.

=

acceleration of p, velocity of u, v=v: a, or acceleration = v2 : a=v2x curvature. Thus the normal acceleration is the same as that of a point moving with the same velocity in the circle of curvature.

This

P

The proposition may be further illustrated by means of the hodograph. Let ou represent the velocity of p, and uc be the velocity of u. may be resolved into um in the direction of ou, which is the rate of change in its magnitude, or v; and mc perpendicular to ou, which is ou multiplied by its angular velocity, or vp, since this perpendicular velocity may be regarded as belonging to

motion in a circle of radius v. Now since the curvature is :v, it follows that vox curvature.

=

This theorem is of great use in determining the curvature of various curves.

LOGARITHMIC MOTION.

A point is said to have logarithmic motion on a straight line when its distance from a fixed point on the line is equally multiplied in equal times.

When a quantity is equally multiplied in equal times, its flux is proportional to the quantity itself. Let mq, nr be two values of such a quantity, at the times represented by m, n; and let mq

= s, nr=

Then if we move mn to the right, keeping it always of the same length, the ratio of s to s, will remain constant; for the different intervals represented by mn will be equal, and the

quantity is equally multiplied in equal times. We shall have therefore s=ks,, where k is this constant multiplier. Therefore sks,, and consequently 8:8, 8:81. Hence we may write s ps, where p is a constant.

=

=

Conversely, when the flux of a quantity is proportional to the quantity itself, it is equally multiplied in equal times. For let s, s, be two values of the quantity, at times

LOGARITHMIC MOTION.

79

separated by a given constant interval. Then we know that s: 81 =s: s1, or ss1- ss1 = 0; that is (p. 65), the flux of the quotient ss, is zero. Now a quantity whose flux is zero does not alter, but remains constant. Therefore sks, where k is constant; so that in any interval equal to the given one the quantity is multiplied by the same number k.

A quantity whose flux is always p times the quantity itself is said to increase at the logarithmic rate p.

If two quantities increase at the same logarithmic rate, their sum and difference increase at the same logarithmic rate. For if u =pu, v = pv, then ù ± v = p (u ± v).

a

b

If a quantity increases at a finite logarithmic rate, it is either never zero or always zero. For let such a quantity be zero at a and have a finite value bq at b. At the middle point c of ab it must have a value which is the geometric mean of zero and bq; that is, zero. Similarly it must be zero at the middle point of be; and by proceeding in this way we may shew that it is zero at a point indefinitely near to any point on the left of b. If we make bd = cb, the value at d is a third proportional to zero and bq; that is, it is infinite. In the same way we may shew that the quantity is infinite at a point indefinitely near to any point on the right of b. It appears therefore that the quantity suddenly jumps from zero to bq and then to infinity; so that at by the rate of increase is infinite. Hence its ratio to bq is infinite, or the logarithmic rate is infinite.

This case corresponds to the case in uniform motion when the velocity is infinite and the point is at a certain finite position at a given instant. At all previous instants it was at an infinite distance behind this position; at all subsequent instants it is at an infinite distance in front of it.

If two quantities increase at the same (finite) logarithmic rate, they are either never equal or always equal. For their difference is either never zero or always zero.

Let P be the result of making unity increase at the logarithmic rate p for one second; then the result of

making it increase at that log. rate for t seconds is P' when t is a whole number, for the quantity is multiplied by P in each second. It is also one value of Pt when t is a commensurable fraction, say m:n. For let x be its value after t seconds, then the value after nt seconds is x", for the quantity is multiplied by x every t seconds. But nt =m, and we know the result of growing for m seconds is Pm. Therefore "P", or x is an nth root of Pm; that is, it is a value of P.

If we spread out the growth in one second over p seconds, the number expressing any velocity must be divided by p; hence if i was ps before, it must now =s. Hence the result of making unity increase at the log. rate p for one second is the same as the result of making it increase at the log. rate 1 for p seconds. Let e be the result of making unity increase at the log. rate 1 for one second; then P is a value of e" whenever p is commensurable.

We now make this definition: the result of making unity grow at the log. rate p for t seconds is denoted by et, and called the exponential of pt. The exponential coincides with one value of e to the power pt when pt is commensurable. Thus a has two values, +/a and -a; but e3 has only one value, the positive square root of the positive quantity e, whatever that is.

If set, then pt is called the logarithm of s. The name logarithmic rate is given to p because it is the rate of increase of the logarithm of s.

We have an example of a quantity which is equally multiplied in equal times in the quantity of light which gets through glass. If of the incident light gets through the first inch, of that will get through the second inch, and so on. Thus the light will be multiplied by for every inch it gets through; and, since it moves with uniform velocity, it is equally multiplied in equal times.

The density of the air as we come down a hill is an example of a quantity which increases at a rate proportional to itself, for the increase of density per foot of descent is due to the weight of that foot-thick layer of air, which is itself proportional to the density.

INFINITE SERIES.

ON SERIES.

We know that when x is less than 1, the series

81;

is of such a nature that the sum of the first n terms can

1

be made as near as we like to

by taking n large

1 Ꮳ

enough. For the sum of the first n terms is

since x is less than 1, x" can be made as small as we like by taking n large enough. The value to which the sum of the first n terms of a series can be made to approach as near as we like by making n large enough. is called the sum of the series. It should be observed that the word sum is here used in a new sense, and we must not assume without proof that what is true of the old sense is true of the new one: e. g. that the sum. is independent of the order of the terms. When a series has a sum it is said to be convergent. When the sum of n terms can be made to exceed any proposed quantity in absolute value by taking n large enough, the series is called divergent.

A series whose terms are all positive is convergent if there is a positive quantity which the sum of the first n terms never surpasses, however large n may be. For consider two quantities, one which the sum surpasses, and one which it does not. All quantities between these two must fall into two groups, those which the sum surpasses when n is taken large enough, and those which it does not. These groups must be separated from one another by a single quantity which is the least of those which the sum does not surpass; for there can be no quantities between the two groups. This single quantity has the property that the sum of the first n terms can be brought as near to it as we please, for it can be made to surpass every less quantity.

The same thing holds when all the terms are negative, if there is a negative quantity which the sum of the first n terms never surpasses in absolute magnitude.

When the terms are all of the same sign, the sum of the series is independent of the order of the terms.

n

For let P be the sum of the first n terms and P the sum of the series, when the terms are arranged in one order; and let Q be the sum of the first n terms and Q the sum of the series, when the terms are arranged in another order. Then P2 cannot exceed Q, nor can Q exceed P; and Pr Qn can be brought as near as we like to P, Q by taking n large enough. Hence P cannot exceed Q, nor can Q exceed P; that is, P=Q.

m

When the terms are of different signs, we may separate the series into two, one consisting of the positive terms and the other of the negative terms. If one of these is divergent and not the other, it is clear that the combined series is divergent. If both are convergent, the combined series has a sum independent of the order of the terms, For let P be the sum of m terms of the positive series, Q, the sum of n terms of the negative series, P, Q, the sums of the two series respectively; and suppose that in the first m+n terms of the compound series there are m positive and n negative terms, so that the sum of those m+n terms is Pm-Qn. Then P-Pm Q-Q can be made as small as we like by taking m, n large enough; therefore P-Q¬ (Pm Q) can be made as small as we like by taking m+n large enough, or P-Q is the sum of the compound series. It is here assumed that by taking sufficient terms of the compound series we can get as many positive and as many negative terms as we like. If, for example, we could not get as many negative terms as we liked, there would be a finite number of negative terms mixed up with an infinite series of positive terms, and the sum would of course be independent of the order.

If, however, the positive and negative series are both. divergent, while the terms in each of them diminish without limit as we advance in the series, it is possible to make the sum of the compound series equal to any arbitrary quantity C by taking the terms in a suitable order. Suppose C positive; take enough positive terms to bring their sum above C, then enough negative terms to bring the sum below C, then enough positive terms to bring the sum again above C, and so on. We can always per