176 QUATERNIONS AND VECTOR ANALYSIS. 



Or, if the problem and solution had been written thus : Required o> 

 in terms of Vo> when SVo> = 0. 



Solution : co = VV ' 2 Vco = V - l V. 



My critic has himself given more than one example of the unfitness 

 of the inverse Nabla for the exact expression of thought. For 

 example, when he says that I have taken "eight distinct steps to 

 prove two equations, which are special cases of 



I do not quite know what he means. If he means that I have taken 

 eight steps to prove Poisson's Equation (which certainly is not ex- 

 pressed by the equation cited, although it may perhaps be associated 

 with it in some minds), I will only say that my proof is not very 

 long, especially as I have aimed at greater rigor than is usually 

 thought necessary. I cannot, however, compare my demonstration 

 with that of quaternionic writers, as I have not been able (doubtless 

 on account of insufficient search) to find any such. 



To show how little foundation there is for the charge that the 

 deficiencies of my system require to be pieced out by these integral 

 operators, I need only say that if I wished to economise operators I 

 might give up New, Lap, and Max, writing for them VPot, VxPot, 

 and V. Pot, and if I wished further to economise in what costs so little, 

 I could give up the potential also by using the notation (V.V)" 1 or 

 V" 2 . That is, I could have used this notation without greater sacrifice 

 of precision than quaternionic writers seem to be willing to make. I 

 much prefer, however, to avoid these inverse operators as essentially 

 indefinite. 



Nevertheless although my critic has greatly obscured the subject 

 by ridiculing operators, which I beg leave to maintain are not worthy 

 of ridicule, and by thoughtlessly asserting that it was necessary for 

 me to use them, whereas they are only necessary for me in the sense 

 in which something of the kind is necessary for the quaternionist also ? 

 if he would use a notation irreproachable on the score of exactness 

 I desire to be perfectly candid. I do not wish to deny that the 

 relations connected with these notations appear a little more simple 

 in the quaternionic form. I had, indeed, this subject principally in 

 mind when I said two years ago in Nature (vol. xliii, p. 512) [this vol. 

 p. 158] : " There are a few formulae in which there is a trifling gain 

 in compactness in the use of the quaternion." Let us see exactly 

 how much this advantage amounts to. 



There is nothing which the most rigid quaternionist need object to 

 in the notation for the potential, or indeed for the Newtonian. These 

 represent respectively the operations by which the potential or the 

 force of gravitation is calculated from the density of matter. A 

 quaternionist would, however, apply the operator New not only to a 



