ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
683 
At this point all the secondary waves from A agree in phase, but the 
waves from a neighbouring point P will arrive at B with discrepancy of 
phase. For a very small interval A P this discrepancy produces no 
practical effect, and A and P will not be separated in the image. The 
question is, to what amount must A P be increased in order that the 
Fig. 104. 
difference of situation may make itself felt in the image ? Simple calcu- 
lation shows that the illumination at B due to P becomes practically 
evanescent when the relative retardation of the extreme rays P L and 
P L' amounts to a wave-length on their arrival at B. The limit of re- 
solution, then, is reached when P L - P L' = X. But since A P is very 
small, A L' - P L' and PL — A L are each equal to A P sin a where a 
is the semi-angular aperture L' A B. Therefore 
A = PL - PL' = 2 AP.sin a, 
and the condition of resolution is that A P or c should exceed J X / sm a. 
In the above discussion the points to be discriminated are supposed 
to be self-luminous. The author considers that the function of the 
condenser in microscopic practice in throwing upon the object the image 
of the lamp-flame “ is to cause the object to behave, at any rate in somo 
degree, as if it were self-luminous, and thus to obviate the sharply 
marked interference-bands which arise when permanent and definito 
phase- relations are permitted to exist between the radiations which issue 
from various points of the object.” 
Preparatory to the actual mathematical calculation of the images in 
the various cases, the author gives the following instantaneous proof of 
Fig. 105. 
Lagrange’s theorem, which is similar to the one given many years ago by 
Hockin.* In fig. 105, A and B are conjugate points on the axis A B ; P 
is a point near A in the plane through A perpendicular to the axis; and 
* This Journal, 1884, p. 337. 
