MATHEMATICAL CLASSIFICATION 



performing thb operation twice on any subject is the well known operation (of 



The discovery of the square root of this operation is due to Hamilton ; 

 but most of the applications here given, and the whole development of tin- 

 theory of this operation, are due to Prof. Tait, and are given in several papers, 

 of which the first is in the Proceedings of the Royal Society of Edu,lir<j1i, 

 April 28, 1 862, and the most complete is that on " Green's and other allied 

 Theorems," Transactions of the Royal Society of Edinburgh, 1869 70. 



And, first, I propose to call the result of V* (Laplace's operation with the 

 negative sign) the Concentration of the quantity to which it is applied. 



For if Q be a quantity, either scalar or vector, which is a function of the 

 position of a point; and if we find the integral of Q taken throughout the 

 volume of a sphere whose radius is r; then, if we divide this by the volume 

 of the sphere, we shall obtain Q, the mean value of Q within the sphere. If 

 Q t is the value of Q at the centre of the sphere, then, when r is small, 



or the value of Q at the centre of the sphere exceeds the mean value of Q 

 within the sphere by a quantity depending on the radius, and on V'Q. Since, 

 therefore, V'Q indicates the excess of the value of Q at the centre above its 

 mean value in the sphere, I shall call it the concentration of Q. 



If Q is a scalar quantity, its concentration is, of course, also scalar. Thus, 

 if Q is an electric potential, VQ is the density of the matter which produces 

 the potential. 



If Q is a vector quantity, then both Q, and Q are vectors, and V'Q is 

 also a vector, indicating the excess of the uniform force Q, applied to the whole 

 substance of the sphere above the resultant of the actual force Q acting on all 

 the parts of the sphere. 



Let us next consider the Hamiltonian operator V. First apply it to a scalar 

 function P. The quantity VP is a vector, indicating the direction in which P 

 decreases most rapidly, and measuring the rate of that decrease. I venture, with 

 much diffidence, to call this the slope of P. Lame* calls the magnitude of W 

 the "first differential parameter" of P, but its direction does not enter into his 





