IN NON-EUCLIDEAN SPACE. BY A. MC AULAY. 17 



II, as above, standing for Ur, and [/>, ;/] for any 

 function linear in each of two quaternions we have 



[e„«j = .i[^, ^«] (3) 



the suffixes and the (^, Z,) having the usual quaternion 

 significations. The following are examples of the use 

 of (3) :— 



eS7^ = J^S?« = — JVz^ 



e . u-^ = — e^ . u-hi^u-^ = J(3S« + Vw). 



Let 5- be a function of position only, not of Tr ; 

 let the tri-linear expression [6^, 0j, q^ imply that 

 the differentiations of each 9 alFect q, but that they 

 do not affect the variable factors r of either ; and let 

 [9p 0j', ^ J imply that, in addition, the variable factor of 

 second 9, namely 9/, is affected by the differentiations 

 of the other 9, namel} 9j. Then we have 



[9,, 9/, ryj = [9,, 9^, q,] + J[^, V^9„ .yj ... (4) 



Q'^q of course means 9(9(/), that is 9j 9/ 7^ ; whereas 

 9^^ q^ of course means 9^ 9^ qy Thus 9^^ (but not 9^) is, 

 like the V^ of quaternions in Euclidean space, a scalar 

 operator. An important special case of (4) is Q'^q 

 = e^^q^ — 2.19(7, or 



and 



Hence 9(9 

 also is (9 + J;" 



<p being any quaternion Unity which is a function 

 of position, the line-surface integral is 



LdX = {L,Vdv(e^ + 2J> (6) 



which is proved by first proving that \rJX = 2.1 r/vl)y 



§7 above, and then proceeding as in the Euclidean case. 

 [^Utility of quaternions in Physics, §6.] Here dv is a 

 rotor element of any surface and d\ a rotor element 



