S'7® Mr. Wilson's Essay on the 
introduced a quadratic surd. Before therefore we can proceed 
to employ the third equation to determine (c), it must be squared 
to clear it of that surd, and of course will then rise to the 6 th 
degree. The solution of such a dimension (if admitted for the 
present to be equally possible) must introduce higher radicals ; 
and, by the intrusion of these superfluous roots at every stage, 
our labour increases, instead of diminishing. This is the diffi- 
culty alluded to before ; and, as we have appropriated already all 
our subordinate equations, we have nothing to oppose it. It 
therefore seems hopeless, to expect to make any general im- 
pression upon indeterminate equations, without more help, be- 
yond the mere knowledge of the constitution of the coefficients. 
11. This difficulty, however, is wholly removed by the least 
circumstance that discloses any particular relation amongst the 
coefficients of an equation, independent of the general law of 
their construction. This, of course, whenever it occurs, fur- 
nishes new conditions and means of comparing the terms. 
Every particularity in the coefficients that gives specific varieties 
to the forms of equations, must, from the nature of their con- 
struction, have its source in some particular relation between 
two or more of the roots, and therefore, as far as that relation 
extends, detects them infallibly. The observation of the forms 
and relations of the coefficients under different species of equa- 
tions, and the correspondent inferences to be drawn, as to the 
connection of their roots, would form a curious and very useful 
part of a complete treatise upon the whole doctrine of equations, 
which is a work much wanted. The most striking of these rela- 
tions will be obvious, or familiar, to the reader who has at all 
considered the nature of the subject; such as, that equations 
deficient in every alternate term arise from pairs of equal roots 
