368 macfarlane: algebra of physics 



ing surface are in a sense unit axes, or unit lines, because the 

 modulus is unity for each. 



It was first shown that for a simple hyperbolic angle in a plane 

 passing thru the axis of revolution, the expression is 



/3 t6 or exp ib(3 w/2 where is a spherical axis. 



Let p denote any spherical axis; it is expressed quite generally 



^ p = cos 8 . h + sin 8 (cos <p . j + sin <p . k); 



and if u denote the circular measure, any spherical angle with p 

 for axis, that is, in the plane normal to p, is expressed by p" or 



7T/2 



exp up . 



In a similar manner any axis, as OP, to the double sheet can be 

 expressed as p = cosh 8 . h + i sinh 8 (cos <p . j -f sin <p . k) ; 

 and if u denote an angle of the circular kind, p u or exp up*' 2 

 expresses an elliptic angle, that is a sector of the elliptic section 

 made by the plane normal to p. 



The axis OP" at right angles to OP, terminated in the single 

 sheet, has the form ip and ip = i { cosh 8 (cos <p . j + sin tp . k) 

 — i sinh 8 . h}. Consequently ip u = p iu = exp iup w/2 expresses 

 a hyperbolic angle; and the plane normal to ip makes a hyper- 

 bolic section. The composition of two general exspherical angles 

 was investigated. 



In the investigation of the elliptic trigonometry, the simplest 

 definitions were not chosen, the sine being defined with respect to 

 OA instead of OB. One of the main difficulties was the want of 

 an expression for the hyperbolic or elliptic arc, the solution of 

 which difficulty I did not then perceive. The trouble is con- 

 nected with the old difficulty of the rectification of such arcs. 



On the analytical treatment of alternating currents. Proc. of 

 the International Electrical Congress, Chicago, 1893, pp. 24-32. 

 In this paper I pointed out that plane algebra was the proper 

 analytical method for dealing with alternating currents. It was 

 read before Section A, Professor Rowland in the chair. Mr. 

 Steinmetz contributed to the same section an elaborate paper 

 to the same effect, entitled "Complex quantities and their use 

 in electrical engineering." Rowland stated that there was no 



