Double Nature of Nabla. 121 



In the most general case of a function of a single 

 quaternion q, we ma}' have under F several ordinary 

 functions; call them Q l5 Q 2 , &c, to which will correspond 

 fa fa , giving 



<*F(Q l5 Q 2 , . . • ) =/i*Qi+/**Q,+ . . . ; . . (24) 



each of the '/'s being obtained by differentiating the given 

 function as if only the corresponding Q were variable *. On 

 the right we may substitute the value of each of the 

 differentials from (15). 



If we have a function of several quaternions, q, r, etc., the 

 formulae (15) and (2-1) will still serve, but the Q's and the 

 f's will contain any or all of these quaternions, When we 

 change d for V, we shall accent only the variables which we 

 write in place of the differentials. As an example, 



d\ogqr = d{qr) . (qr)~ l + dXJYqr . (±TV log qr-TYqr . r^q- 1 ), . (25) 



and changing to V? 

 S/\ogqr = V(qr) . (^^H-V'UV^V. ( + TVlog qr- TYqr . r^q- 1 ), (26) 



in which we may, if we wish, transform TJYqr by the known 

 methods for differentiating a unit vector. 



Most of the formulae of the present section contain, on the 

 right, the differential, or the nabla, of some unit vector. 

 The differential of any unit vector is always at right angles 

 to the unit vector itself t. If the unit vector is normal to a 

 family of surfaces, it is also at right angles to its vector 

 nabla ; in which case the unit vector, its nabla, and the new 

 vector obtained by differentiating the unit vector along its 

 own direction, form a rectangular system J. Thus the term by 

 which the differential, or the nabla, of a quaternion function 

 differs in form from that of a scalar may receive a geometrical 

 interpretation. 



5. In Art. 2, above, we had terms containing the operators 

 SY^V, W</Vj and KK//V: and the terms on the right of 

 (23) can, by the methods of Art. 2, be thrown into similar 

 form. These operators are special cases of the more general 

 expression /V, where/' stands for any linear function ; such 

 functions of nabla are of great use, for example in hydro- 

 dynamics, and in the studv of an anisotropic medium §. 



* Hamilton, Art. 329. t Hamilton, Art. 335. 



X For the proof cf this, with other properties of the rectangular system, 

 see Phil. Mag. j6] vol. iii. p. 581, and iv. p. 187. 



§ Maxwell, -Electricity and Magnetism/ 3rd ed., Art. 101, k. 



