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is parallel to the axis of y. And if we replace h and k, as we

dr

dx

evidently may, by x and y, («r) = 0, and

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are the equations to the two diameters of the conic r(x, y) = 0, which bisect all chords parallel to the axes of x and y respectively.

If (d) and

dh

(d) simultaneously vanish, (52) contains no

term involving the first powers of x and y; and the conic is referred to the centre as origin. The coordinates to the centre are from (53),

x =

2CE-BG
B2-4AC'

y =

2AG-BF
B2-4AC

(54)

We shall have a further illustration of this theorem hereafter, when we come to the consideration of tangents and polars of plane curves.

If in the preceding general formula (51) we make x = 0, y = 0, and then change h and k into x and y, we have

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where we have to replace x and y by the value 0 in all the partial derived-functions, except in those of the last term, where they are to be replaced by ex and Oy; and if this last term decreases without limit, as n increases without limit, then the remainder may be neglected, and the series may be written without it.

142.] If it is required to expand F (x + h, y + k, z +1, ......), then, by a similar process, we have

F(x+h, y+k, z + 1, ......) = F(x, y, z, ......)

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replacing x, y, z,... in the last term by x+0h, y +Ok, z +01, ...

As the equations (51) (55) and (56) stand at present, each side is exactly equal to the other; but if we can assure ourselves that, as n increases without limit, each term, as well as the sum of all the terms, of the part omitted decreases without limit; then the remainders may be neglected, and the equations will be modified accordingly.

Ex. 1. F(x, y, z) = x2 + By2+cz2+2Eyz+2Gzx+2нxy; it is required to expand F(x+h, y +k, z + l).

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.. F(x+h,y+k, z + 1) = ax2 +вy2+cz2+2xyz+2Gzx+2нxу

+2{(ax+GZ+Hy) h + (By+EZ+Hx) k+(cz+EY+GX) l}

+ Ah2+вk2 + cl2 + 2 E kl + 2a lh + 2H hk.

143.] And the result of this last Article may be extended by that method of Derivation, the principles of which have been explained in Art. 95-97, and which has been in those Articles applied to functions of one variable x. I propose now to apply the process to functions of two variables, although it is evidently capable of application to any number of variables. And I will first take the following case. It is required to expand in ascending powers of x and y the function.

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X

x2

У

+ az + bo + by + b2
....
1.2

1

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The first term of the expansion is f(ao, bo); partial derivedfunctions of this are to be taken; in the ao-partial derivedfunctions dao is to be replaced by a1; and in the b。-partial derived-functions db is to be replaced by b1; so that, if ƒ stands for f(ao, bo), the second term of the expansion is

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d2f

da, dbo

the fourth term is

1

dao

(57)

-) α1 b1xy + { (d2) b2 + ( d ) b2 } y2 ] ; (58)

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dbo

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dao3

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+

d3f

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df

dao

df

(db) b3 + 3 (db) b2 b2 + (db) b13 } y3 ]; (59)

and by a similar process other terms may be found; but it is unnecessary to write them at length.

Ex. 1. By the process above explained the product of

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1

+ {aз box3 +3 a2 b1 x2 у+3 a1 b2xy2 + abзy3}
1.2.3

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1.2.3

1

1.2.3

= sin (abo) + (a1 box + аş b1y) cos (ao bọ) ·

(α1box + αb1y)2 sin (a,b)

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{3 (a1box +α b1y) (α2bx2+2a1b1xy + ab2y2)} sin (a。bọ)

Another form is that wherein it is required to expand in ascending powers of the variables a function of variables of the form

1 1.2

f{aoo+α10x+A01 Y + (a20x2+2α11 xy +αo2 y3) + ... }.

The first term of the expansion is evidently f(aoo) let the successive derived-functions of this be symbolized according to Lagrange's notation. As however f is a function of two variables, daoo will be a10 or 01 according as the x- or y-partial differential is taken: and thus we have the preceding function equal to

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+3ƒ" (aoo) (α10x+α01y) (α20x2 +2α11 xy +α02 y2)

+ƒ' (aoo) (α30 x3 +3 а21 x2y +3 α12xy2 + α03 Y3)] +

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CHAPTER VII.

ON THE DETERMINATION OF MAXIMA AND MINIMA

VALUES OF FUNCTIONS.

144.] LET us consider a function of a single variable x; and, to fix our thoughts, let the variable be continuously increased; then the corresponding variation of the function need not always be one of increase or of decrease, but it may increase up to a certain value and afterwards decrease, or vice versa. In the former of these two cases, at the value of the variable when the function ceases to increase, it has attained a greatest value, or what is technically called a maximum state; and in the latter it reaches a least or a minimum state; such singular conditions of a function the principles of Chapter IV enable us to determine. And we have the following definition:

A particular value of a function, which is greater than all its values in the immediate neighbourhood, that is, when the variables are infinitesimally increased or decreased, is said to be a maximum. And the particular value which is less than all its immediately adjacent ones, is called a minimum.

Maxima and minima are therefore terms used not absolutely, but in reference to the values of the functions immediately adjacent to those to which the names are applied.

As a simple illustration, let us consider sin x; and let the radius of the circle be unity; then, when the arc = 0, the sine = 0; but as the arc increases up to , the sine increases and at last be

π

2

comes 1, which is its maximum, for as the arc becomes larger, the sine becomes smaller and continually decreases, passing

through O when the arc, until, when the arc =

3 п

2'

the

sine1; after which it continually increases until, when

5 п

the arc = the sine + 1, which is a maximum, and so on.

, 2

Thus as the arc increases, the sine periodically attains to maxima and minima values.

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