Capacity of the Microscope. By Prof. Ilelmlioltz. 21 
impassable limit to the further extension of microscopic vision, 
which limit, moreover, has been already closely approached in our 
newest and best instruments. 
That diffraction and consequent obscurity of microscopic image 
must necessarily increase with increasing amplifications of the image, 
and this quite independently of any particular construction of the 
instrument, rests as a fact upon a general law which applies to all 
optical apparatus, and which was first formularized by La Grange * 
for combinations of any kind of " infinitely thin " lenses. This 
law has apparently remained almost unknown, perhaps because 
La Grange enunciated it in equations whose coefficients have not 
characters which readily present clear ideas to the mind. In my 
treatise on physiological optics, I have given expression to this law 
in a somewhat more general form, namely, for centred systems of 
refracting curved surfaces with any singly refracting medium 
between them, and have endeavoured to formularize it in readily 
intelligible physical characters. I shall therefore recapitulate as 
briefly as possible this theorem and its demonstration. It holds 
good for every centred system of spherical refracting or reflecting 
surfaces through which rays pass with angles of incidence so fine 
as to form punctiform images of punctiform objects ; that is to say, 
refracts homocentric rays, homocentrically. 
By the term centred system, I designate one in which the 
centres of the curves of each refracting or reflecting* spherical 
surface lie in the same straight line, the " axis " of the system. 
In front of such a system, and situate in its axis, let us suppose a 
luminous point belonging to some object lying in a plane at right 
angles to the axis, and from which rays pass through the system. 
The angle formed between any one of such rays and the axis, we 
shall call the divergence angle of that particular ray. Any plane 
supposed to extend through the axis and along the ray, constitutes 
the incidence plane of that ray at the first refraction, and will 
include, therefore, the same ray after its next refraction, and con- 
sequently after every subsequent refraction. Of this plane, which 
will be divided in crossing the axis into two halves, one half will 
be treated as positive, the other as negative, and in correspondence 
therewith, the divergence angle of the ray as positive or negative, 
according as the ray proceeds towards the positive or negative 
half of the plane. These postulates being settled, the rule may be 
thus stated : — 
Theorem. 
In a centred system of spherical refracting or reflecting surfaces 
the product of the divergence angle of any ray, the refraction 
index of the medium through ivhich that ray passes, and the 
* " Sur une Loi generale d'Optique," ' Memoires de I'Academie de Berlin,' 1803. 
