92 Prof. Tait on the Importance of 



attached to it, and I shall speak of it as Nabla*. We may 

 define it in many ways, all independent of any system of 

 coordinates. Thus we may giye the definition 



meaning that, whatever unit-vector « may be, the resolved 

 part of v parallel to that line gives the rate of increase of a 

 function, per unit of length, along it. From this we recover, 

 at once, Hamilton's original definition : — thus 



V = — a S«v — /3S/?v — 7S7V = *da + fidfi + yd y , 



a, /3, y being any system of mutually rectangular unit-vectors. 

 But, preferably, we may define Nabla once for all by the 

 equation 



— Sdpy = d, 



where d has the meaning already assigned. The very nature 

 of these forms shows at once that Nabla is an Invariant, and 

 therefore that it ought not to be defined with reference to any 

 system of coordinates whatever. 



Either of the above definitions, however, shows at once 

 that the effect of applying y to any scalar function of position 

 is to give its vector-ixite of most rapid change, per unit of 

 length. 



Hence, when it is applied to a potential, it gives in 

 direction and magnitude the force on unit mass ; while from 

 a velocity-potential it derives the vector velocity. From the 

 temperature, or the electric potential, in a conducting body 

 we get (employing the corresponding conductivity as a 

 numerical factor) the vector flux of heat or of electricity. 

 Finally, when applied to the left-hand member of the equation 

 of a series of surfaces 



u = 0, 



it gives the reciprocal of the shortest vector distance from 

 any point of one of the surfaces to the next : what Hamilton 

 called the vector of proximity. 



If we form the square of Nabla directly from Hamilton's 

 original definition, we find 



v~ {(iH|)**(!)'}. 



* Hamilton did not, so far as I know, suggest any name. Clerk- 

 Maxwell was deterred by their vernacular signification, usually ludicrous, 

 from employing such otherwise appropriate terms as Sloper or Grader ; 

 but adopted the word Nabla, suggested by Robertson Smith from the 

 resemblance of v to an ancient Assyrian harp of that name. 



