See 
Sir Wittram Rowan Hamirton’s Researches respecting Quaternions. 295 
2 
‘© Napierian Exponential.—lf f(t) = 1+; + - + &e. (61) 
then, r being = ¥ (#*+y°+42"), &e., 
Sliatjy +hz) = cos r+sin r (i cos $47 sin ¢ cosW-+h sin ¢ sin W) ; (62) 
the modulus of the function f of a pure imaginary ts unity.” 
Tue foregoing is an extract from a letter, hitherto unpublished, which was addressed by 
the author to his friend, Mr. Graves, at the time specified in the date, Two figures have been 
suppressed, as it was thought that the reader would find no difficulty in constructing them 
from the indications given. A fractional symbol in the formula (58) has also been sup- 
pressed, as not entirely harmonizing, under the cireumstances in which it occurs, with a 
notation subsequently adopted. And the reader is reminded by the words ‘ submultipli- 
cation” and ‘* submultiply,” inserted within square brackets, that these words have since 
come to be preferred by the author to the words “division” and ‘ divide,” when it is 
required to mark the return from the product to the multiplicand, in cases when the order 
of the factors is not indifferent to the result: division being (in the text of the present 
paper) defined to be, in such cases, the return from the product to the multiplier. With 
these slight changes, it may be interesting to some readers to see how nearly the author's 
present system, although it has been, since the date of the foregoing letter, in some respects, 
simplified and extended, besides being applied to a great variety of questions in geometry 
and physics, agrees with the formule and constructions for quaternions, which were em- 
ployed by the writer in October, 1843; and were in that month exhibited by this letter to 
a scientific correspondent, and also soon afterwards to a brother of that gentleman, the 
Rey. Charles Graves, before the Meeting of the Academy at which the first public com- 
munication on the subject was made, and of which the date (November 13th, 1843) is 
prefixed to the present series. As that public communication of November, 1843, was in 
great part oral, and as a considerable interval has since elapsed, the author thinks it may 
be not irrelevant to mention expressly here that not only were the fundamental formule 
(1) (2) (3) of the foregoing letter exhibited to the Academy at the date so prefixed, and a 
general sketch given of their relation to spherical trigonometry, but also the theorems 
respecting the connexion established through quaternions between certain spherical quadri- 
laterals, pentagons, and conics, which form the subject of the forty-seventh and forty-eighth 
articles of this paper, were then communicated, and illustrated by diagrams. Those 
theorems have since been printed in the Number of the ‘¢ London, Edinburgh, and Dublin 
Philosophical Magazine” for March, 1845. The fundamental equations between 7, j, k 
received their first printed publication in the Number of that Magazine for July, 1844; 
and other articles on Quaternions, by the present writer, which will probably be continued, 
have appeared in the Numbers of that Magazine for October, 1844; July, August, and 
VOL. XXI. 2k 
