of Edinlurgh, Session 1883-84. 743 
2. Note on Eeference Frames. By Professor Tait. 
As I understand Prof. J. Thomson’s problem (anUy p. 568) it is 
equivalent to the following : — 
A set of points move, Galilei-wise, with reference to a system of 
co-ordinate axes; which may, itself, have any motion whatever. 
From observations of the relative positions of the points, merely, 
to find such co-ordinate axes. 
It is obvious that there is an infinitely infinite number of possible 
solutions ; because, if one origin moves Galilei-wise with respect to 
another, and the axes drawn from the two origins have no relative 
rotation, any point moving Galilei-wise with respect to either set of 
axes will necessarily move Galilei-wise with respect to the other. 
Hence any one solution suffices, for all the others can be deduced 
from it by the above consideration. 
Keferred to any one set of axes which satisfy the conditions, the 
positions of the points are, at time f, given by the vectors 
ttj + fS^t for A , -f p^t for B , &c., &c. 
But it is clear, from what is stated above, that we may look on the 
pair of vectors for any one of the points, say and /Sj for A, as 
being absolutely arbitrary : — though, of course, constant. We will, 
therefore, make each of them vanish. This amounts to taking A as 
the origin of the co-ordinate system. The other expressions, above, 
will then represent the relative positions of B, C, &c., with regard 
to A. 
The observer on A is supposed to be able to measure, at any 
moment, the lengths AB, AC, AD, &c. ; the angles BAG, BAD, 
CAD, &c. ; and also to be able to recognise whether a triangle, such 
as BCD, is gone round positively or negatively when its corners are 
passed through in the order named. What this leaves undetermined, 
at any particular instant, is merely the absolute direction of any 
one line (as AB), and the aspect of any one plane (as ABC) passing 
through that line. These being assumed at random, the simul- 
taneous positions of all the points can be constructed from the per- 
missible observations. But it is interesting to inquire how many 
observations are necessary; and how the ^s depend on the as. 
