Motion of a Rigid System about a Fixed Point. 4-33 



If we assume for convenience ip=7r, and put 



fl d d ,fl 



/tan - =X, m tan- =jw., « tan - =y, sec^ - =x, 



^ ia 'A Ii 



and substitute for ^, >j, ^ their values in terms of a:, y^ », this 

 equation becomes 



{\ ^^ i\^-j[t.-\-h){ix -^-jy -\-'kz){\—i\—j[t.—h) 



^%{i{ax-\- a^y + d^z) -{-j{bx + i^y + b"z) + /i^(c^ + c'j/ + c"z) } . 



The first side, developed, is 



i{{l + \^—l/,^—v^)x + 2{\i/i,—v)y + 2{v\ + iJi)z} 



+i{ (1 +jOt2-v^- A% + 20AV-A)5r + 2(Xnt + v)a;} 



Observing, then, that the equation must subsist indepen- 

 dently of the values of .r, y, z, we have 



xa=l+A2— 1*3— yS xa'=2(X/*— v) xrt" = 2(vX + jx) 



x^ = 2(A/* + y) x&'=l+j*2_^2_;^2 xZ>" = 2(|«,y — A) 



xc=2(yA-|x) xc' = 2(]av + X) x<:"= l+v^— X«— /x^. 



These are the formulae in question, as given by Mr. Cayley 

 in the Cambridge Mathematical Journal*. 



The preceding process affords a striking example of the 

 power of the method of quaternions in reducing complex geo- 

 metrical questions of a particular class to algebraical calcula- 

 tion. The reader will probably have perceived that I consider 

 the quaternion 



{cos •& + sin ^{il+jm + Jcn)]p 



(where Z^ + m^+w^ssl) as representing the rotation opaline 

 or radios vector p through an angle ^ in a plane perpendicular 

 to the axis whose direction cosines are /, m, n. In a future 

 paper I propose, with the editors' permission, to explain the 

 system of interpretation which leads to this result, and to show 

 that it represents the theory of quaternions as a natural and, 

 indeed, necessary extension of the common geometrical algebra 

 of two dimensions. 

 Oxford, April 30, 1850. 



• It is right to state that my knowledge of the contents of M. Rodrigues' 

 memoir is derived solely from the paper just mentioned, as I have not at 

 present an opportunity of referring to the fifth volume of Liouville's 

 journal. 



