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Ex. 2. Required the cube root of 2/7+3√3.

Here A=2√7, B = 3√√3, A-B'=1; hence Q-1, and p=1.

Also √(A+B)2. Q=√(55+12√√/21)×1,

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√7+√3

2

succeed.

the quantity to be tried for the root, which is found to

337. In the operation it is required to find a number Q, such, that (A-B) Q may be a perfect nth power; this will always be the case, if Q be taken equal to (A-B2)"-1; but to find a less number which will answer this condition, let A2-B2 be divisible by a, a, &c....a times; b, b, &c....ß times; c, c, &c,...y times, &c. in succession; that is, let A-B2= ab3c. &c. Also let Q=a'b'c*. &c. then

(A-B').Q=a*+*xb3+Yxc?+*x&c.

which is a perfect nth power, if x, y, z, &c. be so assumed that a+x, B+y, y+x, &c. are respectively equal to n, or some mul-, tiple of n.

Thus, to find a number which multiplied by 180 will produce a perfect cube, divide 180 as often as possible by 2, 3, 5, &c. and it appears that 2.2.3.3.5=180; if, therefore, it be multiplied by 2.3.5.5, it becomes 23.33.53, or (2.3.5)3, which is a perfect cube.

338. If A and B be divided by their greatest common measure, either integer or quadratic surd, in all cases where the nth root can be obtained by this method, Q will either be unity, or some power of 2, less than 2". See Dr. Waring's Med. Alg. Chap. v.*

The proof here omitted, and the reference to it in Dr. Waring's work, may both be safely neglected, as it is of no practical use whatever. ED.

339.

The square root of a multinomial, as a+ √b+ √c+ √d, of which one term is rational, and the rest quadratic surds, may sometimes be found by assuming

√a+√b+ √c+√d=√x + √ÿ + √z,

and proceeding to find x, y, and z, as in Art. 329.

Ex. Required the square root of 21 +6√5+6√7+2√35.
Let √x+√ÿ+√z=√21+6√5+6√7+2√35,

then x+y+z+2√xy+2√xz+2√yz=21+6√5+6√7+2√35;

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Now 2√xy × 2√xx=4x√yz, or 6√5×6√7=4x√35,

x=9, and √x=3.

Also 2√xy×2 √yz=4y √xz, or 6√5×2√35=12y√7,

y=5, and √y=√5.

Again, x+y+x=21, or 9+5+z=21,

..=7, and √√√7.

Hence √x+y+√2=3+√5+√7, the root required.

INDETERMINATE COEFFICIENTS.

340. If A + Bx+Cx2+ &c.=a+bx+cx2+&c. be an identical equation, that is, if it hold for all values whatever of x, then the coefficients of like powers of x are equal to each other, that is, A-a, B=b, C=c, &c.*

For if A+Bx=a+bx, then

A-a+(B-b) x=0,

an equation which admits of one value of x only (Art. 193), unless B-b=0, or B-b, and therefore also A-a-0, or A=a.

* In the proof which is usually given a is assumed equal to 0, and afterwards the equal quantities are divided by x, whereas it is not proved that we may divide any quantity by ☛ when I stands for 0, in the same manner as when it stands for a finite magnitude; and that such a proceeding will in certain cases lead to erroneous results is well known.

Again, if A+Bx+Cx2=a+bx+cx, then

A−a+(B−b)x+(C−c) x2= 0,

a quadratic equation with respect to r which admits of no more than two distinct values of x (Art. 204), unless C-c=0, or C=c, and B-b=0, or B=b, and therefore also A-a=0, or A=a.

Similarly, if any number of terms be taken, or

(A−a)+(B−b)x+(C−c) x2+&c.=0,

there are certain values of x, and none other, which will satisfy the equation as long as it remains an equation with respect to x.

But, by the supposition, the equation must be true for any value whatever which we may please to give to x, and consequently for any number of values of x; and this, therefore, can only be attained by that which is apparently an equation with respect to x ceasing to be such, that is, by the coefficients of the powers of a being separately equal to 0; that is, we must have

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COR. If there be found any power of x on one side of the proposed equation, and no corresponding one on the other, then the whole coefficient of that power is of itself equal to 0. Thus, if A+Bx+Cx2+&c.=0, for all values whatever of x, then A=0, B=0, C=0, &c.

This may also be arrived at in the following manner.

It has been already proved (Arts. 193, 204) that if a simple or a quadratic equation be known to be satisfied by more different values of the unknown quantity than the dimensions of the equation, the coefficients of the several powers of the unknown, and the term independent of it are separately equal to 0. This proposition will be now extended.

-1

If a2+a1x+aqx2+...+a11x"1+ a1x"=0 be satisfied by n+1 different values of x, then a, a,, a,,...a, are each equal to 0.

Let the n+1 values of x be xX1, X2, X3,...Xn+1°

Then we have

a+a12++++...+a+a+=0,

a ̧±à ̧x' +α ̧x13 +...+α-x2-1+a,,x'" =0,

where x' may be equal to any of the n quantities

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a1(x'—Xn+1)+αg(x22 — x2 +1)+...+α-1(x^^-1—x" +¦)+a„(x'”—x+1)=0,

and as '- is not equal to 0, dividing by it we obtain

a ̧+ag(x2+xn+1)+...+a2-1(x2 ́~~2+...+x+?)+a„(x'”-'+...+x*})=0;

or, arranging in powers of x',

(a1+agTM++...)+(a2+.....)x'+...+α„x””-'=0,

where may take any of the n values

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Comparing this with the original equation, we find that it is of one dimension less, and that the coefficient of the highest power of the unknown is the same as before, viz. a,.

By repeating this process we shall have an equation of n-2 dimensions, the coefficient of the highest power of the unknown being still a, and we shall know that it is satisfied by n-1 values of the unknown, viz. X, X.,...-1; and so on. At last we shall arrive at an equation of one dimension, where the coefficient of the unknown will be a, and this will be satisfied by two values, 1, g.

.. by Art. 193, a=0. And our original equation will then become

...

satisfied by more than n−1 values of x;

.. as before, a-1=0,

=

so a-3=0, &c.

This, however, cannot satisfactorily be applied to the case of an infinite series.

341. Otherwise. Let A+Bx+Cx2+...=a+bx+cx2+... be an identical equation, that is, such as will hold for any value whatever of x; then

A~a+(B~b)x+(C~c)x2+...=0,

and, if A-a is not equal to 0, let it be equal to some quantity p; then we have

(B~b)x+(C~c)x2+&c.=−p.

And since A, a, are invariable quantities, their difference p, and .. -p, must be invariable; but-p=(B~b)x+(C~c)x2+... a quantity which may have various values by the variation of r; that is, we have the same quantity (p) proved to be both fixed and variable, which is absurd. Therefore there is no quantity (p) which can express the difference A-a, or, in other words,

A-a=0, and .. A=a.

Also (B~b)x+(C~c)x2+...=0,

or B-b+(C-c)x+...=0. (Art. 82.) Therefore, by what has been proved, B=b. And so on, for the remaining coefficients of like powers of x.

This is open to the same objection as the first proof, in which it is assumed that every equation has only a certain number of values of r which will satisfy it: for as long as a takes the value of any of the roots

of the equation here written, the equality will hold: and as the right-hand side may be an infinite series, there may be an infinite number of values of x that will satisfy it, considered as an equation: the conclusion that p is variable as well as fixed is therefore hardly correct.

The same result may also be deduced from the corollary to Art. 392: for by virtue of it we can make (B-b)x+(C~c)x2+&c. less than -p, by putting for x any quantity less than P where k is the greatest coefficient in the series involving x. But this cannot be unless p=0;

:. A=a.

-p+k'

.. as before, B=b, C=c, &c.

This however supposes, a priori, that the coefficients do not increase without limit.

It will be seen that none of these proofs are altogether satisfactory: the reason of this is, that an identity is treated in them in all respects as an equation. The fact of the matter is, that we predicate of a properties quite distinct from those that belong to the symbols we have hitherto had to deal with. These symbols standing for abstract or concrete quantities considered with reference to number, are obtained by the repetition of certain units which may be as small as ever we please. By that repetition therefore, all numbers, and all quantities that can be represented under any circumstances by numbers generally, are essentially discontinuous. Now in the identity we are dealing with we predicate of x continuity, making it thereby a symbol of quantity in the most general and unrestricted sense it can possibly be conceived in. Now an equation in which appears in an infinite series, certainly has an infinite number of roots: but these roots are symbols of quantity not unrestricted: and the equation is satisfied only when x takes the value of one or another of these roots, and not under any other circumstances. Now, however near these roots may lie to each other, as x passes from one to another of them, it changes discontinuously. But supposing x to possess continuity, which it does in the identity here discussed, it changes by insensible degrees from any one value to any other however near to the former, and consequently the infinitude of the number of values which x takes is of an infinitely higher order than the infinitude of the number of roots of an equation in the form of an infinite series. On this principle, therefore, and not otherwise, in the first proof, we can say that the infinite equality is satisfied by more values of r than the number of its dimensions; and in the second proof we can conclude that p is variable as well as fixed, and by this manner we overcome the objections to those proofs. Or we may proceed to reason as follows:

x

When we say a+bx+cx2+&c.=0 is an identity, for all values of x, we thereby assign to r the property of continuity, and make the series continuous likewise, and therefore capable of taking all manner of different values by the variation of x. These may or may not lie between certain limits, but that is quite another question: what we say is merely that the series does depend upon x for its value, and that therefore it can change

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