238 Prof. J. D. Everett on Dynamical 



will accordingly represent what are called in acoustics 

 " curves of beats," the beats being strongest when A and B 

 (and therefore also k x K and & 2 B) are nearly equal. 



If we start from a condition in which the pendulum x is at 

 rest in the zero position, the equations will be 



x — A(cos £l x t — cos XV), ] 



f = A(A' x cos Q, l t + k 2 cos I2 2 £) J-. . . (25) 



= A (cos H^ -r cos fl 2 t), nearly, J 



When t is such that cos O^ and cos £l 2 t are each sensibly 

 unity, the first pendulum will have sensibly zero excursions, 

 and the second pendulum will have maximum excursions. 

 When one of the cosines is sensibly 1, and the other sensibly 

 — 1, these conditions will be reversed. In fact, we have 



x = 2 A sin —=-z — t . sm ^ — *> rigorously, 



_. OA 12,-12, £t 2 +o, 1 | ' (26) 



£ = 2 A cos — -*q £ . cos » — £, nearly, 



showing that the excursions are 



2 A sin J(X2 2 — Hj.)* and 2 A cos i (I2 2 — fl^, 



each of which in its turn vanishes when the other is 2 A. 

 This can be illustrated by hanging two equal pendulums from 

 the same stand. As regards phases, the comparison of the 

 two factors 



sin \ (fl 2 + Hi) t and cos \ (f2 2 -f X2i) t 



shows that, at first, (• is earlier than x by a quarter period. 



§ 10. Next take the case of one pendulum suspended from 

 another, each consisting of a heavy particle at the end of a 

 weightless thread. 



Upper mass 1, Lower mass s ; 



Length of upper pendulum a, of lower . b ; 

 Displacement of upper mass x, of lower . f ; 



the displacements being measured horizontally from a vertical 

 through the fixed point of support, and being so small that 

 vertical accelerations may be neglected. Then, since the 

 tensions are sg and (l-j-s)y, we have 



t—x ,<. , x x sq^ /s 1-M\ 



