3 00 
Mr. Wilson’s Essay on the 
sums of the combinations of the roots in pairs, that function of 
the roots had been required, consisting of the sums of these 
combinations, into the forming of which no root enters twice r 
only six out of the whole number of combinations of the kind 
would answer that condition ; and those six would be the same 
three repeated, for (ab -(- cd, and cd -f ab &c .) are the same 
quantities. So that the three quantities (ab -f- cd, ac -f bd, 
ad -j- be,) would be the functions required, and all of the kind 
that can be made. Now, there is no proposition in the theory 
of equations more certain, than that the equation of any regu- 
lar function of the roots may always be found by means of the 
known values of the coefficients.* As there are but three func- 
tions in this case, the resulting equation must consequently be 
a cubic; and, by taking the several combinations of the quan- 
tities (ab -f- cd, ac -f bd, ad -}- be,), we may obtain their 
equation, 
viz. (x 3 — qx'-\- P r — 4^ -p t s- j- 4<7-9 — r 1 = o). 
Therefore, the finding the equation of that function of the roots 
of a biquadratic which arises from its combinations by two sum- 
med in pairs, so however that no root shall occur twice in any 
such sum, reduces the biquadratic to a cubic. 
2 g. Another peculiarity of four quantities is also given in 
Art. 14, i. e. that if taken originally so as to have their sum 
equal to nothing, the six quantities formed from their sums in 
pairs, will be the same three quantities taken both affirmatively 
and negatively. Then we know, by the reasoning in Art. 11, 
the equation of those quantities ( though of the sixth degree ) 
will want every alternate term, or be of a cubic form; accord- 
ingly, the equation of the function of the roots formed by sum- 
* Waking’s Med. A/geb. cap. i. p. i. et infra. 
