FROM NUMBER TO QUATERNION. 85 



For, let us suppose for a moment that quantities formed with 

 three linearly irreducible imaginary units ij,k could be introduced 

 in algebra. We must be able to express as a quantity (i, j, k) the 

 product of two of those quantities (i, j, k) and therefore we must 

 have between i, j and k certain relations amongst which 



{i 2 = a Q + a 1 i + a 2 j + a z k 

 ij = h Cf + b 1 i + b 2 j + b s k 

 ik = c Q + c l i + c 2 j + c z k 



a, b, c being scalar quantities. 



The last two equalities can be written 



(b 2 —i)j + b z k = —b —b x i 

 c 2 i + (c 3 — i) k = — c — c 1 i 

 Multiply the first relation by (c 3 — i) the second by — b z and 

 make the sum, we get 



(2) Aj = — (b + b x i)(c 3 — i) + (c + c x i)b z 



where A represents the determinant 



'< b» 



A 



'2 

 ^2 C 3 



Again, multiply the first equality by — c 2 , the second by b 2 — i 

 and make the sum, we find 



(3) Ak= (b + b t i)c a — (c + d i)(b 2 — i) 

 Now multiplying the first of relations (1) by A we obtain 



A i 2 = a A + a x Ai + « 2 Aj + a s A k 

 whence by replacing Aj and A & by their values from (2) and (3) 

 we obtain an equation of fourth degree into i which therefore can 

 be put under the form 



A (i — a x ) (i — a 2 ) (i— a 3 ) (i — a 4 ) = o 

 where a lt a 2 , a 3 , a 4 are ordinary complex quantities and A a 

 constant. And as a product cannot be equal to nought unless 

 one of the factors be equal to nought we must have 



i = ordinary complex quantity 

 and in a similar manner we could demonstrate that 



j = complex quantity. k = complex quantity. 



