594 DR. a. W. ROYSTON-PIGOTT ON A SEARCHER EOR APLANATIC IMAGES. 
Compounding these expressions (1) and (2), 
d d 
m 00 p = P • p ‘ 
The constant ^>=1 ; for if rn be made equal 1G0, the conventional focus F= -yg, when 
d=10, and hence (d being large and F small) approximately, 
( 3 ) 
If (f) be the focal length of a small lens whose thickness is neglected, a very similar 
approximate result can readily he obtained from the optical formula 
For by construction 
and by similar triangles 
-+-=l. 
U 1 V J 
u-\-v=d, 
mv—u. 
Eliminating the unmeasured distances u and v of the object and image from the 
“ centre ” of the lens, it will be found that 
d 
m + 2 + — 
m 
and m being very large in these experiments, 
f=dk 2 near h*> 
or 
( 4 ) 
( 5 ) 
But in the case of the miniature images employed, m is so large that ( — 2) may be neg- 
lected, so that»i=j is sufficiently near for their measurement. 
The thermometer was now placed 100 inches distance from the microscopical focus; 
the one-sixteenth being employed to form the image, d=100; hence 
m=100 -p- 1 i 6 -=lG00 very nearly. 
The divisions on the thermometer would be therefore reduced in the image to a minia- 
ture 1600 times less than the original, or about 40,000 to the inch, whilst the breadth 
of a single line would be only the 400,000th. 
The means being thus obtained of readily estimating the size of images of known 
objects at known distances, the examination of immersion objectives next occupied my 
attention. Double stars were artificially produced in thin brass (two °f an i nc h thick) 
by placing minute apertures (x<roo ni diameter) in front of a brilliant flame, at the 
* This method also gives the focal length of a minute lens, to determine which accurately is attended with 
no little difficulty. 
