844 The Rev. B. Bronwin on the Algebraic 



There are fifteen ways in which this may be done, and con- 

 sequently the equation in a will be of the fifteenth degree. As 

 before, 



&c., and 



= a?! + ^2 + a73 + 0^4 + a?5 + OTg. 



If we eliminate x^, Xc^ &c. from these sixteen equations, we 

 shall have ten resulting equations between a^, «2» &C'j which 

 will give fl!g, ^7, &c. in terms of the first five of these quantities. 

 The equation of the fifteenth degree is therefore reducible to 

 one of the fifth, or 



a^ 4- ma!^ + nc^ -i-pa^ + qa + r=0, 

 where 



— m=ai4-«2 + «3 + «4 + «5=4)A'i. 



The determination of m then will be the same thing as solving 

 the given equation of the sixth degree. And it is easy to see 

 that we shall arrive at results precisely the same in equations 

 of a still higher degree. 



If we resolve the given equation into the factors x^-^ax^ 

 + bx + c and x^—ax'^+Jlr+g, we shall have 



^j — — •*'| ~T" <^2 "1 ill^g, fltg =S i^j -j- ^2 -j- iJ?^, OCC, 



and the equation in a will be of the twentieth degree. But 

 since 0^= — «i, «i2= ~^'2.i &c., the equation in a^ will be only 

 of the tenth degree. The reduced equation however, whether 

 we find by it a or a^, will be of a higher degree than the fifth. 



Let us now return to (1.), or the equation of the fifth de- 

 gree, in order to find Lagrange's final equation of the sixth 

 degree. 



Make 



^, = Q\ + Q\ + fl*3 + a*4, x^ = aS^i + u^\ + et^a + uH\, 



Whence we find 



5Q\=x^ + u'^x^ + ^% + y'*^4 + 8% 



59^2 = -^1 + «% + ^% + y^^4 + ^-^5 

 59^3 = a?i + a^^g + fi% + /^4 + 1% 

 5d*4 = 0?! + (SMJj + /ScVg + yx^ + dx^f 



where 1, a, /3, 7, 8 are the five roots of unity. If we make 



^ = 01.% y^ct^, 8=«% we have 



