WEBSTER. — PROPERTIF:s OF TIIK F.TllKK. 513 



of any two vectors, a and b, will he dcnotcil hy a'b; while the vector 

 proiliiet, 



i (a„bi — a.bv) + j (a,b^ — axb^) + k (a^b,/ - a,,bx), 



will !)(' denoted by axb, where i, j and k are the unit vectors in the 

 directions of .r, //, and z respectively. 



The symbol V will be used for the operator, 



. d . .6 . . d 



1 T- + J .- + k ; 



di dy d~ 



so that Vo = the gradient of scalar a, a vector; V*a = the diver- 

 gence of vector a, a scalar; and Vxa = the curl of vector a, another 

 vector. 



The symbol "Pot" will l)e used for the operation of taking the 

 Newtonian potential of any function, so that 



P"'" = ///f ''-' 



00 



where r is the distance from (h to the point at which we wish to find 

 Pot a. We may apply the operator Pot to a vector as well as a scalar, 

 and, in eit^ier case, Poisson's equation tells us that 



(V-Pot) = — 47r, 



or the application of the operator V" to the Pot of any function gives 

 — 47r times the original function. 



It will be found convenient to indicate differentiation with respect 

 to ct, where c is the velocity of light, by a dot over the letter that 

 stands for the function. Thus 



da, 

 a = 



d(ct) 



For bre\ity let us assume also, unless otherwise stated, that the func- 

 tions used in the following work all vanish at infinity and are finite 

 and continuous throughout space. 



Laws of the Ether. — To write out the laws of the ether in the form 

 that accounts for all the above mentioned phenomena, we must dis- 

 tinguish between the effects due to positive and negative charges, and, 



therefore, it will be convenient to call the density of positive electricity 



+ 

 at any point p, (a ({uantity which is always positive), and that of 



