A 'Treatise of Algebra. 189

I 0 .# A ==«

a°. aA -j- aB — a b

3 0 . ;A + 6B -}• 6L = a + b -f- c

4 0 . 4A -f- 1 2 B -f- 24C 4 * 24D ~*-\-b-\-t-^-d

50^^ A-^- aoB-f-6oC-^-120D—1 2 oE^=æ—J—^-|- t-j-

&c. & c -

Now, the Double of the first of these Equations beingsubtracted from the second; its Triple from the third,and its Quadruple from the fourth, idc. we shall have

* aB = b — a6B -j- 6C= — za + b + c12B-I-24C+ 24D = —3 a bcd

20B -j- 60C+ i2oD-f X2oE = -4sl-}-i-f.f-f d-i-e&c. &c.

Again, if the Triple of the first of these be subtractedfrom the second, and its Sextuple from the third, (dc.we shall next have* 6C — a — 2b + c24C + 24D — z-r — <yb + c + d60C + uoD+ 120E — 6a — yb 4- c 4- d 4 - t.

Moreover, by taking the Quadruple of the first ofthese from the second, idc. we get

* 340 — — a 4- 3 b — 3r 4" and

120D4- iroE —— \a 4 " 11 b — 9 s 4 -^+*»

from the latter of which subtract five times the for-mer, and there will remain

* laoE — & *4 b 4 ” 4 " r.

Now divide each of the Equations mark’d thus, *, bythe Coefficient of its first Term, and there will come outthe very Values of A, B, C, D, ids. above exhibited.Q. E. D.

C O R O L. 1.

If each Term of the proposed Series a , b } c, d, £sV. besubtracted from its succeeding Term, the first of theRemainders, — a 4- b } — b 4- r, — c 4- d, — tdc.

divided