[ 19^ ] 



XIII. 



DEFFEREXTIATIOX IX THE QTJATEEmON ANALYSIS. 

 By ALEXAXDEE MACFARLANE, D.Sc, LL.D. 



[cOilirirxiCATED BY ME. JOLY, EOYAL ASTEOXOilEE OF lEELAXD.] 



[Read June 25, 1900.] 



It is a prevalent belief in tte leamed vroiicl that the quaternion 

 analysis has not taken that place among the methods of research 

 which was predicted for it by its celebrated founder. Eew mathe- 

 maticians, excepting Professor Tait, have been able to bend the bow 

 of Ulysses. "W^riters on the subject appear to share the opinion 

 explicitly enunciated by Professor Hardy^ that the writings of 

 Hamilton contain the suggestion of all that will be done in the 

 way of quaternion research and applications. The very greatness 

 of the ElemenU appears to have had a deterring effect ; for it has 

 been considered a great storehouse of all the results that can be 

 harvested in the quaternion field. It is commonly thought that 

 the field is but a small corner of the mathematical domain, and that 

 Hamilton has gone over it so thoroughly that there is little left to 

 glean after him. However, there are good reasons for believing that 

 the field is not narrow, but is, in all probability, nearly coextensive 

 with the mathematical domain. ITy opinion is that Hamilton's estimate 

 of the importance of the discovery which he communicated to this 

 Academy at the memorable meeting of November, 1843, is under, in- 

 stead of over the mark, and that this will be demonstrated by the course 

 of development of mathematical analysis in the coming century. 



But, first of all, the bow must be examined to find out why it is 

 so difficult to bend. I do not suppose that Hamilton considered that 

 he had arrived at a finality of wisdom on the principles of the analysis. 

 He did not intend to give to the world a dead analysis, however 

 classical. Anyhow, the analysis can receive only good from a free 

 and independent discussion of its principles ; if they are perfect, their 

 perfection will thereby become the more apparent and convincing ; 

 if they are imperfect, it is certainly desirable that the imperfection 



1 In the preface to his Elements of Quaternions. 

 P 2 



