498 Prof. Donkin on the Geometrical Interpretation 



rand a must first be in the plane of the rotation q'. The 

 operation a merely alters its length into «a ; the operation 

 a'q' alters its length into «'a, and also alters its direction. 

 Then (a + a^q')a. is the diagonal of the parallelogram constructed 

 on the two lines represented by aa, a'g^'a; and if a"a be the 

 length of this diagonal, and q" the rotation which would bring 

 the operand into coincidence with it, we have, as before, 



The expression 



a-{-a^q' = a"q". 

 aq + a'g-' + a"*/". 



in which 5-, q\ q" represent any three rotations, and or, «', a" 

 any three numbers, cannot generally be interpreted without 

 the help of conventions similar to those adopted in interpreting 

 products; because unless the planes of the three rotations 

 happen to intersect in the same line, it is impossible so to place 

 the operand that each operation can be performed on it. But 

 we have seen that aq-\-a'q' is equivalent to the symbol of a 

 single determinate rotation in a determinate plane, combined 

 with a determinate alteration of length. Suppose, therefore, 



aq -{■ a' q' ==■ a ^q ^. 



Then we may transport the rotation q^ in its own plane, until 

 its origin coincides with the line of intersection of that plane 

 with the plane of the rotation 5^", so that a^q^-\-a"q" may be 

 interpretable in the same way PS aq-\-a'q'. Thus we shall 

 finally arrive at a single determinate rotation, with a determi- 

 nate alteration of length, as the interpretation of 



{aq-\-a'q^)-\-a^'q". 



The usefulness of the system will depend upon our being able 

 to show (as we shall easily do) that the associative principle 

 holds good in addition, or that 



{aq + a'q') + a"q" = aq + {a'q' + a"q") = {aq -|- a"q") + a'q', 



which will justify us in omitting the brackets. 



The case of subtraction is evidently included in that of ad- 

 dition, since the sign — , preceding any symbol of rotation, 

 may be changed into +, provided we increase the angle of 

 rotation by two right angles. 



We are now in a position to prove the foUowmg funda- 

 mental proposition : — 



Let i,j, k represent, as above explained, quadrantal rota- 

 tions whose positive poles are respectively X, Y, Z, and let 

 /, w, n be the direction cosines of any line r from the origin. 

 Then il+jm-\-kn represents a quadrantal rotation whose po- 



