46 PHILOSOPHICAL TRANSACTIONS. [aNNO ITQI- 



After several other examples of approximation, Dr. W. adds, I shall conclude 

 this paper with 2 theorems of some little use in the doctrine of chances. 



Theor. 1. — H = a + Z'X a -{- — 1 .a + b— l.a-^b — 3...a-\-b 

 — 71 -{• ] z=. a .a — \ . a — 2 . . a — n -\- 1 -\- n . a .a — 1 . a — 2 . . a — ii -\- 2 



X b -\- n . — - — Xa.a— \.a — 2... a — n-{-3Xb.b— \-\-n . 

 n — \ n ■ 



— ^ a.a— \.a-~2..a — n-\-AXbx b — \ . b — 2 -{■ 



„— In — 2 71 — 3 n — I + I , . 

 -|- 71 . — ■;; — . — ■ — . — - — .... J (l) a . a — 1 .a — 2 . a — 3 ... 



a — 71 -\- I + 1 X b . b — \ .b — 2 . . . b — I -\- 1 +... + w. 1^ a . a 



— l.b.b— l.b— 2..b — n+3+na.b.b— l.b — 2...b — 71 + 

 2 + b . b — I . b — 2 . . . b — 71 -{- 1 . 



K (or a + b — I , a + b — 2, a + b — 3, &c., a — \, a — 2, &c., b — 1, 

 b — 2, &c. be substituted respectively a -\- b — x, a -\- b — 2x, a -\- b — 3x, 

 , &c., a — X, a — 2x, a — 3x, &c., b — x, b — 2x, b — 3x, &c., the result- 

 ing equation will equally be just ; and lastly, if for x be substituted O, it will be- 

 come the binomial theorem. 



Cor. If there be 2 different events a and b, of which the numbers are re- 

 spectively a and /', and their chances of happening also as a and b ; and if a's 

 happen, let the whole number (a -|- b) and also the number of a's be diminished 

 by X, and in the same manner of b's happening, and so on ; then will the chance 

 of a's happening n — / times, and b's happening / times in « trials be l X a . 

 a — X . a — 2x . . a — {n — / — l) ar X b . b — x . b — 2x . . b — (/ — 

 \) X divided by h. 



In a similar manner may be found, 1 . the chance of a's happening between h 

 and Ii times ; and, 2. the chance of a's happening (//) to b's happening {k) times; 

 3. of a's and b's happening respectively h and h times more than the other; 4. the 

 chance of a's happening an even to its happening an odd number of times, &c. 

 in {71) trials, &c. &c. &c. 



Theor. 2. — h = a -\- b -\- c -\- d -{■ he. X a + b -\- c -Jf- d -{■ he. — x X 

 aJf.b-\-cJ[-d-\- &c. — 2x . . . .a -^ b + c -\- d -{- he. — (n — 1 ).r = a . 

 a — X . a — 2x . . . a — 7i — \x -\- 7i . a . a — x . a — 2x . . a — 71 — 2x X 



n — 1 

 b -\- c -\- d -{- &c. -f- 71 . ^, . a . a — x . a — 2x . . a — 71 — 3x X (b . b — 



X -^ c X c — X + d . d — X + he. -\- 2bc + 2bd + 2cd + he.) -f- + 



L X {a . a — X . a — 2x . . . a — / — 1 r X b . b — x . b — 2x . . b — t)i — Ix 

 X c . c — X . c — 2x . . c — p — \x X d . d — x . d — 2x . . . d — q — ^x 



X he. = K) -|- &c., where l = n . — -j— . — — . . ^ X n — I . 



n — I — I « — / — '2 « — / — «(+! , n — I — m — 1 n — / — ?« — 2 

 T— •—- 5 •• „, X n-l-m. , . ^ 



