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

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

If (dx) and (d) simultaneously vanish, (52) contains no

dh

dk

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),

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

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 0x 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 r (x+h, y + k, z +l, ......), then, by a similar process, we have

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

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

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) = Ax2 + Bу2 + cz2 + 2 Eyz + 2 6 zx + 2 H xу ; it is required to expand r(x+h, y+k, z +l).

.. (x+h, y+k, z + 1) = ax2 + By2+cz2+2Eуz+2Gzx+2нxу

+2{(Ax+GZ+Hy) h + (By+EZ+HX) k + (cz+EY+Gx) l}

+ sh2+вk2 + cl2 + 2 Ekl+2 & 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

The first term of the expansion is f(ao, bo); partial derivedfunctions of this are to be taken; in the ao-partial derivedfunctions da, is to be replaced by a1; and in the bo-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

d2f

+2

dao

d2f

df

df) a, b, xy + { (das) b2 + ( a ) b2 } y2]; (58)

daodbo

dbo2

dbo

+3

dbo

d3f

daodbo

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

Ex. 2. sin {(a+a11 42 1.2 + ...) (bo + b

x2 + а2

y

1

1.2

1

+

+ b2

= sin (ab。) + (a1box + aş b1y) cos (αobɔ) ·

(α1 box + αob1y)2 sin (abo)

1

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

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 do according as the x- or y-partial differential is taken and thus we have the preceding function equal to

+3ƒ"(αvo) (α10x+α01у) (α20x2+2α11 xy +α02у2)

+ƒ'(α00) (α30 x3 +3 α21 x2y +3α12xу2 +α03 y3)] + (60)

....

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

π

as the arc increases up to the sine increases and at last be

, 2

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

through 0 when the arc, until, when the arc =

3 п

2,

the

sine 1; after which it continually increases until, when

=

the arc =

όπ

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.