258 The Rev. B. Bronwin on a 



has fallen into an error in doing this. If, for example, we have 



and we make 



y-^ y -^ Sic 



and differentiate successively until we arrive at 



p being an integer ; let ^ = AX be the first integral, and we 



shall have 



?/ = A Xj + a + a^x + a^x^ . . . . + a^^KP. 



We must by substitution in the proposed determine o, «,, &c., 

 and we shall have only a particular integral. The supernu- 

 merary arbitraries are always to be thus determined. The 

 two solutions we have found will only give particular integrals, 

 as we shall presently see. 



Change y'(w) into /(n) = o(?i— m), and make ?/=7r~'?^j. 

 With this value (1.) becomes 



and 

 or 



and by the first of (2,), 



By i repetitions of this process, we arrive at a solution, if 



p=f[ii-\)+f{n-^) .... +/(«-^) = «'("-»*-^)- 

 Again, make 2/ = 7r„+i?<p and we have 



Or ifA=iJ +/("), 



and 



which leads to a solution if 



i'= -/('')-/(«+ 1) •... -/('H-^'-l), 



