468 Mr. J. J. Sylvester on the General Solution of 



P denoting a certain system of values of and written in the 

 order of the letters a?, y, 2;, which may always be found by a 

 limited number of trials (provided that the equation admits 

 of any solution). That this is the case is obvious, since we 

 have only to give the Dominant every possible value from the 

 integer next greatest to A* upwards, and combine the values 

 of x^,i/^, Az^ so that none shall ever exceed at each step the 

 cube of such dominant, and we must at last, if there e^/5/ ani/ 

 solution^ arrive at the System of the Least Dominant. 



Now, every system of solution is of one or the other of two 

 characters. Either x and y must be odd and z even, or x 

 and 3/ must be one odd and the other even and z odd. That 

 all three should be odd is inconsistent with the given condi- 

 tions as to A being odd and M even ; and if all three were 

 even, by driving out the common factor we should revert to 

 one or the other of the foregoing cases. 



The systems of solution where z is even may be termed Re- 

 ducible, those where z is odd Irreducible. Let <p denote a 

 certain symbol of transformation hereafter to be explained. 



Then the Reducible systems of the first order may be ex- 

 pressed by 



<pP, f^P, <p^P, ad injinitum ; 



or in general by <p"^ P n^ being absolutely arbitrar}'. I will 

 anticipate by stating that the function <^ involves no variable 

 constants ; that is to say, <p (S) may be found explicitly from 

 S without any reference to the particular equation to which 

 S belongs. Let now \{/ denote another symbol of transforma- 

 tion, also hereafter to be defined, and differing from (p insofar 

 as it does involve as constants the three values of x, y, 2 con- 

 tained in P : then the general representations of Irreducible 

 systems of the first order will be denoted by 4/ (p"'. P. 



It is proper to state here that the symbol -^ is ambiguous ; 

 and \|/ f^iP, when P and n^ are given, will have two values, 

 according to the way in which the terms represented by P 

 are compared with x,y,z in the given equation 



a^ +y + Az^ = Mxyz ; 

 for it is obvious that if <r=flr, y = ^, zz=c satisfies the equation, 

 so likewise will 



xz=.b, y=a, z =c. 



Each however of these values of -^ <f^'P gives a solution of 

 the kind above designated. 



Proceeding in like manner as before, the Reducible system 

 of the second order may be designated by 4>"2 . ■], <p«i . P, the 

 Irreducible by 4/ f "2 . vf; ^«i . P ; and in general every ^possible 



