1891.] Reference to the Discussion of Laplace s Equation. 153 



3. Assuming Boole's result arid writing f'(q) for the quaternion 

 function framed on the model oif(x), we easily obtain 



V/(?) = Sf (q) . VSq - TVf (q) . VTVq + TVf{q) . V UVq 



+ [TVf (q) . VSq + Sf (q) . VTVq] . UVq 

 = {VSq + VTVq . UVq] {Sf (q) + UVq . TVf(q)} 



+ TVf(q).VUVq 

 = V? •/' (q) + VUVq. [TVf{q) - TVq ./ (q)}. 

 In an exactly similar manner we should obtain 



df{q) = dq .f (q) +dUVq . [TVf(q) - TVq ./' (q)}. 



If UVq be constant, i.e. if the vector of q preserve a constant 

 direction, then 



dUVq = and VUVq=0, 



and in that case we have the two relations 



Y/(<Z) = Vq .f (q), df(q) = dq .f (q). 



The equation dUVq = requires that UVq = const., and it is 

 easily established that if q is to be a real quaternion then the 

 equation V UVq = also requires the same condition. 



4. Instead of the elementary solutions of V?* = used in my 

 former paper, we might have taken the simpler pair 



u = y + kx, v = z —joe, 



in which case our theorem would have taken the form 



7 , dr , dr 



dr — du .;r-+dv.~-. 

 oy dz 



Now, if \ and fi be scalar constants 

 UV(Xu+fxv) = 



\k — /jij 



and is, therefore, constant. We also have 



V (\u + fiv) = 0, 



and it, therefore, follows by the preceding article that 



V . e Au +' xy = 0. 



Thus we see that the general integral of the equation Vr = 

 may be written in the symbolical form 



ul + vS- 



r = e f{\, p), 



