XXXVl 



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 

 tioo straight lines, which agree in direction, is to he 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 HI.) 



. 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 



