152 ON THE OBSERVATIONS OF DEFLECTION. 
vertical to a horizontal position, or vice versd ; and for this case, — 
which is far the most important and most frequent in practice, — 
it is easy to execute the operation in such manner that the re- | 
sult may be scarcely sensibly disturbed. It is only necessary to 
take care that the second and third change take place in the 
same manner as the first, so that they may occupy an equal 
interval of time; and to deduct this from the intervals which 
would be otherwise required. If, for example, the logarithmic 
decrement is 0°33570, and the time of vibration 21521439, the 
table shows that the first interval = 108-04, and the second 
= 45-27, If it be found that three seconds are requisite for 
completing a change, begin the first change at 0°; from 3° to 
108 the bar remains in the new position; the second change ~ 
commencing at 10° brings the bar at 13° back to its first posi- 
tion, in which it remains only 14 second, when the third change 
begins, so that 17} seconds are required to complete the whole 
operation. A more extended investigation, too long for insertion 
in this place, shows that if p° passes into p', not per saltum, but 
gradually,—and likewise p! into p®, and p®° into p'—in the second 
and third changes the result continues the same as is given at 
the conclusion of Art. 5, provided only that the three times of 
passage be of equal length,—that the three passages themselves 
proceed by a similar gradation,—and that the calculated inter- 
vals g T, 7 T apply to the commencing moments of change; or, — 
which comes to the same thing, that the two first times of pas-— 
sage be included in the calculation. Chasis 
