83 The Rev. Robert Harley 



on 



u, considered as a function of>, satisfies the linear differential 

 equation. 



(-^D-J?.T [Dr . B 



\n -r n-r/ L J 



= ^¥?\^T r J) -n-r- l ) X U • • • • ( A ') 

 where, 12 as before, 



ax 



Now the substitution 



t r, n, b\ 



\ — r, w. — ?', c/ 



changes the algebraic equation (A) into _ 



ay n + bxy r + c = (?>) 



and the same substitution changes the differential equation 

 (A') into 



\ n nj 



a'c n \ n n J 



which, since 



(-l) b 



is equivalent to 



[D ]^=(-)^r r -D + ^-iTr^ r D- m -iT"^ . . . (bo 



v a r c"\ji n J |_ n n A 



in which ?/ is the m th power of any root of the algebraic 

 equation (B). 



It is hardly necessary to remark that the same substitu- 

 tion will change (B) into (A) and (B') into (A'). 



12 Addendum to Mr. Robert Rawson's paper, "On a New Method, &c." 

 (cited in footnote 7). Proceedings of the London Mathematical Society, Vol. 

 IX., 1878, pp. 216-221. • • - 



" On Certain Linear Differential Equations." British Association Report, 

 for 1878, pp. 466-470. 



