na cos na, 
1 
—u cos na= — K cos na— &c.+G cos 2 na-\- U sin na cos na. 
2 
Then summing from m— 1 to m=2w, we get 
cos na S [u— u'j = (cos 2 na G +sin na cos na U) X 2 n, 
S (u—u^=Zn (G cos na-[- U sin na). 
Here the divisor of S (u—u^ is 2 n instead of n; and these coefficients cannot he 
obtained separately unless « = 0 or in which case the cosine or sine = 0. 
How far the series is to be continued depends on the periodic fluctuations of the ms, 
and may be found by trial, or by Bessel’s expression for the squares of the residual 
errors. In any case it should not be carried further than the order as after that the 
coefficients coalesce. Bessel has shown this for a= 0; and it can easily be proved to 
hold good when a is a submultiple of and b a multiple of a. 
For the horary groups I find the fourth order sufficient. These horary groups might 
be combined in triple sets ; but, as I have said, there is a difficulty in the interpolation 
due to the fact that while u 1 , the mean of any three, is multiplied by cosine or sine of 0, 
0 
the first and third components of it should be multiplied by the same functions of 6— b 
and 6-\-b. This, however, may easily be corrected. Take the case of A : the effect of 
three components to determine this is : — 
