122 Mr. F. L. Hitchcock on the 



Since in Art. 4 use was made of nabla in its capacity as 

 differentiator alone, we may, wherever nabla occurs in any of 

 the formulae of that article, write a linear function of nabla 

 instead of nabla itself. Thus instead of (8) we may write 



/y.Vq-fV'M, (27) 



where /x is the linear function obtained by differentiating ¥</, 

 but / is any linear function whatever. The operator /V, 

 since it includes scalar functions of V, includes ordinary 

 differentiation as a very special case. Therefore (27) virtually 

 includes all the results of the last article. The expansions 

 (15) and (24) may be used whenever one or more ordinary 

 algebraic or transcendental functions enter into F</, which 

 may be any function of q whatever. 



The properties of linear functions afford many ways of 

 transforming the right side of (27). For example, Hamilton 

 showed that any linear function of a quaternion q may be 

 expressed as the sum of several terms of the form tqs, where 

 t and s are quaternions *. Therefore the right of (27) may 

 be expressed as the sum of several terms of the form rXJ f sqt, 

 where r, s, and t are quaternion functions which will in 

 general contain the variable q without accent. Each of these 

 terms may then be transformed so as to bring V next to the 

 operand, as in (1 D) and (1 E), or if we prefer, so as to bring 

 V to the left of the whole, as in (1 F). Such methods are 

 still possible even when the expression to be transformed 

 contains the selective operators S, V, or K, for if a and /3 

 are any two vectors we might put 2Sa/3 = a/3 + /3a, and 

 2Ya/3 = aft — /3a; so that by expanding sufficiently we might 

 always arrive at an expression free from these selective 

 symbols. We have, however, direct methods for handling S 

 and Y too fully illustrated by Hamilton and Tait to need 

 exemplification here. No matter what the form of the 

 original term, we may always transform so as to bring V 

 next to any desired factor of the term. 



Another general method is to put dq = f 2 dp, where / 2 is 

 the linear function obtained by differentiating q. Instead 

 of (27) we then have 



/V.Fg=/V'/i/;/; (28) 



and since it is no longer necessary to separate f x and f 9 . we 

 may more briefly write 



/V.F^/V'/V, (29) 



in which / 3 is the linear function obtained by differentiating 



* ' Elements,' Art. 364. 



