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



that these three last-mentioned functions bear to each other the same ratios as the 

 three cube-roots of unity, 1, p^, p.^ ', we have therefore 



^/3 = y + ?' Pa («. A 7) + r>3^ (a, J8, 7) V 



^y=p' + q'p'('^P,y)+r'pA^P,yy; 



and thus, finally, the six-valued function which has a two- valued cube is found 

 anew to be expressible as follows, 



K A 7) = 3^ K + ft'^/j+Pa^y) ; 



in which the coefficient -^—, is a two-valued function, of the form 



a, b, denoting here some new symmetric functions. 



The theorems obtained incidentally in this last discussion supply us also with 

 another mode of proving that the cube of a three-valued function of three arbi- 

 trary quantities must be Itself three-valued : for if we should suppose 9/^ = ft^/^, 

 and consequently 1/ =i p^y^= p^y^, in which y^ — a + bx^ -\- cx^, and b and o 

 do not both vanish, we should then have relations of the forms 



^a = ;> + ««/a + ry^, 

 ^li=P + 9P3:f/a + ^p3^ya^ 



^y=P+9P3''^a + rp,y^^; 



but these would require that we should have the equation 



^a + P3 ^/3 + /'s^^ = 3 y .y^, 



a condition which it is impossible to fulfil, because the first member has six 

 values, and the second only three. 



[19.] The discussion of the forms of functions of four variables may now be 

 conducted more briefly, than would have been consistent with clearness, if we had 

 not already treated so fully of functions In which the number of the variables is 

 less than four. 



