416 Mr green, ON THE DETERMINATION OF THE 



It will now be easy to obtain the value of V corresponding to 



without integrating the formula (1) No 1, where F is the character- 

 istic of any rational and entire function. In fact it is easy to see from 

 what precedes (No. 9), that we may always expand JF' in a finite series 

 of the form 



F{xl, x-l xl) = bo^o + ii0i' + bo(p2 + 63^3' + &c. 



after a;/, x-J, &c. have been replaced with their values (7). Hence, we 

 immediately get 



p' = „"-«-' . {bo(po' + b,<p! + h(p; + &c.} (29). 



By comparing the formulae (26) and (27) it is clear that any term, 

 as 5,0/ for instance, of the series entering into p, will have for cor- 

 responding term in the required value of V, the quantity 



^ ^ i^„'«/< a:.b.<pM.f-j j^^'"/^ ^ (30): 



''co -'^O "1 Ms (Is 



Ha being a particular value of H satisfying the equation (17), and 

 immediately deducible from (p by the method before explained. 



12. AU that now remains, is to demonstrate that 



V'V0 = VV> (31), 



whatever <p may be. For this purpose let us here resume the value 

 of A0, as immediately deduced from the equation (2") No. 7, viz. 



+ A^2^-A^2lx2i^ (32), 



P /w-1' 



