( 83 ) 
the preceeding Coefficient by the Index of e in that 
Part, and dividing the Produd by e and by the Index 
of jv, in the Term whofe Coefficient is required. 
Lemma XI. If any Equation — A a*”"' + 
Ba’*""* — Ca' 6cc. = o be transformed in the 
fame Manner, by fuppofing ^ = j/ — ^orA; = 
^ -1- j/, and confequently. x" =. e Aa; = 
Ax^+J^r \Ba;” * = Bx^+J^r * ^c. The 
transformed Equation will have this Form, the Order 
of the Terms being inverted, . 
^ . 1 
-F «x xe” = o 
a 
fl M 
* ^ 
+ +« — 2X^ — ^xBe*~+j3’2&c. 
2 
_ c e ^ 3 X C ^ ”“■4^ —FZTJ X I’i&c. 
a 
6cc. &c. tScc.^ 
Where it is manifefl:, 
1. That the firfl: Term + B^ ”■"* 
Ce”^^ Sec. is the CLuantity that arifes by fubftituting 
e in the Place of x in the propofed Equation x" • — 
Aa«”“* +Ba’”-^ — 
2 . That the Coefficient of the fecond Term 
n — i X 2 .xB^”~^ — - 
n — 3 X S>ec. is deduced from the preceeding 
-j- B^”"'^— C^”~* dec. by multi- 
plying each of its Parts by the Index of e in that Part, 
and dividing by 3 . That 
