]^48 UNDULATORY THEORY OV LIGHT. 



may be assumed as established. When tlio body iu its return arrives at 0, it 

 will accordingly be moving with the velocity represented by CD, the radius of 

 the circle, and its inertia will carry it forward iu the dircctiou CB. It will now 

 be resisted by forces similar iu degree but contrary iu direction to those Avhich 

 urged it from A to 0, and its velocity will decrease as it before increased, uutil 

 it is brought to rest again at B, when it will once more return. Supposing no 

 forces or resistances to be called into action but those embraced in our hypothesis, 

 there is no reason why this reciprocating motion should not coutinue indefinitely. 

 We have an approximate illustration of the case under considei-ation in au 

 ordinary pendulum. "Wlum the pendulum is drawn from the vertical position, 

 the component of gi-avity which urges its return is a force very nearly propor- 

 tional, at every instant, to its distance from the position of rest. Were its j)ath 

 a cycloidal instead of a circular arc, this proportionality would be rigorously 

 exact. Its beats are therefore sensibly equal in time, whether it swing through 

 twenty degrees or through only one. 



If Ave suppose the pendulum to be so suspended that its vibrations are not of 

 necessity restricted to a single plane, we shall be able to conceive, without much 

 difficulty, what must happen in another case important to be considered, viz : 

 that in which a body, already in a state of vibration, is acted upon by a second 

 disturbing force, not directed in the same plane with the first. To simplify the 

 •supposition as much as possible, let us imagine that, at the moment when the 

 body, in its return from A, is passing the centre C, it receives an impulse in the 

 direction D, at right angles to its actual movement, capable of giving it, in- 

 stantaneously, the same velocity towards D, which it already has towards B. 

 By the law of the composition of forces, it will take the direc- 

 ^ tion C^I, which is the diagonal of the rectangle formed ujion 

 CB, CD ; and its subsequent vibrations will be represented ia 

 _ff extent and direction by the line NM. It will be seen that the 

 extent of its excursions in the direction AB remains unaltered, 

 since the lines MB and AN are parallel; but it performs, at the 

 " same time, an equal vibration in the direction DP, since DM 

 Fig 2s. j^mi -^p are also parallel. 



Let us suppose, however, that the second impulse takes effect on the body 

 not at the point C, of its greatest velocity, but at A, where its motion is null, 

 and at the instant when it is about setting out on its return to C. It will 

 vibrate, as before, between the parallels NP and MDR ; but it will reach the 

 limits of its vibration in this direction when it is at the middle of the vibration 

 in the other. At the (md of half a vibration, therefore, it will be found at D 

 instead of at C, as in the former case ; at the end of the next l^alf at B ; at the 

 end of the third half at P; and at the end of the fourth, or of a complete double 

 vibration, at A, the point of starting. Apjjarently, therefore, inider these cir- 

 cumstances, the orbit of the body is a circle. We shall see that this is really so. 

 In fact, if we represent by r the radius vector of the body, and by 

 X and y its co-ordinates to the axes AB and DP, we shall have the 

 equation x^ + ]/^^=?-'^. Whence, taking the difiercntial — 



xdx+ydij ^=rdr. (a) 



method, a little further on iu the text, iu regard to the measure of the time elapsed since the 

 beginning of the vibration, at any given moment, and in any position of the vibrating body. 



Since t varies as cos~ -, it is obvious that, if a circle be described on the path of the vibra- 



a 

 ting body as a diameter, and if an ordinate to the circumference be drawn from the position 

 of tlie vibrating body at any moment, the arc of the circle intercepted between the origin and 

 this ordinate will be" the measure of the time elapsed since the vibration began. The arc must 

 be taken always iu the same constant direction around the circumference, and the ordinate 

 must be positive for the advancing and negative for the returning movement. In like man- 

 ner, the arc intercepted between two such ordinates, will measure the time intervening between 

 the moments when the vibrating body occupied the points from which the ordinates are drawn. 



