520 



second fictitious moon, being so placed that the actual moon 

 is midway in the heavens between this fictitious moon and the 

 one which was before considered. These two latter terms of 

 (43) contain the chief laws of the Lunar Variation ; and are 

 easily shown to give the known terms in the expressions of 

 the moon's parallax and longitude, 



1 11 /w^ 



gi = m2cos2(3)-©); §0 = -!-^ sin2( ]) -©). (44) 

 r o 



It may assist some readers to observe here, that when the in- 

 clination of the orbit is neglected, the longitudes of the first 

 and second fictitious moons are, respectively, 



2©— D, and 3 J— 20; (46) 



while those of the first and second fictitious suns, mentioned 

 in a former section of this abstract, are, under the same con- 

 dition, 



2 5-©, and 3©-2]). (47) 



VII. The law and quantity of the regression of the Moon's 

 Node may also be calculated on principles of the kind above 

 stated, but we must content ourselves with writing here the 

 formula for the angular velocity of a planet's node generally, 

 considered as depending on the variable vector of position a, 



the vector of velocity — , and the vector of acceleration -j-^, 



and also on a vector unit A, supposed to be directed towards 



the north pole of a fixed ecliptic. The formula thus referred 



to is the following : 



,_ S.aX.S.d'^ada a ,.„^ 



dS3=-— -——, (48) 



(v.Xv.ada)^ ^ 



where s and v are, as before, the characteristics of the opera- 

 tions of taking the scalar and vector of a quaternion. The 

 author proposes to give a fuller account of his investigations 

 on this class of dynamical questions, when the Third Series 

 of his Researches respecting Quaternions shall come to be 

 printed in the Transactions of the Academy : the Second Se- 



