ERROR IN THE SUM OF WV INDEPENDENT QUANTITIES. 11 
But the series (iii) is equal to 
f (Sty) nr Gr) eo. 7 (iv) 
1 
mn 
the series to be continued as long as the part raised to power # is positive. 
This then is the chance that the error lies between © Sand y, and differentiating with 
respect to y we find that the chance that the error lies Been y and y + dy is 

1G Sep y) =n n (+ ee eee LT G+y ee 2) eau die, dy. (v) 
the series to be continued as in (iv). 
The result in this form would be of very little use except for small values of ». But 
Laplace has obtained approximate values for the above series when y is small, and his 

* This formula may also be proved by induction, for if it be true in the case of any number n of independent 
variables, we shall have for n + 1 variables the chance 
SL's | Gr) Gras) “tee 
ma 
+. 
jn—s |8 
1) n n+l 7. 2 241 mn n mt+l Fe 
at | ; + à) SG Sar = = ocre FA EE + We. 




Ti “Ver Le AE SLT LAVE se 
(Betsey) +n(5+y—3) Saat (—1) Ty s +t) &e. 
el — ES = n+ 
=a | +y) — Ft y—1 mie dust 

Nes LL n + 1 \ ntl a 
( 1) pe=s+lis-1 + <=) (3 HAS + &e. 
(CH) 
which is the same formula as in the case of 7 variables with # + 1 written for n. 


nm +1 n+1 n+1 y 
Ra oe arin 2 +y—s Fie. } 


Now when n =1 the formula (iv) reduces } + y and is therefore obyiously true, Hence it is also true when 
n = 2 and therefore when n = 3 and so on. 
