ON TELESCOPES AND EQUATORIAL MOUNTINGS. 195 
_ (0) = 91) + (2) —9(3) +. +++ 9(™) 
o(%) : 
m being made infinite. If ¢() be n”, so that the series becomes 1—2 +2" 
—a°+z—....., this mean value is (i +54+9+4+13+17+. +(4p+1)) 
+(2p+1)’, which, as p increases, approaches }; and in this case the ana- 
lytical envelopment, which can be found in the shape of a definite integral, 
also gives $4. The doctrine of means gives the same result for any other in- 
teger power of 2; and probably, if the analytical envelopment were found, it 
would give the same value. But if ¢(7) be of the form a”, the doctrine of 
mean values gives as the limit of the series 
e—2 +a) —a@) 4 9) — ey 
not 4, ports. and this circumstance of itself would induce me to require 
1+ = 
a 
to see the analytical envelopment of the series before I pronounced its limit- 
ing value to be 4. There are algebraic considerations which would rather 
tend towards the conclusion that this limit is a function of a; for when a is 
1, its value is 4; and if @ be infinite, its value appears to approximate to 
unity. It is perilous, however, to hazard surmises as to the value of this limit, 
so long as no finite equivalent for the series is produced. 
Report on the Improvement of Telescope and Equatorial Mountings. 
By Tuomas Gruss, M.R.LA. &c. 
Tue labours of the Earl of Rosse, now only perhaps receiving due appre- 
ciation, have placed beyond doubt the practicability of producing specula 
for reflecting telescopes of dimensions equalling, if not exceeding, those 
which the conditions of our atmosphere permit of being used with advan- 
tage, combined with an accuracy of surface and consequent excellence of 
definition which we can scarcely either hope or desire to surpass. 
Meantime the achromatic objective has received but small increment of 
dimension, and is now probably for ever distanced, in this respect, by its 
_ competitor the reflector. The spirited exertions of the Messrs. Chance of 
_ Birmingham have indeed produced a pair of dises suited to the formation of 
an object-glass of about 29 inches diameter, but these exertions have not 
_ been seconded by a corresponding spirit in Great Britain, either public or 
private. A few years since, the possible acquisition of an achromatic tele- 
seope, of corresponding gigantic size, was looked forward to as a national. 
triumph, if ever accomplished; but our Government, retaining its character 
of proverbial supineness (if not apathy) in such matters, has allowed these 
ye lendid dises to be transmitted to a more congenial kingdom ; yet even there 
the work seems to progress but slowly, and I apprehend that their formation 
_ into an object-glass is sti]l a work for the future. Four years have now passed 
‘Since the production of these dises, and nearly three years since, on being 
applied to by Messrs. Chance, I offered to form them into an object-glass. 
Under such circumstances it is desirable that attention should be turned to 
the reflecting form of telescopes as that alone suited for instruments of the 
largest dimensions, and important that these should receive from time to 
time such accessions of improvement as the progressive steps in arts or 
_ science place at our disposal. 
j 02 
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