Double JSature of Nabla. 125 



in Lie's Theory of Transformation Groups. If we have any 

 two linear functions of V, which we may call / V and g\J , 

 they will be amenable to commutation by (3 B) whether or not 

 either of them acts on the constituents of the other, but usually 

 each will act on all that follows it ; if so, we have, by 

 Hamilton, Art. 329, 



and with the operators in inverted order, 



^V./V.7=£/V / ./V.ry + ^/V / ./V / .^ . . (35) 



in both of which identities V, if without accent, acts on all 

 that follows it in the same term, but, if accented, acts only 

 on the accented factor. The terms last on the right of (34) 

 and (35) are amenable to commutation by (3 B), whence the 

 difference of these two terms is 2V\/'\7'V r ^/V / • q' and by 

 subtracting (35) from (34), 



(/ V .rjV.q - : /V ./ V . q) =f V .gV.q -gV ./'V . 7 



+ -2YYjVYgV.q'; . (3G) 



in the last term both V's act on q only, but they both act on 

 it, making this term one of the second order. The last term 

 vanishes (a) if one or both of the functions/' and g is only a 

 scalar function, or {!>) if the vector parts of these two functions 

 are parallel. The " Klammerausdruck " of Jacobi comes, of 

 course, under (a). We might, by using three operators, work 

 out an analogue of the "Jacobian identity." 



8. Finally, we mav, in the general formulae (33) and (36), 



d 

 and in the whole of Art. 4, write V+ i, instead of V ', where 



-t means taking the partial derivative with regard to some 

 dt & i 6 



scalar variable, usually the time t. For these results are 

 consequences of the differential character of nabla, and are 



equally valid if we put the quaternion V + -y- for the vector V- 



If we do so, we shall best follow Prof. Kimura * in reffardino- 



V+ --,-- as an extended nabla, and represent it by a single 



symbol. Its quaternion character makes it amenable to any 

 of the processes used in Art. 3 and Art. 5 of this paper, and 

 the results will differ only slightly from those there obtained. 

 Naturally, in using the extended operator, we imply that the 

 operand is variable both in space and in time, or in some 

 other scalar. 



* Annals of Math. x. p. 127. 



8W3 



