38 
DE. E. S. BALL’S EESEAECHES IN THE DYNAMICS OF A EIGID 
If the two screws 0 2 have equal pitches (g), we must have ^ = — and 
§ =p+m cos20 n 
g 1= =^4-m cos2fl 3 ; 
whence the expression for the volume of the tetrahedron is finally 
If § = 0 we have M. Chasles’s theorem. This generalization might have been deduced 
at once from the original theorem by the remark of [art. 82], that any coordinate system 
of screws is still a coordinate system when the pitches of all the screws have received a 
constant addition. 
Postscript. 
Eeceived January 27, 1874. 
At the time the foregoing paper was read the writer was not aware of the close con- 
nexion between the Theory of Screws and the recent geometrical researches on the 
Linear Complex. His attention was kindly directed to this point by Dr. Felix Klein, 
at the Bradford Meeting of the British Association. 
Plucker, in his ‘Neue Geometrie des Baumes,’ p. 24, thus introduces the word 
“ Dyname : ” — “ Durch den Ausdruck ‘ Dyname ’ habe ich die Ursache einer heliebigen 
Bewegung eines starren Systems, oder, da sich die Natur dieser Ursache wie die Natur 
einer Kraft iiberhaupt, unserem Erkennungsvermogen entzieht, die Bewegung selbst : 
statt der Ursache die Wirkung, bezeichnet.” Although it is not very easy to see the 
precise meaning of this passage, yet it appears that a “ Dyname ” may be either a twist 
or a wrench (to use the language of the present paper). 
On page 25 ( loc . cit.) we read : — “ Dann entschwindet das specifisch Mechanische, und, 
um mich auf eine kurze Andeutung zu beschranken : es treten geometrische Gebilde 
auf, welche zu Dynamen in derselben Beziehung stehen, wie gerade Linien zu Kraften 
und Botationen.” There can be little doubt that the “geometrische Gebilde,” to which 
Plucker refers, are what we have called screws. 
The surface used in the £ Theory of Screws’ (page 161), and also in Phil. Mag. vol. xlii. 
p. 181, under the name of the cylindroid, had been already described by Plucker, p. 97, 
loc. cit. Plucker arrives at this surface by the following considerations: — Let 0=0 
and O'=0 represent two linear complexes of the first degree, then all the complexes 
formed by giving [Jj different values in the expression 0+^0' = 0 form a system of which 
the axes lie on the surface z{x 2 -\-y 2 )~ (k°— 7c 0 )xy=0. The parameter of any complex of 
which the axis makes an angle co with the axis of x is k—k Q cos 2 sin 2 u. The writer 
was informed by Dr. Klein that Plucker had also constructed a model of this surface 
(see note to p. 98). 
Plucker does not appear to have contemplated the mechanical and kinematical pro 
