16 QUATERNIONS APPLIED TO PHYSICS 



Non-Euclidean Space Integrals, Curl, Sfc. 



§10. If '/■ and r + dr are the position quaternions of 

 two neighbouring points then (r + dr)r~^ has been 

 interpreted in §2 as a quaternion whose angle and axis 

 indicate the elementary line-sesment joining the points ; 

 in §§3-8 as a bi-quaternion (in Clifford's sense for 

 elliptic space, in Hamilton's sense for hyperbolic space) 

 indicating the same thing. Tracing the interpretation a 

 step further we may say that in all three cases the rotor 

 element joining the points is 



J-'Vdrr-' = dX= J~^duu-\ 



if u, u + du are the position unitats ; provided J means 

 unity in the real quaternion method, V( — 1) in the 

 complex quaternion method, and E in the Clifford- 

 bi-quaternion method. This extension of the meaning 

 of J will be henceforth understood. A general function 

 of r is a function of position and of an independent 

 scalar T?-. [To fix the ideas take Tr as any given 

 function of the time (such as e'), the same for every 

 point of space.] 



Define the scalars ?r, .r, ?/, z and the operators ^, 6, by 



/• = ,r + .!(/./• + Ji/ + kz), ^ 



^ = D„. — .I-'(^'Dr +.yD^ + ADj, ( ... (1) 



e = — ,)V;-^ ) 



Space differentiations are effected by 0. ^ has been 

 introduced merely to suggest to the reader the definition 

 of 9 and also the proofs of the fundamental properties 

 (2), (3), (4) below of 0. 



For space differentiation we have 



d. = — ^d\Q (2) 



3f is a symbolic quaternion passing through the origin, 

 whose coordinates D,,., etc. may be treated as constants. 

 is a symbolic rotor passing through the point r, whose 

 coordinates may not be treated as constants, because of 

 the r included in the definition of 0. 



