QUATEENIONS AND VECTOR ANALYSIS. 177 



scalar, as I have done, but to a vector also. The vector part of New o> 

 (construed in the quaternionic sense) would be exactly what I have 

 represented by Lap CD, and the scalar part, taken negatively, would be 

 exactly what I have represented by Max to. The quaternionist has 

 here a slight economy in notations, which is of less importance, since 

 all the operators New, Lap, Max may be expressed without 

 ambiguity in terms of the potential, which is therefore the only one 

 necessary for the exact expression of thought. 



But what are the formulae which it is necessary for one to remember 

 who uses my notations ? Evidently only those which contain the 

 operator Pot. For all the others are derived from these by the simple 

 substitutions ISew = V Pot, 



Lap = VxPot, 



Max =V. Pot. 



Whether one is quaternionist or not, one must remember Poisson's 

 Equation, which I write 



V.VPotft)=-47Tft), 



and in quaternionic might be written 



V 2 Pot o> = 4. 



If ft) is a vector, in using my equations one has also to remember the 

 general formulae, 



which as applied to the present case may be united with the preceding 

 in the three-membered equation, 



V.VPotft) = VV. Potft)-VxVxPotft)=-47rft). 



This single equation is absolutely all that there is to burden the 

 memory of the student, except tfcat the symbols of differentiation 

 (V, Vx, V.) may be placed indifferently before or after the symbol 

 for the potential, and that if we choose we may substitute as above 

 New for V Pot, etc. Of course this gives a good many equations, 

 which on account of the importance of the subject (as they might 

 almost be said to give the mathematics of the electro-magnetic field) 

 I have written out more in detail than might seem necessary. I have 

 also called the attention of the student to many things, which perhaps 

 he might be left to himself to see. Prof. Knott says that the quater- 

 nionist obtains similar equations by the simplest transformations. 

 He has failed to observe that the same is true in my Vector Analysis, 

 when once I have proved Poisson's Equation. Perhaps he takes his 

 model of brevity from Prof. Tait, who simplifies the subject, I believe, 

 in his treatise on Quaternions, by taking this theorem for granted. 



Nevertheless, since I am forced so often to disagree with Prof. 

 Knott, I am glad to agree with him when I can. He says in his 



G. II. M 



