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 ( c ) 

 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- 



