THE OBSERVATIONS OF DEFLECTION. 149 
whence it is easy to deduce 
U 0 
sr lela AR ll 8 ' 
ad—a&= 308 6 e** cos (nt! + 9), 
I 0 
hess Cent Resend igo hrc 
i —-P= apg nis (nt! + $), 
-and hence 
Pi Sel) 
ul = uP Paerat e~*¢—4) cos (n (t— t') — 4). 
In like manner we obtain 
dice. F 
a! = a Ff 4 e— *@—#") cos (n (¢— 2") — 9), 
a dln AMT nm 
Pies s ara ¢ (¢—2") cos (n (t —t"")— 4), 
and so on if there are more alterations of the moving force. 
Thus, from the initial movement every succeeding movement 
may be determined. 
5. For the case of the present investigation we should write 
p! = p° and pl = p’. Thence 
ul" = uw — Bo es * (et? cos (n (¢— dt’) — 4) 
cos > 
—e'"” cos (n (¢— 0") — 9) +e” cos (n (t — 2") — $)], 
which formula, if we make 
e~€"—*) cos n(t"! — #1) —1 + & OC’ —”) cosn (i! — t") =f, 
e—* (2) sin n (t" — 2) — ef" —*) sin n (t!"” — 2") = g, 
passes into 
ul! — 0 ~P oe e—'-#”) Lf cos (n (¢ — zt") — 9) 
—gsin (n(¢— 7") — 9]. 
Hence it follows, that if the intervals ¢! — ¢, z!” — ¢’” are so de- 
termined that f= 0, and g = 0, then generally 
ul! = wu’, 
or qil! = a°, TA = 6°, 
Thus, if before the change the needle was in repose in p®, 
after the change it will be in repose in p!: in the opposite case 
the needle, atter the three changes, will have at each instant 
precisely the same velocity and the same position relatively to 
the middle point of its motion p’, which it would have had at the 
