1822.] Col. Beaufoy’s Astronomical Observations. 27 
ArticLe VIII. 
Astronomical Observations, 1822. 
By Col. Beaufoy, FRS. 
Bushey Heath, near Stanmore. 
Latitude 51° 37! 44°3” North. Longitude West in time Lee ost. 
Occultations of small stars by the moon. 
May 23. Immersion... --.+-+-eeeeee8 8n 59! 10-3” 
24, Emersion.....-2eeeeeeseerere 13 45 21-0 $ Siderial Time at Bushey. 
26, Immersion...-.-+-+-2eeeee9° 15 30 O81 
N. B. The observation of the 24th uncertain to five seconds. 
ee eeeiecieieeeteeinaniaiol 
ARTICLE IX. 
“ 
An Investigation of the Method for finding the Sum of all the 
Coefficients in the Expansion of a Multinomial. By Mr. S. 
Jones. 
(To the Editor of the Annals of Philosophy.) 
SIR, Nash Grove, Liverpool, May 29, 1822. 
Ir is a remarkable coincidence that two of our most general 
theorems in analytical calculations should have been published 
by their respective authors, Dr. Brook Taylor, and Sir Isaac 
Newton, without their demonstrations : the utility of the former 
in the differential and integral calculus, vanishing fractions, the 
higher mechanics, Kc. has induced many of the continental, as 
well as several of our own mathematicians, to attempt to demon- 
strate it from first principles; it is, however, generally acknow- 
ledged that its difficulty excludes that conciseness, perspicuity, 
and native simplicity, which all fundamental propositions ought 
to possess ; the latter, or binomial theorem, though less difficult, 
was, at the time of its discovery, no easy task to demonstrate ; 
accordingly we find that it caught the attention of the most 
eminent mathematicians, and has employed the talents of 
Maclaurin, Simpson, Demoivre, Euler, Lagrange, Woodhouse, 
and others, to whose minute researches and amplitude of remark 
it might be supposed that nothing more could be added ; yet 
it is presumed that to demonstrate generally what only one of 
these, Euler has done for a particular case, and that by numeral 
induction, will not be thought an inelegant appendage to this 
beautiful theorem, nor unworthy the attention of the mathemati- 
cians of the present day. 
