190 Prof. Donkiii on the Geometrical Theory of Rotation. 



Let it be required to express p, q, r in terms of X/av and their 

 diflFerential coefficients. 



It is to be observed that I, m, n refer indifferently to either set 

 of axes. Considering them to refer to x, y, z, we have the ex- 

 pression (3.), namely, 



6 . 6 , ., , , 



cos;r — sm — [il+jm + kn), 



for the quaternion representative of the displacement. In the 

 next instant dt, a rotation ivdt takes place round an axis whose 



direction cosines are — , — , — : which will be represented by 

 10 w w 



the quaternion 



wdt . wdt / .p . q 



cos sm — ' - ' " 



\ IV ^ 10 %0 f 



or, neglecting quantities of the second order, by 



And the displacement of the body in its new position may be 

 represented either by the product 



(l- 2" (V^+i'? + ^'')) (cosg - sin 2 (?7+>m + ^«) j, 



or by the quaternion 



& & 



cos - — sin — {iV +jm' + kn'), 



where 6', /',... are put for d + dO, l + dl, ... Equating these two 

 expressions, and then applying each side of the equation as a 

 multiplier to 



cos ^ + sin -^ {il+jm + kn), 



we get 



1 ~ 2~ ^'^ '*"-^^ "^ ^"^ = cos ^ cos ^ (1 — iV —jijJ — kv') (1 + i\ +j/ji, + kv), 



of which the right side, developed (omitting terms of the second 

 order, and observing that 2'lxdX + fi,d/ji, + vdv)=dK), is easily 

 found to be 



1 {i{d\ + vdfj.—/j,dv)+j{d/M+7ulv—vd\)+k{dv+fid\—'\d/u,)}^ 



