352 Astronomical and Nautical Collections. 



^ -' .r^",+. + (,i + l)AS) a-'"--' + (re - 1) JS 5 a-"-"--^ 



+ (2n— 5) D ) 2"--^ 

 + (n - 3) C 5 ^^^^^ 



Hence the new A appears to be (2k + 1) ^ ; and A is always 

 formed by the continual multiplication of the terms 1, 3, 5, 7 . . . 

 of which the last and greatest is 2n — 1 : and if we put m for 

 2w — 1, we shall have A^ = niA. 



From the second term also we have B^-=.mB + nA. Supposing 

 B to be obtained from A by multiplication and division, we may put 



B,t= —A : hence B ~ -^J — — ' vi.A. Consequently ' ' A 

 ' R R, ' R, ^ ^ R 



= 2L^ il + n.A, and [since Q,= Q+Q, that is,=Q+ A Q],^^ 

 R ' R, 



1 "^' . = 2!!_*_ 4- 7j . "In order to reduce this equation to the 



simplest possible terms, I suppose — : — — , that is, _i.:ir_. : 



R^ R R^ R 



hence — Qzzn : But —i is a new value of — , consequently — 

 R,-' R, R ^ R 



is a given quantity and R^zm, and hence Q:=:», ; and by taking the 



integrals of the finite differences, Q = % n^n -^ p: but since 



B ■=! when « =r 0, we have p = 0, and Q = | «, w: whence 



B = "'^. ^ ; and 5^= i^ ^." [It is not very easy, at first 



2m ' n 



sight, to perceive the propriety of making first _x =: — , and then 



R^ R 



R =: m, though it was most consistent with the author's analytical 

 system to adopt these steps, and the next paragraph will show 

 their utility : but it would have been here simpler, and more 



obviously allowable, to suppose at once B ~ —A, then B, =: _ i 



771 ?n^ 



j4, = -Lim^A =: QA^, and Q^=i Q+ n,, whence Q =: ??, as 



