No. III. 



February 10, 1845. {See page 64.) 



The spirit of Sir William Hamilton's communication, which 

 was designed as a further illustration from geometry of the 

 author's theory of algebraic quaternions, consisted in re- 

 garding operations on such quaternions as admitting of being 

 ultimately interpreted as operations on straight lines ; each 

 line being considered as having not only a determinate length, 



but also a determinate direction. The quotient - obtained by 



a 



the division, of one such line (b) by another (a), is, generally, 

 in the author's view, a quaternion ; it depends, in general, on 

 four distinct elements, of which one, namely, the modulus, is 

 a positive or absolute number expressing the relative magni' 

 tude of the dividend and divisor lines, while the three other 

 elements serve jointly to express the relative direction of 

 those two lines. Of the three latter, one is the amplitude, and 

 marks the inclination of one line to the other, or the magni- 

 tude of the angle which they include ; while the two others 

 determine the plane of that angle, and are what have been 

 called, in a former communication, the directional coordi- 

 nate, such as the longitude and colatitude of the quaternion. 

 In this comparatively geometrical view, as in the more alge- 

 braical view which was formerly stated to the Academy, the 

 consideration of these four elements, modulus, amplitude, lon- 

 gitude, and colatitude, presents itself, therefore, naturally. We 

 may also speak of the axis of a quaternion, meaning thereby 

 the axis perpendicular to the plane of the two straight lines of 

 which that quaternion is a quotient; and may say, that such 

 an axis is \t%e\i positive or negative, or that it is taken in the 



c2 



