Royal Astronomical Society. 147 



moon. But the chronological full moon is on the fifteenth day of the 

 moon. Now, half a lunation being, on the average, 14| days, it 

 follows that, unless the mean new moon happen in the first quarter 

 of its day, the mean full moon is on the sixteenth day ; so that, in 

 the long run, the sixteenth is the proper day three times out of four. 

 Hence there is no occasion to increase the epact by 1 , in order to 

 determine the astronomical full moon ; which is as correctly deter- 

 mined as the calendar will do it, by applying the existing epact to the 

 existing hypothesis of the fifteenth day. 



The preceding conclusions as to the probability of truth and error 

 were obtained from the nineteen years 1828-1846; the following 

 are the results for 1851, 1852, and 1853 :— 



New Moon. 



Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. 



1851 2 12+ 1-30+ 30 28+ 28 26 25 24 23 22 



1852 21 19+21- 19 19 17 17 15 14- 13 12- 11 



1853 10- 8 10- 8 8 6 6 4+ 3 2 1*30 30 



Full Moon. 



1851 17 16 17 16- 15 14- 13 12- 10 10 8 8 



1852 6+5 6 5- 4- 3- 2- 1-30-29- 28- 27- 26 



1853 25 23 25 23 23-21 21- 19- 18- 17 16- 15 



Here are exhibited the days of new and full moon by the calendar : 

 when + or — follows the date, the real day is the day after or the 

 day before. And though in this period of three years tjie errors of 

 the full moon much exceed in number those of the new moon, there 

 is no such excess in the long run. The nineteen years 1828-1846 

 gave 140 cases of new moon true to the day, and 141 cases of full 

 moon. 



May 9. — On the Vibration of a Free Pendulum in an Oval differ- 

 ing little from a Straight Line. By G. B. Airy, Esq., Astronomer 

 Royal. 



•' In a paper communicated to this Society several years since, 

 and printed in the eleventh volume of their Memoirs, I investigated 

 the motion of a pendulum in the case in which it describes an oval 

 differing little from a circle ; and I showed that, if the investigation 

 is limited to the first power of ellipticity, and if a is the mean value 

 of the angle made by the pendulum rod with the vertical, then the 

 proportion of the time occupied in passing from one distant apse 

 to the next distant apse, to the mean time of a revolution, is the 

 proportion of 1 to the square root of 4 — 3 sin^ a. When a is 

 small, this proportion is nearly the same as the proportion of \ to 

 1 — ^ sin^ a ; or the time of moving from one distant apse to an- 

 other distant apse is equal to the time of half a revolution divided 

 by 1 — |- sin^ a. . This shows that the major axis of the oval is not 

 stationary, but that its line of apses progresses, and that, while the 

 ellipticity is small, the velocity of progress of the apses is sensibly 

 independent of the ellipticity, and may be assigned in finite terms 

 for any value of the mean inclination of the pendulum-rod. 



