MATHEMATICS: A. EMCH 219 
e 
■2/[fi (fe) 
<r(t - ti) <r(t- t 2 ) 
From this is found 
2 (7 ft) (7 (t 2 ) <j\ (t) 
2 a(h)(r{t 2 ) a\(t) 
These solutions may also be obtained from the differential equations. 
and a similar equation for y. (4) and (5) are uniform analytic functions of t, 
with oo as the only essential irregularity, and they assume real values for 
real values of t. 
Increasing t by 2 mi w h it is found that the motion will be periodic, when 
2»ihi(fl + b) - wfe(a) + ((b)]} = Ikiir, (7) 
k being a positive integer. When t increases by 2 w L , 6 will increase by the 
amount 
$ = - 2*Ma + 6) - ^K(a) + ((b)]} , (8) 
so that in case of a periodic motion, from (7) and (8), 
<£ = 2kw/mi. (9) 
77?e described by the pendulum is now a closed curve intersecting every 
level between the lowest and highest position in 2 mi points and winding k times 
around the z-axis, before it closes. 
Moreover a function-theoretic investigation of the periodic function 
F(t) = a4(t) + M(t) + 7, (10) 
whose zeros determine the intersections of the straight line ax + fty + y = 0 
with the horizontal projection of the curve (x, y), shows that the curve is alge- 
braic and passes through the circular (isotropic) points at infinity (i.e., its hori- 
zontal projection) . From (3) follows that the curve has an m r fold axial symmetry. 
2. In Greenhill's case 1 the pendulum-bob reaches (but does not go above) 
the horizontal plane of suspension with a non-vanishing horizontal velocity, 
and the parametric expressions for the horizontal projection may be written 
in the form: 2 
