the first order, then a2, x3, etc., together with all infinites that have finite ratios to these, will clearly be infinites of the second, third, etc., orders.

4. If 0 and 0 stand for two infinitesimal numbers or quantities of the same kind, then it is clear (from what has been done) that they may be supposed to be connected together by an equation of the form 0xa0, such that if the infinitesimals are of the same order, a shall be a finite number or quantity, which may be constant or variable, according to the nature of the case; noticing that if the dividend in the expression for a is an infinitesimal of a higher order than the divisor, a must be an infinitesimal of a certain order; and that if the dividend is of a lower order of infinitesimals than the divisor, then a must be an infinite of a certain order. Similarly, if two infinite numbers or quantities of the same kind are denoted by and, then they may be connected by the equation ∞ x A∞, or A =

such that A is finite when the infinites are of the same order; noticing that A is an infinitesimal of a certain order when the dividend is of a less order than the divisor, and vice

versa.

(48.) From what has been done, we are brought to the consideration of the following most important

PRINCIPLE.

Supposing the quantity X to be expressed in terms of the (number or) quantity x, if x is changed into a +h, it is clear that X will be changed into the corresponding quantity X+H; in which H results from changing a into a +h, so that H depends on h; now if a and h are such that they may be considered as being indeterminate or arbitrary, the quotient shall (in part at least) be independent of h; so h

H

that when h and I are small, the quotient shall be nearly invariable.

For, put h2h, and use X + H1 to represent the value of X resulting from the substitution of +h, for a in X; then if we substitute x+h, for x in X + H1, and represent the change in H, (resulting from the substitution of h

for x in it) by H'1, X+ H, will become X + 2H1 + H'1; since X becomes X + H1, and that H, becomes H1 + H'1.

Because h2h, and that X becomes X + H by changing into+h, and that X becomes X + 2H + H', by changing x into x + h1 and then changing x in the result into +h, it is clear that we must have H2H1 + H'1; because in each process a has been changed into a + h, and that the same value of X must correspond to the same value of x+h. Dividing the equals H and 2H1 + H', by the equals h and H 2H1 + H'1 H1

2h, we get the equation

h

=

in which, it is clear, if we put h1

1

= + , (1); h1 2h1

2h1

h

for h, that the quotient

2

H1

H' is the variation of

2h1

results from putting x + h, for x, it is plain that must

have a part or a term, which we shall denote by A, such that it shall be different from any term or part of

H'; and 2h

since (1) must clearly be an identical equation, or that the terms of the two members must be identical, so that it may be satisfied independently of h or h1, it follows that A must H H1

be common to and 1; and since is changed into Hi

for h, it is clear that A is independent of

h or h; consequently, the quotient

of h.

H
h

is in part independent

Again, because is a part or a term of the quotient

A
2

results, if A contains a, that the variation

contain a term of the form Bh1, such that B is independent of h; noticing that Bh, results from the substitution of ch for x in A.

Hence, it follows that the right member of (1) must con

B

tain the term Bh=h, and of course, since the equation

must be identical, the first member

H h

must contain a corre

sponding term, which we shall represent by Ah, in which A is independent of h; consequently (agreeably to what

has been shown), if h, is put for h,

term Ah.

H1

must contain the

h1

Because contains the term A1h, if A, contains a, it fol

H'1

lows that 2/1

must contain a term of the form

x,

since A, becomes of the form A1 +Ah1 + etc., in which A' is supposed not to contain h; consequently, since (1) must H be an identical equation, it follows that must contain a

h

corresponding term, which may be expressed by Ah2; in which A, is supposed not to contain h.

H

It may be shown, in like manner, if A, contains æ, that must contain a term of the form A,3, and so on; conseH H quently, must be expressed by the form = A + Agh + h h

h

Agh+Ash+ etc., (2); in which A, A1, A2, etc., are supposed to contain x, and to be independent of h.

It is clear that we may assign particular values to x and h in (2), provided A, A1, A2, etc., do not any of them become infinite; for (2) is clearly true when A, A1, A2, etc., are finite, or one or more of them reduced to 0.

If one or more of the quantities A, A1, A2, A„, etc., as А„, becomes infinite for any particular value of x, it is clear that (2) is true no further than to the term Ah, or to the term which first becomes infinite.

to

If A does not contain a, it is clear that (2) will be reduced

H

h

H

=A, which shows the quotient to be invariable.

h

If A, A1, A2, etc., contain x, and A is not very small in comparison to A1, A2, etc., then if h is small in comparison

to unity, it is clear that the quotient

H
h

will not differ much

from A, when different values are assigned to h; but if A1, A2, A3, etc., are very great in comparison to A, it will be necessary to make h extremely small in comparison to unity (so that the terms Ah, Ah, etc., in (2) may not sensibly affect A), in order that the quotient may be nearly invariable when his varied.

H

Hence, if is treated as a ratio, it results that h may be h

taken so small, that I shall vary very nearly as h.

It may be noticed, that if the two members of (2) are multiplied by h, there results H = Ah + A1h2 + Ah3 + etc.; consequently, if X' = X + H, we get X' = X + Ah + A ̧h2 + A2h3 + etc., (3); in which X' represents the value of X, that corresponds to x+h. If in X' we change a into a+k, and represent the corresponding change in X' by K, X' will be changed into X' + K, and, according to what has been shown, the quotient will be very nearly invariable when

K

k

k is very small. If h is very small, it is clear, from (3), that H K we shall have = (4), very nearly; consequently, if h h k'

and k have like signs, II and K must also have like signs, and conversely.

Remarks.-1. It may not be improper here to notice that any number or quantity whose value is supposed to remain the same during any calculation, is called a constant; while any number or quantity whose value is supposed to change is called a variable; where it may be observed that a constant is usually expressed by one of the first letters of the alphabet, while a variable is expressed by one of the last letters.

2. If any variable, as X, depends on another as x, then X is called a function of x, and X is sometimes called the dependent variable and a the independent variable. If the form of the expression of X in terms of x is known or given, then X is said to be an explicit function of x; but if the form of the expression of X in terms of x is not given, then X is said to be an implicit function of x. Thus, in X = ax + x2,

X is an explicit function of x, while a is an implicit function of X.

To show the use of what has been done, take the following

EXAMPLES.

Ex. 1. Given, =2x+h, to determine whether it has been

H h

derived from a function of x, by the process indicated in (1).

2x + 2h1 =2x+h, and of course, since

2

H1 H',

+ =

么 2h

H H1 H' the

- + "

h = h1 = 2hr

given expression has been derived from a function of ; indeed, we have (x + h)2 − x2 = 2xh + h2 = H, and thence get

H =2x+h, so that the given expression may have been

h

derived from ∞2.

8

Ex. 2. Let s and d denote the sum and difference of two numbers or quantities of the same kind, to find them.

If a represents the less of the two, it is plain that x + d will represent the greater; consequently, if a and b are particular values of a, we have a, a + d, and b, b+d for the corresponding numbers or quantities, and 2a + d, 2b + d will be their corresponding sums.

Hence, considering 2a + d, 2b + d as different particular values of X, we have 2a 2b= 2(a - b) = H; and since a-bh equals the difference of the corresponding values of x, we shall get 2. Because s is the value

Η 2(a - b)

=

h

a

b

of the sum that corresponds to æ, and 2a + d that which cor

-

if we

multiply these equals by x-a, we get 2x2a = 8 — 2a —