of a Number os Observations inpraSHcalAstronomy. 6 7

kind of calculation, from the bare consideration, that thechances for throwing, precisely, the number m, with n dice,

whereof the faces, of each, are numbered — v .— 3,

— 2, — 1, — o, i, -j-2, -j- 3 .... -j- v, must be thevery fame as the chances whereby the positive errors can ex-ceed the negative ones by that precise number: but the formerare, evidently, the same as the chances for throwing preciselythe number v -J- 1 x n -f- m (° r n + q) with the same n dice,when they are numbered in the common way, with the termsof the natural progression 1, 2, 3, 4, 5, and so on; because thenumber upon each face being, here , increased by v -j-1, thewhole increase upon all the («) faces will be expressed by•u -f- 1 x n ; so that there will be, now, the very fame chancesfor the number v -f- 1 x n -f- m, as there was • before for thenumber m ; since the chances for throwing any faces assignedwill continue the same, however those faces are numbered.

PROPOSITION II.

Supposing the respeblive chances for the different errors , whichany jingle observation can admit of , to be exprefed by the terms oj

the series r~ v -j- 2 r'~ v -f- ~ v . -j- v -f- i. r °. y*-*

-j- 2r J ~ l -J- r J (whereof the coefficients, from the middle one (i)-s-i)decrease both ways, according to the terms of an arithmetical pro-gression ) ; it is proposed to find the probability, or odds, that theerror, by taking the mean of a given number (t) of observations ,

exceeds not a given quantity (s)-

Following the method laid down in the preceding propositi-on, the sum, or value of the series here proposed will appear

to be (being the same with the square of the

Jt —r\

geometrical progression r 'l*x i-fr-j-r 1 -)-?' 3 . ~f rV )"

the power thereof whose exponent is t (by making n = 2 /,and w = v -J- i) will therefore be r~™ x i -— iff x i — r\

> lfIZlr z ' !0 ~ tv — &c. into i -j- nr + — .

- nr

ÆO- tV ,

2