CHAPTER III 

 MATHEMATICAL WORK 



IN the preceding chapter Tail's experimental work has been dealt with 

 apart from the other scientific activities of his mind. At no time however 

 did he limit his attention to one problem exclusively ; and while with the aid 

 of his company of voluntary workers he was for the last thirty years of his 

 life busy with experiments in the laboratory, at home in his study he was 

 using his mathematical powers with great effect in all kinds of enquiries. 

 This mathematical and theoretical work may be conveniently classified under 

 three headings : namely, quaternions, mathematics and mathematical physics 

 outside the quaternion method, and the labours incidental to the writing of 

 his more mathematical treatises. The quaternion work will be considered in 

 an appropriate chapter ; another chapter will be devoted to the preparation of 

 Thomson and Tail's Natural Philosophy ; and Tail's other lilerary conlribu- 

 lions in book form will have a similar separale Irealmenl. Here, in a some- 

 whal disconlinuous manner, I propose lo give a general accounl of ihe more 

 malhemalical of his scienlific papers and noles, iracing as far as possible iheir 

 genesis and iheir conneclion wilh olher lines of research. 



Passing over his early qualernionic papers in the Quarterly Journal of 

 Mathematics, the Messenger of Mathematics, and the Proceedings of the Royal 

 Society of Edinburgh, we come in 1865 to a purely mathemalical paper on 

 ihe Law of Frequency of Error ( Trans. R. S. E. Vol. xxiv ; Sci. Pap, Vol. i, 

 p. 47). He was led lo enquire inlo ihe foundalions of ihe iheory of errors 

 when he was wriling ihe article Probabililies for ihe first edilion of Chambers' 

 Encyclopaedia, his aim being lo eslablish the ordinary law of errors by a 

 " natural process " free from ihe malhemalical complicalions which characlerise 

 ihe work of aulhorilies like Laplace and Poisson. Slarling from a simple case 

 of drawing while and black balls from a bag, he deduced ihe well-known 

 exponenlial expression, and ihen generalised ihe demonslralion. If we excepl 

 his much laler papers on ihe kinelic iheory of gases Tail does nol seem to 

 have relurned lo queslions involving ihe iheory of probabililies. 



