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



(1, 2, 3), (2, 3, 1), (3, 1, 2), (2, 1, 3), (3, 2, 1), (1, 3, 2), 

 or, In a more developed notation, 



corresponding to the six different possible arrangements of the three quantities 

 on which it is supposed to depend ; and that these six values of the function f 

 may also be considered as six different but syntypical functions of the same three 

 arbitrary quantities x^,x^,x^, taken in some determined order; which functions 

 may be thus denoted, 



F, (Xj, x^, xj, . . . F^{x^, x^, xj, 



or, more concisely, 



(1,2,3)„...(1,2,3)«. 



For example, the six following values, 



ax^ + bx, + car, = (1, 2, 3) = f(^,, x^, xJ, 



ax, + bx^ + cx^ = (2, 3, 1) = f (x,, x^, x^), 



ax^ + bx^ + cx^ = (3, 1, 2) = f(x^, o:,, orj, 



ax, + bx^ + cx^ = (2, 1, 3) = f(x,, x^y or,), 



ax, + bx, + cx^ = (3, 2, 1) = f(^,, x,, x^, 



ax^ + bx^ + ex, = (1, 3, 2) = F(arp x^, x,), 



of the original or typical function 



may be considered as being six syntypical functions, Fj, f^, F3, F4, f^, Fg, of the 

 three quantities x^, x,, x^. Such also are the six following, 



£l._l_ ^_L_ —A- £i_l_ £l_J_ — I 



Xi "r^3' 3:3+-^'' xi +-^2' X, "T"^'' Xi "T"^'' 0:3 "T*^*' 



which are the values of the function ~-\- x . 



? ' . 

 Now, in general, six such syntypical functions of three arbitrary quantities 



are all unequal among themselves ; nor can any ratio or other relation between 



them be assigned, (except that very relation which constitutes them syntypical,) 



