1

118 A Treatise of Algebra,’

* + y +z —aa2 "P > 2 + 2,1 == ^

and ;

whence by transposing y, in the first Equation, and

squaring both Sides, we shall get 2xz -j- z 1 =: a 2 _.

2ay -j- y z ; from which the second being subtracted, wepext have 2 *z— y 2 — a 2 — iay-\-y 2 — b but 2 *z, bythe third, is = 2y 2 , therefore 2 y 2 — y> is =: a 2 — 2ay -j-

, , , a b

y 2 — b, or a 1 —2 ay — s — 0 ; whence y =-.

2 2 a

Now, to find x and z, y may be look’d upon as a knownQuantity, and then, by the second Step, we shall have**-|-z 2 — I — y 2 ; from which, if 2xz=zzy 2 be subtracted,we shall get the new Equation xx — 2*z -j- 2*= b —whence, by taking thpRoot, x—z = \/b — iy 2 t but,by the first Step, * -j- z = a — y; therefore x =

a — y+cjb —3 y 2 , . a y — Vb —3y 2—-—, and z = —-—.

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PROBLEM LV.

The Sum (a) anchtheSvm of the Squares (b) offur Kum-bers in geometrical Progression being given ; to 'shed theNumbers.

Let half the Sum of the two Means be denoted by x,and half their Difference by y ; then the Numbers them-selves will be x — y and ^ y; whence, by the Natureof geometrical Progressions, it will be *-fy :A —y

x —y :_— , the lesser Extreme; and as * — y : x-\-y

_ * + y

“l-tiL, the greater Extreme ; therefore, by the Ques-A —y

-f and A —/■* + *■+J 1 * + ^; y - z

~r-=.b-, whereof the former, being reduced to one

Dpmom*