550 Mr. G. B. Jerrard's Reflections on the Resolution 



were made to enter into it four at a time in every order of 

 succession, determine the degree of the final equation 

 Xn (Pni Ai, A2, . . A5) = ; Xn representing a rational func- 

 tion. Certain properties of the roots of this equation would 

 also become known. But we shall arrive far more rapidly at 

 the same results, from considering the ten equations in ques- 

 tion ; the remaining eight of which are connected by a re- 

 markable law with those already found. 



8. It is obvious that we should have obtained a different 

 pair of equations if, instead of supposing that v^ = 0, we had 

 equated another of the coefficients v^, v^, vg, v^, to zero. It 

 will not, however, be necessary for us to retrace our steps in 

 order to complete the system. From either of the equations 

 (b.) we may discover all the rest. Thus if we take the first 

 of them, and represent by 



what that equation will become if »' be substituted for j, and 

 the system (a.) remain unaltered ; we shall see that, »' being 

 different from », there will arise a new equation belonging to 

 the system. A difficulty here indeed presents itself. For if 

 we write »" ^' for t and i^^w' for u, it will be evident that we 

 may obtain, corresponding to each of the four expressions 

 a,, a^2, a,3y a,4, five equations of the form in question. We ap- 

 pear therefore at first view to be conducted to twenty and not 

 ten equations in the system. But an examination of the func- 

 tion «/ will, as I proceed to show, point out the relation 



which includes these two conditions, 



9. Reverting to the expression for a,, we see that 



Now a,i considered generally as a rational function of t, 

 may evidently be included under the form 



«, = C4 + C3 » + Cg i^ + Ci j^ + Cq »'* ; 

 where ^4, Cg, . . Cq do not involve ». 

 Hence we find 



l-, = Ci-C3 4-(Co-C2)< + (C4-C,).2 + (<:3-^o)»^+(^2-^4)*^ 



which will be satisfied independently of », if 



Cj ^3=1, Cq <^2^ ^' 

 C4-Ci = 0, Cg— Co=0, Cg — C4 = 0. 



We thus perceive that 



