110 Proceedings of the Royal Society of Edinburgh. [Sess. 
Its effect in the first term of R will evidently be as indicated, qualifying 
the escapement error by addition of a negative portion. 
Fig. 11. 
It would be easy to multiply the applications of this method, but the 
examples given above will suffice to illustrate it. I shall now return to 
the three standard clocks, and to conclude the present paper I shall show 
how far its formulae succeed in explaining the observed numerical features 
of the clocks. 
It should be remarked that the comparisons that follow are not meant 
to be final ; the numbers quoted cannot be ascertained without disturbing 
the running of the clocks, and especially for the clock Riefler, which is in 
continuous service as the standard mean time clock of the city. This is 
inconvenient to do ; the data have been found as occasion offered, and their 
final determination has been postponed to a later occasion. Subject, how- 
ever, to revision, they are believed to be fairly reliable. The purpose of the 
present study being chiefly theoretical, they will serve to show how far the 
theory is sufficient and how far it is demonstrably incomplete. 
Take first the arc; by the energy equation (19), p. 101, we have, when 
a steady state has been obtained, 
where 
since the equation 
[E] = [H]/* 
E = i(x ' 2 + M 2 x 2 ), H = Rx': 
x + kx + n 2 x = R 
represents (7) of p. 88, supplemented by frictional terms, viz. 
M { d 2 + . . . + k 2 }6" + . . . +M g{d+ . . . }0 = P(7i. . .) . (7) 
If we write x = 0, and a is the full semi-arc described, we have 
E = JwV 
and 
M{d 2 + . . . + tf}'Rx=M{d 2 + . . . + k 2 } H 
= work done by the maintenance upon the pendulum in 1 sec. = W, say, 
or, using the equation 7r/w = 1 sec., the equation for the arc is 
-M {d 2 + . . . + h 2 }n 2 a 2 = W(n/ K ). 
2 
I shall now calculate the arcs of the different clocks from this formula. 
