and the difference between the rise from W to 2 and the rise from 1 to N is

But by § 170 these differences equal each other. Cancelling out the common product Sn.dw, we have the equality

W does not involve n, and similarly
W'

Using the nomenclature of the end of § 172, since

Thus it is indifferent whether the n or the w differentiation be taken first, and whether F"nw(nw) or F'wn(nw) be used as symbol. Although these second-gradients, calculated in these two different ways, have the same value, they represent two perfectly distinct physical phenomena. The one is the northward rate of change of the westward gradient of h. The other is the westward rate of change of the northward gradient of h. That these are equal, whatever kinds of physical quantities be represented by h, n and w, is a proposition of mathematical physics that is most interesting and fertile in its various concrete applications.

175. Second x,y-Gradient.-When two functions of x and y fulfil the condition of being the partial x and y gradients of one and the same function, then the function formed by adding the products of these functions by dc and dy respectively, is said to be ax Οχ a complete differential, and be the functions, ду

Thus if

дх

ascertained to be the partial x and y gradients of the same function

is a "complete differential," and this latter is said to be "integrable." If this has been found, by accurate deduction from correct observation of physical fact, to be the increment of a real

physical quantity, then it is certain that the function is theoretically, integrable (although the integration may be impracticably difficult) and that its two parts will fulfil the condition of § 174. Of course, it is easy for the pure mathematician to invent functions of this sort that are not integrable, and incorrect physical observation or inaccurate deduction from physical investigation may lead to differentials that are not integrable; but such have no real physical meaning.

176. Double Integration by dx and dy.-Conversely, if any function of two independent variables, x, y, be twice integrated first by dx and then by dy, the result will be the same as if first integrated by dy and then by dx, being in either case the sum of a function dependent on both x and y and of two other functions depending separately, one of them on x alone and the other on alone.

These two latter functions are introduced by the integrations in the same way as constants are introduced by integrations with respect to one variable; the one function being a constant with respect to one variable, and the other being a constant with respect to the other variable.

where the laws of the functions X and Y must be determined by "limiting conditions.”

The finding of x from the given value of

d2x is called the

dx dy

double integration of this function, and is symbolised by

d2x_dx dy

dx dy

or [p(xy)dx dy if p(xy) be the given functional form of

or

d2x

dx dy'

177. Graphic representation of Double Integration by da and dy. The meaning of the double integration of (xy) may be represented graphically in the following different manner.

Let (xy) be represented by the height of a surface from a datum plane, the co-ordinates parallel to this plane being x and y.

Then the first integration [p(xy)da may be considered as extending along a section perpendicular to the datum plane and parallel to the x-axis, in this integration y being a constant. The result of this integration is a general formula for the area of any such section. Two such sections at the very small distance dy apart will inclose between them, under the surface and above the datum plane, a volume equal to dy multiplied by the area of the y-constant section

at the middle of dy. This volume is, therefore, {[(y)dx}. dv,

and the whole volume under the surface and above the datum

plane therefore properly represents [[p(xy)dxdy. [[p(xy)dady. This geometric

conception is more easily grasped if the integration be taken between limits.

178. Connection between Problems concerning One Independent Variable and those concerning Two Independent Variables.—In an investigation concerning two mutually dependent variables, such as those in Chapters I. to VII., the two variables may always be represented by the co-ordinates to a plane curve. This curve may be looked on as a plane section of a surface, the three co-ordinates to the points upon which are related to each other by the more general kind of law dealt with in this chapter. Thus the former problems may always be conceived of as the partial solutions of more general laws connecting three variables with only one specific relation between them. The problems of Chapters I. to VII. may thus be considered special cases of more general problems of the kind now dealt with, and each of them might be deduced by specialising from a more general

theorem.

CHAPTER IX.

MAXIMA AND MINIMA.

179. General Criterions.-In fig. 1, at the parts C, E, H, K, R, S, U, the l-gradient of h is zero. The points C and K are places where h rises to a maximum, the maximum K being greater than the maximum C, but the phrase "maximum" being understood to mean a value greater than any neighbouring value on either side. E is a place where h falls to a minimum.

Thus the gradient falls to zero wherever there is either a maximum or a minimum value.

At the maxima points, C and K, the forward gradient passes through zero by changing from positive to negative, that is, the increase of the gradient is negative at these places.

At the minimum point E, the forward gradient changes from negative to positive, so that its increase is positive.

Thus the criterion for distinguishing between a maximum and a minimum is, that at the former the second gradient or second differential coefficient is negative, while at a minimum point it is positive.

It is not always, however, necessary to find the sign of the second gradient in order to make sure whether the point is a maximum or a minimum. For instance, if it be known that at the place where the first gradient is zero, the value of h is positive, and if it be also known that at two points near and on either side of this place the value of h becomes zero, or of any positively less value than at this place of zero gradient, then evidently this place gives a maximum.

At the place H, fig. 1, the second gradient is zero, because to the left of H it is negative, while to the right of H it is positive. This case of zero second-gradient occurring along with zero firstgradient is the limiting case coming in between the two previous ones, giving respectively maxima and minima; and it gives neither a maximum nor a minimum. This includes the case of the dead level RS, where also both first and second gradients are zero.

Usually one's general knowledge of the physical phenomenon being investigated is sufficient, without need of evaluating the second gradient, to indicate whether or not there is any such point as H. That is, the practical man who thinks of what he is working at, and does not follow blindly mere mathematical formulas, runs substantially no risk of mistaking such a point as H for either a maximum or a minimum point.

180. Symmetry.-In very many practical problems conditions of symmetry show clearly where a maximum or minimum occurs without the need of investigating either first or second gradient. Thus, if a beam be symmetrically supported, symmetrically loaded, and have a symmetrical variation of section on either side of a certain point of its length, which point is then properly called its centre, then the bending moment and the deflection each reach a maximum at this centre. Such considerations are to be utilised wherever possible, and their use is sometimes more profitable in practical result than the more strictly mathematical process.

181. Importance of Maxima in Practical work.—As examples

of the utility of these theorems may be cited the finding of the positions and magnitudes of maximum bending moments, of maximum stresses, of maximum deflections, of maximum velocities, of maximum accelerations of momentum, of the positions of rolling load on bridges to give maximum stress in any given member of the bridge, etc., etc. All these things are of special importance in the practical theory of engineering. In the jointing of pieces together in machines and static structures, it is never possible to obtain uniform stress over the various important sections of the joint. It is of the greatest importance to find the maximum intensities of stress on such sections, because the safety of the construction depends on the maximum, hardly ever upon the average, stress. The average stress on the section is found by dividing the whole load on the section by the whole area of the section. Such average stresses are often very different from the maximum stress, and no reliance ought to be placed upon them as measures of strength and safety.

Another class of technical problems in which maxima points are of paramount importance is that in which two or more sets of variable driving efforts, or of variable resistances, are superimposed in a machine. Thus a first approximation to the turning moment on the crank shaft of a steam engine of one cylinder, makes this moment vary as sin a, where a is the angle at which the crank stands from the dead point. If there be two cylinders in which the total steam pressures are P, and P2, constant throughout the stroke, and the two cranks, keyed on the same shaft, stand apart by an angle A; then a being the angle from dead point of one crank, (a+A) is that of the other. A remains constant while a varies. If S, and S2 be the two strokes, the total turning moment on the shaft is

1

2

1

which reaches a maximum when its a-gradient is zero; that is, when

S2P2
S1P1

2

This ratio is minus unity when S,P2 = SP1; and if, further, A = 90°, then a = 45° at the maximum.

182. Connecting Rod Bending Moments.-The connecting rod of an engine is at each instant bent by transverse accelerations of momentum, which, taken per inch length, would increase uniformly from zero at the crosshead to a certain amount at the crank end if the section of the rod were uniform. The