Sir Witi1AM Rowan Hamitton’s Researches respecting Quaternions. 291 
coincide with r’; also » tends to become = w. If, therefore, a quaternion be multiplied 
by a positive real quantity, w=, the effect is only to multiply its modulus by that quan- 
tity, without changing the amplitude or direction. But if @ tend to 7, then yu tends to 
—w,; R” tends to become diametrically opposite to Rr’; and 6” tends to become supple- 
mentary to @’. Ifa quaternion be multiplied by a real negative, w= —p, the effect is to 
multiply the modulus, y’, by the real positive, —w=; to change the amplitude @’ to 
a—O0’; the colatitude, ¢’, to 7—q’; and the longitude, y/, to 7+. Accordingly, by 
inspection of the second line of the expressions marked (16), we see that these changes 
are equivalent to multiplying each of the four constituents, w’, a’, y’, 2’, of the proposed 
quaternion, by —y. In each of these two cases of multiplication by a real, the residual 
triplet disappears by (39), because sin @ vanishes. 
‘* Multiplication of a Real by a Quaternion.—We have only to suppose that 6’ tends 
to 0 orto. ‘The residual vanishes, and the order of multiplication is indifferent. 
“* Multiplication of two pure Imaginaries.—Here 0= 6’ = >. war, p=r’'; KR” coincides 
with r,’, that is, with the positive pole of rr’; the direction of the product is perpendi- 
cular to the plane of the factors; and the amplitude of the product is the supplement of the 
inclination of those two factors to each other. Introducing the consideration of the 
. . T 
reduced product and residual, since R’/R//=5, we have; by; (80), 7/=0; 7/"=r"; the 
reduced product is a pure real, namely, the real part of the complete product; and the 
residual is equal to the imaginary part. The amplitude of the reduced product is = 7, 
. . . . . Tv r| 
or = 0, according as the inclination of the factors is less or greater than 3: such, then, is 
the condition which decides whether the real part of the product of two pure imaginaries, 
taken in either order, shall be negative or positive. The real part itself = yp’ cos 6” = — 
rr’ cosrr’= the product of the radii of the factors multiplied by the cosine of the supple- 
ment of their mutual inclination. The radius of the residual =77 sin77’= the product 
of the same radii of the factors multiplied by the sine of their inclination to each other. 
The product is a pure imaginary, if the factors be mutually rectangular; but a pure 
real negative, if the factors coincide in direction ; and a pure real positive, if their direc- 
tions be exactly opposite. 
‘© Squaring of a Quaternion.—As nk’ tends to coincide with r, and 6! to become equal 
to @, Rr” tends to coincide likewise with r, and 9” to become double of 6, at least if @ be 
less than >" But if @ be greater than = then r” tends to coincide with the point diame- 
trically opposite to r, and 9” tends to become equal to the double of the supplement of 0. 
If é= > then Rr” tends to become distant by 5 from Rr, but in an indeterminate direction, 
which is, however, unimportant, because 9” tends to become = 7, and the square (of a pure 
imaginary triplet) is thus found to be a pure real negative ; which agrees with the recent 
