574 MR TALBOT ON SOME MATHEMATICAL RESEARCHES. 



S x m , m being any whole number, in terms of the given coefficients p, q ) r, &c. 

 (Wood's Algebra, sixth edition, p. 192) ; and from these the values of binary 

 compounds like S x m y m , and ternary like S x m y m z m can be computed, provided 

 each root has the same index m. 



I shall now return to cubic equations, which are the more immediate subject 

 of this paper. The value of Cardan's rule, when it can be applied, consists in 

 this, that it gives an accurate result by a direct process, without guesses or ten- 

 tative trials. Its inconvenience is, that the calculation is often very prolix, 

 requiring two extractions of the cube root, although the root may be a whole 

 number. Thus, in the example chosen by Wood {Algebra, p. 172), viz., x 3 + 6x 

 — 20 = 0, it is necessary first to extract the cube root of 10 + a/108, and then 

 that of 10 — a/108, and to add these partial results together; which being done, 

 their sum is found to be 2 ; not, however, without some rather refined argu- 

 ments (see p. 132) to prove that it is exactly 2. Now, from the mere inspec- 

 tion of the equation a? + 6 x — 20 = 0, an arithmetician would not be long in 

 perceiving that he could solve it by supposing x = 2, since 8 + 12 = 20 ; a 

 process so much shorter, that it is worth while to explain why it must be dis- 

 allowed. It was long ago perceived that if one of the roots of an equation was a 

 whole number, it would necessarily be found among the divisors of the last term ; 

 and could, therefore, with more or less trouble, be found. But though this is no 

 doubt the fact, still it cannot be admitted among the scientific modes of solving 

 the equation. For, though it succeeds perfectly in an easy equation like the last, 

 in which the last term 20 has only 2 and 5 for its prime divisors, yet in other 

 equations the number of divisors may be so great that it would be nearly im- 

 possible to try them. For example, suppose the last term of a cubic to be 30 100 , 

 and that the roots are known to be whole numbers from the nature of the question 

 which produced them. Then, since 30 is the product of the primes 2, 3, 5, it is 

 certain that each root is a number of the form 2" 3 6 5 C . But how many trials 

 would it not require before one of the roots, which we will suppose, for instance, 

 to be 2 37 . 3 51 . 5 19 , would be hit upon? 



For this reason, tentative processes are regarded as of doubtful value. A 

 direct and unerring process, however long, is required, if the solution is to be 

 regarded as a scientific one. Now, it will readily be conceded, that if ever an 

 accurate solution is effected of equations of the 5th and higher degrees, the value 

 of their roots will be expressed by radicals of great complication. 



Nevertheless, this will not be considered to detract from the merits of the solu- 

 tion. The importance of such a problem is purely theoretical ; and, therefore, 

 provided only that the process be direct and unerring, its length has nothing 

 whatever to do with the question ; it is not intended to be used in practice, but 

 is merely a speculation of the mind. Something similar to this is seen in the 

 famous theorem called Wilson's Theorem, which gives a direct and certain 



