of Algebraic Equations of the Fifth Degree, 571 



(H - Bf) (H - H,) (H - S,) "1 



X (H - %(^.)) (S ^ S^(^ ,)) (B - S,(s . )) = * ; / ^^ ^-^ 



Sy, S„, . . S^/Ss) representing rational functions of Py, P^, . . 



42. Comparing the equations (ab,) and (ac), we are hOW 

 conducted to an equation of the form 



in which r is expressive of a rational function ; that ISj we find 



a„ + aio Py + ag P/ + . . + Oq Py" = b^^ + b,o P^^^,^ 



+ ^«^V) + '- + ^«^Vo' 



a^, ai,„ . . ao, b^, bjo, . . bg being symmetric functions of «■,, 



And comparing this equation witli 



V/^ + B, V/' + B2 PyW + . . + B„ = 0, 

 there will result 



P/ =■'•(?/(,.)); 



where 'r will represent a rational function. 



We must also have, since in the theorem (a a.) we are per- 

 mitted to write f(P,f) instead of ^ 



and therefore 



Py-H'er(Py)). 



43. Similarly, on considering that F/v may be expressed as 

 a rational function of Py, and Py/iss) of P///3 A (^l*)) we shall 

 see that 



Py=,R(Py+Py,)==,R(WyO, 



Py = 2R (Py(/3 s) + P/'(/3 .)) = 2^ ( Wy^^.p J 



and thence 



,R(W»=2R(Wy<(/^,,))^ 



^R and ^R representing rational functions. 



* The equation (ac.) must, itt fact, be capable of coinciding with the 

 celebrated eiiuation of the sixth degree, by which Vandermonde and La- 

 grange were stopped in their researches on the 'solution of algebraic equa- 

 tions of the fifth degree. 



