,0N 



', tJ 



168 Professor Baker, On the transformation of the equations of 



By the laws above given we shall then find 



E^ =^E, EI = IE = I, etc. 

 p^j2 = K2^_E, JK = -KJ = I, 

 KI = - IK = J, IJ^-JI=^ K, 



so that /, J, K, E behave precisely as the units of the theory of 

 quaternions. Other binary squares can be found with the same 

 results of composition; for instance the work below suggests that 

 /'= — /, J'= — K, K'= — J might have been taken. Adopting 

 however the above, the usual quaternion xl + yj + zK + tE is 

 represented by 



0, x\ + fey, ON 



X, 0/ V 0, - eyj 



or ( ^'^ ^^' 



V- a; - ez, 



and the formula of multiplication of two quaternions may be 

 obtained by multiplying two such squares according to the rule 

 given above. 



Now consider squares of four rows and columns. But for 

 brevity, instead of writing four rows and columns, introduce 

 symbols to stand as above each for a square of two rows and 

 columns. Namely, using the letter for the square of two rows 

 and columns of which each element is zero, let 



0, -E\, i^fO, I\, j=/0, J\, Jc^/0, K\, u=fE,0\, 

 ,E, 0) \I, O) \J, 0) \K, O) \0, E) 



where E, I, J, K are as above, E being the unit square of two rows 

 and columns, and u the unit square of four rows and columns. 

 Then by the rules above given it is easy to compute that 



gS _ ^2 _ j2. _ ^2 ^ _ ^^ 



/, 0\, ej = -je= f-J,0\, ek=-ke= /- K, 

 0, l) \ 0,J) \ 0, K 



jk =-kj= fl, 0\, ki=- ik = fJ, 0\ , ij = - ji = (K, 0\ , 



\o, I J \o, J J [o, id 



together with 



ijk = / 0,-E\, ejk = /O, - /\ , eki = fO, - J\, eij = fO, - K 



\~E, 0) [l, O) U O) [k, 



and eijk = /E, 



\0, ~ Ej 



and, the multiplication being associative, it is not necessary to 

 remark such equations as 



ijk = — jik = jki «^ — ikj, etc. 



