XXXVI 
and especially should be conducted anew to those two rules, or 
principles, which presented themselves to the author in ‘his 
earliest researches on quaternions (as described in the printed 
letter already referred to), and which he still regards as fun- 
damental in their theory: namely, first; that the product of 
two straight lines, which agree in direction, is to be considered 
as a negative number, namely, as the product of their two 
lengths taken negatively ; and, secondly, that the product of 
two rectangular lines is to be regarded as a third line perpen- 
dicular to both, of which the length represents the product 
of their lengths, and to which the rotation, from the multipli- 
cand line, round the multiplier line, is positive. The para- 
doxical, or, at least, unusual appearance of these two funda- 
mental rules, combined with the variety of the applications of 
which the author has found them susceptible, induce him to 
hope that he shall be pardoned for thus offering new confirma- 
tions or new illustrations of them, derived from considerations 
of the manner in which they present themselves from various 
points of view. 
July 14 and 21, 1845. (See pages 110 and 111.) 
. The following is the substance of the communications made 
to the Academy by Sir William Hamilton, on the application 
of the method of Quaternions to some dynamical questions : 
The author stated that, during a visit which he had lately 
made to England, Sir John Herschel suggested to him that 
the internal character (if it may be so called) of the method 
of quaternions, or of vectors, as applied to algebraical geo- 
metry,—that character by which it is independent of any 
foreign and arbitrary axes of coordinates,—might make it 
useful in researches respecting the attractions of a system of 
bodies. A beginning of such a research had been made by 
Sir William Hamilton in October, 1844, which went so far, 
but only so far, as the deducing of the constancy of the plane 
ve 
