MACFARLANE : ALGEBRA OF PHYSICS 397 



and V 2 is said to be 



d 2 h 2 ft 2 



dx 2 dy 2 dz 



from which is derived by reduction 



52 52 . 52 



+ - — 7 + 



dx 2 dy 2 dz 2 



But in the multiplication the vector part has been cancelled out 

 due to an artificial order of the factors, whereas the real order 

 is a cyclical order of the terms. When the parts cancelled are 

 restored, the square has the additional term 



f ?\2 7\2 >\2 ) 



I dxdy dydz dzdx 



It was shown also in the case of polar coordinates, that the restor- 

 ation of the cancelled terms gave the complete square. 



Unification and development of the principles of the algebra of 

 space. Bulletin of the Quaternion Association. October, 1910, 

 pp. 41-92. As regards unification the following position is held: 



We have before us three forms of space-analysis: the scalar, founded 

 by Descartes, which makes use of axes, but provides no explicit notation 

 for directed quantities whether line or angle; the quaternionic, founded 

 by Hamilton, which is characterized by a notation for versors or angles 

 in space; the vectorial, founded by Grassmann, which is built on vector- 

 units and compound units derived from them. For the past half cen- 

 tury the masters of these several forms have been engaged in a trian- 

 gular fight: much has been written on vectors versus quaternions; and 

 we have heard of a thirty years' war between one who could bend the 

 bow of Hamilton and one equally skilled in the more ancient weapon of 

 Descartes. It will surely be admitted that each branch contains part 

 of the truth; it is therefore highly probable that no one of them contains 

 the whole truth, and that each has a part of the truth which the others 

 have not. It has for long seemed to me that what is wanted is an analy- 

 sis which will harmonize all three, and present itself as the space-general- 

 ization of algebra. As to this conception of the oneness of the algebra 

 of space, I may quote Sylvester's declaration that he would as soon 

 acknowledge a plurality of gods as a plurality of algebras. Likewise, 

 Gibbs at the close of his address to the Mathematics Section of the Ameri- 

 can Association, said we begin with multiple algebras and end with mul- 

 tiple algebra. 



