M^^ of Qitaternions. 497 



The associative principle, so far as the products of the qua- 

 drantal rotations represented by i,j, k are concerned, is now 

 established, since 



{ij)k = F = -- 1 , i[jk) = ?2 = - 1 , &c. 



Hitherto we have been considering the use of the symbol 

 + r as representing rotation. We must now proceed to ex- 

 amine the principles on which the symbols + and — are to 

 be employed in the present system, as indicating the addition 

 and subtraction of directed lines. 



Let (7, g' denote any two rotations, and a, a! two numbers. 

 What is the interpretation of aq-\-a!q''^ In the first place, 

 aq represents the compound operation of turning the operand 

 line through the rotation </, and altering its length in the ratio 

 of a to I; and a'q' has a similar meaning. Next, in order 

 that aq, a'q' may both refer to the same operand line, that line 

 must at first coincide with the intersection of the planes of the 

 two rotations y, q'. Let a be the operand so placed ; then 

 aga, «'</'a represent two lines whose lengths are ata, «'a, and 

 whose directions are determined by the rotations g, q'. Their 

 sum aqsi + a^q^a, represents the diagonal of the parallelogram 

 constructed upon them. Let a^o. be the length of this dia- 

 gonal, and q^' the rotation which would bring the original 

 operand into coincidence with it ; then we have 



aqa. + a'^a = a"q"a , 



or omitting the symbol of the operand, 



aq + a'q'=:a"q". 



Thus the complex symbol of operation, aq + a'q', represents a 

 single determinate rotation combined with a determinate alte- 

 ration of length. It is easy to illustrate this by a diagram, ^ 

 thus : — 39;- 



Let ABC be any spherical triangle, and let <7, q' represent 

 the rotations AB, AC. Then {aq^a'q')a represents the dia- 

 gonal of a parallelogram constructed on two radii vectores 

 which cut the sphere in B and C, and whose lengths are aa, 

 a'a. This diagonal will cut the sphere in a point D of the 

 arc BC, such that 



sin BD : sin CD :: a' :a. 



Join AD; then, in the equation aq-\-a'^=a"q"^ we have ob- 

 viously 

 Kp^PH' I ™ ,ur,a!'^=a^ + a''' + 2aa' cos BC, ^a& 



and q" is the symbol of the determinate rotation AD. 



The expression a-\-a'q\ which is merely a particular case 

 of the above, is easily interpreted in the same way. The ope- 



