534 Mr. S. Roberts on the Order of the Conditions that 

 Therefore 



M=2m; 



and the required order is 



2(m — 2){m—3)\(\ + mci)-\ - — - . -or. 



(3) Four equal roots. 

 The order is of the form 



(m-3){4\ 3 + MX« 2 + N\« 2 + m. (m-l)(m-2)a 3 } 



which must be the same as 



{m-3) \4{\-\-ctm) 3 — M{\ + am)u 



+ N(M-aw)a 9 — m(m — l)(m — 2)a 3 }. 

 Therefore 



4?s¥-MfflV + Nm« 3 = 2m(m — 1) (m — 2)a 3 , 

 4.3m 2 a 2 -2Mm« 2 = 0, 

 giving 



M = 2.3m, 



N=2(m-l)(m-2)+2m 2 . 



The order required is consequently 



4(m-3)X 3 + 6m(?7i-3)Xa+2((m-l)(m-2)+m 2 )\a 2 



+ m(m— 1) (m— 2)(m-3)a 3 . 



(4) A triple root and a double root. 

 The order is of the form 



(m-3)(m-4){3.2X 3 + MX 2 a + NX« 2 + m (m-1) (m-2)« 3 }, 



which must be identical with 



(m-3)(m-4){3.2(X + w2«) 3 -M(\ + «m) 2 « + N(A-j-?/?a)a 2 

 —m(m — l)(m — 2)a 3 }. 



Therefore 



6m 3 a 3 — MmV + N ma 3 — 2m (m — 1 ) (m — 2) « 3 , 



6.3m 2 « 2 -2M™a 2 = 0, 

 giving 



M = 9w, 



N = 2(m-l)(m-2)-f3™ 2 . 



