EQUATIONS IN FINITE DIFFERENCES. 99 



h + Ai + b, + kc. = n 

 2ft, + 3^2 + &c. = « - 1, 



Avhicli I have given in my Memoir on the Resolution of Algebraic 



Equations.* 



With respect to the inverse function in this particular example, 

 we haA'e 



and putting for x successively 1, 2, 3 ...a; we get 



It-. = \/{ff + U's/{a- + 2h^{a- + ... + 2hy/{(t' + 9.hu,)\, 



tlie number of roots being x; and for the value of this successive 

 function we have 



-' "y + ^y ^ 2*7(7 -1) + 2*7(7 - l)(y- 1) ^''• 



Thus to find the value of ■^{20 + ^{20 + -^/{20 + ...} *• times, 

 we have a" = 20, 2i = 1, 7 = 10, 



.«--!+ 10' +2.9.IO-+2.9.99.IO- ^'^■ 



and B is very readily found by putting .r = 1, and thence the required 

 value of ?/_, is obtained in a series extremely convergent. 



As another numerical example, let ?/^+, = 2?// - 1. 

 Here 7 = 4, Z<7 = 1, and therefore 



u 



:, = 1 + ^ . 2^' + ;r77^r^^-rT + 



2(2= - 1) 2 (a'' - 1)(2' - 1) 



9^^2-' 



^ 2{T - 1)(2' - 1)(2^' - 1) "^ 



for li write , and this series becomes 



1 . 2 



* Trans, of Camb. Phil. Soc. Vol iv. p. 144. 

 n2 



