558 



NATURE 



[Oct. 13, 1887 



follows : — Look at the successive lines of figures in the 

 table and write 



1 + O--*" + o.-f- + o.A-^ + o-i^ + o-^ + = Fq 



' x-\- .r^+ A-3+ ;tr*+ ^r^ + =iFo = Fj 



2^2 _,. 3,-3 _^ 4^4 _^ 5,.5 + =iF^ = Y^ 



.1-2+ 5.-K^+ 9.1-4+ I4;^5_(_ =lp^ 



6.*3 + 15.^-4 _[_ 29.r'> + =2F2 = F3 



SA'3 + 2ijr4+ 5o.r^+ =iF3 



4^-3 + 26.1^ + 76.r« + = 2F3 



3-*"^ + 30-1^ + io6.i-^ + =^F3 



2^'^ + 33-1^ + 139-1^ + =^^3 



,1-3 + 35;i-4+ 174^5+ =5F3 



36.r* + 2io.i-^ + =«F3 = F4 



and so on. 

 Then evidently 



IF,. = (!-.!-)-' F, - x' 

 and in general 



>+ ^F,-= (I - ,r) - ^J Yi - x' 



and consequently calling the zth number in the hypc- 

 tenusal series i, i, 2, 6, 36, «., and bearing in mind 



that "'F^. = F,. ^ ^, we shall have 



F,+ ,-(l-;i-)-"'F, 

 = -;r'{i+(i -.r)"' + (i -.r)-'+ . . . + (i - ;i-)-''^+'j 



= x \{\ - x") - (I - x) ] 



which obviously' regarded as an equation in differences of 

 the 1st order in F^-, gives the means of expressing F,- , ^ 

 as a function of x and a^, a^ _ ^ . . . . cJq, and consequently 

 must enable us to express all the coefficients in F^- , j, of 

 which the first is the hypotenusal number a^- , ^, in 

 terms of all the hypotenusal numbers of lower order. 

 But what is surprising and unexpected is that, as we 

 shall in a moment see, the relation obtained is expressed 

 by an immediate equation between the sums of the 

 hypotenusals i, i, 2, 6, 36, . . . ., each increased by 2, 

 z'.e. by an equation between the ipsissimi numbers of 

 Hamilton augmented by unity. 



In fact, multiplying each side of the equation by 



i\- x) ' + ', where 



S,- .^. J = ^0 + rt^i + <^2 + . . ■ • + ai, 



(so that Si = I, S, = 2, S3 = 4, S4 = 10, S,5 = 46, . . . .), 

 it becomes 



(i-^-f'-'^F.^^-CT-.rf^-F, 



= x'"^(i -A-){(i -xf^^^ -(I -xfi) 



which equation, it may be noticed, proved for all values 

 of i down to i may be extended also to i = o, provided 

 we make Sq = o. 



Accordingly, giving i all values down to o inclusive, 

 we shall easily obtain by addition 



(l-.r)^'F,= l + ;i-'"'(l -xf^^' - x~\i -X) 



+ /-^ (I - xf^-^^'^J-' (I - .rf'-^ + ^ + . . . . 



+ .r (I - -r) , 



or which is the same thing 



(l - A-) ' F^- - 2 + jr - X (i - .1-) ' 

 = X {\ - x) "■ -\- X (i - a-) ' - ' + . . . . 

 + ,r {i - x) 



Hfe'nce, equating the coefficients of x^ and using ^^q in 



general to signify 



Q^q - i) 



{q- /&+I) 



, if we call 



1 + 82 = H,. we obtain 



a,+ S,+ i = H,+ , 

 = ^2(H, + I) - ^3(H._ ^ + ,) +/3XH,_ , + I) - . . . 



,2 + I V'+ I 



+ (-)' 



And on calling i + H • = 



(Hx+i) 

 E^. this equation becomes 



I - ^E,+ , +^2E, - ^3E._ ^ +^4E,_, - . . . . 



+ (-) ^ Ei=o. 



This relation between the sharpened Hamiltonian num- 

 bers {i.e. these numbers increased by a unit) is in a 

 slightly different form from the result obtained by 

 Hammond. By aid of this formula the values of the 

 successive numbers can be calculated with wonderful 

 facility. The series of them commencing with i, which 

 although not properly speaking a Hamiltonian number, 

 belongs to the class S^-+ i, have been found to be 



I, 2, 3, 5, II, 47, 923, 409619, 83763206255, 



35081 25906290858798 17 1, 



6153473687096578758448522809275077520433167, 



Thus ex. gr. having found Hg = 923 and all the Hamil- 

 tonian numbers inferior to it, we have 



H, = 



924.923 



47-46 



1 . 2 

 = 426426 - 

 = 409619. 



I .2.3 

 17296 



I 12. II. 10.9 



+ 



1.2.3.4 

 495 - 6 



6.5-4- 3-2 

 1.2.3.4.5 



I have alluded to Bernouilli's numbers as a parallel 

 case to that of Hamilton's in so far as they too are 

 subject to a scale of relation by which each can be 

 expressed in terms of those of a lower order than itself. 

 If we use 



Bi, B2, B3, . , . . 

 to signify as usual the BernouUlian numbers 

 I I I 

 6' ^' ^^"•'• 

 and write 



Go = - I, G, - - I, G2 = (- 4)Bi, G3 = o, 

 G4 = (-4)%, G, = 0, G, = (-4)%, 



and so on, the well-known scale of relation for Bernouilli's 

 numbers may (provided only that n be odd) be written 

 under the form 



s:::(--)V«.G._.-o. 



I f in this formula we suppress the n which intervenes between 

 the operative symbol /3^' and G„_^, so that the former 

 is brought into juxtaposition with and acts on the latter, 

 it becomes identical with that which we have found f( 

 the sharpened Hamiltonian numbers. 



Those who wish to pursue the subject further may 

 consult my memoir " On the so-called Tschirnhausen 

 Transformation" {Crelle, vol. c. pp. 465-86), another "On 

 Hamilton's Numbers," by Mr. Hammond and myself con- 

 jointly, just published in the Philosophical Transactions, 

 and an addition thereto about to be presented to the 

 Royal Society, in which a large generalization of thetheorj^— j 

 discussed by Hamilton in his Report to the British Associ^B 

 tion for 1836, but not brought by him to perfection, i^"' 

 resolved with a completeness which leaves nothing to be 

 desired. J. J. Sylvester. 



New College, Oxford, October i. 



i 



