EQUATIONS IN FINITE DIFFERENCES. 97 



II Wlien the equation u.^, = ^{u.) is of the «- degree, the enua- 

 t.on for finchng A is of the same degree, and therefore \4 has . 

 values; then y = ^' {A) has also n corresponding values, which, being 

 lepresented by ^„ y,, ... ^„, and putting for abridgment F{By\ for 

 the series above found for «„ we have v 7 ; roi 



«x - F{B,y,') = 0, >,^ - F{B.,y.f) = 0, &c. ; 

 and the complete solution is found by taking the product of the 

 members on the left side and equating to zero: the result will only 

 contam one arbitrary constant, since ^„ B., &c. are all found in terms 

 or u„, as before shewn. 



III. «„ is a known function of B, n. is the same function of By. 



IV. Since 0^0...... {x times} («„) = «^^„; 



•■• ^^0 {^ times} (m_,) = «„. 



Let ./)-■ be the function which is inverse to rf>, that is such that 

 <^-'0(«) = «, then it follows that ""^ 



The same formula therefore which represents the x-^ successive direct 

 funcion of «„, will also give the x^- successive inverse function by 

 merely writing - .r in place of x. ^ 



^'- P"t 7' = ^, or .• = -^, hence 



^ log. y 



•^•^^ { - 1^ times}(«„) =A + l-,c, + c^ + c,, &c., 



which is a known numerical quantity and may be represented by a^; 

 ^^"^^' ^'Pi^ {1^ times} («„) = ,,„. 



Thus the number of times it is necessary to take the successive 



functions cp of ./„ to arrive at «„ as a result, determines j^gi^^ 3^^ 



since 7 is known, B may be thus also determined. °^' ^ 



Vol. VI. Paht I. N 



