variables are said to depend on, and to be functions of, each other: and the equation which expresses the mode of dependence is said to be a function of such variables. If one variable is involved in such an expression, it is said to be a function of one variable; if two variables are involved, to be a function of two variables; and so on. Thus sin x, e", log a1, \/(a2—x2), are functions of one variable, viz. x; eax+b>', tan (ax -f by), xy are functions of two variables, x and y; xyz, x2+y2 + z2 are functions of three variables: similarly we may have functions of more variables. Functions are designated by the symbols F,/, (f>, &c. Thus r(x) means a fuuetion of one variable x, combined or not with constants as the case may be; T(x2) means a fuuetion of x2; <p(x, y) symbolizes a function of two variables; \ji(x, y,z) a function of three variables: thus these functional symbols are general, and the specific forms of them are the particular functions which arise from operations in algebra, trigonometry, &c. Thus if F (x) = cos x, F is the general symbol of an operation of which cos is the specific instance; similarly would tana:, \ogx, >/(a2 — x2), be all represented by r(x); and \og(x + y) would be represented by f(x,y). Now as such equations represent the mode of mutual interdependence of two or more variables in their symbolized state, so in their unsymbolized state they express the relation between, and the law of, certain causes and effects. Suppose a mass of metal to have been heated to a certain temperature, and that we have to find the temperature at any subsequent time; this latter quantity will depend on (say) three circumstances, viz. the original temperature, the law of radiation of heat, and the . length of intervening time. Moreover suppose the law of relation of these four circumstances to be known, (which it is,) and it to be possible to express that law in a symbolical form; then the equation of dependence will involve four variables, viz. the original temperature, the time elapsed, the law of radiation, and the present temperature, and thus will be a function of four variables; but we shall also say, that the present temperature is a function of three other variables, and write it as follows: Present temperature = F (original temperature, time, law of radiation). Now it is possible that any one of the last three variables may vary without involving any change of the other two, in which case however the " present temperature" must vary also; and as a similar variation of any other of the three may take place, there may he three separate variations of it due to the separate variations of each of the three variables on which it depends. On this account it is called a dependent variable, and each of the others is called an independent variable. 13.] Functions are said to be implicit and explicit, according as they assume the form of one or the other of those of the last Article. When by any artifice or operation, as, for instance, by the algebraical solution of an equation, one variable is expressed in terms of all the others, then it is said to be an explicit function of them; but when it is not solved, and all the variables remain involved in one expression, then the function is said to be implicit. Thus the illustrating case of the last Article will be an implicit function of four variables, if the quantities are combined in the form, r (original temperature, time, law of radiation, present temperature) = 0 and the present temperature becomes an explicit function of three variables, if it is written in the form, Present temperature = F (original temperature, time, law of radiation). Thus x2 + y2—as = 0 is an implicit function of two variables, but y = (a2 —x2)i is an explicit function of one variable, of which y is the dependent and x the independent variable; and y = /(*) is the general form of such explicit functions, and r (x, y) = c (c being a constant) is the general form of an implicit function of two variables: and in these forms x and y are called the subjects of the functional symbols / and F. So again: JP^ >/2 &l —j + H—^ = 1 is an implicit function of three variables of fl ( x* w2)4 the form F (x, y,z) = c; whereas z=z c <l — — ^> , which is of the form z=f(x,y), is an explicit function of two variables. Implicit functions are often written in the form, u=r(x,y,z,...) = c, or = 0, as the case may be. The terms dependent and independent variables have reference to explicit functions. When functions are implicit, there arc no general marks whereby to determine the variable, which may first most conveniently change value. 14.] Functions have again been divided into two classes, algebraical and transcendental: the former being those functions which involve the operations of addition, subtraction, multiplication, division, involution, and evolution, or the algebraical sum of many such functions; the latter those wherein the operations symbolized are such as ex, logex, sinx, sec-1 a?; that is, where they are either exponential, logarithmic, or circular. This however is a division not necessary to our present purpose. Functions again may be simple or compound; that is, according as one or many operations, the results of which are the functions in question, are involved. Thus y = sinx, y = logax, are simple functions of x; but y = log sin x, y = eUaiajc are compound functions; compound functions are thus functions of functions. It is necessary to observe, that, if two functions are represented by the same functional symbol, they are formed in the same manner by means of the variables which they involve. Thus if f(x) = sin *, f(y) = smy; if /(*) = e>>*, f{y) = e<*>. 15.] Functions may be either continuous or discontinuous. A continuous function is subject to the two following conditions: 1st. As the variable gradually changes, the function must gradually change. 2nd. The law symbolized by the functional character must not abruptly change. When these two conditions are not satisfied, the function is discontinuous. Thus, for instance, both conditions are fulfilled in the functions y = ax + b, y = sinx; in which, as the variable x changes, the value of the function also changes, but changes gradually, and there is no abrupt passage from one value to another; and the law symbolized by the functional character does not change, but always remains the same: but if the function were such as to express a line of the form in fig. 1, so that Ba should be a continuous curve drawn after some determinate law, but at A the law suddenly should change, and the curve, from being, say, a circle, become a straight line, then the second of the above conditions is not satisfied, and the function is discontinuous, A is called a point of discontinuity. As an instance of a function of this description the following may be mentioned. Replacing the circular quantities by their exponential values, it may easily be proved that sin(a-|) cos a + cos (a + 0) -(-cos (a + 2/3)+ ... ad infin. = . a 2sinl Suppose that a = ^, then the series becomes cos a + cos 3a + cos 5a+... ad infin. = — —: 2 sin a but if a = any multiple of it, the sum of the series assumes the indeterminate form ^; hence we have this remarkable result, each term of the series varies continuously with o, but the sum of the series varies discontinuously, being always zero, except when a passes through some multiple of it, when the sum of the series suddenly and abruptly becomes ;that is, some indeterminate quantity; thus we have a series of points of discontinuity. It is of continuous functions of continuous variables generally that we shall treat; and if discontinuous functions are introduced, they will be considered only for those values of the variables for which they are continuous. 16.] There are two different modes of viewing such continuous functions and variables, both of which will be convenient for the future purposes of the treatise. Firstly, suppose Xi and x3 to be two definite values of a variable number x, of which Xi is the larger; and suppose the difference x2—xx to be finite, and to be resolved into an infinite number of equal parts, each of which is therefore an infinitesimal; then the passage from Xi to Xi may be made by the successive addition of such infinitesimal elements, the whole sum of which is of course the finite quantity. Secondly, the idea of motion or continuous growth may be introduced, and we may conceive number to be in a gradually increasing state; and thus, if the rate of increase be finite, the increment due to a finite time will be finite, and that due to an infinitesimal interval of time will be an infinitesimal. The former mode we have hitherto invariably considered, except in the illustrations of Art. 6, but as we have now to deduce infinitesimals from finite quantities, and the latter method is more convenient for that purpose, we shall apply it. Both manifestly lead to identical quantities and results; for whereas in the one we consider a quantity during the process of generation, and the infinitesimal elements as they are successively produced; so in the other, we resolve the finite quantity when generated into its infinitesimal elements: in the latter then we arrive at the finite quantity from the elements, in the former we derive the elements from the finite quantity. The latter idea is the more complex, inasmuch as it involves motion and perhaps time, but adapts itself more readily to mechanical questions; and the former is undoubtedly best suited to geometry. The former of these two ideas is that on which Leibnitz conducted his investigations; the latter is for the most part that which sir Isaac Newton has embodied in the Principia, although some of the lemmas involve the former. Thus suppose we consider the arc of a quadrant of a circle of radius a; its length is which if we resolve into infinitesimal elements, each element will be the distance between two consecutive points: and the two points will be taken so near together, that the line joining them must be considered straight; and thus must the circle be conceived to be made up of an infinity of infinitesimal straight lines, and the tangent at any point is the line which coincides with the straight line joining the point and its consecutive point; that is, the tangent is the element produced. In elementary treatises on conies, the tangent to a conic is defined to be the straight line passing through two points on a curve infinitesimally near to each other, and its equation is derived from this definition. Similarly must all continuous curves be considered as composed of infinitesimal straight lines, and all surfaces of infinitesimal plane areas, and all solids of infinitesimal elements, and all concrete bodies as made up of infinitesimal corpuscles. Or if we consider the quadrant to be generated by a point moving according to a given law, and the motion to be carried on during a finite time, and the time to be resolved into very short instants, then the space passed over in one of these instants is the infinitesimal increment of the curve; and the direction in which the point is moving at the time of generating the element is that of the tangent of the circle at the point. In this view points generate lines, lines generate surfaces, and surfaces generate solids. |