76 
SUMMARY OF CURRENT RESEARCHES RELATING TO 
or, in words — g is proportional to r, on the same scale on which G is 
represented, by R. For two beams in the same meridian which are 
focused in the image x at the points p and p' we have (see ante) 
or 
XXX 
9 + 9' 
E (r + r) 
where d is the distance between p and p\ 
The equation 
v ,G 
<t = A/ — a is 
K 
proved to be equally true for two beams which are not in the same 
meridian. 
If a radius of the image x be divided into a scale (the X scale) of 
equal parts, with zero at the centre, and on which the number G is at 
the outer end, then G becomes = R, and we have o- = X/d. Hence o- is 
equal to X divided by the number represented by d on the X scale ; and, 
further, it can be proved that the ruling of which <r is the spacing has its 
luminous bars perpendicular to the line d. 
This is a very important proposition in the interpretation of micro- 
scopical phenomena. As an illustration of the use of the above formulae 
and of the assistance which the Abbe theory atfords in microscopical 
work, the successive steps in the arrangement of the microscopic appa- 
ratus for the resolution of such a difficult “ test-object ” as Ampliipleura 
pellucida are minutely described. 
The author gives the following explanation of the bright specks 
becoming dark on change of focus, and of detail appearing to shift upon 
an object. 
Fig. 5. 
The finer detail in image C is formed by the interlacing of beams 
inclined at a wide angle. Let u and u' be two such beams, and let 
the unbroken lines of fig. 5 represent wave-surfaces which at the instant 
