130 
On the Probable Errors of Frequency Constants 
predetermined dichotomic ordinates to approach these as far as the material permits. 
As a rule hg (and hq) will have the opposite sign to h,. (and h.^,), but it will not be 
possible to make them of equal magnitude. 
It will probably be desirable to work with several pairs of dichotomic lines. In 
this case a^j.,^. can be written down at once from the value of a„^^ by proper inter- 
change of subscripts. But we need [S(j,.4So-,.y]. If we suppose the r' and s' 
dichotomic ordinates to fall on the median sides of r and s respectively, we easily 
find in the manner of earlier work in this paper : 
100^ 
[So-„8o-,.'s'] ■■ 
- h j^ (L - h,) {h, - h,) [H,H, N 
n,. 
1 - 
1 "r 
1 - 
N, 
.(47). 
Further any range like to be found from 
Xr - = 100 {h, - K)l{hp - h,j) . 
and the error in x,. — Ws will arise from our using h,., It^, h^,, Ji,j instead of h.,., hs, hp, 
8 {Xy — a;^) _ SA,- — 8/is ZJip — hh(j 
.(48), 
h,^. In other words 
h, - hs 
hp - h,j 
or (^\xr-x,) = {hr - hsf {o-Vp^ + o--,,, - 2 [S(7^„^So-,,,]{. 
The terms in the curled brackets are all known from equations (46) and (47) 
above, and we can thus determine the probable error of each portion x,- — x^ of our 
notace scale as found from equation (48). 
We can now consider the position of the mean. The distance of the mean from 
the rth dichotomic line is 
hrlOO 
hr. 
and accordingly 
8xr _ Sh,. Shp — Shq 
*V Ti, hp — hy 
or 8xy - aSIi,- + x,.h(jp,jj(j. 
Thus o-'-^y = ctV-/,^ + 'i;-o--cpj + 2.r,.a [8h,8ap,j] 
Now this may be written : 
' + 2h. 
'hh,^^i 
a 
(49). 
.(49 bis). 
