608 Proceedings of the Royal Society 
1870, § 17, it is shown to be generally expressible in the form of 
a i d a + Pi d (3 + 7i d y> 
where a, /3 , y, are any three unit vectors (not necessarily rectangu- 
lar), and a v /3 V y v any three vectors whatever. The scalar and 
vector parts of the result of its operation on a vector-function, cr-, of p 
are first considered — with various interpretations, especially as to dis- 
tortions, condensations, &c., in a group of points — then it is exhi- 
bited in its applications to various questions ; especially to Physical 
Strain, to Heat, and to Electricity. By making the constituents 
of <p variable, we have a means of Deformation specially applicable 
to problems such as that of Orthogonal Isothermal Surfaces. 
4. Note on Pendulum Motion. By Professor Tait. 
Mr Sang’s papers in recent parts of the Transactions of the 
Society have reminded me of some geometrical constructions which 
are to a certain extent indicated in Tait and Steeles Dynamics of a 
Particle (1856). Some of these were suggested to me by a beautiful 
construction given (I believe by 
Clerk-Maxwell) in the Cambridge 
and Dublin Math. Journal , Feb. 
1854, the others by a very simple 
process which occurred to me for 
the treatment of oscillations in 
cycloidal arcs. The former en- 
ables us easily to divide the arc of 
oscillation of a pendulum, or the 
whole circumference if the motion 
be continuous, into two, four, 
eight, &c., parts, which are de- 
scribed in equal times; also to 
solve by simple geometrical con- 
structions problems such as the 
following : — Given any three 
points in a circle, find how it 
must be placed that a heavy 
of them, may take twice as long 
