44 Proceedings of the Royal Irish Academy. 



IV. — Applications of Quaternions to the Theory of Scrcivs. 



One of the most useful discoveries of Hamilton, at least in so far as our 

 present subject is concerned, is contained in the following proposition* : — 



" Any infinitely small change in the position of a rigid body is equivalent 

 to the alteration of each of its vectors a to another of the form 



(1 + Sa = a + £ + Viu, (42) 



£ and i being two arbitrary but infinitesimal vectors which do not vary in 

 the passage from one point of the body to another." 



This formula is the representation by vector-analysis of the fundamental 

 principles on which the Theory of Screws is based. The vector-conception 

 of a twist which is here indicated shows by its conciseness, elegance, and 

 lucidity that there must be intimate relationship between Quaternions and 

 the Theory of Screws. 



The lat€ Professor Charles J. Joly, the editor of " Hamilton's Elements 

 of Quaternions," has contributed an admirable series of original memoirs 

 on Quaternions to the Transactions of the Koyal Irish Academy .f These 

 memoirs relate very largely to the Theor}' of Screws. A concise account of 

 Joly's researches on the application of the Theory of Quaternions to the 

 Theory of Screws has been given by him in the appendix to vol. ii. of his 

 edition of Hamilton's " Elements of Quaternions," pp. 390-397 (1901). But 

 the subject has been much more fully dealt with in Joly's " Manual of 

 Quaternions," 1905, a most instructive and useful book, to which, as already 

 indicated, we refer briefly as Joly's " Manual." 



The proof by vector-analysis that the most general displacement of a 

 rigid syst«m must be in all cases what we understand as a twist about a 



'"Hamilton's £lemeoU," toI. ii., p. 287. 



t(l) "TheTheory of Linear Vector FuncUoiu": Traiu. B. I. A., vol.xxx., pp. 897-047 (1894)- 



(2) '• Scalar InTariants of two Linear Vector Functions" : Trans. R. 1. A., vol. xxx., pp. 709- 



728 (189.5). 



(3) " The Interpretation of a Quaternion as a Poinl-syinbol " : Trant. It. I. .\., vol. xx«ii., 



pp. 1-16 (1901). 



(4) " Quaternion Arrays" : Trans. R. I. A., vol. xxxii., pp. 17-30 (1901). 



(5) " Reprewntation of Screws by Weighted PoinU": Trans. R. I. A., vol. xxxii., pp. 61-92 



(1962). 



(6) "The Quadratic Screw System : a Study of a Family of Quadratic Complexes" : Trans. 



R.I. A., vol. xxxii., pp. 15.5-238 (1903). 



(7) " The Geometry of a Three-system of Screws" : Trans. R. I. A., vol. xxxii., pp. 239-270 



(1903). 



(8) " Quaternions and Projective Geometry," communicated by Joly to the Boyal Society, and 



published in the Philosophical Transactions, Series A, toI. 201, pp. 223-327 (1903). 



