and each gradient would represent bs. to a certain scale. Continuing the above example, a tangent or gradient measuring unity on the paper, i.e., the tangent of 45°, would mean

This is the scale to which gradients from axis of x are to be measured; or

Gradients measured from the axis of X have a reciprocal interpretation and are to be measured to a reciprocal scale. Thus x/X means ft./ft.-lbs., or 1/lbs., that is, the reciprocal of a number of lbs.

35. Differential Coefficient, x-Gradient or X'.-The gradient from axis of of the curve at any point x, X, is called the "Differential Coefficient of X with respect to x," and is symbolised by

The phrase "Differential Coefficient of X with respect to x" is a cumbrous one. A shorter phrase is

the a-gradient of X;

and as this phrase is very easily understood and definitely descriptive, it is used in this treatise.

The gradient of the curve from the axis of X is the reciprocal of the above. It is called

the Differential Coefficient of x with respect to X, or the X-gradient of x;

36. Scale of X'.--The scales of X and x are in general different; and that of X' must always be different from either of these. The numerical relation between these scales and that of

dx

may be

dX

thus expressed:

Let

the scale of the x's be 1′′= =s units of the x kind or quality;

X's "

1" S

X

=

37. Sign of X'.-The sign of X' is

when the slope of the curve is such as to make both X and x increase positively at the

same time; it is when it makes one increase while the other

dx dX

decreases. Evidently must always have the same sign as X'.

The possible variations of X' and

dx dX

are very fully illustrated in

fig. 3.

In fig. 3, +x is measured towards the right and +X upwards; negative x's are measured towards the left and negative X's downwards.

The student should follow out the variations from + through 0

dx dX

to of both X' and throughout the lengths of all the four

curves A, B, C, and D in each of the four quadrants.

38. Subtangent and Subnormal.-In fig. 2 there are drawn three lines from a point xX of the curve, viz., a vertical, a tangent, and a normal. These intercept on the axis of x the lengths marked T and N, on the diagram. T, is called the subtangent and N, the subnormal.

Since (by definition) the tangent has the same gradient as the curve at its touching point, evidently

Here T measured to the x-scale, and interpreted as being of the same kind as the x's, is a true graphic representation of X/X'.

dx dx

But is of the same kind as x/X, and, therefore, would be

measured to the x-scale, Thus in order that N

not of the same kind as X/N, if N, were and interpreted as of the same kind as x. may be used as a true graphic representation of X'X, which is of the same kind as X2/x, care must be taken not to measure it to the x-scale, and not to interpret it as the same kind of thing

as x.

If the diagram were replotted, leaving the x-scale unaltered, and making the X-scale more open, the paper-height of X would be increased, and the paper-gradient X' would be increased in the same proportion. It can easily be shown that the paper-length of To would remain unaltered, while that of N, would be increased in a ratio which is the square of that in which X is increased. Similarly if, while the X-scale is maintained unchanged, the x-scale were altered so as to increase the paper-length of x, then the paper-gradient of the curve would be decreased in the same proportion as x is increased; T, would be increased in the same proportion as x; N, would be decreased in the same proportion.

Thus N in order to be a true graphic representation of X'X, a

quantity whose dimensions are those of X2/x, must be measured to the scale

may

In fig. 2, T, and N are taken upon the x-axis, and be termed the x-subtangent and x-subnormal. If the curve-touching line and the normal be prolonged to cut the X-axis, they and the horizontal through the touching-point will give intercepts on the X-axis, which may be termed the X-subtangent and X-subnormal, and may be written Tx and Nx. They are shown on fig. 2, and their proper scales are given below.

39. Scale of Diagram Areas.-An area enclosed by any set of lines upon such a diagram may be taken as the graphic representation of a quantity of the same kind as the product Xx, and must be measured to the scale, 1 sq. inch = (Ss) units of the (Xx) kind.

40. Table of Scales.-The following is a table of interpretations of the diagram. This diagram will be constantly used hereafter for both illustrations and proofs, which latter cannot be accepted as legitimate unless the whole nature of this manner of symbolic expression be intimately understood.

TABLE OF INTERPRETATIONS AND SCALES OF DIAGRAMMATIC OR GRAPHIC REPRESENTATIONS OF DERIVATIVE AND DERIVED FUNCTIONS.

41. Increments.-In going forward from a point on the curve a little way, a rise occurs if the gradient be upwards. The short distance measured along the sloping curve may be resolved into two parts, one parallel to axis of x, the other parallel to axis of X. These two parts are the projections of the sloping length upon the two axes. They constitute the differences of the pairs of x and X co-ordinates at the beginning and the end of the short sloping length. These differences are designated by the Greek 8; thus, see fig. 4,

Since the gradient X' is the ratio of rise to horizontal distance throughout a short length, it is evident that

SX = X'Sx.

If I and L be the co-ordinates, and if the gradient be called L', then this would be written

SL = L'Sl.

If y and Y were the co-ordinates, the gradient being called Y',

then

SY=Y'Sy.

42. Increment in Infinite Gradient.-These are the direct selfevident results of the definition of gradient, or differential coefficient. They do not, of course, apply to points where there is no. gradient, that is, to sharp corners in a diagram, where the direction of the diagram line changes abruptly.

If the diagram line run exactly vertical at any part, then for that part X' becomes infinite, and the equation appears in the form