297 



beginning to decay, on the 15th. These three month dates refer, of 

 course, to the conjunction, first appearance, and plenitude of the moon. 

 The 16th of the lunar month was oftener than the 15th regarded as 

 the day of full moon ; and two monthly festivals, celebrated on the 

 1st and 16th, have been rightly referred by Lepsius to the lunar year. 



Dates of the lunar year are, according to Dr. Hincks, occasionally 

 met with on the monuments. Such dates are, he thinks, those of the 

 1st and 16th of Athyr, of the 11th of Amenhotap III., mentioned on a 

 scarabseus, so as to imply that the ISTile was then rising, and near its 

 height. The date of the Exodus in the month Abib, presumably 

 Epiphi, is also referred to a lunar year. The month Abib is identi- 

 fied with what was afterwards the first month of the Israelites, and was, 

 therefore, like this, a lunar month; and we know that it was the 

 month which began at the new moon following the vernal equinox. 

 Epiphi was the eleventh Egyptian month ; the lunar Epiphi would, 

 therefore, begin more than 295 days after the solstice, while the vernal 

 equinox was about 271 days after it. Erom this it follows that the first 

 Hebrew month would in general coincide with Payni, the tenth Egyp- 

 tian month; but that it would occasionally coincide with Epiphi — 

 namely, when the new moon followed the summer solstice very closely. 

 This would furnish a means of determining the year of the Exodus accu- 

 rately, if it were known approximately ; for example, 1491 B. C. could 

 not be the year of the Exodus, but 1494 B. C. might. But, what is of 

 more consequence, the remark respecting the month Abib is a very strong 

 argument in favour of the genuineness of the Biblical account of the 

 Exodus, which has been recently called in question. No forger of a 

 later age, and who had not lived in Egypt, could have thought of 

 making such a statement. 



The President read a paper, by Professor Sylvester, u On the De- 

 monstration of Newton's Theorem respecting the Imaginary Eoots of 

 Equations." 



The President read the following paper, with a Note by the late 

 Sir W. E. Hamilton, LL. D. :— 



On a Theoeem eelatino to the Binomial Coeeeicients. 



Towaeds the end of March, I communicated the following theorem to 

 Sir William Rowan Hamilton : — 



Putting s 0 = n 0 + n 3 + n 6 + &c, 

 = + % + n n + &c, 

 s 2 = n 2 + n 5 + n 8 + &c, 



where n 0t n u &c, are the coefficients of the development 



(1 + x) n = n Q x° + + n 2 x 2 + &c, 



and n is a positive whole number ; 



