92 Mr. Davies Gilbert on the 



I'ouadk to diminish considerably, the apparent irregularities iu 

 the motion of our best time-pieces. 



A variation in temperature of about 16° of Fahr. thermo- 

 meter ( = — I would produce an equal chansre with 



V 480 30 / ^ ^ ^ 



one inch of the barometer ; but in the opposite direction from 

 expansion : this, however, is obviously included as a part, in the 

 general compensation for heat and cold. 



Such as these investigations may prove to be, I place them 

 in your hands ; and it will be highly gratifying to me if I am 

 allowed to see them honoured by a place in your Journal. 



1st. The Descent through Free Space. Fig. I. 

 Let the line AB = 2, represent the height through which a 

 body is supposed to fall, 

 T =r the time. 

 When the part x remains to be described, the velocity will 



be 2— a;* . consequently 2 — x^x T = — i orT=:2 — .c 

 X — X T=2.2-a-^ when x — 2 the equation vanishes 

 when j; ::= OT =r 2^2. 



2d. The Semi-vibration in the Arc of a Cycloid. Fig. II. 



Let CP the length of the pendulum = 4, applying itself to 

 cycloidal cheeks CA and CB. 



Let the diameter of the generating circle DP be ■=. 2. 



Let a = the length of the chord in the generating circle, cor- 

 responding with the cycloidal Arc Pjo, through which the pen- 

 dulum is supposed to vibrate ; x — the length of the chord in 

 the generating circle corresponding with the Arc Pw remaining 

 to be described. 



Then will the velocity at the point w = aP- 6P^= g" — x'^ 



And, the cycloidal Arc being double to the chord of the gene- 

 I'ating circle, 



t = - 2i or T =: 2V 2 X a' - x' X — x 



