744 
Proceedings of the Poiyiil Society 
Tims, at time t^ whatever be the mode of measurement of time, 
we have equations such as follows:— 
^a-al + 2Sa2/g2' ^ + 
- & = Sa2tt3 S(a2/?g + ^ 2 ^ 3 ) • ^ + S^2^3 - 
^^ = o| + 2Sa3/33,if + ^P^ 
For any one value of t we have n equations of each of the 1st and 
3rd of these types, and n{n l)/2 of the 2nd, w + 1 being the whole 
number of points. In all, n{n+ l)/2 equations. 
The scalar unknowns involved in these equations are (1) the 
values of f ; (2) a|, ag, &c. j (3) &c.; (4) &c.; (5) 
(®) Sa 2 ^ 2 , S« 3 ^ 3 , &c.; and (7) S{a^s + ^^a^), &c. 
Their numbers are, for (2), (3), (6), each ; for (4), {5), (7), 
n{n-l)/2 each; in all 3?z(?z + 1)2. Suppose that observations are 
made on m successive occasions. Since our origin, and our unit, of 
time are alike arbitrary, we may put ^ = 0 for the first observation, 
and merge the value of t at the second observation in the tensors of 
/? 2 j &c. This amounts to taking the interval between the first 
two sets of observations as unit of time. Thus the unknowns of 
the form (1) are m - 2 in number. There are therefore 
mn{n + l)/2 equations and 2>n{n + l)/2 + m- 2 unknowns. 
Thus TO = 3 gives an insufficient amount of information, but to = 4 
gives a superfluity. 
In particular, if there be three points only, which is in general 
sufficient, 3 complete observations give 
9 equations with 10 unknowns; 
while 4 complete observations give 
12 equations with 11 unknowns. 
Thus we need take only two of the three possible measurements, at 
the fourth instant of observation. 
The solution of the equations, supposed to be efiected, gives us 
among other things, a| , ug , and Sa 2 a 3 . Any direction may be 
assumed for and any plane as that of and a^. Fom these 
assumptions, and the three numerical quantities just named, the 
co-ordinate system required can be at once deduced. 
