360 Mr. J. Cockle on the Theory of Equations 



And, importing into these results the relations of VI. (art. 13), 

 we find that, of the thirty binary interchanges which can be ap- 

 plied to 0, twenty-four are reducible to single interchanges. 



18. Of the remaining six binary interchanges four are simi- 

 larly reducible. For from = 61 we obtain 



and the interchanges on the left, being independent, may be 

 transposed. 



19. Lastly, P does not affect 6. 



20. Hence the binary, and of course the higher, interchanges 

 produce no other effect on 6 than the single ones ; so that by 

 single interchanges we can evolve all the values of which 6 is 

 susceptible. 



21. Those values are six only, viz. 



for 



e, «P), «(!f), »(?■?). »('f), e(^), 



«=W' gives »(?^)=«(*f). 



23. Consequently depends upon an equation of the sixth 

 degree, the coefficients of which are symmetric functions of a, b, 

 c and d. 



Scholium. The proposition of art. 16 cannot be applied to 

 the equations 



Z\fi=Z\v and Z/i=Zv, 

 neither of which is a necessary consequence of the other. 



Section III. 

 23. Designating the equations (b) in the order in which they 

 occur by 



re=0, r«=0, ?, = 0, r^ = and ^„=0, 



and omitting the suffix of f„ let us, for convenience, make 



?6 =?=»/« + ^2/13 + cyy + dT/s- 





Assume 



and 



then 



identically, and ^ is indeterminate. Hence, if 



^■^ + b^^ + c^ + d=0, 

 we have ?=?'— !?"> 



