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71. If ds be an element of the boundary of any portion of the plane xy which is occupied wholly by moving liquid, and if dr be an element of the normal to ds drawn inwards, we have, by Art. 44,

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the integration extending round the whole boundary. If this boundary be a circle, and if r, ✪ be polar co-ordinates referred to the centre P of this circle as origin, the equation (8) may be written

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Hence the value of the integral føde, i.e. the mean-value of

over a circle of centre P, and radius r, is independent of the value of r, and therefore remains unaltered when r is diminished without limit, in which case it becomes the value of at P.

If the region occupied by the fluid be periphractic, and we apply (8) to the space enclosed between one of the internal boundaries and a circle with centre P and radius r surrounding this boundary, and lying wholly in the fluid, we have

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where the integration in the first member extends over the circle only, and 2πM denotes the flux into the region across the internal boundary. Hence

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i.e. the mean value of 4 over a circle with centre P and radius r is equal to Mlogr + C, where C is independent of r but may vary

with the position of P. This formula holds of course only so long as the circle embraces the same internal boundary, and lies itself wholly in the fluid.

If the region be unlimited externally, and if the circle embrace the whole of the internal boundaries, and if further the velocity be everywhere zero at infinity, then C is an absolute constant; as is seen by reasoning similar to that of Art. 46.

It may then be shewn, exactly as in Art. 51, that the value of at a very great distance r from the internal boundary tends to the value M log r+C. In the particular case when M = 0 the limit to which tends at infinity is finite; in all other cases it is infinite, and of the same sign as M.

We infer, as in Art. 52, that there is only one single-valued function which (a) satisfies the equation (7) at every point of the plane xy external to a given system of closed curves, (b) makes

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the value of equal to an arbitrarily given quantity at every dn point of these curves, and (c) has its first differential coefficients all zero at infinity.

72. The kinetic energy of a portion of fluid bounded by a cylindrical surface whose generating lines are parallel to the axis of 2, and two planes perpendicular to the axis of z at unit distance, is given by the formula

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where the surface-integral is taken over the portion of the plane xy cut off by the cylindrical surface, and the line-integral round the boundary of this portion.

If the cylindrical part of the boundary consist of two or more separate portions one of which embraces all the rest, the enclosed region is multiply-connected, and the equation (10) needs a correction, which may be applied exactly as in Art. 66.

If we attempt, by a process similar to that of Art. 65, to calculate the energy in the case where the region extends to infinity, we find that its value is infinite, except when M = 0.

For if we introduce a circle of great radius r as the external boundary of the portion of the plane xy considered, we find that the corresponding part of the integral on the right-hand side of (10) tends, as r increases, to the value πpM (M logr+ C), and is therefore ultimately infinite. The only exception is when M=0, in which case we may suppose the line-integral in (10) to extend over the internal boundary only.

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These are the conditions that +i, where i stands for √− 1, should be a function of the 'complex' variable x + iy. For if

+ i =ƒ (x + iy)..

(12),

we have

d

d

dy ($ + i¥) = if'(x + iy) = i († + i¥) .........................(13),

dx

whence, equating separately the real and the imaginary parts, we obtain (11).

Hence any assumption of the form (12) gives a possible case of irrotational motion. The curves = const. are the curves of equal velocity-potential, and the curves const. are the stream-lines. Since, by (11),

=

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we see that these two systems of curves cut one another at right angles, as we have already proved. See Art. 27. Since the relations (11) are unaltered when we write for 4, and 6 for y, we may, if we choose, look upon the curves const. as the equipotential curves, and the curves const. as the stream-lines; so that every assumption of the kind indicated gives us two possible cases of irrotational motion.

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74. The fundamental property of a function of a complex variable, from which all others flow, is that it has a differential coefficient with respect to that variable. If p, denote any functions whatever of a and y, then corresponding to every value

of x+iy there must be one or more definite values of +i; but the ratio of the differential of this function to that of x+iy, viz.

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depends in general on the ratio de: dy. The condition that it should be the same for all values of this ratio is

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We may therefore take this property as the definition of a function of the complex variable x+iy; viz. such a function must have, for every assigned value of the variable, not only a definite value or system of values, but also for each of these values a definite differential coefficient. The advantage of this definition is that it is quite independent of the existence of an analytical expression for the function.

The theory of functions of this kind has received considerable development at the hands of Cauchy, Riemann, and others; and has grown into an important branch of mathematical analysis. We give here only such elementary notions connected with the subject as are of immediate hydrodynamical interest *.

75. We assume the student to be acquainted with the method of representing the symbol x+iy by a vector drawn from the origin of rectangular co-ordinates to the point (x, y).

In this method the sum of two vectors is defined to be the vector drawn from the origin to the opposite corner of the parallelogram of which they form adjacent sides.

The effect of multiplying one vector x+iy by another a + ib is to increase its length in the ratio r 1, and to turn it in the

* The reader who wishes for an elementary exposition of the analytical theory may consult Durège: Elemente der Theorie der Functionen einer complexen veränderlichen Grösse. 2nd ed., Leipzig, 1873.

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positive direction (i.e. from x to y) through an angle 0, where b r = √(a2 + b2), and 0 is the least positive value of arc tan With respect to the expression a +ib, r is called the 'modulus,' and the 'amplitude.'

The meanings of subtraction and division of vectors follow at once from the considerations that they are the operations inverse to those of addition and multiplication, respectively.

With these conventions, the addition, multiplication, &c., of vectors are performed according to the same laws of operation as in common algebra.

76. For shortness we denote the complex quantities + if, and xiy by the letters w, and z, respectively. These symbols not being required at present in their former meanings may without inconvenience have these new ones assigned to them. Then w being any function of z, according to the definition of Art. 74, we have corresponding to any point P of the plane xy (which we may call the plane of the variable z) one or more definite values of w. Let us choose any one of these, and denote it by a point P' of which, are the rectangular co-ordinates in a second plane (the plane of the function w). If P trace out any curve in the plane of z, P' will trace out a corresponding curve in the plane of w. By mapping out the positions of P' corresponding to the various points P of the plane xy, we may exhibit graphically all the properties of the function w.

Let now be a point infinitely near to P, and let Q' be the corresponding point infinitely near to P'. We may denote PQ by dz, P'Q' by dw. The vector P'Q' may be obtained from the

dw

vector PQ by multiplying it by the differential coefficient dz' whose value is by definition dependent only on the position of P, and not on the direction of the element dz (PQ). Now the effect dw of multiplying any vector by the complex quantity is to dz increase its length in some definite ratio, and to turn it in the positive direction through some definite angle. Hence, in the

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