FROM NUMBBR TO QUATERNION. 89 



quaternions — everyone in its plane — so that their origins shall be 

 at the same point of the intersection of their two planes. 



Addition, — Two quaternions being given and rendered collinear, 

 turn them round their common origin O and alter, if necessary, 

 the lengths of their initial vectors so that these initial vectors 

 coincide. (Fig. 13.) Then the quaternions 

 occupy the positions AOB _ AOC. Con- 

 structing the parallelogram on OB and 

 00, and drawing the diagonal OD _ 

 p . 13 AOD is said to be the sum of the quatern- 



ions AOB _ AOO. That definition is 

 extended easily to any number of terms and is justified by the 

 fact that it coincides with the definition given for complex 

 quantities when the quaternions are co-planar. 



It is readily seen that the addition of quaternions has the same 

 four fundamental properties as the addition of plane complex 

 quantities i.e., uniformity, commutativity, associativity and 

 AOB + o = AOB. 



From that definition, we are able to decompose any quaternion 

 AOB into the sum of a scalar quantity and a quadrantal 

 quaternion or vector. (Fig. 14.) Construct the rectangle OCBD 

 having OB as diagonal and OA as direction for one of its sides, 



00 



AOB = qj + AOD 



= a + i a x + j a 2 + ka z 

 Now let 



QT> ^ 



T. AOB = « = OA A0B - a 



and U . AOD = A* 



OC 

 We have OC = OB cos a whence — — = a cos a and similarly 



T. AOD = 7y-r- = a si n a s0 ^ na * 



* T . AOB and U . AOD are the usual notations for tensor of AOB and 

 versor of AOD. 



