JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. [I, SEPTEMBER 19, 1912 No. 15 



MATHEMATICS. — Account of researches in the algebra of 

 physics. 1 II. A. Macfarlane. 



The fundamental theorems of analysis generalised for space. 

 Read before the New York Mathematical Society, May 7, 1892. 

 Separately printed. First of all the generalised DeMoivre's 

 Theorem was considered. Let a a , /3 5 , y c denote three independ- 

 ent spherical angles ; then : 



a a fib y c _ ( CO g a _|_ gm a q, 71 "/ 2 ) ( cog q _|_ gm Q flf/ 2 ^ 



(cos c + sin c . y* /2 ) = cos a cos b cos c 

 + cos a cos b sin c . y w/2 -f- cos b cos c sin a . a 12 

 + cos c cos a sin 6 . /3 7172 



I • 1. ^iH"/2 7T/2 , 7 • 7T/2 7T/2 



-f- cos a sin b sin c . /3 ' y ' + cos 6 sin c sin a . a ' y 



r 7, tt/2 rjr/2 , • • r ■ 7r/2 „7r/2 tt/2 



-f- cos c sin a sin 6 . a ' fi ' + sin a sin 6 sin c . a ' /3 7 



The compound angles are expanded by means of the principle 

 07T/2 7 *72 _ _ cog ^ _ gin ^ [ 1 g 7 ] 7r/2 . As the angles are inde- 

 pendent of one another, the theorem has its general meaning in 

 the composition of rotations. 



The main object of the paper was the demonstration of the 

 generalised Exponential Theorem; from it the other theorems 

 follow as consequences. To prove that 



.„7r/2 TT/2 _ .,71-/2 , TT/2 



gb/3 gCT _ gb/3 ' +cy ' 



1 Read before the Philosophical Society of Washington, April 20, 1912. See 

 this Journal 2:331-337. 1912. 



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