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For since by hypothesis the two expressions are equal whatever be the value of x, they must be equal when x=0; but in this case all the terms of each vanish, except the first ; thus we have A=A'. Therefore taking away these equal quantities from the general expressions, we have
Br + Cr? + Dr?.. =B'r+C'r? + D'x'.. and dividing byr,
B+Cr+D.ro.... =B' +. C'x+D'r?.. And as this quality must by hypothesis subsist, whatever be the value of x, let us again suppose x=0; and we get B=B'. By continuing to reason in this way, it will appear in like manner that C=C, D=D', &c. and so on, whatever be the number of terms.
If we bring all the terms of the two series to one side, so that the equation may stand thus, A--A' +'B-B'x+(C-Cr? +(D-D'.x =0 then we must have A-A=0, B-B'=0, C-C'=0, &c.
1 Let it be proposed to develope the fraction 1-2cr + q* into an infinite series by the method of indeterminate coefficients.
We assume the proposed expression equal to a series with indeterminate coefficients, thus
1-2ct +x=A+ Bx +C.ro +Dr+Er*+&c. where A, B, C, D, denote quantities independent of r.
We now multiply both sides of the equation by 1-2c3+3*, the denominator of the fraction, to take away that denominator ; then, bringing all the terms to one side, we get
+A +B Hence, to determine the quantities A, B, C, &c. we have, by the foregoing theorem, the following series of equations,
A-'=0, C-2cB + A=0,
&c. From which we obtain
&c. ar 1 here the law of the series, or the manner in which each term is
weduced from the two preceding it is very evident. Thus it appears that 1
=1+2cx+ (4co - 1)x?+ (8c-4c)... + (166-12c + 1) 1-2cr + 1+&c.
2. Let it be required to develope v(a% +) into a series by the method of indeterminate coefficients.
In this case we might assume the series A+ Bx+Cx2 + D.x3 + &c. for the root, but as we should find that the coefficients of the odd powers of x' are each =0, we rather assume
via + x)=A+Br? +Cx* + D.ro + &c. By squaring each side of this equation, and transposing the terms on the left-hand side of the result to the right, and putting the whole equal to 0, we have
+2BC Therefore, by the principle laid down in section 314, we have Aa=0.
-BC 1 2AD+2BC=0,
&c Hence it appears, that
x2 v(a + x2)=at
2a 8a3 16a" agreeing with the result obtained by a different method.
Definitions. 1. A function is the result which an algebraic .operation produces on a variable quantity.
Thus, if y=a+bx+cx', or if y=v(ux-xo) or if y=a*, and so on, then in each case y is called a function of x; but the quantities a, b, s, being constant, are not considered, x alone is supposed to vary.
2. A quantity which depends on two or more variable quantities, is also called a function of these quantities. Thus, if y=arl + bz* then y is a function of x and z.
3. A quantity y is an explicit function of another quantity x, when the value of y is given directly by that of x, without the resolution of an equation. Thus, if y=a+bx+cx*+&c then y is an explicit function of r.
4. A quantity y is called an implicit function of another quantity, when it is necessary that the equation should be resolved in order to discover the value of y. Thus, in the equation y' + ax=bxy, y is an implicit function of x. 5. The quantities a, b, c, &c. are called constants.
Notation. One of the letters f, F, or when prefixed to a variable quantity, indicates any function of that quantity; the letter thus prefixed, must therefore not be understood as a factor or coefficient, at least in this species of analysis.
If the same letter be prefixed to each one, of two or more different quantities, the letter thus prefixed indicates that the same operation is to be performed on each of the quantities, whatever that operation may be : whether addition or subtraction, multiplication or division, involution or evolution, or indeed any combination of these operations.
Thus, if pr=rn, then will o(k+x=(k+lx)"
c(k+lx) +d(k+lr) + &c. or if pr=ar, then will, ok+lx)=(k+lr)", and so on; but fr and Fx, represent two different operations upon the quantity t; that is, two different functions of x.
0(x, y) indicates some function of x and y, as xy--yo, again P(x, y, z) indicates a function of x,y and %, as ary+by®z.
In order to show the general application of this doctrine, it will be necessary to prove the following particulars.
(142.) If the series i+b'x+b"r+1"**+&c. be multiplied by another series, 1+c'r+d"x2 +0" x® +&c. of the same form, the product will be 1 + B'r+B"xo + P'x3+ &c. still of the same form.
For by actual multiplication
1+Vx+ 1"x" + 1"*+&c.
dll.r+ &c. the produet is 1 + B'r +B"z® +B" x + &c. by making B'='+ B=6" +CV +8" &c.; and since the product is of the same form as each of the factors ; it is evident, that if any number of factors m, of the same form, are multiplied together, the product must still be of the same form as each of the factors.
Hence we may observe that in every new product, the letter which is the coefficient of the second term of the new multiplier, which produces that product, will always be added to the sum of the coefficients of the second terms of all the preceding multipliers ; and therefore if the number of factors be m, the coefficient of the second term of the product will be the sum of all the m coefficients of these factors.
wnence if the coefficients of all the factors are the saine, the coefficients of all their second terms will be the same.
Now, suppose the coefficient of the second term of every factor to be V', and the coefficient of the second term of their product will be VIV'; whence the first and second terms of their product will be i nd mVr.
If the series B+B'r+"x2 + 8''x' + &c. be raised to any power m, the expansion will be of the form
B + B'r+B"x? +BM++ &c.
-29 + &c.
= 21, &c. then will ē B = (3+B'x+ß"8+""'x+&c.)>= m(1 +Vx+l're + L'"zo+&c.).
=Cm(1 + mb'x+A"x? +A"X3 + &c
+ &c. That is, the first and second terms of the expansion will be sm and mm-1B'r and the remaining terms of the form B"x? +B"r+&c.
If B+I had been raised to the mth power instead of the series B+b'r+B"x2+ß"'x*+&c. the first and second terms of the expansion would have been, fm and mßm-rx, and the remaining terms of the form Bor? +B"x3+ &c.; that is, the whole expansion of the form B+B'r+ Boz? + B'" x + &c.
(144.) If the series po+px+p"ro+px' +&c. be divided by the series 1-Ur-6":"&c. the quotient will be of the (form q+q'r+ q*re+q"r*+&c.
For let the operation be performed, so that q may be substituted for P, q' for the sum of the parts which forms the coefficient of the first term of the first remainder, q" for the sum of the parts which forms the coefficient of the first term of the second remainder, and so ou, as follows
+9". +&c. 1-47-07-0"--&c.)q+p'r +p"ro +p" x3 + &c.(q+q'r+q"z*
q'x+(" +91;")r + (pl" tq1."!)r*+&c. q'um
9'6" x*+&c. q"x2 + (p!" +91'"+q'0"2S+&c. q"a— -9'"l'x' +&c:
q'' 7+&c. q"x*+&c.
Then the co-efficients of the terms of the quotient may be thus derived, viz.
9. = 1
=pl +90 +961 +9"V
&c. Therefore the co-efficients q, q', q", q'" &c. of the terms of the quotient are functions of the co-efficients, of the terms of the divisor and dividend, and the quotient is of the form asserted.
Hence, also, the quotient arising by dividing the dividend by the product of any number of series must always be of the same form.
(145.) If i be divided by any given power m of the series B+B'z+ B"z2 +811.38 +&c. the quotient will be of the form C+Clx+C"x® + C"x+&c. and the first two terms B and B'x will be B-,
B-m-b'c. For the first two terms of the mth power of the series B+B'3+ B"x® +ß"zo+&c. are 3* +m3*-*ßlx, and the remaining terms are of the form B".ro+B".X + &c. ; whence the quotient arising by dividing I by the series B+B'r + B^2 + B'" x® + &c. will be of the form Cť Clx+C"r? + C'"* + &c. and with respect to the absolute values of the first two terms of the form C,+C'x, it will only be necessary to divide 1 by 2 + m Bm="b'x, the first two terms of the series B+B'r+B"x+ &c. thus ß"+mBm-'B'x)1
(-mßß'x, quotient. 1+mB-- B'x -MB-B's
-m-' B'x-&c. If the series B + B'r+B"22+ &c. had been the expansion of the mth power of the binomial B+x, the first two terms would have been B" + Smer.
2146.) Therefore whether the operation be that of involving a binomial or a whole series to any power m, or of dividing unity by the mth power of the binomial or mth power of the whole series, the expansion will always be of the form A + A'x+A'zo +A!" x-*+&c. of which the two first terms will be Bm Em Bm- p'x or 6-7 m3---1B'I, according as the expansion arises from multiplication or division, and if the root be a binomial, then B=1; and consequently ß' will disappear in the second term. (147.) In each case the co-efficient of the first term is the same.
Dt the sign of the exponent, which in the expansion arising by