28 Sir W. R. Hamilton's Theorem comiected with the 



most slow and cautious manner. It is probably from this 

 cause that the water has come to be overestimated in the case 

 of the alums. But they stand a low red heat without decom- 

 position, if first made quite anhydrous. - 



VIII. Second Theorem of Algebraic Elimination, connected 

 with the Question of the Possibility of resolving, in finite 

 Terms, Equations of the Fifth Degree. By Professor 

 Sir William Rowan Hamilton, Astronomer Royal of 

 Ireland. 



(In continuation of a Communication in the last volume, p. 538.) 



Theorem II. 1 F a- be eliminated between a proposed equa- 

 ■*■ tion of the fifth degree, 



rHA.r4 + Br3 + Cx°- + D:r+E = 0, {55.) 



and an assumed equation, of the form 



^ = Q.r+/(.r), (2.) 



in whichy(.i') denotes any rational function of x, 



„. . Waf' + W'af"+... .,,. 



fix) = ; 7, ; (3.) 



K'.r" + K".r' + ... 



and if the constants of this function be such as to reduce the 

 result of the elimination to the form 



y^ + B'y + D'3/ + E' = 0, (56.) 



independently of Q: then not only must we have 



A = 0, C = 0, (57.) 



so that the proposed equation of the fifth degree must be of 

 the form 



.^5 + B.r3+ D.r + E = 0, (58.) 



but also the function y(.2) must be of the form 



f{x) = qx+{x^ + Bx^ + Tix+Y.).<^{x), ... (59.) 



q being some constant multiplier, and (f) {x) some rational 

 function of x, which does not contain the polynome x^ + B .r^ 

 + D 0.' + E as one of the factors of its denominator ; unless 

 we have either, first, 



E=0, (60.) 



or else, secondly, 



5D = B% (61.) 



or, as the third and only remaining case of exception, 



5^ E'* + r- B E^ (2^ 5^ D-- 32 5- B^ D + 3^ B') 



+ 2*D3(2'»D--2=B2D + B*) = (62.) 



