142 Transactions of the Royal Microscojpieal Society. 
the index of refraction of a sphere of which 0 is the centre and 
0 B the radius, a ray Q E proceeding from a point Q within the 
medium would after refraction proceed along q E produced. The 
limiting position of E is when ^EO is a right angle, or ^E 
a tangent to the circle, which is when E Q 0 is a right angle, since 
then the sine in the angle of incidence = QE:EO = l : fx, so 
that Q E 0 is the critical angle. 
Fig. 2. 
V . — On the Results of a Computation relating to Tolies* ^ Objective. 
By Professor E. Keith. 
(Read before the Royal Microscopical Society, June 5, 1878.) 
Having received from Mr. Tolies, at the request of Mr. Mayall, jun., the elements * 
of the ^-inch immersion objective made by him for Mr. Frank Crisp, I have 
made a computation of its angular aperture (Plate VII.), and present a figure 
accurately representing its different lenses and their distances apart, and also 
the path of a ray of light, emanating from a focal point 10 inches behind the 
objective, and coming to a conjugate focus, free from aberration. O’ 01620 of an 
inch before the front lens. 
As a result of the computation, I find that when the objective is used with 
the thickest cover possible making balsam contact with the front lens, its 
aperture is 110° 11' 40". Under the same conditions the focal distance of the 
outside rays is 0*01620 of an inch, and of those near the axis 0*01618 of an inch, 
showing practically no aberration. The computation is made with more precision 
than is warranted by the nature of the elements, which are necessarily given to 
only two or three places of decimals, accurate enough, however, for the main 
purpose. 
As the objective is thus shown practically free from aberration at the same 
time that its balsam aperture is far beyond that which corresponds to 180° of air- 
angle, the only impropriety in calling its air-angle 180°, for the purpose of com- 
parison, is in the fact that such a statement does not do justice to the objective. 
* With the computation Professor Keith wrote : — “ Mr. Tolies has been liberal 
of his time in making the elements sure in every point ; going so far as to make, 
from his memorandum in the case of Mr. Crisp’s lens, a new objective in order to 
be more sure of the distances of the lenses apart and the final focal distance. 
These are not necessary as elements of the computation, but afford a very 
decisive confirmation of the correctness of the figuring.” 
