1816.] Scientific Inielligence. 475 
As the Repertory is extensively circulated, it is scarcely necessary 
to say that the lenses above-mentioned are equally cylindrical on 
both sides (whether convex or concave), and that the axes of the 
cylinders stand at right angles to each other. As these are said to 
produce no aberration of the rays of light, and give a correct re- 
presentation of objects at every point of their surfaces, optical in- 
struments may, by their means, possess a much greater degree of 
light than formerly, and a vastly larger field of view. Telescopes and 
microscopes of a given length may also bear eye-glasses of a much 
greater power than formerly. 
Your mention of this subject, in whatever manner you may 
judge proper, will greatly oblige 
Newry, April 3, 1816. A Constant READER, 
I noticed these glasses already in the Annals of Philosophy, vol. 
vii. p. 324. The trials made of them in this country did not 
promise much; but I believe all the lenses of the kind tried were 
too thick to give a fair idea of the value of the inventon. It can 
hardly be expected that common spectacle grinders can make such 
lenses, because they are not provided with the necessary utensils. I 
am sorry that I cannot supply my Correspondent with the requisite 
information about grinding glasses, having never myself witnessed 
the methods, though they may be seen both here and in Birming- 
ham, and probably in many other places.—T. 
X. Demonstration of a curious Relation between the various Orders 
of Differences. 
(To Dr, Thomson.) 
SIR, a 
In the year 1772 the celebrated Lagrange announced in the 
Memoires de Berlin that a remarkable relation existed between the 
various orders of differences and their corresponding powers. Thus 
if e represent the number whose logarithm is unity, then 
dy 
Any = (ce * — 1)’, 
provided that be any positive number, and that the exponents of 
the powers of dy be transferred in the developement to the cha- 
racteristic d. 
The following demonstration of this important formula owes its 
origin to the perusal of an excellent paper by Dr. Brinkley on the 
same subject in the Philosophical Transactions for 1807. It is here 
offered as an inductive proof, and is perhaps better adapted to the 
comprehension of the young analyst than the one alluded to. The 
demonstrations given by Arbogast, Art. 399, Du Calcul des Deriva- 
tions, and by Lacroix, Art. 864, Traité des Differences, are distin- 
guished for their perspicuity and elegance. 
Your obedient servant, 
Plymouth, May 4, 1816. * Grorcr Harvey. 
2u 2 
