10 QUATERNIONS APPLIED TO PHYSICS 



IS 



X and y being real scalars, xi is a rotor ; Jj/i 

 rotor-couple ; (x + Jy) i is a motor, [See §7 below.] 

 Clearly multiplication by x + Ji/ completes our inter- 

 pretations. Without entering into details, it will be 

 evident to the reader, that we have a unique geometric 

 interpretation in hyperbolic space of the general com- 

 plex quaternion as the ratio of two motors. 



(10) q{ ) q~^ where g- is a complex quaternion 

 shifts any motor (or complex quaternion) by a finite 

 twist whose axis is that of q and whose angle and 

 advance are twice those of q. [For proof take the 

 motor as w + Jo- Avbere w and a are rotors through the 

 point of intersection of the rotors i, j, k ; and take q as 

 [X -r Jy)e(« + J«''^'.] 



(11 \ The rate of increase of a motor a, fixed in a 

 rigid body can be expressed as Yya where -y is a motor 

 expressing the rate of displacement of the rigid body, 

 [7 = 2Yqq-^ where q is as in (10)]. 



(12) If a, /3, 7 are three motors such that a + j3 

 + 7 = one straight line intersects all three axes 

 perpendicularly (for VjSa"^ + ¥70"^ = 0). If A is a 

 unit rotor along this line, and i any unit rotor inter- 

 secting k perpendicularly, then 



Ua = e^'H, U/3 = e^"-i, U7 = e''-i 

 where a, Z>, c are complex scalars satisfying the sine 

 formula, 

 Ta'sin {c — b) = T/3 sin (a. — c) ^ T7/siu {b — a). 



[Interpret the condition Saj37 = 0,] 



If a and j3 are two motors ; then if Saj3 = they 

 inteisect perpendicularly, and conversely : and if Va(5 

 = they are co-axial, and conversely, 



(13) If (u, (T are rotors through the origin, required 

 the axis, pitch, etc., of the motor w + Jo-, First obtain 

 the co-axial unit rotor by dividing by T((u + 'Tcr), that 

 is, by s/(— w^ — J^a^ — 2JSw(t). Let this unit rotor 

 be (jJq + Jffo where (Uq, <ro ^^'^ rotors through the origin j 

 (for which 



Wo" + 'I'tTo^ = — 1, Sa»o(To = 0). 



Then put Wo + •^0-0 = ^-O' + J'')'.-/ and deduce 

 .I(7o wo~' = ^ tan t)b. 



