122 Mr. R. F. Gwyther on 



where the several functions^ <p,\p on the right are functions 

 of x and r only. 



These are the sole conditions which arise out of the 

 permanency of form condition, and, although reasons can 

 be given why \p h \p 2 and f 2 should vanish, as this is not my 

 present theme, I proceed to the more general cases to 

 which this is an introduction. 



2. I shall consider in the first place the conditions that 

 functions of a scalar or vector function of a point may 

 themselves be scalar or vector functions, and, when I use 

 these expressions which are now so common, and which 

 were introduced by Hamilton with his quaternions, it 

 seems a proper place to note the origin of the simplicity 

 which attends the equations of physics when expressed in 

 the forms introduced by Hamilton. This simplicity arises, I 

 believe, from the fact that the vectors and vector functions 

 such as 



7 . . 7 . d .d d 



ix + iy + kz, iu + w + lew, %—r- + j-r- + /c-r, etc. 

 17 ax " ay dz 



possess the permanency of form with which I propose to 

 deal, while their cartesian components x, y, z, etc., taken 

 separately, do not. In the general equations of physics we 

 meet with expressions which demand this permanency of 

 form, and, therefore, Hamilton's notation is a most suitable 

 mode of expressing them. 



Let u, v, w be the components, along a set of axes, of a 

 vector function of a point in a continuous medium, and let 

 be some scalar function, and consider that these functions 

 are not related to any directions fixed in the medium but 

 only to the axes. 



Let some new set of axes be chosen, and let the new 

 co-ordinates of the point be X, Y, Z, then the new com- 

 ponents U, V, W, of the vector function and the new scalar 

 function $ can be found from the original values by 

 substituting for x in terms of X, Y, Z, and the three angles 



