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XXVIII. On the Application of Quaternions to the Theory of 

 Rotation. By Arthur Cayley, Esq.* 



JN a paper published in the Philosophical Magazine, Fe- 

 bruary 1 845, I showed how some formula? of M. Olinde 

 Rodrigues relating to the rotation of a solid body might be 

 expressed in a very simple form by means of Sir W. Hamilton's 

 theory of quaternions. The property in question may be thus 

 stated. Suppose a solid body which revolves through an angle 

 round an axis passing through the origin and inclined to the 

 axes of co-ordinates at angles a, b, c. Let 



A=tan-0cosa, /x= tan - cos b, v— tan -0 cose, 



Z m 2 



and write 



A=l +i\-\-jfj. + kv. 



Let x, y, z be the co-ordinates of a point in the body pre- 

 vious to the rotation, x v y v z x those of the same point after 

 the rotation, and suppose 



Tl — iw+jy + kz 



rii=^ 1 +j> 1 +/^ 1 . 



The co-ordinates after the rotation may be determined by the 

 formula 



n^AIIA- 1 ; 



viz. developing the second side of this equation, 



Tl x = i.{ctx + fiy + yz) 

 +j{u'x + P'y + y'z) 

 + k(ct"x + P"y + y"z), 

 where, putting to abbreviate x=l -|-X 2 +ft. 2 -f-v 2 , we have 

 xa=l+A 2 — ju, 2 — v 2 , xa' = 2(Ajtt + v), xa" = 2(Av — jw,), 

 x/3 = 2(A J «,-v), x/3'=l-A 2 -fj* 2 -v 2 , x/3"=2(jw + A), 

 xy = 2(Av + j«,), xy'=2(ju,v — A), xy"= 1 — A 2 — j«, 2 -f v 2 ; 



these values satisfying identically the well-known system of 

 equations connecting the quantities «, /3, y, a', /3', -/, a", /3", y". 

 The quantities a, b, c, being immediately known when 

 A, ,a, v are known, these last quantities completely determine 

 the direction and magnitude of the rotation, and may therefore 

 be termed the co-ordinates of the rotation — A will be the qua- 

 ternion of the rotation. I propose here to develope a few of 

 the consequences which may be deduced from the preceding 

 formulae. 



* Communicated by the Author. 



