DEFINITION OF QUATERNIONS BY INDEPENDENT POSTULATES 259 



=[aibi — a2b2 — a^b^ — a^b^+aiCi — a2C2 — ajCj—a4C^ , ajb2+a2bj+ajb^—aj)j 

 + a^Cs + OjCj + a-f^ — a^c^ , aj)^ — 02^4 + ajb^ + ^462 + a^c^ — asC^ + a^Cj^ 

 +a^C2 , ajb^+a2b^ — a^b2+aJ)i+aiC4+a2C3 — ajC2-\-a^Cj] 



= [ai6, — fla^a — ^3^3 — C-Jf^. , Oi^a + «2&i + ^^^3^4 " ^4^3 > <^i^3 " ^264 + 0361 

 + aj)a . ai&4+a2&3— <l3&a + «4^i]®[^i<^i"~<^2<^2 — 03^3 — ^4^4 , CiCj + ajCi 



•{■a^c^—a^Cj, ajCj—asC^+a^Ci+a^Cj, o,C4+a2f3 — fl3C2+a4Ci] 

 = (a(^b)®(a<S)c) . 



Theorem 4. — Multiplication is associative. 



C(8)(6 (S)f) = [ai,a2,«3, ^4]® [^i^i — 62^2 — &3<^3 — &4<^4, 6iC2+&2<^i+&3<^4 — ^4^3 , 



&1C3 — 62C4+63C1 + 64C2 , 61C4+62C3— &3C2 + 64C1] 



= [<^i(^it"l — &2C2-*3<^3 — V4)~^2(^l<^2+ ^2^1+^4 — V3) — <^3(*i^3~^2^4+ ^3^1 

 + V2) — ^4(^iQ + ^2f3-V2 + ^4^i) ' Oi(6iC2 +62C1 +63C4 — &4C3) + a2(6iCi 



— 62^2 — &3C3 — 64C4) +03(^1^4 + 62^3 — ^3^2 +^4^1) — 04(^1^3 — ^2^4 + ^3^1 + ^4^2) ' 

 ^1(^1^3 — ^2^4 + ^3^1 + 64C2) — ^2(^1^4 + b2C3, — 63C2 + 64^1) — <l2(6i<^4 + ^2^3 



— ^3^2 + b^Cj) + 03(61^1 — ^>2C2 — ^3^3 — ^4^4) + a^{b^C2 + &2C1 + 63^4 — V3) ' 

 a^{b^c^-^b2C3 — bj^C2 + b^c-,)-\-a2{b^C3— 62C4+63C1 + 64C2) — a3(6i<:2 + 62^1 + b^c^ 



— 64C3) + a4(6iCi — 62C2 — ^3^3 — &4C4)] 



= [(ai&i — «2^2 — 03^3 — 0464)^1— (ai&2 + 02^1 +a3&4 — <l4&3)C2—(ai&3 — 02^4 + ^3^1 

 + 0462)^3 — (0164+ 02^3 — 0362+ C461V4 5 (Oi*2+ 02^1 + 03^4 — <^4^3)^i + (^1*1 



— ^2^2 — O363 — a4^4)<^2 + (<^i^4 + <^2&3 — ^3^2 + 04^1)^3 — (01^3 — «2^4 + ^3^1 

 + 04^2)^4 , (C163 — a2^4 + 03&i + «4^2)Ci— («i^4+ ^2^3 — <^3^2 + «4^i)C2+ (fli^l 



— ^2^2 — a^b^ — a464)c3 + (0162 + ^2*1 +<i3^4 — ^463)^4 , {a-ib^ + 5263 — ^362 

 + 04^1)^1 + (01^3— 02^4+ 0361+ a4&2)<:2 — (ai&2+ 02^1 + 03*4— 04^3)^3 + (<^i^i 

 a2b2 — ajb3—aj)^c^ 



=\(iib^ — ^2^2 ~ <^3&3 — cij>\ > <^i^2 + <^2^i + O364 — 04^3 , a^bj — a^b^ + 0361 

 +0462 , C164+C2&3 — a3&2+«4&i]<8)[ci, C2, C3, C4]=(a(8)6)®c. 



To make quaternions four dimensional we add a fifth postulate. 



Postulate V. — If t^, t^, T3, t^ are elements of D, such that T^a^+r2a2 

 + r^a^ + r^a^ = o for every quarternion a, then Tj=T2=T3 = t^ = o. 



Theorem 5. — There are four quarternions, e^ = [a,-i, ««> ^»3> ^14] such 

 that J Uij I 4= o. 



By postulate V there is at least one quaternion different from 

 zero. Let ^j = [a,,, a, 2, ^13, ajJ=t=o be a quaternion where in 

 particular a^^^o. Consider Cj and also the quaternion €^ = [02^, a22» 



