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If (tt) aQd (4t) simultaneously vanish, (52) contains no

is parallel to the axis of y. And if we replace h and k, as we evidently may, by x and y, = 0, and = 0, or

2kx + By + E = 0,1
2cy + nx + o = 0, J

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.

'© - (a

term involving the first powers of a1 and y; and the conic is referred to the centre as origin. The coordinates to the centre are from (53),

2ce Bo 2ag Bp

x = — , y - — . (54)

B24ac' " B2 4ac v'

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

-tor)+[(£)• +(g)r]§

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-+(^)d^;(55)

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 6x and 6y; 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, s + l ),

then, by a similar process, we have

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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

f(a0 + a1X + a1~+ b0 + b, | + b2^ +....). (56)

The first term of the expansion is /(a0, b0); partial derivedfunctions of this are to be taken; in the a0-partial derivedfunctions rfo0 is to be replaced by ay; and in the A0-partial derived-functions db0 is to be replaced by b\; so that, if / stands for f(a0, bo), the second term of the expansion is

(£)*•+(£)•>». <m

the third term is the fourth term is

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

ao-f a, y + a2 j~2 + • • and 6o + *i| + *«Y2 + " is Price, Vol. 1. H h

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CHAPTER VII.

ON THE DETERMINATION OF MAXIMA AND MINIMA
VALUES OP 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 versd. 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

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

3 Tt

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

sine = — 1; after which it continually increases until, when

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

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

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