oj Edinburgh, Session 1883 - 84 . 
575 
attached to the surface of the earth, or it may have been firmly 
attached to the floor and side walls of the cabin of a ship sailing in 
devious courses over the sea and tossing on the waves. ISTotwith- 
standing any such motions, or any motions whatever, belonging to 
the original reference frame, the mutual motions of the points will 
possess the character that they admit of having reference frames, as 
many as we please, relative to which they will be rectilinear and 
mutually proportional (or, in other words, they will be uniform 
rectilinear motions, by mutual reference without reference to time). 
If the moving points alone were available to us for progressive 
observation or measurement it might be a difficult, perhaps an 
extremely difficult, geometrical or Idnematical problem* to find 
from them a reference frame accomplishing the stated condition ; 
but this does not hinder us from easily and distinctly understanding 
that such a frame is geometrically or kinematically possible. On 
the other hand, for a set of points moving at random like flies in 
the air, or for a set of points having uniform rectilinear motion as 
already described, together with others revolving like satellites 
round some of them, no reference frame to accomplish the condi- 
tions stated would be possible. For a single fly moving anyhow, 
reference frames would be possible, relative to any one of which the 
motion of the fly would be rectilinear, and would be uniform in 
rate of progress relatively to true time, or to any assumed standard 
whatever for rate of progress ; but for two flies, or any greater 
number, no such frame would be possible. Eeasons for this are so 
obvious as scarcely to require statement. Briefly, however, it may 
be mentioned that any two flies might in their mutual motions 
come into contact once and then separate, and then come into con- 
tact again ; but no second meeting could occur with points moving 
* Postscript Note, May 1884. — On the evening of the reading of the paper 
(March 3, 1884), just after the close of the meeting of the Society, the author 
inquired of Professor Tait whether he could see how the jn-oblem referred to 
here in the paper as being perhaps extremely difficult, could be solved. Pro- 
fessor Tait replied that he could solve it very briefly by use of quaternions. 
The author, not being at all acquainted with quaternions, has since seen his 
way clearly to the solution by an easy method of mechanical adaptations. The 
mechanical method is merely for intellectual use, not for practical application. 
The ideal mechanism can serve as an instrument for use in reasoning, thou«-h 
friction, and elasticity of materials, &c., might render it incapable of complete 
practical realisation. 
