208 Proceedings of the Royal Irish Academy. 



therefore d'R = {cPr - r{(Wy - r{sm ed^f]p 



+ {2drde + rd^d - r sin ^ cos (dcfiy} -^ 



+ ( 2drd<ii + rd-4> + 2rdddcji cot 6)~- 



Here E does not denote a radius vector merely; it denotes any circular 

 vector whatever. The ordinary direct process of deducing these com- 

 ponents is very long and cumhrous. See Price's Infinitesimal Calculus, 

 vol. in., p, 433. 



Consider now that important and mysterious operation peculiar to 

 the calculus of Quaternions which is denoted by v- The original 

 definition of this symbol, as given by Hamilton, is purely symbolic ; 

 namely, ^9 -^ s.].^ 



dx dy ' dz 



Here the i, j, k are written in the numerator, and before their respec- 

 tive differential operators ; why should they not be written after the 

 operator, and in the denominator ? As they form a kind of differential 

 operator, it is natural to suppose that they should appear in the 

 denominator, in order that the homogeneousness of the dimensions of the 

 expressions may be preserved. In fact, if u denote a scalar function, v^ 

 is said to be the rate of change of tt per unit of length in the direction 

 of the most rapid change ; from which it may be inferred that the axis 

 naturally appears in the denominator. The i is written to the left of 



the --, probably because Hamilton chose for the standard order that 

 ox 



from right to left ; where the natural order of writing is followed, the 



i ought to be written to the right of the — • 



ox 



The following may be taken as a preliminary definition. By the 

 nabla (v) of a function of several variables is meant the sum of the 

 partial derivatives, each multiplied by the nabla of its variable. Let 

 a denote a Hamiltonian vector, having a magnitude r and an imaginary 

 axis p equivalent to ^y- 1 po- ^oi' any function of r and p, 



Apply it to H itself „ 3(rp) 9(rp) 



= pvr + ^Vp- 

 * ( ) indicates the place for the function. 



