25. Hyperbolic Functions.

By analogy with the exponential values of the sine, cosine, tangent, etc., the exponential functions

and called the hyperbolic sine, cosine, tangent, etc., of 0, and as a class are styled hyperbolic functions.

φ

2) cosh’0−sinh0=1;

(3) sin(0+c)=sin 0 cosh + cos 0 sinh ¢ ;

with many other formulae analogous to, and easily deducible from, the common formulae of Trigonometry.

an inverse hyperbolic function of x analogous to the inverse trigonometrical function sin1.

This species of function however is merely logarithmic;

while corresponding results hold for cosh 'a, tanh 'x, etc.

2. Show that the co-ordinates of any point on the rectangular hyperbola x2-y2=a2 may be denoted by a cosh 0, a sinh 0.

26. All measurable quantities are estimated by the ratios which they bear to certain fixed but arbitrary units of their own kind. The whole measure of a quantity thus consists of two factors--the unit itself and an abstract number which represents the ratio of the measured quantity to the unit. The magnitude of the unit should be chosen as something comparable with the quantity to be measured, otherwise the abstract number which measures the ratio of the quantity to the unit will be too large or too small to lie within the limits of comprehension. For instance, the radius of the earth is conveniently estimated in miles (roughly 4,000); the moon's distance in earth's radii (about 60); the sun's distance in moon's distances (about 400); the distance of Sirius in sun's distances (at least 200,000). Again, for such relatively small quantities as the wave-length of a particular kind of light, one ten-millionth of an inch is found to be a sufficiently large unit: the wave-length for light from the red

B

end of the spectrum being about 266, that from the violet end 167 such units (Lloyd, "Wave Theory of Light,” p. 18).

27. Any comparison of two quantities is equivalent to an estimate of how many times the one is contained in or contains the other; that is, the one quantity is estimated in terms of the other as a unit, and according as the number expressing their ratio is very large compared with unity or a very small fraction, the one is said to be very large or very small in comparison with the other. The terms great and small are therefore purely relative.

The standard of smallness is vague and arbitrary. An error of measurement which, centuries ago, would have been reckoned small would now be considered enormous. The accuracy of observation, and therefore the smallness of allowable errors of observation, increases with the continual improvement in the construction of instruments and methods of measurement.

28. Orders of Smallness.

part

A 1012

If we conceive any magnitude A divided up into any large number of equal parts, say a billion (1012), then each is extremely small, and for all practical purposes negligible, in comparison with A. If this part be again subdivided into a billion equal parts, each= each of

A

10240

A

these last is extremely small in comparison with

and

1012

so on.

We thus obtain a series of magnitudes, A,

A A A

1012, 1024 1036›

each of which is excessively small in

comparison with the one which precedes it, but very large compared with the one which follows it. This furnishes us with what we may designate a scale of smallness.

More generally, if we agree to consider any given fraction f as being small in comparison with unity, then fA will be small in comparison with A, and we may term the expressions ƒA, ƒ2A, ƒ3A, ..., small quantities of the first, second, third, etc., orders; and the numerical quantities f, f2, f3, ..., ,..., may be called small fractions of

3

the first, second, third, etc., orders.

Thus, supposing A to be any given finite magnitude, any given fraction of A is at our choice to designate a small quantity of the first order in comparison with A. When this is chosen, any quantity which has to this small quantity of the first order a ratio which is a small fraction of the first order, is itself a small quantity of the second order. Similarly, any quantity whose ratio to a small quantity of the second order is a small fraction of the first order is a small quantity of the third order, and So on. So that generally, if a small quantity be such that its ratio to a small quantity of the pth order be a small fraction of the qth order, it is itself termed a small quantity of the (p+q)th order.

29. Infinitesimals.

If these small quantities Af, Af2, Aƒ3, ..., be all quantities whose limits are zero, then, supposing f made smaller than any assignable quantity by sufficiently increasing its denominator, these small quantities of the first, second, third, etc., orders are termed infinitesimals of the first, second, third, etc., orders.

From the nature of an infinitesimal it is clear that, if

any equation contain finite quantities and infinitesimals, the infinitesimals may be rejected.

30. PROP. In any equation between infinitesimals of different orders, none but those of the lowest order need be retained.

Suppose, for instance, the equation to be

A1 + B1 + C1 + D2+ E2+F3+... = 0,

1

1

1

2

2

(i.)

31. PROP. In any equation connecting infinitesimals we may substitute for any one of the quantities involved any other which differs from it by a quantity of higher order.

f, denoting an infinitesimal of higher order than F, we

have

F1+B1+C1+f+D2+... = 0,

2