A ‘treatise of Algebra. 41

na n ~'b z=. 6 a 5 b, T Ll 1 -. a n ~ l b 1 = ijct 4 ^ 2 , &c. and

r . 2

consequently a-\-E 6 , (or a b raised to the sixth Power)equal to a 6 -\- 6 a S b 15 a*b z -\- zoalbl 4* i^a z b*-{-

4. i 6 , which is the very same as was above determinedby continual Multiplication.

Lastly, let it be proposed to involve cc 4- xy to thefourth Power.

Here a must stand for cc, b for xy, and n for 4; then,by substituting these Values instead of a, b and n in thegeneral Expression, it will become c 8 -j- 4 c&x y +

-j- 4 .c 2 *3} ,j -j- x'y* for the Value sought in this Cafe.

Note. From the preceding Operations, it may be ob-served, that the Unciæ or Coefficients increase till theIndices of the two Letters a and b become, equal, orchange Values, and t^en return or decrease again ac-cording to the same Order: Therefore it appears thatwe need only find the Qpefficients of the first Half ofthe Terms, in this Vsanner, since, from these, the restare given,

SECT. VII.Of Evolution .

TO Volution, or the Extraction of Roots, being direflly thecontrary to Involution, or raising os Powers, is per-Jormed by converse Operations, viz. by the Division os In-dices, as Involution was ly their Multiplication.

Thus the Cube Root of a or a', by dividing (1) the

Index of a, by 3, will be represented by a : In likeManner, the Square Root of a 4- x will be represented by

a 4. x 11 ; and that of a -|- <s X y -|- b by a -f- v'xy +T 12 .Moreover the Square Root of aa or a z will be represented

by a 1 or a-, and that of a 2 (or Va) by dividing the In-

dex (.) by 2, will be represented by a So likewise

the