120 
SUMMARY OF CURRENT RESEARCHES RELATINO TO 
If in the determination of the lengths p and q we make errors of 
measurement, respectively A p and A q, there will result an error of A / 
in the calculation of the focal length, having a value determined by the 
equation 
(f+ A/) 2 = O + Ap) (q + Ag), (2) 
or 
/ 2 + 2/A/+ A 2 f = pq + p Aq + q Ap + Ap Aq. (3) 
Subtracting (1) from (3) and neglecting small quantities of the 
second order, we have : — 
2fAf = pAq + qAp. (4) 
Henco, divided by/ 2 — pq we get 
y=i( A f + a f)' < 5 > 
or the percentage error in / is the mean of the percentage errors in p 
and q. Hence, since A p and A q are obviously of the same order of 
magnitude, if we write A m for the arithmetical mean of them, and assume 
that each of them is equal to this value, we get from (4) 
( 6 ) 
which shows that for a given mean error A m and a given focal length, 
the error made in determining this focal length wjjl be proportional to 
p -f- q. Hence those values ofp and q which make p + q a minimum 
will make the error A/ a minimum. And, as p q = f 2 is a constant for a 
given lens, it is obvious that the case of minimum value of p + q is 
wdien p = q; this being the case when the conjugate points are at the 
symmetric points. 
The assiimption made above that the experimental difficulty of deter- 
mining the position of a conjugate focus is equal for conjugate foci in 
any position, is, however, hardly justified in practice, for in all labora- 
tory experience it is admitted that it is more difficult to ascertain with 
precision the position of an image (real) which is remote from a lens than 
that of one near the lens. In fact, the experimental location of the 
image is mainly delimited by the sharpness of the crossing of the rays, 
and the tangent of the angle at which the extreme rays cross is inversely 
proportional to the distance from the lens. The aperture of the lens 
then limits the accuracy of determination of foci at great distances. 
The larger the aperture the more accurately (assuming spherical aberra- 
tion above) will be the delimitation of the foci ; but the larger the 
aperture, the greater do spherical aberrations become. The error in 
determining q may arise at either end of the measurement ; it is more 
likely to occur at the end most distant from the lens than at the principal 
focus. If it be assumed that the probable magnitude of an error A q 
made in estimating the value of q is jiroportional to the distance of this 
focus from the lens, then we may write A q as proportional to q + /, and 
similarly A p as proportional to p + /. Substituting these values in (4) 
we got 
2/A/ oc p (q +/) + q(p +/) 
