118 Theory of Derivative Points of Curves of the Third Degree. 



the geometry of the question, the quantities belonging to the «th 

 derivative being in fact a known algebraical function of n ! I 

 was led to the discovery of this surprising and unique law by a 

 statement of a friend, not since verified, and which, for aught that 

 has yet been shown, may or may not be true, that the number 5 

 could be divided into two rational cubes : assuming this to be 

 the fact, it necessitated (by virtue of my investigations) the coin- 

 cidence to a factor pres of two functions obtained by apparently 

 independent algebraical processes, which coincidence by actual 

 comparison of the functions I found to obtain. 



AYith reference to the connexion of this theory of derivation 

 with the arithmetic of equations of the third degree between 

 three variables with integer coefficients, it is after this kind. 

 Fermat has taught us that a certain class of such equations, viz. 

 the equation a^-\-y^ + ^-=Q, is absolutely insoluble in integers 

 (abstraction made of the trivial solutions of the type a? = 0, 

 y + z ■=()). I have greatly multiplied the classes of such known 

 insoluble equations, as may be seen by a communication from 

 me to Tortolini's Annali in 1856. But over and above such 

 equations I have ascertained the existence of a large class of 

 equations, soluble, or possibly so, it is true, but enjoying the 

 property that all their solutions in integers, when they exist, 

 are monobasic ; that is to say, all their solutions are known func- 

 tions of one of them, which I term the base, and which is charac- 

 terized by this propertj', — that of all the solutions possible it is 

 the one for which the greatest of the three variables is the smallest 

 number possible. If this solution be laid down as a point in the 

 curve corresponding to the given cubic, all the other solutions 

 possible in integers \n\\ be represented by points in this curve, 

 which are derivatives (in the sense previously employed in this 

 note) to the given point, having coordinates respectively of the 

 4th, 9th, 16th, &c. degrees, in respect of the coordinates of the 

 basic point*. 



If my memory serves me truly, I have found (as a particular 

 case) that all cubic equations in numbers of the form 



sfi-\-'^^ + z^^= imxyz, 



where i is 1 or 3 or 6 (I cannot at the moment remember which), 

 are either insoluble or monobasic. The case of im = S must of 

 course be exceptional, being satisfied by x + y + g=0. This 



* This theorem is analogous to that relating to the integer solutions of 

 a;* — Ay'=l, in so far as there is a basic solution to this equation in inte- 

 gers of which all the other solutions are derivatives, and not more than 

 one such derivative exists of any given degree, but with the diflferenoe that 

 there does exist one of every degree, and not merely (as in my theor^in for 

 cubic forms) of every square degree. 



