68 PROCEEDINGS OF THE AMERICAN ACADEMY. 



This procedure gives the equation 



2 k(e% + e\) - 3(# 2 o - eh) = o, (13) 



which is satisfied when 6 = — 6q and from this we may find, by aid 

 of a second development, the very approximate result, 



0! = O - § !c6 2 . (14) 



If terms of the fourth order are kept, we may obtain the expressions 



6 X = 6 - I \k6% + H- 2 ^ 3 o, (15) 



but for most practical purposes (14) is quite accurate enough. 



After the swinging system has come momentarily to rest at the 

 elongation — do, it moves in the positive direction with an angular 

 velocity which increases to a maximum at a position determined by 

 the constants of the motion, and has the value o) when 6 = 0. 



It is easy to see from (3) that 



wo 2 = 2 c + m, (16) 



and from (9) that 



2c = -m0. + k6 )er- k \ (17) 



so that 



W0 2 = rn - m (i + kdo)e- ke ° ; (18) 



and it is evident that fc>o is greater, other things being given, the greater 

 the amplitude of the motion; that is, the greater the value of 6q. 

 Equation (16) shows, however, that the greatest value which <wo can 

 have is <y/ra, and it is interesting to determine what elongation on the 

 positive side of the zero point corresponds to this angular velocity at 

 = 0. 



If in (12) we suppose to grow large without limit, 6\ approaches 

 the limit 1/k, and it appears that however great the angle through 

 which the swinging system may have been turned out of the position 

 of equilibrium at the outset, the amplitude of the next elongation can- 

 not be greater than l/kth of a radian, and the next turning point to 

 this (on the same side of the zero as the original disturbance) must 

 come at an angular distance from the position of equilibrium not 

 greater than about 0.594//c radians. The subsequent swings decrease 

 regularly in amplitude in such a manner as to make the logarithmic 

 decrement decrease towards zero. At any time during the motion the 

 determination of two successive amplitudes serves to determine k 

 through (12), for it is easy to solve the transcendental equation to any 

 desired accuracy. 



