pure source of misfortune to him who remembers it. It is infinitely more important to cultivate the faculty of cautious and yet ready use of formulas than to have the whole range of mathematical formulas at one's finger ends; and this is also of immensely greater importance to the practical man than to keep in mind the proofs of the formulas.

To obviate the necessity of such memorisation the "Classified Reference List of Integrals" has been constructed in the manner thought most likely to facilitate the rapid finding of whatever may be sought for. The results are not tabulated in "rational" order, that is, in the order in which one may be logically deduced from preceding ones. They are classified, firstly, according to subject, e.g., Algebraical, Trigonometrical, etc., etc., and under each subject they are arranged in the order of simplicity and of most frequent utility. A somewhat detailed classification has been found desirable in order to facilitate cross-references, the free use of which greatly diminishes the bulk of the whole list. The shorter such a list is made, the easier is it to make use of.

10. Scope of Prefatory Treatise.—This treatise does not prove all the results tabulated in the "Reference List." The latter has been made as complete as was consistent with moderate bulk, and includes all that is needed for what may be described as ordinary work, that is, excluding such higher difficult work as is never attempted by engineers or by undergraduate students of physics. The treatise aims at giving a very thorough understanding of the principles and methods employed in finding the results stated concisely in the "Reference List"; proofs of all the fundamentally important results; and, above all, familiarity with the practical uses of these results, so as to give the student confidence in his own independent powers of putting them to practical use. The last chapter on the Integration of Differential Equations ought to aid greatly in pointing out the methods of dealing with various classes of problems. The ninth chapter, on Maxima and Minima, is perhaps more illustrative than any other of the great variety of very important practical problems that can be solved correctly only by the aid of the Calculus.

CHAPTER II.

GENERAL IDEAS AND PRINCIPLES-ALGEBRAIC AND GRAPHIC

SYMBOLISM.

11. Meaning of a "Function."-Suppose that a section be made through a hilly bit of country for some engineering purpose, such as the making of a highway, or a railway, or a canal. The levels of the different points along the section are obtained by the use of the Engineer's Level, and the horizontal distances by one or other of the ordinary surveying methods. Let fig. 1 be the plot

Το

NRS

FIG. 1.

ting on paper of the section. According to ordinary practice, the heights would be plotted to a much opener scale than the horizontal distances; but in order to avoid complication in a first illustration, we will assume that in fig. 1 heights and distances are plotted to the same scale.

Each point P on this section is defined strictly by its level h and its horizontal position 7. The former is measured from some conveniently chosen datum level. The latter is measured from any convenient starting-point. These two are called by mathematicians the co-ordinates of the point P on the curve ABC, etc.

For each ordinate 7 there is one defined value of the co-ordinate h; except throughout the stretch MN, where a break in the curve occurs. Putting aside this exception, the height h is, when this strictly definite relation exists, called in mathematical language "function" of l; or

а

Height Function of Horizontal Distance,

=

or, more simply written in mathematical shorthand,

h = F(1).

12. Ambiguous Cases.-As seen from the dotted line drawn horizontally through P, there are three points on the section at the same level. Thus the statement that

or

Distance Function of Height

l = f(h)

must be understood in a somewhat different sense from the first equation namely, in the sense that, although for each height there correspond particular and exactly defined distances, still two or more such distances correspond to one and the same height, so that, if nothing but the height of a point were given, it would remain doubtful which of two or three horizontal positions it occupied. This ambiguity can only be cleared away by supplying special information concerning the point beyond that contained in the equation.

and

13. Inverse Functions. The two formulas

h = F(l)
l=f(h)

are simply two different forms into which the relation between h and 1, or the equation to the curve, can be thrown. The first form may be called the solution of the equation for h; the second the solution of the equation for 7.

The functions F( ) and ƒ( ) are said each to be the "inverse" of the other. An inverse function is frequently indicated by the symbol 1 put in the place of an index. For example, if s be sin a, then the angle a may be written sin -1s. Or if 7 be the logarithm of a number N, or = log N; then N = log-17, which expression means that "N is the number whose log is l."

14. Indefiniteness of a Function in Special Cases.-The stretch of ground from R to S is level. Here the value of h corresponds to a continuously varying range of values of 7. For this particular value of h, therefore, we have between certain limits indefiniteness in the solution for ĺ.

If there were under the point J a stretch of perfectly vertical cliff, then for the one value of to this cliff the solution for would be similarly indefinite between the limits of level at the foot and at the top of the cliff.

15. Discontinuity.-From M to N there is a break in the curve. In such a case mathematicians say that h is a discontinuous function of; the discontinuity ranging from M to N.

16. Maxima and Minima. From A to C the ground rises; from C to E it falls. At C we have a summit, or a maximum value of h. This maximum necessarily comes at the end of a rising and the beginning of a falling part of the section. Evidently the converse is also true, viz., that after a rising and before the following falling part there is necessarily a maximum, provided there be no discontinuity between these two parts. There is another maximum or summit at K.

The ground falls continuously from C to E, and then rises again from E onwards. There is no discontinuity here, and E gives, therefore, a lowest or minimum value of h. This necessarily comes after a falling and before a rising part of the section; and between such parts there necessarily occurs a minimum, if there be no discontinuity.

We have here assumed the forward direction along the section to be from A towards the right hand. But it is indifferent whether we call this or the reverse the forward direction as regards the distinction between maximum and minimum points.

17. Gradient or Differential Coefficient. Each small length of the section has a definite slope or gradient. Engineers always take as the measure of the gradient the ratio of the rise of the ground between two points near each other to the horizontal distance between, the same points. This must be carefully distinguished from the ratio of the rise to distance measured along the sloping surface. This latter is the sine of the angle of inclination of the surface to the horizontal; whereas the gradient is the tangent of the same angle of inclination. This gradient is the rate at which h increases with 7. It is, in the present case, what is called a space rate, or length rate, or linear rate, because the increase of h is compared with the increase of a length / (not because h is a length, but because 7 is a length).

If at the point Q the dotted line Qq be drawn touching the section curve at Q, then the gradient at Q is the tangent of the angle Qq0. The touching line at point P on the downward slope cuts 00' at p, and the tangent of PpO' is negative. It equals the tangent of PPO with sign reversed.

In the language of the Calculus this gradient is called the Differential Coefficient of h with respect to 7. Taking the forward direction as from A towards the right hand, the gradient is upward or positive from A to the summit C; downward or negative from C to the minimum point E; positive again from E to K, and negative from K to M. From N to R it is positive, and along RS

it is zero.

18. Gradients at Maxima and Minima.-At each maximum

and minimum point (C, E, K) the gradient is zero.

At each maximum point (C, K) it passes through zero from positive to negative. At a minimum point (E) it passes through zero from negative to positive.

At H there is also level ground, or zero gradient. Here, however, there is neither maximum nor minimum value of h. This point comes between two rising parts of the section: there is a positive gradient both before and after it. Although, therefore, we find zero gradient at every maximum and at every minimum point, it is not true that we necessarily find either a maximum or a minimum wherever there is zero gradient.

19. Change of Gradient. On the rising part of the ground it becomes gradually steeper from A to B; that is, the upward gradient increases. Otherwise expressed, there is a positive increase of gradient. From B to C, however, the steepness decreases; there is a decrease of gradient, or the variation of gradient is negative (the gradient itself being still positive).

Thus the variation of gradient being positive from A to B and negative from B to C, passes through the value zero at B, the point where the gradient itself is a maximum.

From C to D the gradient is negative, and becomes gradually steeper; that is, its negative value increases, or, otherwise expressed, its variation is negative. From D to E the gradient is negative, but its negative value is decreasing, that is, its variation is positive. Thus at Ď the variation, or rate of change of gradient, changes from negative to positive by passing through zero, and at this point D we have the steepest negative gradient on this whole slope CE. The steepest negative gradient, of course, means its lowest value. Thus at D we have a minimum value of the gradient.

20. Zero Gradients.-The distinction between the three parts C, E, and H, at all of which the gradient is zero, becomes now clear. At C the variation of the gradient is negative, and this gives a maximum height. At E this variation is positive, and here there is a minimum height. At H this rate of variation of the gradient is zero, and here, although the gradient be zero, there is neither maximum nor minimum height.

21. Discontinuity or Break of Gradient. Wherever there is a sharp corner in the outline of the section, as at I, J, R, S, T, U, there is a sudden change or break of gradient. This means that at each of these points there is discontinuity of gradient; and the above laws will not apply to such points.

Wherever there are points of discontinuity, either in the curve itself or in its gradient, special methods must be adopted in any investigations that may be undertaken in regard to the character