XXXil 
positive or in the negative direction, according as it is the axis 
of a positive or a negative rotation, from the divisor to the di- 
vidend line. Quaternions may be said to be coaxal when their 
axes coincide, or only differ in sign. A quaternion is not 
altered in value when the two lines of which it is the quotient 
are transferred, without altering their directions, to any other 
positions in space; or when their lengths are both changed 
together in any common ratio; or when they are both made 
to revolve together, through any common amount of rotation, 
round the axis of the quaternion, without ceasing to be still 
in (or parallel to) the same common plane as before. It is, 
therefore, always possible to prepare any two proposed qua- 
ternions, or geometrical quotients or fractions of the kind 
above described, so as to have one common denominator or 
divisor line ; and then the addition or subtraction of those 
two quaternions is effected, by retaining that common line as 
the denominator or divisor of the new quaternion, and by 
adding or subtracting the numerator lines, in order to obtain 
the new numerator of the same new quaternion, that is to say, 
of the sum or difference of the two old quaternions ; addition 
and subtraction of straight lines (when those lines are supposed 
to have not only lengths but also directions) being performed 
according to the rules which have already been proposed by 
several writers, and which correspond to compositions and 
decompositions of rectilinear motions (or of forces). Multipli- 
cation of two quaternions may be effected by preparing them 
so, that the denominator (b) of the multiplier, may be equal 
to, or the same line with, the numerator (b) of the multi- 
plicand (lines being equal when their directions as well as their 
lengths are the same), and by then treating the numerator (¢ ) 
of the multiplier as the numerator of the product, and the de- 
nominator (a) of the multiplicand as the denominator of the 
product: and division may be regarded as the return to the 
multiplier, from a given product and multiplicand. 
With this view of multiplication, it is evident that the pro- 
>to 
