Double Nature of Nabla. 119 



and as the differentials stand at the left of the terms we have 

 immediately, by (8), 



\/q n = Vq . nf^ + VUVg . (±TYq*-TYq . nq n ^). (14) 



There is nothing in the reasoning employed which will not 

 apply equally well if, instead of q n , we take any other function 

 of q containing neither quaternion constants nor any of the 

 selective symbols S, V, &c. ; in other words, if the functional 

 form be any of those contemplated in the Newtonian, as 

 distinguished from the Hamiltonian, differential calculus. 

 For if we do not introduce any vector except Yq into the 

 functional operation, the result can contain no other vector, 

 or scalar multioles, and the vector part, axis, and plane of the 

 function will be parallel to those of q itself. All such functions 

 of a single quaternion q must therefore be commutative with 

 one another. They will also be commutative with the result 

 of d upon them, since that operator also can yield no other 

 vector except scalar multiples of JJYq. 



It will be convenient to call any such function, agreeing 

 in form with those treated in the usual text-books on differen- 

 tiation, or, better, any function which we can construct by 

 the employment of scalars only, an ordinary function. For 

 example, sin (nq) is an ordinary function of the quaternion q 

 if n is a scalar, but not if n is a vector or a quaternion. 



It will also be convenient to represent such a function by 

 the symbol Q, and to write DQ for the result of differentiating 

 it with regard to q as if q were a scalar. Thus if Q be any 

 ordinary function of q we shall always have 



tfQ = d Q+<ZiQ 



= r/ u7 .DQ±TVQ.^UV( 7 



= d q . DQ+dTJYq . ( + TYQ-TV? . .DQ), (15) 



of which (14) is evidently a particular case*. We note that the 

 oulv differentiation to be performed upon Qis that denoted by 

 D, namely ordinary differentiation as if q were a scalar. The 

 differentials dq and dJJYq pertain to q only, depend therefore 

 on the distribution of q with respect to the variables of which 

 it may be a function, and have nothing to do with the 

 relation of Q to q. Thus we may, by (15), at once write 

 down the differentials o!' any of the functions which Hamilton 

 has defined in the Elements^ or any which we might ourselves 

 define in case we use forms not treated by him (e.g. elliptic 



* We mav write + t^- instead of +TVQ in (15). A similar remark 

 vxq 

 applies to equations (16)-(26). 



