258 UNIVERSITY OF COLORADO STUDIES 



Postulate I. — If a and b are any two quaternions, then ^ = [^1 + 6,, 

 dj + b^, ^3 + 63, a^ + fij is a quaternion. 



Definition. — Addition of quaternions is defined by a®b = s. 



Postulate II. — o = [o, o, o, o] is a quaternion. 



Postulate III. — If o is a quaternion, then to any quaternion a cor- 

 responds a quaternion a' such that a®a' =0. 



Theorem i. — Quaternions form a commutative system under addi- 

 tion. For, by postulate I, quaternions form an additive group. Addition 

 is commutative because 



a®b=[ai + bi, a^ + b^, a^ + b^, O4+6J 

 =[bi + ai, bj + a^, b^ + a^, 64-1-04] 



Postulate IV. — a and b being any two quaternions a<^b = p=[p^, 

 Pij p3f PJ is a quaternion, where 



Pi = aibi — a2b2 — a^b^ — aJ)^ , 

 p2 = aib2 + a2bi+aj)^—a^b^ , 

 p3 = aibi-a2b^^-a^bi + aj)2 , 

 P^=aib^+a2b^ — a^b2 + a^bi , 

 if the Pi are in D. 



Definition. — The product of two quaternions is defined by a <S)b = p. 

 Theorem 2. — Multiph'cation is not C9mmutative. For, by postulate 

 IV, a®b = 

 [ajbj — a^bs—a^bj — a^b^ , atb2+a2bi+aj)^—aj)j , 



0163—0264-1-036,-1-0462 , 0,64-1-0263 — 0362-1-0461] 

 and 



60o = [oi6, — O262 — O363 — O464 , 026,-t-Oi62-fo463 — O364 , 



0361-0462-1-0x63 + 0264 , 046,-1-0362 — 0263-1-0,64] . 

 .-. o(8)6=}=6®o . 



Theorem j. — Multiplication is distributive. 

 For, 

 o®(6 + c)=[o,, 02, O3, 04](8)[6,-|-c, , 62-^^2, 63-f-C3, 64-l-cJ = 



[0,(6,-|-C,)-02(62-|-C2)-03(63-|-C3)-04(64+C4) , 



Oi(62+C2)-ho2(6, -l-c,) -1-03(64 -I-C4) — 04(63 -I-C3) , 



0,(63 +<^3)- ^2(^4 +^4) +'^3(^1 +^i) +^4(^2 +C2) , 

 0,(64 -I-C4) +^2(63 +C3) -03(62 -hCa) +04(6, +C,)] 



