54 H. PARTICULAR MOTIONS OF A POINT. 



Accordingly we have 



therefore *P > > as above stated. 1 ) 



If in the integral 70) we substitute for the factor 



the greatest and least values that it takes during the motion, namely 

 = a and z = ft, we shall get closer limits between which ty lies. 

 If we then make a and /3 approach I, *P will approach a right angle, 

 so that the horizontal projection tends to be a closed curve. 



This case may also be treated directly. Our equations 40) were 



Now we have =~J/Z 2 (x* + / 2 ) = l( 1 - j^-J and developing 

 by the binomial theorem, 



.- 



If now x and y are small with respect to I and we neglect small 



d* z 

 quantities of the second order, z is constant. Then -^ = 0, and 



from the third equation above, 



i _ 9 



T 



Inserting this value of "k in the first two gives 



_ 

 dt* ' I > dt* 



the integrals of which are 



where a, 6, a, /3 are arbitrary constants, giving elliptic harmonic 

 motion of the same period as that of the small plane harmonic motion. 

 Another important case is that in which the two roots a and /3 

 are equal. We then have 2 and & constant, and 



1) This treatment is taken from Appell, Mecanique Eationelle. The proof 



that *F > -is due to Puiseux. 



