398 REV. T. p. KIRKMAN ON THE THEORY OF 



which is to be true for all values of x. This requires that 

 the coefficients of the three powers of /Q' in this equation 

 written U = o shall be zeros. And by these three con- 

 ditions we can determine p, c and s in terms of q and r. 

 Wherefore it is evident that 



tp- TO- -^ -JIT p 



when r>i; and if W^.— W^, the same thing is demon- 

 strated by beginning with i ^ -8* + i ; and if Wq=.'¥^, we 

 still obtain three equations {a){b)[c), by which we can 

 determine p, c ?cadi s. 



22. It is necessary, in the next place, to demonstrate 

 that 



that is, that 



j /3-(-+^)((i-g) + g) g ^\ip,i + c, N[ 



_(pi + c 'M](^-^''+''\i-k)+k k N 



i ^Nj I ^%x-k)+k ' N ' A: 

 and that^i, c^ and q are given in terms oi p, c and k. 

 'heti = l3''{i-k)+k. 

 The effect of the substitution V^^Wf. is to change i into 



and then to change this into 



p{^-^'^+''\i-k)+k)+c. 

 Let 



p,{^-^\i -k)+k)+c, = /3~~{i-q)+q, {a) 



which can be satisfied by a value of z, whatever be a\ 



The effect of the substitution W^V,,^ is to change i into 



;3-(~~+l)(l_g)+g; 



SO that 



pll3-(-+^)(^i-k) + k)+C = (3-^'^'\i-q)+q. {b) 

 The equations (a) {b) give 

 {p,mi-k)+Ic)+c,-q}{^p{^-^'^'+\i-k)+k) +^{c-q)} 



which is to be true for all values of x. This requires that 

 the coefficients of the three powers )8' should be zeros, 



