342 The Rev. B. Bronwin on the Algebraic 



To which we may add, 



= ^1 +.r2 + a?3 + ^4 + ^5- 



By eliminating .Tp Xc^, &c. from these, we find 



3a5=— 2a4— 2cf3 + a2 + «iJ 3^6=— 2a4 + ag— SflTg + ^u 

 3fl7=«4— 2^3— 2a2 + «iJ 3«8= — 2a4 + fl'3 + «2~2^i> 

 3a9=C4— 2«3 + fl2~2ai, 3^10=^4+ ^3~2a2—2ai, 



which may be verified by putting for a„ «2, &c. their values 

 in a-'i, .^2, &c. Therefore six of the roots «i, ag, &c. are linear 

 functions of the remaining four, and the equation in a of the 

 tenth degree is reducible to one of the fourth. 



We also find 



^l=-3 (^l + '^2 + «3 + «4)» •^•2=-3-(2«l — «2-^3 — «4)> I 

 ^3=3- (-«l + 2Cf2-«3-«4)» •^4=3-(-«l-% + 2^3-«4)> K^O 



^^5= -(~a,-fl2-«3+2«4)- 



Now let the reduced equation in a be 



a'^ + mc^-\-tia'^+pa + r=0, .... (4'.) 

 the roots of which are «i, «2» ^3> ^4? ^"^ therefore 



— »2 = 2(«x), « = S(a?i«2)> — J^=^(^i«2^3)j r—a^acfi^a^. 

 Consequently, — m = 3a?i by (3.), 



w = 6a?2 + 2.ri (372 + ^3 + ^4 + ^5) + '^[Xy':^ — ^x\ + '^{XyX^ 



because S(^i) = 0; 



—7; = ^x\ + 2.r2 (a;2 + .^3 + X4 + 075) + ^i2 (a^i^-g) + 2 (a^i^a-^s) 



2 1 



= 2^3 + ^iSCj^-ja^g) -\-X{x^x^^ — — — n^— — mk—V>\ 



r = x\-\- x''^{x^x^ + 2(^13^2^3.^4) = — »i4 + _ ^^2^ ^ c^ 



Hence w, 7?, and /- are given in terms of 'm, and m=^ —Zx^ can 

 only be found by solving (1.); or the resolution of the pro- 

 posed into factors, one of which is of the second degree, de- 

 pends upon the solution of the proposed itself. 



We may introduce fifth roots if we please; thus, let 



A'*+^^3-f^X^ + ^X + /=0, .... (5.) 



