8 QUATERNIONS APPLIED TO PHYSICS 



througli O, to the point whose position quaternion is cf' 

 they become 



i' = qiq-\ f = qjq-\ // = qkq-\ 



If i, /, k are called real we must call i\ j', k' complex 

 because q is complex. But the calling one set real and 

 the other complex is a mere naming of the two sets and 

 does not imply any difference of contained meanings. 

 All properties of the set i, j, k in the first place and of 

 i\ j\ li in the second are based on 



P =/ = ^2 ^ ^-y^ ^ _i 

 and P — p = k'- ^i'j'k' = — 1 



and no alteration would occur in any application if 

 we called i\ j, k' real and therefore ?, j, k complex. 



(7) When the origin is shifted to the point Kq, we 

 have seen that the position unitat y of a ])oint changes to 

 vVq~'^. The same rule holds for the position unitat v of 

 a plane ; it likewise changes to v'iio~^ ; for the equation 

 of the plane, v given, is ^ia/~^ = 0, or 



Although a change occurs in the position unitats 

 themselves, no change occurs in the ratio of any two 

 of them. Hence in interpreting the meaning of any 

 such ratio we may take the origin wherever we 

 please. Thus if vu~^ ^ vu~^ where m, v, u, v' are the 

 position unitats of four points, how are the points 

 related ? Take the point u for origin so that u becomes 

 unity and v is of the form cos Jc" + t" sin Jc" where t" 

 passes through the origin and may be considered real. 

 Thus 



v'u'~^ := COS Jc" + t" sin Jc". 



Now when using t" above we saw that this implied that 

 the points ?/, w' and the origin were collinear. Thus the 

 original three v\ ?/, 7/ are collinear, and similarly v. u' 

 and V are collinear. Hence regarding u and /" as given, 

 when vu'-'^ = vu~'^ the points u, v' are both on the given 

 straight line joining the points ?<, v. And their distance 

 apart is the given distance of u, v apart ; because 

 S?/«'~^ = Sy?/~^ This discussion shows that our 

 present method is primarily a calculus of lines and not 

 one of points and planes. To adapt it to points and 

 planes an origin has to be selected. 



