282 
MR. R. S. HEATH ON THE DYNAMICS OF 
pair of rotations about any two lines which are conjugate to each other in a certain 
linear complex. The surface corresponding to the cylindroid is found to be of the 
fourth order with a pair of nodal lines. Lastly, the condition of equivalence of any 
number of twists about given screws is investigated. 
In kinetics, the measure of force is deduced from Newton’s second law of motion, 
and the laws of combination and resolution are proved. The consideration of the 
whole momentum of a body suggests the idea of moments of inertia, and a few of 
their properties are investigated. The general equations of motion referred to any 
moving axes are then found, and in a particular case they reduce to a form corres¬ 
ponding to Euler’s equations ; these are of the type 
— (B—H)w £ w 6 —(G—C> 5 a> 3 =Q x . 
The last part is occupied in the solution of these equations when no forces act, in 
terms of the theta-functions of two variables. A solution is obtained in the form 
&a(x; y) 
OJ j Cl . ) 
diaO, y) 
r d 7 (r, y) 
y )’ 
7 &i(z,y) 
y)’ 
_ y) 
0,5 3 'aj*,v)’ 
n y ) 
W3 $u(*,y)’ 
7. y) 
'Su&y)’ 
where x=nt-\-a and y is arbitrary. But in order that these values may satisfy the 
equations, a relation among the parameters of the theta-functions must be satisfied. 
This is 
C G C 10 C 5 C 9+ C 1 C 13 C 2 C 14 = 0* 
The solution is not complete, because after satisfying the equations of motion only 
four constants remain to express the initial conditions, whereas six constants are 
required. 
Introduction. 
A concise review of the characteristics of the different kinds of generalised space will 
be found in the introduction to Professor Clifford’s mathematical works by the late 
Professor H. J. S. Smith (Introduction, p. xxxix.), together with an analysis of 
Clifford’s numerous memoirs relating to this subject. Further information may be 
found in the following papers :— 
Dr. Ball, “On the Noil-Euclidean Geometry,” ‘Hermathena,’ vol. iii. 
Professor Cayley, “A Sixth Memoir on Qualities," Phil. Trans., 1859. 
Professor Lindemann, “ Projectivische Behandlung der Meckanik starrer Korper,” 
Math. Annalen. Bd. vii., 1874. 
