740 
Proceedings of the Royal Society 
There is yet a curious matter which I have to adduce for conside- 
ration. Many matters of interest may indeed remain for further 
consideration or for mathematical investigation in connection with 
this subject ; hut I do not at all profess to exhaust the subject, and 
I shall only he glad if it he found that I escape from offering in- 
advertently any importantly erroneous views. 
What I do propose now further to put forward is some scrutinies 
as to whether the problem entered on, in some of its varieties, 
admits of duplicate or multiple results for accomplishing its condi- 
tions ; and to open out some views in relation to this matter which 
appear to be true and to be of interest. 
It is to be noticed that when we begin putting together our templets 
for the kinematical model or mechanism for three original moving 
points A, B, and C, we have no means available for knowing on 
which side of templet 1 we are to place the point Ag of our templet 
2, and on which side of the same we are to place the point B 2 of 
our templet 2. Further, it is to be noticed that, under the restric- 
tion of our measurements being confined to three of the original 
points only, we have no means whatever for making the extra 
measurements, or taking the extra observations that would supply 
us with means for choosing one side rather than the other of tem- 
plet 1, as that on which we ought to place either Ag or B 2 . To 
help our conceptions let us imagine among the original moving 
points a reference frame relative to which the original point C shall 
be at rest and which shall have no rotation relative to the original 
secret frame ; and let us name this as the vice-original frame and 
designate it briefly by the letter d>. Let us imagine the original 
triangular plane ACB as having one of its two faces red and the 
other blue ; and imagine its face which at the point A is anterior in 
its motion relative to the vice-original frame to be red, and the one 
which at that point is posterior to be blue. In respect to this it is to 
be noticed that the red face thus specified though anterior at A may 
happen to be the posterior one at B : but this need not give us 
trouble, and for brevity we may speak of the red face which at A is 
anterior as being the anterior face. Let the faces of all the templet 
triangles be coloured red and blue correspondingly. 
By going forward with considerations readily suggested by what 
has just been set forth, we may obviously find that the process of solv- 
