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, 2, 2, y’, 2’; &c., by means 
of relations of the forms 
axtatjythz, a’ xix’ +jy’+hk2’, &e.; (7) 
where 77k are the three original and coordinate vector units 
of Sir William Hamilton’s theory of quaternions, and satisfy 
the fundamental equations 
ay ah ee | 
Y= ho Mh ky RES RS | (8) 
ji=—h, hj=—i, tha=—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. X XI. Part 2.) 
II. Itis evident, from inspection of the equations above re- 
capitulated, that every transformation of the vector function, 
o(a)=a7"(—a")# (9) 
which represents, in direction and amount, the attraction ex- 
erted by one mass-unit, situated at the beginning of the vector 
