1 90 Ofi a discovery in the Tlieory of Numbers. 



Second General Theorem of Transformation, 

 The equation f^oi^-\-^y^-{-h^z^=^^xyz . . . (2.) 

 may always be made to depend upon an equation of the form 



Aw^ -f Btfi + Cwr^ = Tiuim, 

 where ABC=:R3-S3 



D = 3R; 

 and u.v.vo =. some factor o^fx-^gy + hz. 

 R representing K + 6fgh 

 S ... K-3/^A. 



I have not leisure to show the consequences of this theorem 

 ef transformation in connexion with the one first given, but 

 shall content myself with a single numerical example of its 

 applications: x^-{-'y^-{-z^=—Qxyz 



may be made to depend on the equation 



and is therefore insoluble. 



It is moreover apparent that the Determinant of equation 

 (2.) transformed is in general — 27R% and is therefore always 

 a Pure Factorial, and consequently the equation 



f^a^ +g^y^ + h^i^ = ^xyz 

 will be itself insoluble, being convertible into an insoluble form, 

 provided that K + Gfgh is divisible by 9, and provided further 

 that {K + Qfghf — {K — Sfghf belongs to the form m^.Q, 

 where Q is of the form 9«+ 1, and also of one or the other of 

 the two forms ^Ip^'^^^ 4;>^'*>, p being any prime number what- 

 ever. 



Pressing avocations prevent me from entering into further 

 developments or simplifications at this present time. 



It remains for me to state my reasons for putting forward 

 these discoveries in so imperfect a shape. They occurred to 

 me in the course of a rapid tour on the continent, and the 

 results were communicated by me to my illustrious friend M. 

 Sturm in Paris, who kindly undertook to make them known 

 on my part to the Institute. 



Unfortunately, in the heat of invention I got confused about 



27"25 is a Pure Factorial : consequently if the solution be possible, since 

 in this case the transformed must be identical with the given equation, this 

 latter must be capable of being satisfied by making x and y positive or ne- 

 gative units. Upon trial we find that x=.\ i/=.\ z=2 will satisfy the equa- 

 tion. I believe, but have not fully gone through the work of verification, 

 that these are the only possible values (prime to one another) which will 

 satisfy the equation. Should they not be so, my method will infallibly 

 enable me to discover and to give the law for the formation of all the others. 

 Here, then, under any circumstances, is an example, the first on record, 

 of the complete resolution of a numerical equation of the third degree be- 

 tween three variables. 



