356 Mr. J. Cockle on the Theory of Equations 



9. Since B,( = Ci = S .y) must be equated to zero, we have 



again, 



and, eliminating pc^, there results 



yr = Xr"^ - ^Xx^ + [Xr - ytx)p^ ', 



consequently, any one of the five equations of arc. 8 will enable 

 us to express jOj as a rational function of the roots of the equa- 

 tion in X. 



10. Add 0=X(y. + ?/^ + ?/.^ + j/s + j/.) 

 to 



divide the sum by /i« + X, and designate '" by Vr; then 



Now, in order that <1> may be diflFerent from /iaSy, X must 

 admit of being determined so as to make one at least of the v's 

 vanish without any of the others becoming infinite. 



First, let v^ = 0; 



then, expressing y in terms of t, u and v, there will arise 



where 



and if 



H'oL—H; 



N„=:l + i% + f3vy + i2»/4, 

 N„=l+i% + iVy + 2Vi; 



N,=0, N„ = 0, N„=0, 



the expression for will in vanishing introduce no relation 



among t, u and v. 



The last three equations will be satisfied provided 



v^ = l + i^ Vy=—i—r and vj=— t^ 

 Whenever it is necessary to distinguish this from the other 

 corresponding sets of equations, I change v into ^v. 

 Secondly, let 



v«=0. 



We need not retrace our steps. The change of i into i^ produces 

 the same effect on the system (a) as the substitution of y, e, /3, 8 



