146 Royal Astronomical Society. 



that the difference between the two suras in question is very small 

 if the stations be equally elevated above the surface. It is not, how- 

 ever (nor can be geometrically), shown that this difference is not 

 greater than the probable error of observation, and therefore it may 

 be useful to find an expression for this small difference in terms of 

 the latitudes and longitudes of the stations, in order to see whether 

 it may be in any case greater than the probable errors of observation, 

 and large enough to be worth taking into account." 



The author then investigates by accurate formulae of analytical 

 geometry, as applied to the co-ordinates of points which satisfy the 

 spheroidal equation, the expressions for the tangents of the angles 

 of reciprocal azimuths of two stations, and forms the accurate ex- 

 pression for the tangent of the sum of azimuths, and for the tangent 

 of the excess of this sum above the sum of corresponding spherical 

 azimuths. The expression is then cautiously reduced, and it is 

 found, at length, that the value of this excess is insensibly small ; 

 amounting only to 0"*000003 X m^n, where m is the number of de- 

 grees in the distances of the stations, and n the number of degrees 

 in the difference of latitude. Then the influence of difference of 

 heights is computed ; and it is shown that, though (in cases which 

 may arise in practice) it is greater than what has just been found, 

 yet that it also will be insensible. 



At the close of the meeting, Mr. De Morgan made some remarks 

 upon the Gregorian Calendar, as an instrument for determining the 

 moon's phases with sufficient accuracy to settle the question of 

 moonlight. Having been led to examine it in this point of view, for 

 the purposes of a collection of almanacs which he is preparing for 

 publication (and which has since been published), he found that it 

 may be made to give the day of new moon or of full moon right in 

 three cases out of five, and with an error of only one day in almost 

 all the other cases ; the error of two days occurring only about once 

 in 120 results. In order to obtain this amount of accuracy, the rule 

 is : — Use the Gregorian epact to determine full moons, and that 

 epact increased by 1 to determine new moons ; both with the well, 

 known epact-table which appears in all extensive works or articles 

 on the calendar. 



The reason of this rule is as follows : — Clavius constructed the 

 Gregorian Calendar expressly in such manner that the moon of his 

 calendar should be always, as well as it could be managed, one day 

 younger than the moon of the heavens ; the object being, that the 

 fourteenth day, by which Easter is determined, should follow the 

 day on which the Jews keep the Passover. And as this was done 

 with good success, it follows that one day added to the age of the 

 calendar moon at the beginning of the year (that is, to the Gregorian 

 epact), gives the same degree of success to the calendar, as a means 

 of determining the day of astronomical new moon. 



If the chronological full moon had been correctly laid down, this 

 same addition of 1 would have been equally successful as to the full 



