Quaternions in Physics. 93 



simply the negative of what has been called Laplace's Opera- 

 tor : — that which derives from a potential the corresponding 

 distribution of matter, electricity, &c. 



Thus Laplace's equation for spherical harmonics &c. is 

 merely 



V 2 ^=0 ; 



and, as 1/T(p— a) is evidently a special integral, an indefinite 

 series of others can be formed from it by operating with 

 scalar functions of v? which are commutative with y, such as 

 S/3v, e~^ v , &c. In passing, we may remark that if ft be a 

 unit vector, il+jrn + kn, we have 



an 7 ^ . d . d 



iXiAj (JbU U/Z 



This is the answer to the question proposed a little ago. 



The geometrical applications of Nabla do not belong to my 

 subject, and they have been very fully given by Hamilton. 

 But, for its applications to physical problems, certain funda- 

 mental theorems are required, of which I will take only 

 three of the more important ; — an analytical, a kinematical, 

 and a physical one. 



I. The analytical theorem is very simple, but it has most 

 important bearings upon change of independent variables, and 

 other allied questions in tridimensional space. Few of you, 

 without the aid of quaternions or of immediately previous 

 preparation, would promptly transform the independent vari- 

 ables in a partial differential equation from x, y, z to r, 6, cf> : — 

 and you would certainly require some time to recover the 

 expressions in generalized (orthogonal) coordinates. But 

 Nabla does it at once, Thus, let 



. d . . d . 7 d 



V° = t dl + 3T v + k W 



where 



<r = ifj+jy + k£, 



f , rj, £ being any assigned functions of x, y, z. Further, let 



da = <p dp, 



where <j>, in consequence of the above data, is a definite linear 

 and vector function. Then, from the mere definition of 

 "Nabla, 



which gives at once 



8 . ^>^ / oy cr = 8 .dpfiy^&dpy. 



