510 



dices already cited, to the Proceedings of the above-mentioned 

 dates, yet it is hoped that, in consideration of the importance 

 and difficulty of the subject, and the novelty of the processes 

 employed, the Academy will not be displeased at having had 

 this brief recapitulation laid before them, as preparatory to a 

 sketch of some additional developments and applications of 

 the same general view, which have since been made by the 

 author. It may, for the same reason, be not improper here to 

 state again, what was stated on former occasions, that all ex- 

 pressions involving vectors, a, a', &c., such as are considered 

 in this new sort of algebraical geometry, and enter into the 

 foregoing equations, admit of being translated into others, 

 which shall involve, instead of those vectors, three times as 

 many rectangular coordinates, x, y, z, x', y', z', &c., by means 

 of relations of the forms 



a - ix +jy + kz, a' = ix' +Jy' + kz', &c. ; (7) 



where ij k are the three original and coordinate vector units 

 of Sir William Hamilton's theory of quaternions, and satisfy 

 the fundamental equations 



ij=kjk = i, kizzj; I (8) 



ji =—k,kj=— i, ih = —j ;J 



which were communicated to the Royal Irish Academy at the 



Meeting of the 13th November, 1843. (See the Proceedings 

 of that date, and the author's First Series of Researches re- 

 specting Quaternions, which Series has lately been printed in 

 the Transactions of the Academy, Vol. XXI. Part 2.) 



II. It is evident, from inspection of the equations above re- 

 capitulated, that every transformation of the vector function, 



0(a) = a-X-a^)-* (9) 



which represents, in direction and amount, the attraction ex- 

 erted by one mass-unit, situated at the beginning of the vector 



