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we reduce (12) to this form :


Theorem. 2ιπξ

2ιπξ $() = 4, +=1 A, cos + B, sin

(14), P

P which is the general expression of an arbitrary function in terms of a series of cosines and of sines. Or if we take


P.=(4,+B;), and tan e =

A we have $(6) = 4, + &** P. cos Come

Cine --)



which is the general expression in a series of single simple harmonic terms of the successive multiple periods.

Each of the equations and comparisons (2), (7), (8), (10), and Convers(11) is a true arithmetical expression, and may be verified by actual Fourier's calculation of the numbers, for any particular case; provided only that F (2) has no infinite value in its period. Hence, with this exception, (12) or either of its equivalents, (14), (16), is a true arithmetical expression ; and the series which it involves is therefore convergent. Hence we may with perfect rigour conclude that even the extreme case in which the arbitrary function experiences an abrupt finite change in its value when the independent variable, increasing continuously, passes through some particular value or values, is included in the general theorem. In such a case, if any value be given to the independent variable differing however little from one which corresponds to an abrupt change in the value of the function, the series must, as we may infer from the preceding investigation, converge and give a definite value for the function. But if exactly the critical value is assigned to the independent variable, the series cannot converge to any definite value. The consideration of the limiting values shown in the comparison (11) does away with all difficulty in understanding how the series (12) gives definite values having a finite difference for two particular values of the independent variable on the two sides of a critical value, but differing intinitely little from one another. If the differential coefficient


is finite for every value of

within the period, it too is arithmetically expressible by a series of harmonic terms, which cannot be other than the series obtained by differentiating the series for $($). Hence

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and we may




2iπξ ency of

*iP, sin

...(17), Fourier's


P series. and this series is convergent;

therefore conclude that the series for $($) is more convergent than a harmonic series with

1, 3, 5, etc.,

d'$($) for its coefficients. If has no infinite values within the

d2 period, we may differentiate both members of (17) and still have an equation arithmetically true; and so on. We conclude that if the nth differential coefficient of $($) has no infinite values, the harmonic series for $($) must converge more rapidly than a harmonic series with

1 1 1 1,

2" 3n: 4") for its coefficients. Displace- 78. We now pass to the consideration of the displacement rigid body. of a rigid body or group of points whose relative positions are

unalterable. The simplest case we can consider is that of the motion of a plane figure in its own plane, and this, as far as kinematics is concerned, is entirely summed up in the result of

the next section. Displace- 79. If a plane figure be displaced in any way in its own plane figure plane, there is always (with an exception treated in $ 81) one in its plane.

point of it common to any two positions; that is, it may be moved from any one position to any other by rotation in its own plane about one point held fixed.

To prove this, let A, B be any two points of the plane figure in its first position, A', B the positions of the same two after

a displacement. The lines A A', BB' will B'

not be parallel, except in one case to be presently considered. Hence the line equidistant from A and A' will meet that equidistant from B and B' in some point 0. Join (A, OB, OA', OB'. Then, evidently, because OA' = 0A, OB = OB and AB

= AB, the triangles OA' B' and OAB are B

equal and similar. Hence O is similarly situated with regard to A'B' and AB, and is therefore one and


the same point of the plane figure in its two positions. If, for Displace. the sake of illustration, we actually trace the triangle OAB upon plane figure

in its plane. the plane, it becomes 0 A'B' in the second position of the figure.

80. If from the equal angles A'OB', AOB of these similar triangles we take the common part A'OB, we have the remaining angles AOA', BOB' equal, and each of them is clearly equal to the angle through which the figure must have turned round the point 0 to bring it from the first to the second position.

The preceding simple construction therefore enables us not only to demonstrate the general proposition, $ 79, but also to determine from the two positions of one terminated line AB, A'B' of the figure the common centre and the amount of the angle of rotation.

81. The lines equidistant from A and A', and from B and B', are parallel if AB is parallel to A'B' ; and therefore the construction fails, the point 0 being

B' infinitely distant, and the theorem becomes nugatory. In this case the motion is in fact a simple trans


B lation of the figure in its own plane without rotation-since, AB being parallel and equal to A'B', we bave AA' parallel and equal to BB'; and instead of there being one point of the figure common to both positions, the lines joining the two successive positions of all points in the figure are equal and parallel.

82. It is not necessary to suppose the figure to be a mere flat disc or plane—for the preceding statements apply to any one of a set of parallel planes in a rigid body, moving in any way subject to the condition that the points of any one plane in it remain always in a fixed plane in space.

83. There is yet a case in which the construction in § 79 is nugatory—that is when AA' is paral

B lel to BB', but the lines of AB and 1 AB intersect. In this case, however, the point of intersection is the A point O required, although the former

B' method would not have enabled us to find it.


Examples 84. Very many interesting applications of this principle may of displacement in one be made, of which, however, few belong strictly to our subject, plane. and we shall therefore give only an example or two.

Thus we know that if a line of given length AB move with its extremities

always in two fixed lines 0A, OB,

any point in it as P describes an B В

ellipse. It is required to find the Q direction of motion of P at any in

stant, i.e., to draw a tangent to the

ellipse. BA will pass to its next Р

position by rotating about the point Q; found by the method of § 79

by drawing perpendiculars to OA and OB at A and B. Hence P for the instant revolves about Q, and thus its direction of motion, or the tangent to the ellipse, is perpendicular to QP. Also AB in its motion always touches a curve (called in geometry its envelop); and the same principle enables us to find the point of the envelop which lies in AB, for the motion of that point must evidently be ultimately (that is for a very small displacement) along AB, and the only point which so moves is the intersection of AB with the perpendicular to it from Q. Thus our construction would enable us to trace the envelop by points. (For more on this subject see § 91.)

85. Again, suppose AB to be the beam of a stationary engine having a reciprocating motion about A, and by a link BD turning a crank CD about C. Determine the relation between the angular velocities of AB and CD in any position. Evidently the instantaneous direction of motion of B is transverse to AB, and of D transverse to CD-hence if AB, CD produced meet in 0, the motion of BD is for an instant as if

it turned about 0. From this А

it may be easily seen that if
the angular velocity of AB be


w, that of CD is


is of course applicable to any combination of machinery, and we shall find it


W. A

very convenient when we come to consider various dynamical Examples

of displace. problems connected with virtual velocities.


ment in one

of rotations


86. Since in general any movement of a plane figure in its Composition plane may be considered as a rotation about one point, it is about evident that two such rotations may in general be compounded axes. into one; and therefore, of course, the same may be done with any number of rotations. Thus let A and B be the points of the figure about which in succession the rotations are to take place. By a rotation about A, B is brought say to B', and by a rotation about B, A is brought to A'. The construction of $79 gives us at once the point 0 and the amount of rotation about it which singly gives the same effect as those about A and B in succession. But there is one case of exception, viz., when the rotations about A and B are of equal

B amount and in opposite directions. In this case A'B' is evidently parallel to AB, and therefore the compound result is a translation only. That is, if a body

B' revolve in succession through equal angles, but in opposite directions, about two parallel axes, it finally takes a position to which it could have been brought by a simple translation perpendicular to the lines of the body in its initial or final position, which were successively made axes of rotation; and inclined to their plane at an angle equal to half the supplement of the common angle of rotation.

87. Hence to compound into an equivalent rotation a rota- Composition tion and a translation, the latter being effected parallel to the and translaplane of the former, we may decompose the translation into two plane. rotations of equal amount and opposite direction, compound one of them with the given rotation by § 86, and then compound the other with the resultant rotation by the same process. we may adopt the following far

А simpler method. Let 0 A be the


C translation common to all points in the plane, and let BOC be the angle of rotation about O, BO being drawn so that OA bisects the exterior angle COB. Take



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