1890.] to the Discussion of Laplace 's Equation. 123 



Similarly we should obtain 



da = (dp . T- 1 - dq . v~ i ) ( ut~ i - wv~ l y\ 



Substituting these two values for dp and da in equation (2), we 

 see that it takes the form 



dr=dp.P + dq.Q (3), 



where 



P=U~\ (TU~ l - VW 1 )- 1 .R + T-'.iUT' 1 - WV' 1 )-' . 8, 

 and 



Q = W' 1 . (VW- 1 - TU'Y .R+V' 1 . (WV 1 - UT' 1 )- 1 . s. 



4. It now only remains to remark that equation (3) is to 

 be regarded as the three-dimensional analogue of the relation 



dw =f (z) dz, 



where w=f(z)=f(x + iy); which relation expresses that the 

 ratio dw : dz depends only on the origin from which the vector 

 dz is drawn, and not upon its direction. If we have two complex 

 variables u=oc + cy and v = z + it, and if w =f(u, v), then we have 

 a relation of the form 



dw = ~- du + ~- dv, 

 ou ov 



where the values of df/du and df/dv depend only on x, y, z, t 

 and not upon the values of the ratios dx : dy and dz : dt. 



In the quaternion theory the order of factors in a product is 

 material, and it turns out that the differential factors must be 

 placed before the finite ones. 



The existence of the relation expressed by equation (3), seems 

 to point to the necessity of a discussion of quaternion functions 

 of two variables, i.e. involving two variable quaternions. This 

 I hope to be able to furnish in a future communication to the 



Society. 



I am not at present prepared to give a geometrical inter- 

 pretation of equation (3). Our work furnishes us with materials 

 for calculating P and Q in terms of the differential coefficients 

 of the elements of p, q and r, but the working out of the values 

 by this method would be very tedious. It is possible that when 

 a geometrical interpretation is discovered for equation (3), this 

 may suggest some shorter method of obtaining the required 

 values. 



