ESSAY-REVIEWS 651 



intersection with the curve of the tangent at the point P. Similarly, we can 

 proceed with the point Q ; or, again, we might take the third point of inter- 

 section of the line PQ and the curve. Hence, in general, an infinite number 

 of rational solutions can be found from one known one. 



The analytical interpretation is also very simple. The co-ordinates of 

 any point on the curve can be written as 



X =p{u),y =p'{u) 



in the usual notation of elliptic functions. But the point {p{nu), p'{nu)) 

 is also on the curve, and if •w is an integer, p{nu) and p'{nu) can be expressed 

 rationally in terms of p{u) and p'{u), giving in general an infinite number 

 of solutions corresponding to « = i, 2, 3 . . , . 



The question, of course, arises, " Can we find thus all the rational solutions 

 from an initial one ? " For some equations this can be done, as was proved 

 by Sylvester, but it does not appear to be true in general. In any case, 

 how can we find an initial solution, let alone a general one ? The prospect 

 of any immediate solution of such problems appears almost as remote now 

 as that of discovering any knowledge concerning the chemical constitution 

 of the stars must have appeared, say, in 1800. Perhaps our difl&culty is 

 comparable with that of the persons who, in the Middle Ages, tried to solve 



x^ ■\- y^ = z^ 

 by means of parameters. 



Some account of indeterminate equations of the types just mentioned, 

 was usually given in Algebras pubUshed some 125 years ago. The classical 

 example is the Algebra of Euler, to whom Diophantine Analysis is so much 

 indebted. The subject seems to have gone out of fashion ; perhaps because 

 many of its devotees have overlooked its connection with the Theory of 

 Numbers, and have concerned themselves with particular solutions of isolated 

 problems from which little advance in knowledge could be expected. This 

 is to be regretted, as few things are more productive of astonishment, and 

 more stimulating to the student, than to be brought quickly and intelligently 

 to an impenetrable mathematical frontier. 



The reader will now see, not only interesting results by Sylvester, Pepin, 

 and Lucas, but also that the general rational solution of the ternary 

 cubic / {x, y, z) =0 requires a knowledge of only one solution, and the 

 rational solution of 



y^ = ^x'' - g^x - g„ 



where g^ and g^ are numerical multiples of the fundamental invariants of the 

 cubic. Further, despite an assertion by Fermat that seems almost to the 

 contrary, the equation 



yi = x^ + b 



has at most a finite number of integer solutions if A is given. {The same 

 theorem appHes to the equation 



ey' = ax^ + bx^ -{- ex -{• d 



where a, h, c, d, e are given, and the right-hand side has no squared factors in x.) 

 For equations of higher degrees, the difficulties are considerably greater. 

 Thus, for 



y% ^ ax^ -^ bx^ -\- . . . 



then, even if one rational solution is given, no method is known whereby 

 from it we can find another. Nevertheless, there are some very noteworthy 

 results for equations of higher degrees, such as the one by Thue, that if 

 / {x, y) is an irreducible binary quantic of degree greater than the second, 

 then the equation 



