TRANSACTIONS OF SECTION A. 337 
the coordinates of the star-image referred to the black réseau may be represented 
by the following simple expressions :— 
hich, cetapicrh 
web p(h— hy)". "89 | 
and y=n-Uv—v ee eal 
(a) 
where the numerical value of the quantities € and y cannot in any case exceed 15. 
Also h-h,, v—v, will always be less than unity, and none of the quantities p, ¢, 
hy — hy, vi —v, h,—h,, and v,—v, will ordinarily exceed more than a few units in 
the third place of decimals, The reduction is therefore of a very simple character. 
These formulz take account of the ‘run’ of the screws, a possible error of inclina- 
tion of the two systems of lines, a possible inequality between the horizontal and 
vertical scales of the white réseau, and an inequality in size between the black and 
the white réseau squares. They do not correct for errors of division of either réseau 
nor for a possible error of inclination in the lines of the black réseau. Errors of 
division, if sensible, must in all cases be applied separately to the original 
measures. The errors of division of the white réseau do not affect the readings 
ho ¥o 3 2,, v,; nor hy, vs, since we are of course at liberty, when determining them, 
to make the errors of the extreme lines zéro, But they do affect h and v. They 
may, however, be very simply tabulated with m or x as argument. On the other 
hand, if the errors of the black réseau are sensible they must be applied to the 
readings h,, v,, &c., and with these corrections our measures will then be referred 
to an ideally perfect réseau. 
In the method of measurement here described the following advantages are 
claimed :— 
(1) No absolute perfection nor constancy in the adjustments is postulated. If 
the lines of the black and the white réséaux are not absolutely parallel, or if the two 
squares are not exactly the same size, the errors thus introduced are completely 
corrected by the third and fourth terms on the right of equation (a). 
(2) The sharp bright lines of the white réseau can be set with remarkable 
accuracy on the black réseau lines or the black star-images. 
(3) The measures are referred to the intersections only of the lines of the black 
réseau. This has the double advantage that these are the very points for which 
the errors of the réseau are determined, and that when these intersections only are 
required the rest of the réseau might, if desired, be completely masked out in 
printing. 
(4) The principal advantage, however, lies in the reduction which is effected 
in the number of settings when more than one star-image falls within a réseau 
square. With a simple eyepiece scale subdivided by micrometer screws, as em- 
ployed at Cambridge and other places, six settings are the winimum required to 
determine the position of each individual star-image. Thus, if there are fifty 
stars within the square, at least 300 settings must be made. In the same case, 
with the Oxford instrument, only 106 are required. The enormous advantage thus 
gained in measuring plates with large numbers of stars upon them is obvious. 
Plates of the size used in the astrographic survey contain 900 réseau squares. A 
rich plate may easily afford 9,000 stars, or, on the average, ten per square. To 
determine the position of each of these without duplicating any of the measures 
would, with the Cambridge type of instrument, entail 900 x 60 = 54,000 settings. 
With the Radcliffe measuring instrument we should require only 900 x 26 = 23,400, 
a reduction of 30,600 settings, or 57 per cent. of the whole ! 
1905, Zz 
