particular Differential Equation. 257 



Vihevep^=p—f{m+\). After z transformations, this becomes 



Pi=P-f{m + l)-f{m + 2) .... -f(m + i). 

 Ifp, = 0, we find, making/{m) = a{n — m), 



p = taln — m ■ — I. 



If/J has this vaUie, we have 



1 n m + t m + i— 1 • • • • ",„ -<»» 



and 



Before I make any observations on this, I will proceed to 

 what Mr. Boole calls the conjugate solution. 

 Make u^ti-Hi^, and (1.) becomes 



or 





and 



•^Jh +P^K>'^^i = ^> Pi =P +f{m), 

 which, by operating with 7r,„_, on all the terms, becomes 



or 



Tm-l'Tn?^! +7^i^?<i = T„j_i X. 



Tiierefore, by repeating the operations, 



Pi=P+f{m)+f{m-\) .... +/(w-/ + l). 

 If;>, = 0, and/(/«) as before, we have 



p = ia\vi—n j. 



< a m-i + 1 "m-i + 2 • • • • "nj-i -^ 



By tlie nature of these operations, the integer i is necessa- 

 rily positive. The two solutions obtained are solutions of two 

 different e(|ualioMs, the quantity p being dilferent in the two 

 cases. In the latter of tiiem, when X = 0, the factors ^„,-, + , 

 i"«.-,+2 • • . . Tm-1 niay be omitted ; but in the former, the fiic- 

 ^"•"^ KiXi-x ''m H-2 • • • • Ki^ '"I'st not be omitted. Mr. Boole 



Phil. Mag. S. 3. Vol. 82. No. 215. April i^iS. S 



