JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. II, AUGUST 19, 1912 No. 14 



MATHEMATICS. — Account of researches in the algebra of 

 physics: 1 I. A. Macfarlane. 



As these researches extend over thirty years and the results 

 have appeared in a variety of publications, I am glad to accept 

 the invitation of the Philosophical Society to give a connected 

 account of the principal results, more especially because I .have 

 reason to believe that I have solved the main difficulties of the 

 problem investigated. 



Note on plane algebra. Proc. Roy. Soc. Edinb., 1883, pp. 184- 

 186. I began by studying plane algebra as a logical generalisa- 

 tion of ordinary algebra. The algebraic symbol used was a 

 small Roman letter, which denoted a length combined with an 

 angle : thus a = a. a b=6./3 c = c . y r = r . 6. In this nota- 

 tion V — 1 r = r . tt/2 and — r = r . <k . The sum was obtained 

 by the parallelogram construction. The product ab was defined 

 as the product of the lengths combined with the sum of the angles; 



that is ab = ab . a + (3, giving b 2 = b- . 2/3. The quotient ~ b 



was defined as the quotient of the lengths combined with the 



difference of the angles ; thus 



/ 



-b = « + /3; giving - b = 10 



a a b 



The product of three quantities abc = abc . a + /3 + y, giving 



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



331 



