ON THE SPECIAL PROBLEMS OF DYNAMICS. 223 
all divided by the common denominator 1+)?+ y?+¥ 7. Conversely, if the 
coefficients of transformation are as usual represented by 
2 B Bay? Me 
s »B, Y's 
hers ce y'; 
then d’, nu’, v”, A, w, v are respectively equal to 
1+a—,'—y", 1l—a+p'—y", 1l—a—f'+y", 
Fig aap eo) pi es 
each of them divided by 1+a+f'+y’". 
The memoir contains yery elegant formule for the composition of finite 
rotations, and it will be again referred to in speaking of the kinematics of a 
solid body. 
140. Sir W. R. Hamilton’s first papers on the theory of quaternions were 
published in the years 1843 and 1844: the fundamental idea consists in the 
employment of the imaginaries 2, 7, k, which are such that 
P=Pp=P=—1, jk=—kj=i, i=—tka=j, y= —p=k, 
whence also 
(w+iat+jy + kz) (w' +22! +7y' + kz') 
= ww'—xu' —yy'—2z2' 
fi(we'+w'e+yz2'—y'2) 
4j(wy' +w'y + 20'—z2'2) 
+kh(w2' +w'z+axy'—ay) ; 
so that representing the right-hand side by 
W+iX4+jY+kZ, 
we have identically 
W4XC4+ V4 2=(w?+a*+y?+2) (w?+u?+y?+2"), 
It is hardly necessary to remark that Sir W. R. Hamilton in his various 
publications on the subject, and in the ‘ Lectures on Quaternions,’ Dublin, 
1853, has developed the theory in detail, and has made the most interesting 
applications of it to geometrical and dynamical questions ; and although the 
first explicit application of it to the present question may have been made in 
my own paper next referred to, it seems clear that the whole theory was in 
its original conception intimately connected with the notion of rotation. 
141. Cayley, ‘‘ On certain Results relating to Quaternions” (1845). —It is 
shown that Rodrigues’ transformation formula may be expressed in a very 
simple manner by means of quaternions ; viz., we have 
tx jy +ke=(1 ++ juthvy)-\iX4+7Y +kZ) (1+i4+jut+hy), 
where developing the function on the right-hand side, and equating the coeffi- 
cients of 7,7, k, we obtain the formule in question. A subsequent paper, 
Cayley, ‘‘On the application of Quaternions to the Theory of Rotation’”’ (1848), 
relates to the composition of rotations. 
Principal Aes, and Moments of Inertia. Article Nos. 142-163. 
142, The theorem of principal axes consists herein, that at any point of a 
solid body there exists a system of axes Ox, Oy, Oz, such that 
Syzdn=0, Jzxdm=0, JS xydm=0. 
