470 On the equation j:^+^' + A«^= M^n/^, S^c, 



in number and infinite to the power of infinity in variety of de- 

 gree, above given: this is not strictly true, unless we understand 

 that all systems of solution are considered to beequivalent which 

 differ only in a multiplier common to all three terms of each; 

 that is to say, which may be rendered identical by the expul- 

 sion of a common factor. So that 7nu, m^, my as a system is 

 treated as identical with a, /3, 7, which of course substantially 

 it is ; and it should be remarked that there is nothing to pre- 

 vent the operations denoted by <p and vf/ introducing a com- 

 mon factor into the systems which they serve to generate, and 

 the latter in particular will have a strong tendency so to do. 



I believe that this theorem may be extended with scarcely 

 anymodification to the case where A, instead of being a prime, 

 is any power of the same, and to suppositions still more gene- 

 ral. I believe also that, subject to certain very limited restric- 

 tions, the theorem may prove to apply to the case where the 

 determinant 27 A — M^ becomes negative. 



The peculiarity of this case which distinguishes it from the 

 former, is that it admits of all the three variables x, 3/, z in the 

 equation 



x^ + j/^ + A^ = Ma^z 



having the same sign, which is impossible when the determi- 

 nant is positive; or in other words, the curve of the third 



M 



degree represented by the equation Y^+X^+l=-rT XY 



(in which I call the coefficient of XY the characteristic), 

 which, as long as the quantity last named is less than 3, is 

 a single continuous curve extending on both sides to infinity, 

 as soon as the characteristic becomes equal to 3 assumes to 

 itself an isolated point, the germ of an oval or closed branch, 

 which continues to swell out (always lying apart from the 

 infinite branch) as the characteristic continues indefinitely to 

 increase. 



I ought not to omit to call attention to the fact that the 

 theorem above detailed is always applicable to the case of the 

 equation 



^+y+A23=o, 



when A is awj/ power of a prime number 7iot of the form 

 6i+l; in other words, the above always belongs to the class 

 of equations having Monogenous solutions, which for the sake 

 of brevity may be termed themselves Monogenous Equations *. 



* Thus the equation j:'^+j/^4-A2^=0 alluded to by Legende is Monoge- 

 nous, and the Primitive system of solution is x=l i/=2 z= — 1, from which 

 every other possible solution in Integers may be deduced. 



