matter propofed ; which I fhall confider in the fol- 
Suppofing that the feveral chances for the different 
errors that any lingle obfervation can admit of, are 
expreffed by the terms of the progreffion r~^ 
r-3, r\ r\ r"' (where the 
exponents denote the quantities and qualities of 
the particular errors, and the terms themfelves the 
refpedtive chances for their happening) : "tis pro- 
pofed to determine the probability, or odds, that 
the error, by taking the Mean of a given num- 
ber (n) of obfervations, exceeds not a given quan- 
It is evident, from the laws of chance, that, if 
the given feries, r~^ -j- r“"3 » _j_ i _j_ ^.o- 
'r ^ j- expreffing all the chances- 
in one obfervation, be raifed to the «th power, the 
terms of the feries thence arifing will truly exhibit 
all the different chances in all the propofed («) ob- 
fervations. In order to raile this power, with the. 
greateft facility, our given expreflion may be reduced 
tor ^x— : whereof the power (making 
i), will be r~”'^ x i — r’")" x i j 
lowing propofitions. 
Proposition I. 
+ I 
which, expanded, becomes 
» « — 1 nv 
ft » — I n — 2 2‘tv—nv 
multiplied into 
Now, 
