Double Nature of Nabla. 117 



in a fixed plane, their axes, or vector parts, are parallel to 

 each other and to the vector parts of their differentials and 

 the last term vanishes. The expansion then reduces to what 

 it would be if \/ were the symbol of ordinary differentiation 

 and q and r were scalars. That coplanar quaternions obey all 

 the laws of ordinary algebra is well known. 



These results may be extended. Thus for three quaternions, 

 q, r, and s, by writing qr for q in (1 F), and s for r, 



\/(qrs) = V(qr) . 5 + Vs . qr-2\/ , Y\s'\qr 



= Vq • rs-rVr . qs + Vs . qr~2VYYr'Yq . s 



-2V'VV6-'V$r, . (5) 



which may be transformed in a multiplicity of ways. 

 As a special case of (1 F) let q = r, then 



Vq* = 2Vq.q-2VYYq , Yq, . ... (6) 



where the last term does not vanish unless q and dq are 

 coplanar. 



1. The result just obtained for the effect of nabla on the 

 square of any quaternion suggests the question whether a 

 simple general method exists for applying nabla to any 

 function of a function. The answer is that, as has already 

 been illustrated, the accents which we apply to V and to the 

 operand may be treated in any way that the symbol d of 

 ordinary differentiation can be treated. Thus (1 A) is 

 true because 



d(qr) = dq . r + qdr , 



and we may similarly, in any differential identity, replace d 

 by an accent applied to the letter before which d stands, and 

 write V at the left of the whole : this is the first of the two 

 characteristic properties of nabla mentioned in Art. 1. 

 Therefore, if Fq be any function of q and we write 



dFg=fdq, (7) 



where /is the linear function obtained by differentiating Fq 

 completely, it at once follows that 



VF 7 = VVy (8) 



which is the fundamental formula for applying nabla to a 

 function of a function. The constituents of /will in general 

 contain unaccented q as well as constants. 



As an illustration, let us obtain (6) by this method. 



