of Edinburgh , Session 1867 - 68 . 
431 
given, partly after Hamilton ; and in it the main object sought is 
usually the quaternion, q , on which depends the operator 
1 ( ) 2 _1 
which turns the body from any initial position whatever to its 
position at time t. The investigation of the axis and amount of 
the single rotation by which the body may be thus changed in 
position was first suggested by Euler, but it was greatly simpli- 
fied and extended by Rodrigues and Cayley. The fundamental 
kinematical formula of the present paper, which connects the 
quaternion, q, with the instantaneous axis of rotation, e, is 
e = 2Y qq~', 
and had been obtained by Cayley, though not in this very simple 
form, as a quaternion translation from some of his Cartesian 
results. 
From this equation the formulae, connecting the angular velocities 
about the principal axes with the various sets of three angular 
co-ordinates which have been employed to determine the position 
of the body at time t , are deduced, mainly to show how complex 
are these systems as compared with those suggested at once by 
quaternions. 
Hamilton has pointed out that, if be the vector of an element 
m of the mass, the whole kinetic properties of the motion are con- 
tained in the equation (which is really that of Lagrange) 
zv[zs- — \^) = 0 , 
where $ is the vectorex pressing the applied force on unit of mass 
at m. He has also given the kinematical relation 
= Ye 
3T . 
By means of this he obtains 
. mzu'Y ezv — y , 
where y is a constant vector if no forces act, otherwise it is the 
time-integral of the vector-couple 
3 K 
VOL. VI. 
