260 The Rev. B. Bionwin on a 



Make <p=X, and we have a = \" ; and integrating 



X= -ax'^ + bx + c, 

 and therefore 



T,„ = ( i fl;r2 + ^>^ + c j D + m{ax + i) +/(.r) . 



This will give an infinite number of equations, to which the 

 solutions found will apply. But in (3.) make <p = /3x; then 



a=/3D(,6x') = ^'/3f +/3^A". 



Whence by assuming either X or /3 at pleasure, the other may 

 be fount! by integrating a linear equation of the first order ; 

 and thus we shall have an infinite number of integrable equa- 

 tions, the coeflicients being, or not being, integer functions of*'. 

 Now, instead of (2.), let the conditions be 



Th.tt,, =7r„7r,„ +/(»0^> T,„X = X7r,„_i. . . (4'.) 



The first of these by substitution gives as before, and the se- 

 cond f,„ —(pm-i=0; or <p,„=<?5„=ip, a function of .r, without 

 m or n. Also 



<p\' -}- X(5,„ - 6,„_ ,) = 0, Afl,„_ 1 =-<?-', 



x' x' 



L-x=-{m-l)f - +f{x), Q,n = -?«<?- +/(.r)- 

 MakeJ'(m)r=a[m — 7i), and we have 



as before. 



To apply (4.) to the solution of (1.), make u = 7r,n-iU^t and 

 we have successively 



7r,„ 7r„ TTrn^iU^ +pX7r^-itli = X, 7r,„ x„ 7r„_ ,7<, +;"r,„ Xf/j = X, 



T„ir„_iMi +jy>^?'iT-' X, Tm.iTT^e^i +PiXKi = 7r-iX, 

 ;ji=;j-/(»2-l). 

 And therefore 



'r„-i'r„«,=^;i<+»'r,;i,+2 • • • • 'f;;'x, 



if^ji=0, or 

 p=f{m-l)+f{m-2) . . . .+f{m-i) = ai(^m-7i- ^j, 



