CLIFFORD'S "DYNAMIC" 271 



and the three first integrals above referred to as the immediate deduction from this in 



the form 



Vpp = a. 



Take again Gauss's expression for the work done in carrying a unit magnetic pole 

 round any closed curve under the action of a unit current in any other closed circuit. 

 As originally given, it was [a long unwieldy expression in x, y, z, x', y', d~\. With the 

 aid of the quaternion symbols this unwieldy expression takes the compact form 



i . pdpdp' 



The meanings of the two expressions are identical, and the comparative simplicity of 

 the second is due solely to the fact that it takes space of three dimensions as it finds 

 it ; and does not introduce the cumbrous artificiality of the Cartesian coordinates 

 in questions such as this where we can do much better without them. 



In most cases at all analogous to those we have just brought forward, Prof. 

 Clifford avails himself fully of the simplification afforded by quaternions. It is to be 

 regretted, therefore, that in somewhat higher cases, where even greater simplification is 

 attainable by the help of quaternions, he has reproduced the old and cumbrous notations. 

 Having gone so far, why not adopt the whole ? 



Perhaps the most valuable (so far at least as physics is concerned) of all the 

 quaternion novelties of notation is the symbol 



.9 .9 .3 



V = I + J~~ + K^~, 



dx J dy dz 

 whose square is the negative of Laplace's operator : i.e. 



A glance at it is sufficient to show of what extraordinary value it cannot fail to be in 

 the theories of Heat, Electricity, and Fluid Motion. Yet, though Prof. Clifford discusses 

 Vortex Motion, the Equation of Continuity, etc., we have not observed in his book 

 a single V. There seems to be a strange want of consistency here, in coming back 

 to such " beggarly elements " as 



Sftt + ty + &,iv 



instead of - SVtr, 



especially when, throughout the investigation, we have a used for 



ui+vj+ wk, 



and when, in dealing with strains, the Linear and Vector Function is quite freely used. 

 Again, for the vector axis of instantaneous rotation of the element at x, y, z (p), when 

 the displacement at that point is <r = in +jv + kw, we have the cumbersome form 



instead of the vastly simpler and more expressive 



