an Algebraical Equation may have a set of Multiple Roots. 535 



The order required is 



6(m-3)(m~4)X, 3 + 9m(m-3)(m~4)X 5 « 



+ (2(m-l)(m-2)+3m 2 )Xa 2 + m(m-l)(m-2)(m-3)(?w~4) a 3 - 



(5) Three pairs of equal roots. 

 The order is of the form 



{m— 3)(m-4)(m — 5) 



which must be identical with 

 (m-3)(m-4)(ro-5) 



/|(\ + ™«) 3 -M(\ + «™) 2 a + ^^ 

 Therefore 



4 S q Hit 2 3 . AT 3 ™(w — l)(m — 2) o 



o 6 



4m 2 a 2 -2Mwa 2 = 0, 

 giving 



M = 2m, 



N _ (m-l)(m-2)-f2m 2 

 3 



The order required is consequently 



|(m-3)(m-4)(m-5)X 3 + 2??z(m-3)(m-4Xm-5U 2 « 



(m— l)(m-2)-f2m 2 w AU . 2 



+ ^-s — (wi — 3) (m — 4)(??i — 5)A,or 



o 



m(??z — l)(m — 2)(m — 3)(m — 4)(m — 5) ^3 

 + : 2~3 



The foregoing process, however, is not sufficient to determine 

 all the terms in higher cases. 



(1) and (2) were given by Dr. Salmon (Quarterly Journal of 

 Mathematics, vol. i. pp. 255 & 256). 



2N2 



