186 PHILLIPS AND MOORE. 



If Ml and M2 are self-complementary Mi + M2 is also and so 

 /• (Ml + il/2) •/• (Ml + il/2) = - I (Ml + 3/2) • fil/i + M2) I. 



Since 



(/•Mi)-(7-Mi)= -k(Mi-Mi)I, 



(/•M2)-(/-il/2) = - I (il/2-M2) 7. 



by subtraction we get 



(I- Ml)- (I- 1/2) + (7 • 3/2) • (7 • Ml) = -(Ml- M2) I. (52) 



also 



(7 -Ml) -(7 -Mo) - (7-il72)-(7-il7i) = 7- (il7i * il/2), 



whence by addition we get 



(7- Ml) ■ {I-M2) = i [7- (Ml * M2 - (Ml ■ il/2) /]. (53) 



Let 



Ml- Ml = M2-M2 = 2. 



Then 



4>i = c^'^-^' = Icosqi-\- I-Misinqi, 



02 = e^'^-^' = I cos q-2 + 7- 3/2 sin 92- 

 Multiplying and using (53) we get 



0i</>2 = 7 [cos gi cos 92 — 2 (il/r il/2) ?in gi sin 92] (54) 



+ /■ [il/i sin qi cos ^-2 + il/2 cos qi sin 92 + 2 -^/i * il/2 sin qi sin Q'2. 



Let 



Qi = Ml tan qi, • Q2 = il/2 tan q^. 



The corresponding quantity for (^1 (^2 is 



il/i sin gi cos 92 + il/2 cos qi sin 92 + 2 (-^-^i * ^^^2) sin qi sin q^ 



Q = 



cos gi cos 92 — 2 il/i'il/2 sin gi sin q^ 

 Qi + Q2 + I Qi * Qo 



1 — 2 Qi'Q^ 



(55) 



The quantities (?i, (?2, Q are analogous to Gibbs' vector semi-tangent 

 except that the whole angles qi, q^, q are used instead of the half angles. 



