Dr. Waring on 
170 
n 
a - 1 • b . b — 
tf + 3 + M . b . b — 1 • 
JZ~ 2 . . . b-n+ z+b . b- 1 ./> - 2 
If for & b — i, a b — 2, a +b- 3, &c., r?- 1, r? - 2, 
&c., £ — 1, b~2,Scc. be fubftituted refpeaively a + b-x , 
„ + V-2x/fl + ^-3-- n-x, a- 2x, a - 3 x, &c., *-*, 
2*, b - 3*, &c., the refulting equation will equally be juft ; 
and, laftiy,'" if for x be fubftituted o, it will became the bino- 
mial theorem. 
Cor. If there are two different events A and B, of which 
the numbers are refpeaively a and b, and their chances 
of happening alfo as a and b ; and if A’s happen, let the 
whole number 0 +£) and alfo the number of A’s be dimt- 
nilhed by x, and in the fame manner of B’s happening, and 
fo on ; then will the chance of A’s happening n -l tim es, 
and B’s happening / times in n tri als be L X a • a -x . a- zx . , 
a -(»-/- ijx X b . />'- * . b - 2X . . b - (/ - 1 )x divided by H. 
In a fimilar manner may be found, 1. the chance of As 
happening between h and k times; and, 2. the chance of A’s 
happening (b) to B’s happening 0 times; 3. of A’s and B’s 
happening refpeaively b and k times more than the other; 
4. the chance of A’s happening an even to its happening an 
odd number of times, &c. in ( n ) trials, &c. &c. &c. 
theorem II. 
H 4- c + d + &c^ xa-\- 1 > + c d &c. - x X ~a + b + c + d 
+ &C.-2X . .7 + ^ + C + < / +&C.-(«-I> = g . *-*'«- 2X 
t . ,~a^n-\x +n .a . a-x . a-zx..a~n -zxxb + c + d+ See. 
