428 Mr. O'BRIEN ON A NEW NOTATION, ETC. 



It appears from this equation tiiat the north pole has two velocities, namely 



\ sin w cos sr sin'nV, parallel to a, i.e. perpendicular to the solstitial colure ; 



n 



Sn"' . . , , , . 



and \ sin ■ar sin n < cosw <, parallel to ^, i.e. in the plane of the solstitial colure, and paral- 

 lel to the equatorial plane. 



Hence the length of the path described parallel to a in any time t is 



Sn 



■ X sin •jsr cos •ar (n't - i cos 2«'<), 

 2w ■^ 



and the path parallel to /3' is 



3n' . 



— \ sin 73" cos 2n t. 

 in 



Which ai'e the well-known values of the solar precession and nutation of the pole. 

 The calculation of Lunar Nutation may be effected very simply by the above method; in fact 

 the equation 



dv Sm' , „ , V 



-f = -^,X(Am'.7)(Dm'.7); 

 dt nr" 



still iiolds, and we have only to make the proper substitution for u to suit the Moon's motions, and 



then integrate as above. 



M. O'BRIEN. 

 Upper Norwood, Surrey, 

 Nov. 1846. 



(Note.) In a series of papers on Symbolical Geometry by Sir W. Hamilton, which are at present being 

 published in the Cambridge and Dublin Mathematical Journal, a very remarkable interpretation is given 

 to the product of two symbols. According to this interpretation ^(bu'+ u'u) means the same thing as A?< . w 

 in the present paper, and ^ {iiii — u'u) means the same thing as Du . u'. 



