60 Proceedings of the Royal IrkJi Academy. 



By using the equations r = 1, &c., we get the follo-vring multipli- 

 cation tables : — 



Multiplier 



i 



J 



h 



h 





i 



j 



h 



li 



Teh =ji 



3 



- i 



-h 



I 



wJtk — Wjl 



j 



-i 



h 



-h 



ill = Ij 



-li 



h 



-J 



i 



wfti - wZy 



h 



I 



-j 



- i 



jh = ih 



-h 



-h 



i 



J 



whj = wih 



-I 



h 



i 



-j- 



Here the w simply denotes that expressions like w/Y operate according 

 to the B system. 



Xow, consider the efiect of multiplying jY into 



xi + yj ■\- %k -r wh : 

 we get + ^j - yi - zA + wTi, 



or the projections of the vector are turned by right angles in each of 

 the coordinate planes of ij and M ; o>;V gives 



■V xj - yi + zA - wh^ 



which differs from the last result in that, in the plane of Wi, the rota- 

 tion is in the opposite direction. Hence 



(cos & + sin B'iiiji) (cos B + sin dj%) 



will turn a vector by the angle 6^-6' in the plane of ij, and by the 

 angle 6 - 0' \n. the perpendicular plane of kh. Hence jV can be repre- 

 sented as a plane pair — that of _;V and of kh. 



^ow i, y, k, Ji are any unit lines mutually perpendicular in space : 

 hence this symbolism is perfectly general ; and, introducing the six 

 coordinates which define any plane, we have enough constants to 

 determine the next more general rotation — a rotation, namely, in the 

 plane of X1X2, where Aj and X^ are any perpendicular unit vectors, by a 

 given angle, and in the plane perpendicular to XA^ by another given 

 angle. 



The plane X^Xr, has direction cosines ya^, yalfj' subject to the two 

 equations 



a= -f ^^ + 7^ + a'- + ^'U y'2 = 1, 

 and 



ac! + fi(i' + yy' = 0, 

 Avhich give also 



(a ^ aj + (^ + (iJ + (y + yj = 1. 

 Writing X1A2 as 



y;V + akj + pik + y'kh + a'ih + fi'jh, 



we see that, by the equations derived from ijkh = + 1, it becomes 

 . (y + y')>-+(a + a')X7 + (/3 + /S')^/^, 



