TKANSACTIONS OF THE SECTIONS. lu 



/, 2(N-a") X u+l)N + («-l)a'' 



"I ^(«-l)N+(n+l)a'7 (n-l)N+(« + l)a» "' ' ' ^ ) 



which is the elegant formula of Dr. Hutton, first given in his Fourth Tract in the 

 year a.d. 178G. 



Now, to estimate the degree of accuracy attained by each new application of this 

 elegant formula, let N = a», so that a is the correct wth root of N, and a-\-x the 

 assumed or guessed root whose error is x ; then the rule being applied, gives the 

 corrected root 



_ (n+l)a"+(n-l)(.+.r J/, , .N _ ^"^^^ (^ + f) +^"-^> (^+S"^ ^^ 

 (n-\)a" + {n+V){a+xT ^ ^' „_l+(«+l) (l+^)" 



^^(iV-^.:-:) nearly/ 



Now if the first two or three figures to the left be correct in a+x then the 



relative error - will be <,-^, and .-. ±- <____; and as -— - is m general 



a 100 a^ lOOOUOO 12 



small when n is not a large integer (it is <1 when n=2 or 3), .•. the new corrected 

 root will be true in its first six figures (to the left) ; and if the assumed root a+x 

 a"-ree with the true root a in its first three or foiu- figures to the left, then its rela- 



^ 1 2 13 



tive error - must plainly be < —p-^, and therefore the relative error ^-r^ • '^ of 



the corrected root will be <; j^qqoqqoqqq ^^®" m<10; so that its first nine 



figures at the left-hand side must be correct ; and hence, in general, each operation 

 or new application of this formula trebles the number of correct figures in the 

 assumed root. 



On the Evciltml'ioh in Series of certain Definite Integrals. 

 By J. "W. L. Glaishek, B.A., F.R.A.S. 



It is a well-known result (due originally to Laplace) that 



r 



so that bv continued operation with the symbol - — or its reciprocal, the value 



, ?! 

 can be found of f "^ v'"' e~" " ""' dv. The result is 



r 2t _^2_l „ i _2i-2 -tf2-^ , 



1 17 e ^-dv = » \ V e ^"do 



Jo Jo 



_ 1.3...(«-2) , fi,!^2*-P^^=lK^?-i):^ + ...le-2% . . . (1) 

 P^TTI) '^^l^^n-l ^(n-l)(n-2) (^ J 



in which n is written for 2i+l (so that 2i—n — \) ; the series is to terminate when 

 the factor zero appears first in the numerator of a term. There are several ways 

 in which (1) can be proved ; but it is unnecessary to enter into details, as it is 

 only a case of a more general formula proved below. The identity of the two in- 



teoi-als in (1) is obvious, since each is deducible from the other by taking u=-. 



V 



(1 — 1 — B- — ^ 



But although (1) gives the value of ^ v e " "'dv when n-1 is of the 



3* 



