18 



REPORT — 1872. 



the expression in q[uestion, wliicli gives a value for Bn when n is fractional. In 

 all cases, therefore, the formula is 



2r( 2«+l) 5, , 1 _^A , I (1) 



B„=- 



(2n) 



2ft 



jl+22» 



■^1^.+- 



The first four Bernoulli's numbers are I, ^^, ^^, ^^, after which they increase 

 rapidly, so that there is a minimuni between Bj and B^. As this minimum point 

 is the only intrinsic point of interest on the curve 3/ = Bx, the following Table was 

 calculated of values of B^. in its vicinity : — 



From Lagrange's formula, that if A, B, C be three values corresponding to ar- 

 guments a, b, c, then 



X=A (^~^)(^~c ) +B (a?— c)(a;-a ) i p {x—a){x—b) 

 (a~b)(a-c) (b-c)(b-a) (c-a)(c-by 



it follows that if A, B, C are three values in the neighbourhood of the minimum, 

 then X, the argument of the minimum, 



_ (b^-c'').\+(c"'-a^)B + (a--P)G . 

 {b-c)A+(^c-a)B+(a-b)C ' 



and by deducing the value of x from 2-8, 2-9, and 3-0, and also from 2-9, 3-0, and 

 3-1, it is found that the minimum corresponds to a?=2-93. .. ; and therefore, by the 

 usual interpolation-formula, the minimum value =-02377. . . . 



The values in the Table were calculated from the formula (1) expressed in the 

 modified form 



2r(2Hhl)._J_/ J_ J_ \ 



(2-f* i__LV 3''"*"6''■^•••;• 

 For ar=2-l it was necessary to include terms as far as /^— j ^ , for 2-2 as far aa 



(35) ''' ^^^ ultimately for a;=4 only as far as (^Y". The calculation was per- 

 formed in duplicate, and the accuracy of the values is apparent on forming the 

 5th diiferences. The values of log r(x) were deduced from Legendre's Tables. 

 It may be noted that, by means of the formula 



somewhat different form may be obtained for B„ ; for wo have 



B,= - 



Bn= 



2r(2M+l) 2^"-. 3^". 



)2n 



(2;r)^" (2"'-l){3"*-l) 



