12 REPORT 1861. 



simplification of which the calculations are susceptible is effected by supposing that 

 the product Uy, u.,,.^. u^ has been reduced to its canonical form, Va.x^+y^-^(\x+fi.yy. 

 Then the diflerential equation which V must satisfy takes the form 



where ^ is a constant. 



Developing the diiferential equation and equating the coefficients of the powers 

 and products of .r, y in the two sides, we have a series of linear equations for deter- 

 mining the ratios of a : /3 : y . . . ; from the solution of which it turns out that each 

 of the coefficients of the covariant V is equal to the product of\P fii, tmiltiplied by a 

 rational integral function of X% /i% independeiit of all intermediate j)oicers of X, /x. 



On the Involution of Axes of Rotation. 

 By Professor Sylvester, M.A., F.B.S. 



After a brief statement as to the most general mode of representing the displace- 

 ment of a rigid body in space by means of angular rotations about six distinct axes 

 fixed in position, it was shown that under pecidiar conditions the six axes woidd 

 become insufficient, being, in fact, equivalent to a smaller number, in which case 

 they would be said to form a system in involution. Various constructions for re- 

 presenting such and similar systems were stated, and the remarkable conclusion 

 presented, that the necessary aud sufficient condition for three, four, five, or six lines 

 being thus mutually, as it were, implicated and involved consists in their lying in 

 rided sm-faces of the first, second, third, and fom-th orders respectively. The theoiy 

 of involution originated with Prof Mobius, by whom, however, it had been left in 

 an imperfect contlition. The author referred for further infoi-mation on the subject 

 to some recent notes by himself in the ' Comptes Eendus ' of the Academy of 

 Sciences of Paris, and to certain masterly geometrical investigations of M. Chasles 

 and Mr. Cayley, to which these had given rise. 



ASTBONOMT. 



On tlie Almanac. By M. N". Adlee. 



In this paper the author gave easy methods for finding, by a dii'ect mental process, 

 the fundamental points requisite in forming the almanac for any year. 



RemarTfs onDv. Hincks's Paper on the Acceleration of the Moon^s Mean Mo- 

 tion as indicated by the Records of Ancient Eclipses. By the Asteonojcek 

 Royal. 



The author stated his unaltered conviction that the Tables of Hansen gave the 

 date of the gi'eat solar eclipse which temiinated the Lydian war, as all the most 

 reliable records of antiquity fixed it, in the year 585 B.C. He said he must first 

 recall to their remembrance some geogi-aphical facts, and he sketched on the 

 board a rough plan of Asia Minor, Upper Asia, the Black Sea, and the Mediter- 

 ranean. An impassable moimtain bamer, which the ancients called Mount Taurus, 

 stretches across Detween Asia Minor and Upper Asia, lea\Tiig only two passes at all 

 practicable for an anny : one to the north, along the shore of the Black Sea, cele- 

 arated for the well-known retreat of the Ten Thousand Greeks, as chronicled by 

 Xenophon, but so extremely difficidt that only one anny besides had ever traversed 

 it; the other to the south-east of Asia IMinor, through which, all the circumstances 

 rendered it highly probable, the invading Asspian anny entered Asia Minor, as it 

 was certain the anny of Alexander the Great passed through it in the opposite 

 direction when he invaded Syria, Egypt, and Upper Asia ; and every other recorded 

 march between Asia Minor and Upper Asia had been made through the same pass. 

 Now, in the line between this pass aud the capital of Lydia it was nearly certain 

 the decisive battle was fought, and calculation from the Tables showed that at the 

 date assigned to the eclipse, commonly called the Eclipse of Thales, because pre- 



I 



