Macfarlane — Differentiation in the Quaternion Analysis. 211 

 If we introduce, in the result, the reduction p^ = - 1 , then 



^3_ n ) 2a( ) 



9'r^ r dr ' 



which agrees with the well known result, excepting that it introduces 

 the negative sign. The remaining terms are 



a^c )4 3^( ) 4 ay 2 



3p2 ^2 2,^gp ^p g^ ^3p' 



which exist when the function involves p, and which are, in general, 

 vector in nature. 



I shall next apply the above principles of differentiation to find y 

 and y- for any function expressed in terms of the spherical coordi- 

 nates r, 6, 4>. According to the definition adopted, by the nabla of a 

 function is meant the sum of the several derivatives of the function 

 each multiplied by the nabla of its variable. Hence for a function 

 of r, e, cf>, 



Apply this operator to the function rp, in which case we know 

 that the nabla of the function is 3. We have 



V {rp) = pvr + ^ g^ V^ + »^ g^ V</> = 3. 



"We know that pv^ = 1 5 hence 



from which I infer that 



^^=_L and vc^= '^; 

 r^r^ r ^r- 



. n a( )i a( ) 1 a( ) 1 



wherefore v = ^ - + ^ "T + "^ T" 



or p oB op ocf> rop 



