90 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Integrating this value of — D,y, and supplying a constant, we 
have: 
(¢ ae b) (t{—e)=a, 
in which the constants a + be, e and — J, are easy to calculate by 
least squares. 
To show the agreement of this formula with observation, take | 
Mr. Pierce’s “mean swing” at three European stations (U.S. Coast 
Survey Report for 1876, appendix 15, pages 232, 271) and apply 
b = 29'.2, e = —7632°, a = 756847, in see ¢ from t. Hence 
the following table: 
t. g, obs’d. ¢, cale’d. Residuals (1st). Residuals (2d). 
— 2880° 130’ 130’.07 —0’.07 —0'.16 
—2187 110 109 .80 +0.20 +0.13 
—1779 100 100.11 —0.11 —0.04 
—706 80 80 .08 —0.08 +0 .24 
0 70 69 .97 +0 .03 +0 .40 
+1927 50 49 .98 +0 .02 0.00 
+3304 40 * 40.01 —0.01 —0 .66 
The agreement (in column “ residuals, Ist”) is as close as could 
be desired. The equation is that of the equilateral hyperbola, with 
asymptotes parallel to the axes of g andi. This agreement can 
be made still closer by inclining one of the asymptotes, a term 
—c(t—e)* being added. There are thus four constants to com- 
pute; but this form of equation has the advantage of having its 
constants directly deducible by least square reduction. With the 
additional term, a perfect agreement between theory and the most 
precise observations hitherto made can be attained. As an instance, 
the thirty-five observations of amplitude, from over 2° down to 10’, 
given by Prof. Oppolzer in the Proceedings of the Vienna Academy 
for October, 1882, were compared with the formula 
(¢ + 60'.6) (¢ + 10.8) — 0.5 (¢ + 10.8)? = 2178.1 
(the unit of ¢ being an interval of about 5.7) and of the residuals, 
which need not be given in detail, the largest was 0’.8. A similar 
accordance was found in a set of observations extending over six 
hours, the pendulum swinging under less than half an inch of 
atmospheric pressure. (See Mr. Pierce’s report, page 248, last two 
columns combined.) In this formula, 
