286 Sir Wittram Rowan Hamirton’s Researches respecting Quaternions. 
therefore, since 
gl! = EM SER AA y” sy 0 +y,/"5 2! = 2, +z,,", (1 1) 
we have 
" : way! 4 Zr anit $y)? 42/P4+2,/P4y,/P +2,'". (12) 
Again, 
(wa! + yy + zz'P 4a [Pry (P42 /P= (a? +y? +27) (a? +y? +2’), (13) 
—2ww' (vx! +yy! +22!) +0/?+y/?4+2,/2= warty? 42”) + wre? +y?4+2"); (14) 
therefore, 
wl? +a! py! 4.2/7 = (w? +2? +y? +27) (w? +07 +y +2"). (15) 
Let 
uw =u cos0; a =p sinf cosp; y =p Sin sing cosy; z =p sinO sing sin); 
w' =p' cosh’; 2! =! sin’ cos¢’; y/ =p’ sin’ sing’ cosy’; 2’ =p’ sinO! sing’ sind’; -(16) 
w= p"eosh!’; a” =p'sinOcosp” sy’ =psinO”singcosp"; 2/’= wsinO”sing’sinb”; 
and let u, sin 0, and sin ¢, be treated as positive (or, at least, not negative) quantities ; 
we shall then have 
p= pp’; (17) 
which may be enunciated by saying that the modulus of the product of two quaternions 
is the product of the moduli of those two factors. 
‘* At the same time we shall have 
r=psin 0, if we make r= (a? +y? +2?) ; (18) 
and may call this quantity, 7, the modulus of the pure imaginary triplet, ix+jy+hz. We 
may also call it the radius of the imaginary part of the quaternion w+7x+jy +z, or 
even the radius of the quaternion itself; and may speak of the inclination of one such 
radius to another, the cosine of this inclination being 
cos . 77’ = cos ¢ cos ¢' +8in @ sin ¢/ cos (if’ —p). (19) 
The angle ¢ may be called the colatitude, and yf the longitude, of the radius, or triplet, 
or quaternion. And @ may be called the amplitude of the quaternion ; so that the real 
part, multiplied by the tangent of the amplitude, produces the radius of the quaternion, 
or of its imaginary part, 
wtan@=r. (20) 
The amplitude, 0, may be supposed to range only from 0 to 7. It vanishes ‘fora pure, 
real, positive quantity, and becomes = 3 for a pure imaginary; it is = 7 fora pure real 
negative. 
*« The equation (5), combined with (16) and (17), gives 
cos 6 = cos # cos 6’—sin 0 sin 6’{ cos ¢ cos p’+sin ¢ sin @’ cos (yf —wW)t; (21) 
if, therefore, we construct a spherical triangle, of which one side is the inclination of the 
