JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. II, OCTOBER 4, 1912 No. 16 



MATHEMATICS. — Account of researches in the algebra of 

 physics. 1 III. A. Macfarlane. 



Differentiation in the quaternion analysis. Read before the 

 Royal Irish Academy, June 25, 1900. Proc. (3) VI: no. 2. The 

 object of the paper was stated as follows: There are two places 

 in the Elements of Quaternions where further investigation seems 

 desirable. The quaternion analysis is intended to be applicable 

 to space of three dimensions, but at these two places Hamilton 

 restricts the analysis to the plane. The first place is in the treat- 

 ment of logarithms. He says in Art. 316: 



In the present theory of diplanar quaternions we cannot expect to 

 find that the sum of the logarithms of any two proposed factors shall be 

 generally equal to the logarithm of the product; but for the simpler 

 and earlier case of complanar quaternions that algebraic property may be 

 considered to exist with due modification for multiplicity of value. 



The other place is in the treatment of differentiation. He says 

 in Art. 333. 



The functions of quaternions, which have been lately differentiated, 

 may be said to be of algebraic form; the following are a few examples of 

 differentials of what may be called, by contrast, transcendental func- 

 tions of quaternions; the condition of complanarity being, however, 

 here supposed to be satisfied, in order that the expressions may not 

 become too complex. 



I then proceeded to show that the source of the difficulty in 

 both cases was one and the same, namely, the erroneous use of 



^ead before the Philosophical Society of Washington, April 20, 1912. See 

 this Journal 2: 331-337 and 363-372. 1912. 



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