of Edinburgh, Session 1866-67. 
165 
6. Some Mathematical Researches. By II. Fox Talbot, Esq. 
7. On the Small Oscillations of Heterogeneous Interpene- 
trating Systems of Particles, which Act upon each other 
according to Laws dependent only on their Mutual Dis- 
taries. By A. R. Catton, M.A., Fellow of St John’s 
College, Cambridge. 
8. Note on a celebrated Geometrical Problem. By 
Professor Tait. 
The following problem, originally proposed by Fermat to Torri- 
celli, To find the point the sum of ivhose distances from three given 
points is the least possible , seems to have given considerable trouble 
to the older mathematicians, and even in modern times (see Gregory's 
Examples , p. 126) to have been solved in a very tedious manner. 
Simpler solutions have since been given (ey., Cambridge and Dublin 
Mathematical Journal , viii. p. 92), but none, to my knowledge, so 
direct as that indicated by Quaternions. The object of this note is 
to show the simplicity of the quaternion method. 
If a, (3 be the vectors of two of the given points, the origin being 
the third, and if p be the vector of the required point, we must have 
(by the conditions of the problem) 
Tp + T(a-p) + T(/3-p) a minimum. 
Hence S [Up — U(a — p) — UQ3 — p)] dp = 0, 
for all values of U dp. Hence the versor sum in square brackets 
must vanish identically. The immediate interpretation is, that 
lines parallel to p, p — a, p — (3 : form an equilateral triangle. The 
required point is therefore in the same plane as the three given 
points; and their distances, two and two, subtend equal angles at 
it, which is the well-known solution. 
Equally simple is the quaternion solution of the same problem 
if more points than three be given. Let their vectors, to any 
