EQUATIONS IN FINITE DIFFERENCES. 105 



Stances; but there is a method at once common to algebraic equations, 

 differential equations, and those of finite differences, which leads from 

 particular solutions to the general: this, however, more properly be- 

 longs to the 



THIRD CLASS OF EQUATIONS. 



The most general form of an equation in finite differences of any 

 order and of any degree is represented by 



<p {x, ?/„ ?<,.+,, u,,^i M^+4 = 0. 



Put II r =f{Y) =/{z), and wherever a; enters, let its value °^' ^ be 



log. 7 



substituted, z consequently entering in a different form from x, the 



transformed function may be represented by 



^{^. /(^). /(7^), /(7^~) /(7"^)l = 0. 



Putting z = 0, F{0, /(O), /(O) /(O)} = 0, 



from whence y(0) is known. 



Differentiating relative to x, and then putting s = 0, the result is 

 manifestly of the form 



F, + F,.f'iO) + F.yf'(0) + F„^,Yf'{0) = 0, 



from whence ^(0) is known in terms of the indeterminate quantity 7, 

 unless Fn = when 7 becomes known, and y'(0) remains the indeter- 

 minate constant. 



The successive differentiations putting s = after each, will determine 

 /"'(O), ,/"'(0), and thence by Maclaurin's Theorem, 



2= 



A^) =/(0) +/'(0) . « +/"(o) . Y72 &c 



7 



K =/(0) +/'(0) . Y +/"(0) . f^ &c. 

 Vol. VI. Part I. O 



