396 macfarlane: algebra of physics 



conjugate powers instead of the complete powers. For example, 

 in the case of the square, the use of 



A 2 + AB + BA + B 2 

 which gives rnerely the square of the resultant, instead of 



A 2 + 2AB + B 2 

 which is the complete square of the complex A + B. 



Hyperbolic quaternions. Read before the Royal Society of 

 Edinburgh, July 16, 1900. Proc. 23: pp. 169-180. In this 

 paper the hyperbolic analogue to the spherical quaternion was 

 further considered, the surface corresponding to the sphere being 

 the equilateral hyperboloid comprising all three sheets. Lord 

 Kelvin was in the chair; from his well known antipathy to the 

 spherical quaternion it required some courage to attempt the 

 elucidation of the hyperbolic one, even with the aid of models 

 made of wire. 



Application of space-analysis to curvilinear coordinates. Deux- 

 ieme congres international des mathematiciens, pp. 305-311. 

 Paris, 1900. An expression is given for V in the case of equi- 

 lateral-hyperboloid coordinates. Let R denote a vector to the 

 positive double sheet of such a surface; then R = rp h where r 

 denotes the hyperbolic modulus and pi the hyperbolic unit. 



Now pi = cosh d-h-\- i sinh 6 (cos <p j + sin <p-k); 



p 2 = — = sinh 9-h + i cosh 8 (cos <p j + sin <p ■ k) ; 

 dd 



and P3 = — sin <p-j + cos <p-k; 



hence V = — / p + — / rp 2 + — / ri sinh 9 ■ p 3 . 

 dr dd d<p 



On the square of Ha?nilton , -s delta. Atti del IV coiigresso inter- 

 nazionale dei matematici, 3: pp. 153-157. Roma, 1908. The 

 common way of forming the square of the sum of three successive 

 vectors gives merely the scalar part of the complete square. Does, 

 it was asked, a similar distinction hold in forming the square of 

 Hamilton's operator V, which is well-known to be a kind of sym- 

 bolic vector? In the case of rectangular coordinates 



dx dy dz 



