Anglin — Mathematical Notes. 291 



Eepeating this process, we shall find by the application of (I) — 



x^ = h-i.x? + {qho + rhi + s) .XT + {rli-i + sJi-^ .x + s.h^, 



where A„ is the sum of the homogeneous products of roots of 

 xi^ = pa? + qx^ -\- TX -^ s of n dimensions. 



Finally, it is shown by the process of Mathematical Induction, 

 and the repeated application of (I), that 



a;" = 7^„_3 . X? + {jiK-i + r/^„-5 + s . 7i„_6) . X' 



+ (r . A„_4 + s.7z„_5) .a; + s.h„_i. 



The general case is established in a similar way — 



X"^ = 2h • X"^'^ +'P2.X''^~^ + . . . -^ Pm- 



Multiplying both sides of this equation by x, and arranging the 

 terms, we shall find 



a;"' + l = Jl2 ' X"'~'^ + iP2 h +P3) • *''"'^ + . . . + {P,n-1 . h +Pm) • « 



+ p,„.hi' 

 Eepeating this process, we shall find by the application of (I), 

 a;'»+2 = hs.x'"-'^ + {po7i2 +pzhi +pi).x"'-' + . . . 



+ {Pm-l ■ h + Pm • h) 'X + i?,„ . ^2- 



Finally, it is shown by the process of Mathematical Induction, 

 and the repeated application of (I), that — 



If a;™ = pi . x'"^'^ + P2 . x"'-~ + . . . + p,n, then 



X"- {n > m) = K-,n + i . X'"-'^ + {p. . h„_,„ + p^ . /«„_',n_i +Pi . K-m-2 + • • 



+ {PZ • K-m -^Pi • K-,n-l + . . • +P,n ' K-im + z) • ^'"'^ 



+ + iPm~l ■ K-m + Pm • K-m-l) ' ^ + p„, . h„.„„ 



where h,i is the sum of the homogeneous products of roots of 



X'" = Px . «"'"^ +P2,X"'-^ + . . . +p,n, 



of n dimensions. 



Z2 



