of Edinburgh, Session 1883-84. 
741 
ing the stated problem for the case of only three original points by the 
kinematical mechanism can bring out two straight lines AA and BB 
really at rest relatively to the vice-original frame, and consequently 
having all their points either at rest or moving rectilinearly and 
mutuo-proportionally in relation to the original secret frame : and 
further that the same process of solving the stated problem can 
bring out another real true solution in finding two straight lines 
which we may call A' A' and B'B' which will be the images of AA 
and BB in a plane mirror whose plane always passes through 
the three points A, B, and C. The two straight lines A' A' , B'B' 
so found may be taken as lines fixed in a frame which we may 
designate as and which will rotate relatively to the vice- 
original frame <E>, as also relatively to the original secret frame. 
Now, as the motion of the original points goes on making their 
distances apart increase unlimitedly, this relative rotation between 
the frames <h' and $ will be becoming evanescent, and the two 
frames will be approaching unlimitedly towards relative rest. So 
the solution which brings out the frame 4>' approaches ultimately 
to identification with that which brings out the frame ^ which is in 
agreement with the original secret frame. 
If now instead of using the three points C, A, and B, we use a 
different group of three points C, A, and D, these will bring out 
for us as solutions two frames $ and ^>", of which the one <I> will 
be identical with the frame ^ already found by the three points C, 
A, and B. It follows from this (and it seems very obvious that it 
could be brought out in various other ways), that for four original 
points no frame in general could be brought out as a solution except 
one in agreement with the original secret frame ; that is to say, a 
frame either at rest relatively to that original frame, or having all 
its points moving rectilinearly and mutuo-proportionally relatively 
to that original frame. One reason which seems very decisive in 
favour of this conclusion is : — that if any three of all the original 
points be retaining their distances apart unchanging, then these 
three will themselves constitute a frame <1> which will be in agree- 
ment with the original secret frame : and then for any other one or 
more of the original points taken along with these three, no frame 
will be possible to serve as a solution except such as shall be in 
agreement with that one. 
