5^6 Mr. G. B. Jerrard's Reflections on the Resolution 



where x^. x^^ . . x^ are the roots o^ x^ -[■ A^x'^ + . . = ; y„ 

 2/2> "I/s^^y^ + ^\l/^ + • • = 9 ; and where g^, q.^, . . q^, y^, 

 Va • • Vb are all of them expressible as known functions of ^i, 



Can then ^j, P^^Pq be found? To bring this question to a 

 decision, we must enter upon an inquiry of great length and 

 intricacy. But it is in the nature of such discussions to un- 

 fold in their pi'ogress theories which bear upon remote and 

 seemingly unconnected departments of knowledge: and of this 

 truth we shall find in what follows some remarkable exem- 

 plifications. 



Section I. 



1. Now Pj, jo2» P3 must be such that x^ + PiX^ -\- p^x + p^ 

 may become a root of an equation of the form 



y^+A\y^ + ~A\^y + A\ = 0; 

 or that the expressions 



^1^ +Pi^i^ +P^2^■^ +Psi 

 x^ + Px xi + i'a'^g + Pa> 



may become the five roots of that equation. For r^, x^s . . x^ 

 enter symmetrically into the calculus, and there is consequently 

 nothing to connect one of them rather than another with the 

 X of the expression x^ + /», x~ -^ p^x + p^. 



If then we consider that the roots of the equation for y 

 must, as De Moivre has shown, be generally expressible by 



t + u, */+»"* u, «2 / + <^ w, 4=^ t + t^u, I'^t + I w, 

 I, i% »^, i^ denoting the imaginary roots of the binomial equa- 

 tion p^— 1 = 0, we shall be conducted to a system of equa- 

 tions 



^/ + Px x^ ^■p^3c^^-p3-t-\- II, 



■ ^/ + Pi ^H^ + ^2 ^/} + i?3 = '^ + ^'^^h 



Xy + Px !^y + PciXy + Jp3 — <^^ + 1^?/, \ . . (a.) 



^f + Px xf + pci_xi + Pq = i^t+i^ w, 



^i^ + Pi x^ + PciXt+P^ = i^t+i u; .^ 

 in which a, j3, 7, 8, s represent, in an undetermined or arbi- 

 trary order of succession, the five indices 1, 2, 3, 4, 5. 



2. From this system there will arise, as we know from the 

 theory of permutations, 1 . 2 . 3 . 4< . 5 systems ; if instead of 



