30 Sir W. R. Hamilton's Theorem connected isoith the 



And from these equations (of which some occurred in the 

 investio-ation of the former theorem, but are now for greater 

 clearness repeated,) we see that we must have 



/i. = 0, //2 = 0, h^ = 0, 7/4 = 0, (29.) 



and therefore, by (9.), 



/(xi) - qx^, f{xc,) = <Z^2» /(^a) = ?^3> 



/K) = 9^4>f{^5) = (Z^ss (4-5.) 



unless we have either 



^1 + ^4 = (65.) 



or else 



(a:, +^4) {x,-x^f + {xc, + Xs) {x^-x^y = 0, (30.) 



or at least some one of those other relations into which (65.) 

 and (30.) may be changed, by changing the arrangement of 

 the roots of the proposed equation of the fifth degree. 



The alternative (65.), combined with (57.), gives evidently 



E = 0; (60.) 



but the meaning of the alternative (30.) is a litde less easy to 

 examine, now that we do not suppose the coefficient B to 

 vanish, as we did in the investigation of the former theorem. 

 However, the following process is tolerably simple. We may 

 conceive that x^ x^ .1-3 x^ are roots of a certain biquadratic 



equation, 



x'^^ax^ + bx''+ ex +d = 0, (66.) 



and may express, by means of its coefficients abed, the sym- 

 metric functions of ^1 x^x^x^ which enter into the develop- 

 ment of the product formed by multiplying together the con- 

 dition (30.), and these two other similar conditions, 



(^1 + ^3) (^1-^3)'+ (-^2 + ^4) G'^s-'^J"- = 0, ... (67.) 

 {x, + x^){x,-x^f + {x^ + x;){x^-x,y = 0. ... (68.) 



If we put, for abridgement, 



x.^ + x^^ + x^^ + x^" = f, {69.) 



-X^Xc,{Xi-\-Xc^)-XsX4{x._i + X4) = g, ... (70.) 

 — Xi Xq {Xi +X3) —^2 ^4 (-^2 + -^4) 



-x,X4{x, + X4)-Xc^Xs(xc, + Xs) =h, (71.) 



{X^Xs{Xi+X3)+XciX4 {X^ + X^)] 



{x,X4{xi + x^) + x\x.^{xo + Xs)} = i, ... (72.) 

 the condition (68.) will become 



f+g = 0, (73.) 



