VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 537 



of distinct characters to be determined will proportionally be diminished : and as 

 the number of subordinate equations furnished by the co-efficients remains always 

 the same, while the dimension of the proposed equation is unaltered, more of them 

 may be used together to discover the related roots, and their investigation be pro- 

 portionably facilitated. This single observation, in the hands of a skilful analyst, is 

 sufficient for the reduction, if not the solution, of any particular numeral equation 

 whatever, and the more so the larger its dimension : for, from the endless variety 

 of relations numbers bear to each other, hardly any set of them can occur, as the 

 co-efficients of an equation, or perhaps exist, that, on being compared, do not ex- 

 hibit some peculiarity, of greater or less extent, sufficient to afford a clue to the cor- 

 respondent relation in their roots. And if no such clue is immediately given by the 

 equation itself, taking the equation of the differences or sums in pairs, or of the 

 squares, &c. of the roots, will soon find one. But, as peculiarities of that sort, 

 though never so frequent, may be deemed always accidental, and evidently no gene- 

 ral method can be founded on them, even where the co- efficients are given, it may 

 be asked, how any use can be made of them in cases of indeterminate equations ? 



14. To this I answer, that there are some properties of quantities that depend 

 only on the index of the equation, without any regard to the value of its co-effici- 

 ents ; or, in other words, there are some peculiar properties which merely depend 

 on the number of any set of quantities, abstracted from all consideration of their 

 nature and values. For example, 2 quantities a and b have their differences the 

 same quantity a — b, only taken both affirmatively and negatively, a — b and b — a; 

 when squared, these differences become equal ; a 2 — lab + b 2 is the square of 

 both : therefore, let the quantities themselves be chosen as they may, the equation 

 of the squares of their differences must have both equal roots, and consequently be 

 reducible by the reasoning in art. 11, 12, and 13. Again, 3 quantities, however 

 distinct in themselves, give a set of differences marked with a peculiar relation, any 

 1 of them being equal to the 3d ; a, b, c, being 3 quantities, (a — b) + (b — c) 

 = a — c. Also, if the 3 quantities be so chosen originally as to have their sum 

 equal to nothing, one of them must necessarily be equal in magnitude to the sum of 

 the remaining two ; and therefore, whether taken simply or- summed in pairs, their 

 relative magnitudes must remain the same. Again, 4 quantities, of any sort what- 

 ever, may be pursued to a constant relation, though somewhat , more remote, and 

 grounded on very different causes ; viz. a, b, c, d, being 4 quantities, from their 

 combinations by pairs, ab, ac, ad, be, bd, cd, 6 in number, added together two by 

 two, thus, ab -f- cd, ac + bd, ad -f- be, 3 quantities are formed, sufficiently dis- 

 tinguished from the group of similar combinations to be found separately, as will be 

 shown hereafter. And also, if the 4 quantities are originally so taken as to have 

 their sum equal to nothing, their sums in pairs, though 6 in number, will be re- 

 duced to 3 in effect; for, if a + b + c -j- d = O, by transposition, a + £ = — c — d, 

 a -\- c =■ — b — d, a-\-d= — b — c, i. e. 3 of the 6 must be merely the nega- 

 tives of the other 3 ; which relation, if they are squared, will become equality, so 



VOL. XVIII. 3 Z 



