ee, U. S. COAST AND GEODETIC SURVEY 
The mean value of y for a group of consecutive ordinates from 7/2 
hours before to 7/2 hours after any given time ¢, 7 being the number 
of solar hours covered by the group, is 
Ali i cos (at-+-oat—| 2° 2 sin (a tea) | 
_ 2 *| sin Gas (atta-¥ )| 
360 A 360 
aa eign COS (at+ a) sin 5=78 sin FA cos (at+ a) (306) 
212. Since the true value of y at any time ¢, is equal to A cos (at+a) 
by (305), it is evident that the relation of this true value to the mean 
value (306) for the group 7 hours in length is 
A cos (at+a) rr TAT 
am sin 5 — A cos (at+ a) 360 sin 5 Gu 
The quantity — StS isthe augmenting factor which is to be 
360 sin—,— 
applied to the mean ordinate to obtain the true ordinate. In the use 
of this factor it is assumed that all the consecutive ordinates within 
the time 7/2 hours before to 7/2 hours after the given time have been 
used in obtaining the mean. This assumption is, of course, only 
approximately realized in the summation for any constituent, but the 
longer the series of observations the more nearly to the ‘truth it 
approaches. 
213. According to the usual summations with the primary stencils, 
the hourly heights included in a single group may be distributed over 
an interval from one-half of a constitutent hour before to one-half of a 
constituent hour after the hour to be represented. In this case 7 
é 15, 
equals one constituent hour, or = solar hours. 
Substituting this in (307), the 
Tp 
. 1ldp 
which is the formula generally adopted for the short-period constituents 
and is the one used in the calculation of the augmenting factors in 
Form 194. For the long-period constituents special factors are 
necessary which will be explained later. 
214. If the second system of distribution of the hourly heights as 
described on page 53 is adopted, 7 equals one solar hour and formula 
(307) becomes 
augmenting factor= 
Ta 
360 sin 5 
augmenting factor= (309) 
