210 Sir William R. Hamilton on the Argument of Abel, 



c, retaining its recent meaning ; so that in each case it would be superfluous to 

 perform any new extraction of a cube-root or other radical in order to calculate 

 a.2', after 0/ and a/' had been calculated ; and consequently no such other 

 radical a/' of the second order can enter into the composition of the irreducible 



function h . If then that function be itself of the second order, it must be 

 capable of being put under the form 



Ki ^I'i K' being functions of the forms 



V = (V)o+(V)i<. 



in which the radicals a/ and a/' have the forms lately found, and (b^\, . . . (5/), 

 are rational functions of a,, a^, O3. And on the same supposition, the three roots 

 s^, s^, X3, of that equation must, in some arrangement or other, be represented 

 by the three expressions, 



^p = K"=w+hW<+p^'b2ar, 



P3 retaining here its recent value : which expressions reciprocally will be true, if 

 the following relations, 



can be made to hold good, by any suitable arrangement of the roots s , a: , a: , 



and by any suitable selection of those rational functions of a,, a^, a^, which have 

 hitherto l)een left undetermined. Now, for this purpose it is necessary and 

 sufficient that the arrangement of the roots x , ^ , x , should coincide with one 



or other of the three following arrangements, namely ar^, x^, X3, or x^, x^, .tr,. 



