of-1 in the imaginary terms. When the quantity is real, it has for conjugate an equal quantity, and the modulus is nothing else than the quantity itself, abstraction being made of the sign.

Now I shall proceed to establish two propositions relating to moduli, which may be often useful.

PROPOSITION I.—The sum and difference of any two quantities whatever have a modulus comprehended between the sum and the difference of their moduli.

Let there be two expressions a+b √ −1, a'+b' √-1. Calling r and r' their moduli, we have r2=a+b2, r'2=a22+b2o. Naming R the modulus of

their sum, we have evidently

R2=(a+a')2+(b+b')2

=a2+a+b2+b2+2(aa'+bb')

=r2+r'3+2(aa'+bb').

But multiplying r2 by r'2, it is easy to see that

r2r'2=a2a22+b2b'2+a2b22+a”2ba

=(aa'+bb')2+(ab' —ba')2;

then the numerical value of aa'+bb' is less than, or at least equal to, rr'. Consequently, it is clear that R is comprehended between the two quantities r2+r22+2rr' and r2+r22—2rr', or, what is the same thing, between (r+r')2 and (r—r')2. Then the modulus R is comprehended between the sum and the difference of the moduli r and r'.

The demonstration is precisely the same where, instead of the sum of the imaginary expressions, we consider their difference.

PROPOSITION II.—The product of two quantities has for modulus the product of the moduli of these quantities.

In fact, multiplication gives

(a+b√ −1)(a'+b' √ −1)=aa'—bb'+(ab'+ba') √ —1; and if we take the modulus of this product, we find, conformably to the enunciation,

√(aa' bb')+(ab'+ba')2= √a2a'2+b2b'2+a2b22+b2a'2

= √(a2+b2)(a+b'3).

Corollary. Then the product of any number of factors whatever must have for modulus the product q, the moduli of all the factors. Then the nth power of an imaginary expression has for modulus the nth power of the modulus of that expression.

The above nomenclature and propositions are from Cauchy, who exhibits in a remarkable manner the efficiency of imaginary expressions as instruments in the investigation of the properties of real quantities. The following is a specimen :

If two numbers, of which each is the sum of two squares, be multiplied together, the product must also be the sum of two squares.

Let the two numbers be

a+b2 and a+b.

The first of these may be considered as the product of the factors

a+b-1 and a-b√ -1,

and the second as the product of the factors,

a+b'1 and a'-b'√1;

so that the product of the proposed numbers will be the product of the four factors

a+b√1, a-b√—1, a'+b' √ —1, a'-b' √ -1. Actually multiplying the first and third, and then the second and fourth, we have the following pair of conjugate expressions, viz.,

(aa'-bb')+(ab'+ba') √ —1, (aa' —bb')—(ab'+ba') √1,

of which the product is

(aa'—bb')2+(ab'+ba')3,

which is, therefore, the product of the original numbers, and proves that that product must, like each of the proposed factors, be the sum of two squares.

If we interchange the numbers a and b, or the numbers a', b', the terms of the product just deduced will be different; thus, putting a' for b', and b' for a', which produces no essential change in the proposed numbers, we have

(a2+b2) (a'2+b'2)=(aa' — bb')2+(ab'+ba')2=(ab' —ba')2+(aa'+bb')2. Consequently there are two ways of expressing by the sum of two squares the products of two numbers, each of which is itself the sum of two squares; thus,

METHOD PROPOSED BY MOUREY FOR AVOIDING IMAGINARY QUANTITIES. 198. Objections have been made to results obtained by the calculus of imaginary expressions. The rules observed in the calculus, it is said, have only been demonstrated for real magnitudes; it is by mere analogy that they are extended to the case of imaginary quantities; we may, therefore, raise reasonable doubts as to the exactitude of the results thus deduced.

M. Mourey, who has been much occupied with these difficulties, has sought to free analysis from them entirely, in a work published in 1828, entitled the True Theory of Negative Quantities and of the so-called Imaginary Quantities. Without entering into long details, we shall endeavor here to give an idea of the methods proposed by this author.

Let us resume the expression a+b√ −1, and give it, at first, the form

If we take the sum of the squares of the fractions, which are between the brackets, we find that this sum is equal to 1; and from thence we conclude that these two fractions can be regarded as being the sine and cosine of a same angle a. Designate also the modulus √a+b by A; the imaginary expression can be put under the form A(cos a+ √−1 sin a). Considering that this expression contains really but two quantities, the modulus A and the angle a, M. Mourey proposes to regard the modulus A as expressing the length of a right line O A, and a as being the angle A O X, which this line makes with a fixed axis OX. In other words, the modulus A represents a line of a certain length, which at first lay upon the axis O X, and which, by making a move

X'

*

A

-X

To understand this, a knowledge of the first principles of Trigonometry is necessary.

ment round the origin O upward, has departed from this axis by an angle a. M. Mourey gives the name verser to this angle, or, rather, to the arc which measures it; and then, instead of the imaginary expression, he writes simply Aa, a notation very suitable to recall at the same time the modulus A and the verser a. He proposes even to give the name route, or way, to the length O A, placed in its true position with regard to OX, so that A verser a, or Aa, is the route from O toward A.

As a line can make around the origin O as many revolutions as we please, and that, also, as well by commencing its rotation below as well as above O X, it follows that the verser may pass through all states of magnitude, and be as well negative as positive. It will be positive when the movement of the line shall have commenced above; it will be negative when the movement commenced below. From this it follows that the same route can be represented with a verser which is positive, or one which is negative, provided that the sum of the versers, abstraction being made of the signs, is 360°.

From the preceding conventions it results that a way can be represented by giving to the length A an infinity of different versers. Suppose, to fix the ideas, that OA should be a determinate way, and that then the verser A OX should be an acute angle a; it is evident that the position of OA will undergo no change if we add or subtract from a any number whatever of entire circumferences. Thus is established this important remark, that if we designate by 2 an entire circumference, or 360°, and by n any whole number whatever, positive or negative, the expression A2n+a will represent the same route as Aa; this is expressed by the equality

А2n+a=Aa.

When we give to A a verser equal to zero, the length A lies upon the line OX. When the verser is equal to π or 180°, this length is found in the opposite direction, OX'; then it is nothing else than the negative quantity — A. Thus we ought to regard as altogether equivalent the two expressions - A and Ал.

After these preliminaries, M. Mourey establishes the rules of algebraic calculus; then he passes to equations, and reconstructs algebra thus entirely. I shall not follow this author in all his details; I shall confine myself to the developments necessary to explain here what sense the new algebra attaches to the old imaginary expression √-A. I shall seek, first, the rule to be followed in the multiplication of any two quantities whatever, Aa and B3. Here the two factors are the magnitudes A and B, measured upon two lines

A

OA and OB, which make, with a fixed axis OX, angles A OX, BOX, represented by the A' versers a and B. It is necessary, then, first of all, to give to the definition of multiplication the extension suitable to render it applicable to the case in question. But, considering that the multiplier Bß indicates a line B, which departs from the fixed line O X by an angle equal to ß, M. Mourey regards multiplication as having for its object to take at first the length A in its actual direction as many times as there are units in B. and to turn the new line O A' around the point O, to depart from this direc

В

X

tion by an angle equal to 3, and to give it the position OC. From this it follows that, in designating by A B the product of the two magnitudes, obstraction being made of all idea of position, the product sought will be (AB)a+ß. Thus we have

AaxB3=(AB)+3;

that is to say, we multiply the moduli according to the ordinary rules of arithmetic, and take the sum of the versers.

If the two versers are equal to π or 180°, we shall have ATX Вл=(АВ)2π. But A and B are nothing else than -A and -B, and (AB)2π is the same thing as +AB; then -AX-B=+AB. This is the known rule, — by — gives +.

According to this rule, the square of Aa will be (A3)2a; that is to say, we take the square of the modulus and double the verser. Then, reciprocally, the square root is obtained by extracting the square root of the modulus without regarding the verser; then take half the verser.

Let us come now to the interpretation of the imaginary expression √—A2. For this purpose, let us observe, first, that it is equivalent to √(A2)2nπ+π; then extracting the square root,

IP

√−A2=Anπ+1′′.

If n is even, the verser n+1 places the length A in the same position as ; that is to say, in the position OP, perpendicular to O X. If n is uneven, the verser n+ will place the length A in a position O P', perpendicular to OX, but below. Thus, in X the system of M. Mourey, the expression √-A2 offers no longer to the mind any idea of impossibility. It represents two routes, OP and O P', equal and opposite, both perpendicular to the fixed axis OX.

X'

PERMUTATIONS AND COMBINATIONS.

199. THE Permutations of any number of quantities are the changes which these quantities may undergo with respect to their order.

Thus, if we take the quantities a, b, c; then abc, acb, bac, bea, cab, cha are the permutations of these three quantities taken all together; ab, ac, ba, be, ca, cb are the permutations of these quantities taken two and two; a, b, c are the permutations of these quantities taken singly, or one and one, &c. The problem which we propose to resolve is,

200. To find the number of the permutations of n quantities, taken p and p together.

Let a, b, c, d, ..................... k, be the n quantities.

The number of the permutations of these n quantities taken singly, or one and one, is manifestly n.

The number of the permutations of these n quantities, taken two and two together, will be n(n-1). For, since there are n quantities,

a, b, c, d, . . . . . . . k.

If we remove a there will remain (n-1) quantities,

Writing a before each of these (n−1) quantities, we shall have

ab, ac, ad, . . . . . . . ak;

that is, (n-1) permutations of the n quantities taken two and two, in which a stands first. Reasoning in the same manner for b, we shall have (n—1) permutations of the n quantities taken two and two, in which b stands first, and so on for each of the n quantities in succession; hence the whole number of permutations will be

n(n-1).

The number of the permutations of n quantities, taken three and three together, is n(n-1)(n-2). For since there are n quantities, if we remove a there will remain (n-1) quantities; but, by the last case, writing (n-1) for n, the number of the permutations of (n—1) quantities, taken two and two, is (n-1)(n-2); writing a before each of these (n−1)(n−2) permutations, we shall have (n−1)(n—2) permutations of the n quantities, taken three and three, in which a stands first. Reasoning in the same manner for b, we shall have (n-1)(n-2) permutations of the n quantities, taken three and three, in which b stands first, and so on for each of the n quantities in succession; hence the whole number of permutations will be

n(n-1)(n-2).

In like manner, we can prove that the number of permutations of n quantities, taken four and four, will be

n(n-1)(n-2)(n-3).

Upon examining the above results, we readily perceive that a certain relation exists between the numerical part of the expressions and the class of permutations to which they correspond.

Thus the number of permutations of n quantities, taken two and two, is n(n−1), which may be written under the form n(n−2+1).

Taken three and three, it is

n(n−1)(n—2), which may be written under the form n(n−1)(n−3+1). Taken four and four, it is

n(n−1)(n—2)(n—3), which may be written under the form.n(n—1)(n—2) (n-4+1).

Hence, from analogy, we may conclude that the number of permutations of n things, taken p and p together, will be

In order to demonstrate this, we shall employ the same species of proof already exemplified in (Arts. 23 and 78), and show that, if the above law be assumed to hold good for any one class of permutations, it must necessarily hold good for the class next superior.

Let us suppose, then, that the expression for the number of the permutations of n quantities, taken (p-1) and (p-1) together, is

n(n − 1)(n—2)(n−3). . . {n—(p−1)+1} . . . (A)

It is required to prove that the expression for the number of the permutations of n quantities, taken p and p together, will be

n(n−1)(n—2)(n—3) ..

(n−p+1).

Remove a, one of the n quantities a, b, c, d................................k, then, by the expression (A), writing (n-1) for n, the number of the permutations of the (n-1) quantities b, c, d............k, taken (p-1) and (p-1), will be