istics of the law connecting the ordinates. The methods applicable to the continuous parts of the curve may, and usually do, give erroneous results if applied to discontinuous points. 22. Infinite Gradient.-Under J the face being vertical, the gradient is commonly said to be "infinite." At each of the sharp points I, J, R, S, T, U, the variation of gradient being sudden, the rate of variation of gradient becomes "infinite." More correctly expressed, there exists no gradient at J; and at I, J, R, etc., there are no rates of variation of gradient. 23. Meaning of a "Function."-The symbolic statement h = F(1) is not intended to assert that the relation between 1 and h is expressible by any already investigated mathematical formula, whether simple or complicated. For example, in fig. 1 the said relation would be extremely difficult to express by any algebraic or trigonometric formula. Equally complicated would be the law expressing the continuous variation of, for example, the horsepower of a steam-engine on, say, a week's intermittent running; or that connecting the electric out-put of a dynamo when connected on to a circuit of variable and, perhaps, intermittent conductivity. Yet separate short ranges of these laws may in many cases be approximated to by known mathematical methods; and even when this is not possible, many very interesting, important, and practically useful special features of the general law may be investigated by mathematical means, without any exact knowledge of the full and complete law. Thus without making any reference to, or any use of, the exact form of the function F() in the equation applicable to fig. 1, we have already been able to point out many important features of the law it represents. 24. Horse-power as a Function of Pressure.-Again, although the actual running of, say, a steam-engine from minute to minute varies with many changes of condition, still, if we choose to investigate the separate effect of one only of these changes, for instance, change of initial pressure, it may be found fairly simple. Thus we may write or Horse-power Function of Initial Pressure, = HP = $(p), where p is the pressure. This means that any change of pressure changes the horse-power; and to investigate the separate effect of change of pressure on horse-power, we consider all the other con ditions to remain (if possible) constant, while the pressure changes. Some other conditions may themselves necessarily depend on the pressure, and these, of course, cannot be assumed to remain constant. For example, the cut-off may be supposed to remain constant. But the amount of initial steam condensation in the cylinder depends partly on the initial pressure, and it cannot, therefore, be assumed a constant in the equation HP=4(p). Similarly, the HP may be considered as a function of the speed, it alone being varied while all other things are kept constant. Or the HP may be taken as a function of the cut-off, the initial pressure, the speed, and everything else being kept constant, while the cut-off is varied. 25. Function Symbols. When different laws connecting certain varying quantities have to be considered at the same time, different symbols, such as F(1), f(l), (l), 4(1), are used to indicate the different functions of l referred to. 26. Choice of Letter-Symbols.—In fig. 1 we have used 7 to represent a distance, because it is the first letter in the word "length," and similarly h to represent "height." It is very desirable when letter-symbols have to be used, to use such as readily call to mind the nature of the thing symbolised. Especially in practical applications of mathematics, and more particularly when there is any degree of complication in the expressions involved, is the adherence to this rule to be strongly recommended. By keeping the mind alive to the nature of the things being dealt with, error is safeguarded against, and the true physical meaning of the mathematical operations and of their results are more easily grasped. Without a complete understanding of the physical meaning of the result, not only is the result useless to the practical man, but its correctness cannot be judged of. If, on the other hand, the physical meaning be fully grasped, any possible error that may have crept in in the mathematical process of finding the result, is likely to be detected and its source discovered without great difficulty. 27. Particular and General Symbols. But many mathematical rules and processes have such wide application to so many entirely different physical conditions, that, in order the more clearly to demonstrate the generality of their application, mathematicians prefer to use letter-symbols chosen purposely so as to suggest only with difficulty anything endowed with special characteristics; such as x, y, z, symbols which do not suggest to the mind any idea whatever except that of absolute blankness. It is doubtful whether this is a desirable habit in mathematical training. It seems probable that a course of reasoning might be more firmly established in the mind of the student if he were first led through it in its concrete and particular aspect-the mind being kept riveted on one special set of concrete meanings to be attached to his symbols-and then afterwards, if need be, he may go through it again once, twice, or, if necessary, a dozen times, in order to discover (if or when this be true) that the general form of the result will remain the same whatever particular concrete meanings be attached to his symbols. 28. x, y, and z.-There is one feature in the use ordinarily made of x, y, z in mathematical books which the writer thinks is a real evil. In his earlier chapters the orthodox mathematician establishes a habit of using y to indicate a function of x: he constantly writes y=f(x): that is, he takes y to represent a thing dependent on x, and which necessarily changes in quantitative value when changes. But in his later chapters he uses y and r as two independent variables, that is, as two quantities having no sort of mutual dependence on each other, the variation of either one of which has no effect whatever upon the other. This is apt to, and does, produce confusion of mind; especially as regards the true meaning of different sets of formulas very similar in appearance, one referring to y and x as mutually dependent quantities, the other referring to y and x as independent variables. 29. Functions of x.-When x is used to indicate a variable quantity, any other quantity whose value varies in a definite way with the varying values of x, may be symbolically represented in any of the following ways: F(x), f(x), p(x), 4(x), x(x), and X, X or E. The last forms, X, etc., are for shortness and compactness as convenient as y, and are more expressive. They will be used chiefly in connection with x in the following pages. 30. X dependent on x.-X may mean a function which is capable of being also changed by changing the values of one or more other quantities besides ; but in so far as it is considered as a function of x, consideration of these other possible changes is eliminated by supposing them not to occur. This is legitimate because these other elements which go to the building up of X do not necessarily change with x. All elements involved in X, which necessarily change with x, are to be expressed in terms of x, and their variation is thus taken account of in calculating the variation of X. 31. Nature of Derived Functions. In dealing with functions of this kind, mathematicians call x the "independent variable," a somewhat unhappy nomenclature. X and x are in physical reality mutually dependent one on the other. In the mathematical formula, however, X being expressed in terms of z, it is considered as being derived from, or dependent upon, z; the various values of X being calculated from those of x, and the changes in X being calculated from the changes in . Thus it should be borne in mind that the dependence of the one on the other suggested in the commonly used phrase "independent variable ” is purely a matter of method of calculation, and not one of physical reality. 32. Variation of a Function.—Similarly Y may be used to indicate a derived or "dependent" function whose value depends only upon constants and upon the variable y. Or L may be made to denote a derived function depending only on constants and on the variable /. 33. Scales for Graphic Symbolism.-Those readers of this treatise who are engineers must, from practice of the art of Graphic Calculation, be familiar with the device of representing quantities of all kinds by the lengths of lines drawn upon paper, these lengths being plotted and measured to a suitable Scale. So long as the quantity of a function is its only characteristic with which we are concerned, each quantity can always be represented by the length of a line drawn in any position and in any direction on a sheet of paper, the scale being such that 1 inch or 1 millimetre of length represents a convenient number of units of the kind to be represented. In "Graphic Calculation" we very commonly represent on the paper also the two other characteristics of position and direction of the things dealt with; but in the Differential and Integral Calculus, so far as it is dealt with in this treatise, we are concerned only with quantity. It is convenient to draw all lines representing the various values of the same kind of thing in one direction on the paper. Thus we may plot off all the x's horizontally and the corresponding X's vertically. If, when the magnitude of x is varied continuously (i.e., without break or gap), the change of X be also gradual and continuous, there is obtained by this process a continuous curve which is a complete graphic representation of the law connecting X and x. The student ought at the outset to understand fully the nature of this kind of representation. It is clear that it is in its essence as wholly conventional and symbolic as is the lettersymbolism of ordinary algebra. Spoken words, written words, and written numbers are in the same way conventional; they also constitute systems of arbitrary symbolism. Graphic diagram representation is neither more nor less symbolic and arbitrary than ordinary language. 34. Ratios in Graphic Delineation.-The curve in fig. 2 is such a graphic delineation of a law of mutual dependence between X and x. If X be of a different kind from x, it is impossible to form any numerical ratio between the two scales to which X and x are plotted. For instance, if the X's are ft.-lbs. and the x's feet, then the vertical scale may be, perhaps, 1" 10,000 ft.-lbs., and the 10,000 horizontal scale 1" 10 ft.; but the number = = 2 10 or 1000, is not a pure ratio between the two scales. But in physics we have relations between things of different kinds, which are called physical ratios. It is only by use of such physical ratios that derived quantities are obtained. Thus the physical ratio between a number of ft.-lbs. and a number of feet, or ft.-lbs./ft., is a number of lbs. The ratio is of an altogether different kind, in this case lbs., from either that of the dividend or that of the divisor. Now the ratio between a height and a horizontal distance on the paper is a gradient measured from the horizontal. In this example, then, a gradient would mean a number of lbs., |