nrvoLtmoN .] 



MATHEMATICS. ALGEBRA. 



461 



denoted by n'+ /, where n and n' represent numbers 

 exactly divisible by 9 ; and r and r 1 numbers less than 

 9. You see by the above expression for the product of 

 the two proposed numbers n-\-r and w'-f- r 1 , that the 

 whole of it is necessarily divisible by 9, except the part 

 r r f , because n or n' is a factor of every other term : you 

 may, therefore, at once conclude that if you divide a 

 multiplicand and its multiplier each by 9 (or indeed by 

 any other number), and note the remainders (r, r'), and 

 then divide the product of these remainders by 9 (or the 

 other number), the remainder arising from this last 

 division must be the very same as the remainder arising 

 from dividing the product of multiplier and multiplicand 

 by 9 (or the other number). And thus you have the 

 principle of the method of proving multiplication by casting 

 out nines. (Arith. , p. 437). The reason that 9 is chosen 

 for divisor is, because that for the divisor 9, the re- 

 mainder is the same whether the number itself, or only 

 the sum of the figures composing it, be divided ; and it 

 is easier to sum up the figures, and reject tie nines, than 

 to perform the division on the number. 



EXAMPLES FOR EXERCISE. 

 Multiply 2<w - Z6x 2 by 3* 3 2x 

 Multiply 4oy + Six 4c by 34y + 2e 

 Multiply out (5.r 3 4*= + 3x 2)(2* s x+1) 

 Multiply out (*f 2) (x 2) (*+3) (x 3) 

 Multiply out (x-f-a) (x a) (x 1 a') 



ultiply out (2x -\- 3o) (4* 5a) (x + o) 

 (Sax 4) (4ar + c)(5<u: 3) 

 (4ax -f Zky 1) (2ajc by + 2) 



(j. y)'(jr + yj 10. j* (a-H)x-fcj (x-c) 

 3(x 2 a')' 12. (2* + 3) (2* 3)(4* 2 



1. 

 2. 

 3. 

 4. 

 5. 



7. 



8. 



9. 

 11. 

 13. 

 14. 



INVOLUTION. 



Involution is nothing more than multiplication : it is 

 a term employed to sijjnify that the factors multiplied 

 together are all equal, the product or result being a 

 power. Example 4, for instance, page 4CO, is a case of 

 involution ; for x -J- y is raised, as it is called, or involved, 

 to the third power. Involution is thus the operation of 

 raising a proposed quantity to a proposed power, and 

 this operation is multiplication. The following are ex- 

 amples of the involution of simple quantities. 



)'=_a"x l y , &c. 



The RULE for obtaining the powers in such cases as 

 these is pretty obvious. 



To the power of the coefficient annex the letters, with 

 tLcir several exponents multiplied by the exponent of the 

 power. 



The rule of signs must of course be attended to. If 

 the quantity to be raised or involved be negative, the 

 of every even power must be positive that of every 

 odd power negative. See following examples : 

 I. (Za-x^Y 2. (7V) 4 



3. ( 4a 2 4 3 *) s 4. (****}' 



5. ( Say*') 5 6. ( Sfx'y'y 



7. (aa'AV) 1 8. ( 2W ; y) 1 



When the quantity to be involved is a compound quan- 

 tity, the proposed power of it is to be found by multiply- 

 ing the quantity the requisite number of times by itself, 

 as in the examples below : 

 1. To find the cube of a 1x. 



a- lax 



(a 2a) 5 = a- lax + 4 x- 

 a 2x 



2a 2 x 



.-. (a 2x 3 ) = a' 6a*x 



See Example I, p. 460. 



2. To find the cube of Sax 4y. 

 3ox 4y 

 3 ox 4y 



9a"j? 2 12a,ry 



(3ax 4y)== 9a-x- 

 3ox 4y 



2/Vx 3 Tla? 



36a-x"y + 9Ga.ry- 



.-. (Sax ly) s =27a.r 3 108a 2 x 2 y-f 144<ujr 



4. 



6. 



8. 

 10. 



(a 



EXAMPLES FOR EXERCISE. 



2. (Sax 4y) 3 3. (a -f b + 

 f 5)' 5. (a-- ' 



(a *)} 7. {( 



9- (( 



11. 



{(x 1 



The following is a table of the powers of (a -j- *) <^- 

 veloped in order, from (a -(- x) 1 up to (a + x) 8 : the result 

 of the actual multiplication is called the development, 

 and sometimes the expansion of a power. The table 

 exhibits the developments of the powers of an expression, 

 consisting of two simple terms, a and x. Every expres- 

 sion of two terms is called a binomial. 



Table of the developments of the powers of a binomial. 



= o + tax + it 

 + So 

 -f 6o.r +4or> + i 



= a + 6ai 

 = a' 4. 7oJ! 

 = o+8o'i 



This table, which may be carried to any extent, shows 

 that the coefficients of the terms in the development of a 

 binomial follow one another according to a remarkable 

 law, by observing which, they may be derived, each from 

 that which precedes it, with very little trouble, so that 

 the actual involution of the binomial may always be dis- 

 pensed with. By examining the several rows of co- 

 efficients, you will discover the law to be this ; namely 

 If the coefficient of any term be multiplied by the ex- 

 ponent of o in that term, and the product be divided by 

 the number which marks the place of the term in tho 

 row (as the 2nd place, 3rd place, <tc.), the quotient will 

 be the coefficient of the next term. Thus, look at the 

 development of (a+x) 7 . Tho first term in that develop- 

 ment we might know to be a 7 , without actual multipli- 

 cation. The coefficient of this first term is 1 ; and 



7x1 



I = 7 is the next coefficient, the complete term being 



7 e ,<;. This is the second term ; and 



the 



next coefficient, the third term being 21a 6 3 ; and from 



tliis we get -,; = 35, the next coefficient, the fourth 



3 



term being 35a*x 3 . In like manner, from this we find 

 = 35, the next coefficient; and in this way all tho 



coefficients may be found, one after another, without 

 involution. As to the letters connected with these co- 

 efficients, the writing of them involves no trouble at all. 

 The first term is the first term of the binomial, with the 

 exponent of the power over it ; the last term is the other 

 term of the binomial, with the same exponent over it ; 

 this is the highest exponent in the row of terms ; in each 

 intermediate term, both the letters occur in conjunction ; 

 and, in proceeding from term to term, the exponents of 

 a regularly descend in value, and the exponents of a; as 

 regufarly ascend ; that is, the powers of a are 

 a 7 a a 5 a* a 3 a 2 a 1 



and those of x, x 1 x* x 3 x* a: 5 x X T 



the turn of the exponents of both letters always making 



