348 Sir William Rowan Hamilton on Equations of the Fifth Degree. 



as a function of the same five roots, can be decomposed into more than three ra- 

 tional factors, nor that any of these can be depressed below the fifth dimension. 



20. Confining ourselves then for the present to the consideration of h^, we 

 have, by (42) and (38), the following expression for the square-root of that 

 quantity : 



/h, = \{>^- X') {^' - ^kPKk — P (/.' + X') } ; (60) 



and, therefore, by (59), and by the same relations between *-, \, and e, rj, i, which 

 were used in deducing the formula of the sixteenth article, we obtain the follow- 

 ing expression for the quantity h^ itself, considered as a function of a, e, rj, i: 



H, = 2'»5'« { {rf - e^ + Secf - 4i'} l' ; (6l ) 



in which we have made, for abridgment, 



L = /t' - Sifiu-" + (ri" -e' + del) i^, (62) 



and 



;x = (-5+f)(5a^ + f^)+(ll + ^)ae,. = 4(2+f)«. (63) 



Now, without yet entering on the actual process of substituting, in the expression 

 (61), the values (16) for a, e, rj, t; or of otherwise eliminating those four quan- 

 tities by means of the equations (15), in order to express h^ as a function of or', 

 p, q, r, from which j/ is afterwards to be eliminated, as far as possible, by the 

 equation of the fifth degree ; we see that, in agreement with the remarks made 

 in the last article, this expression (61) contains (besides its numerical coefficient) 

 one factor, namely, 



(^2_e3_l-3et)2_4t^= (;r3_V)^ (64) 



which is of the twelfth dimension ; and another, namely, l*, which is indeed it- 

 self of the eighteenth, but is the square of a function (62), which is only of the 

 ninth dimension : because a, e, i], i, are to be considered as being respectively of 

 the first, second, third, and fourth dimensions ; and, therefore, fi is to be re- 

 garded as being of the third, and v of the first dimension. 



21. Again, on examining the factor (64), we see that it is the square of 

 another function, namely, a-' — X^ which is itself of the sixth dimension, and 

 is rational with respect to y, x'", x'^, x'^, though not with respect to a, e, t], i, 

 nor with respect to x\ p, q, r. This function k^ — X^ may even be decomposed 

 into six linear factors ; for first, we have, by ( 11 ), 



