
Summation of Series. 413 
dimenfions refpectively. In demonftrating thefe propofitions 
he ufes one (amongft others) before given by me (a7. if an 
equation of # dimenfions contains m unknown quantities, the 
number of different terms which may be contained in it will 
So Se =) . Inthe Medit. 1770 there is 
2 3 m 
given a method of finding in many cafes the dimenfions of the 
equation, whofe root is x or y, &c.; from which one, if not 
bew+1.. 
both, of the above-mentioned cafes may more eafily be deduced, 
and others added. 15. In the Medit. 1570 is obferved, that 
if there be # equations containing m unknown quantities, 
where z is greater than m, there will be 2—m equations of 
conditions, &c. 16. In the Mifcell. 1s given and demon- 
{trated the fubfequent propofition; wx. if two equations con- 
tain two unknown quantities » and y, in which « and y are 
fimilarly involved; the equation, whofe root is x or y will 
have twice the number of roots which the equation, whofe 
root is x+y, «°+y, &c. has. In the Medit. 1770 the fame 
reafoning is applied to equations, which have two, three, four, 
&c. quantities fimilarly involved. 17. Mr. pe LA Grance has 
done me the honour to demontftrate my method of finding the 
number of affirmative and negative roots contained in a biqua- 
dratic equation. A demonftration of my rule for finding the 
number of affirmative, negative, and impoflible roots contained 
in the equation x”+ Ax"+B=01s alfo omitted, on account of its 
eafe and length. From the Medit. the inveftigation of finding the 
true number of affirmative and negative roots appearsto be as diffi- 
eult-a problem as the finding the true number of impoflible roots 5 
and it further appears, that the common methods in both cafes 
can feldom be depended on. But their faults lie on different fides, 
4 the 
