EARLY EFFORTS IN QUATERNIONS 127 



and treated i, j, k as imaginaries (like *J i) though of course according to their 

 proper laws of combination. For fun I extract this 



Much of course could not have been made of this, and accordingly on my return to 

 Cambridge I set to read other things, and to write my recently published Treatise 

 on Particle Dynamics. The Theories of Heat, Electricity and Light have since 

 occupied much of my spare time, and it was only in August last that I suddenly 

 bethought me of certain formulae I had admired years ago at p. 610 of your 

 Lectures and which I thought (and still think) likely to serve my purpose exactly. 

 [The matter which more immediately suggested this to me was a paper of Helm- 

 holtz's in Crelle's Journal (Vol. LV) which I was reading in July last as soon as we 

 received it, and which put the subject of Potentials before me in a very clear light. 

 The title (in German) I forget but an MS translation of my own which I have 

 now beside me is headed "Vortex Motion 1 ." It refers to the integration of the general 

 equations in Hydrodynamics, when udx + vdy + wdz is not a perfect differential.]... 

 So far from having any assistance, save what you have so kindly given me, I am 

 not even acquainted with any one who knows aught about quaternions (except 

 Boole of Cork with whom however I have not exchanged a remark on the subject, 

 and who, I suspect, looks on them in their analytical capacity only). 



So you see that, if there is any credit in my progress, it is entirely to your 

 Lectures and Letters that it is due. 



Hamilton's letter xiv, which was begun on Nov. 17, and continued 

 at fairly short but irregular intervals till Feb. 5, 1859, when it reached 

 88 closely written pages, ran on till April 3, in the form of eight postscripts. 

 There seems to be no later reference to Tail's confession of how he began 

 the study of Quaternions ; but various sections call for quotation because of 

 the bearing they have on the subsequent history. 



In his letter 19, of date Jan. 3, 1859, Tait wrote as follows of 

 Quaternions in general : 



About quaternions in general I may remark (as indeed I very frequently feel) 

 that the processes are sometimes perplexingly easy by which I mean that one is 

 often led in a step or two and without (at once) knowing it to the solution of what 

 would be by ordinary methods a work not so much of difficulty as of labour. This 

 however I take it must form one of its great excellencies in the hands of a person 

 very well acquainted with it. A drawback to a beginner, but (as I am gradually 

 being led to perceive) an immense advantage to one well skilled in the analysis, is 

 the enormous variety of transformations of which even the simplest formulae are 

 susceptible ; a variety fully justifying a remark of yours (Lectures Art. 504) which 

 not many months ago used somewhat to puzzle me. If I had gained nothing more 



1 The translation was published in Phil. Mag. July 1867. 



