VIBRATION. ^49 



Now, by the law of velocities above given, if a be put for tlie maximum ve- 

 locity, or that which the body has in passing C, and if (p be put for the arc of 

 the circle on AB which i,s included between the origin, A, and y, the velocity 

 in the direction AB, will be a^mcp ; and that in the direction CD, aco^w. 

 Ilcnce, for the differentials of the co-ordinates x and y, we have, putting t for 

 the time — 



Jx-= — aa'mcdt; a^nd dy = acof-(fdt. 



But by construction x=^rcos^, and ?/=:?-sin^. 

 Substituting these values in ecpation (a) there results — 



— crsin^cos^d'^ + crsin^cos^-cZ/! ^ rdr. 



dr 

 Or, ^ = 0. 

 dt 



From which it appears that the radius vector is constant and the orbit a circle. 

 Also the motion in the circle is uuifbrm. For if dc> be the increment of the 

 arc, 



de^^^dsi^ + dy'^. 



And sitbstituting the values of dx and dy, given above, we have, 



da^ = a^sin^^'fZi;^ + a'^co&"<fdl^ = ci^di^. 



A 1 ^^ 

 And-,-=t'^a. 



dt 



That is to say, the velocity of the movement in the circle will be uniform, and 

 will be equal to the maximum velocity of the plane vibration. 



Hence it follows that if, at the moment when a body, vibrating in a recti- 

 linear path, is at the limit of its movement, a second body sets out from the same 

 point, in a circle of which the path of the first is a diameter, with a uniform ve- 

 locity equal to the maximum velocity of the first, the line which joins tlie two 

 will move parallel to itself, and vrill always be perpendicular to the path of the 

 plane vibration. 



We hence obtain a convenient measure of the time which has elapsed since 

 the beginning of vibration, when the body is at any point, as H, Fig. 27, of its 

 path. For, taking as the unit of time the duration of a complete double vibra- 

 tion, and employing the ordinary symbol, 2-, to denote the circumference of a 

 circle whose radius is 1, 2-t may express the entire space passed over by a body 

 making its revolutions in such a circle isochrouously with the movements of the 

 vibrating body, and t (whether its value be integral or fractional) will then de- 

 note at once the number of vibrations which have taken place and the number 

 of units of time which have elapsed since the beginning. Assuming the starting 

 point to be at the commencement of a double vibration, every integral value of i 

 will denote a return to the original position, and every fractional excess will de- 

 note a corresponding progress in a new vibration. Thus if the body be at II, 

 the fractional part of 2-t will be the arc AG, and this will have the same ratio 

 to the entire circle which the time of describing the portion of jjath, AH, has 

 to the total time of double vibration. 



Let us now suppose that the second impulse (still normal to the first) is not 

 equal to the first, but greater or smaller; and that tlie vibration which it would 

 independently produce has an amplitude (measured from the centre) represented 

 by a'. The figure of the orbit will, in this case, be an ellipse, with the greater 

 or lesser axis in the direction of the original vibration, according as. a is greater 

 or less than a'. The velocities in the direction of a will still, as before, be 

 represented by asin2-rt, while those in the direction of a' will be represented 

 by a'cos2-t; and these expressions will also stand- for the ordinates of the 

 orbit, y aud x. 2-t here takes the place of the former symbol, f . 



