4 REPORT—1857. 
particles of the electrical fluid acted on each other according to any such law, 
except that of nature, free electricity, instead of residing as it does entirely on the 
surface, would be dispersed through the entire of the changed body. 
These theorems have been hitherto (so far as the author is aware) known only for 
a spherical surface. 
After alluding to a well-known theorem of Chasles, he stated as a generalization, 
that if any two bodies have one external equilibrium surface common, then attrac- 
tion at any point of external space will be in the same direction, and proportional to 
the masses of the bodies, the law of force being that of nature. 
On certain Properties of the Radii of Curvature of Curves and Surfaces, 
and their Application to the Method of Polar Reciprocation. By T. 
Martin, 4.B., T.C.D. 
The author drew the attention of the Association to certain equations connecting 
the principal radii of curvature of a surface and its polar reciprocal at corresponding 
points, and of their inverse surfaces, and deduced the analogous equations for curves. 
He then pointed out the peculiar power of these equations in transforming theorems 
of quantity, which he illustrated by selecting some of the most familiar properties of 
curves and surfaces of the second order, thereby with facility and despatch arriving 
at some novel and elegant conclusions. 
A Demonstration that the Three Angles of every Triangle are equal to Two 
Right Angles. By B. A. Murray. 
On the Surface of Centres of an Ellipsoid. 
By the Rev. G. Satmon, D.D., MRA. 
Licut, Orrican INSTRUMENTS. 
On the Centring of the Lenses of the Compound Object-Glasses of Micro- 
scopes. By Sir Davin Brewster, A.A, LL.D. FLRS. LS BE. 
The author said,—In studying the subject of diffraction, as observed through the 
microscope, I was led to believe that in the best object-glasses now made the axes of 
the individual lenses are not coincident. I have no means of learning by what process 
the optician centres his Jenses and groups of lenses, but it must be a very delicate 
one, when we consider the small size of the lenses, and the great depth of their 
curves; and I have no doubt that, however imperfect, it is one which is anxiously 
and carefully applied. You are, no doubt, acquainted with Dr. Wollaston’s 
interesting paper, ‘On the Concentric Adjustment of a Triple Object-Glass”’ (Phil. 
Trans., 1822, p. 32), 45 inches in focal length, executed by the celebrated John 
Dollond, and regarded as one of his best works. By a process which he has 
described, Dr. Wollaston found that it was very imperfectly centred; and, con- 
trary to the advice of his friends he separated the lenses, and by applying two pairs 
of adjusting screws to the edges of each lens, he placed their axes in the same line, 
and to use his own words, ‘‘he restored his object-glass to such correct per- 
formance,”’ that it was “ capable of either separating very small and nearly equal 
stars, as those of 44 Bootis and o Corone, or of exhibiting the minute secondaries 
of 8 Orionis and 24 Aquilz, with as much distinctness as the state of the air would 
admit.”” Dr. Wollaston adds, ‘ that the actual limit to its powers cannot be fully 
ascertained, excepting under such favourable conditions of the atmosphere as do but 
rarely occur.’ If such a distinguished artist as Dollond failed in centring a group 
of three lenses, about 4 inches in diameter, and with comparatively flat curves, 
how much more difficult must it be to centre the six minute lenses of an achromatic © 
object-glass one-eighth or one-twelfth of an inch in focal length; and if such 
