148 GAUSS ON A METHOD OF FACILITATING 
e the base of the hyperbolic, and m the modulus of Briggs’s 
logarithms, 7 the proportion of the circumference to the diame- 
ter. Let 
a 
T 
spas LE = 
Then the position 2, for the time ¢, will be expressed by the for- 
mula 
vz=p+Ae—* sin (nt—B), 
to which we may also give the form 
v=p+ae—** cosnt+ be*' sinnt, 
where p expresses the position of equilibrium, and the coefficients 
a, b, continue constant as long as p is the same. The velocity of 
the movement is hence found 
a's 
di 
—e—‘" (na sinnt+eacosnt—nbcos nt+eb sinnt); 
or, if we introduce an auxiliary angle ¢$, so that = = tang ¢, 
Etec uni Mee 
dt —cosd 
For ae~‘’ cosnt + be-** sinnt we may write u, so that 
L=ptu. 
Now let ¢', 2", ¢!", be the particular values of ¢, in which an al- 
teration of the acting force has been made ; further, let the par- 
ticular values of p, a, b, in the different portions of time be the 
following :— 
p®, a°, &° before Z! p', a, &" from z" to ¢!" 
py, a, U from Z to t! pl", all, O" after f!", 
Lastly, let the general expression of u become v°, w', u", wl", when 
the particular values are substituted for a@ and 4, so that before 
the first change 2 = p® + u°, from thence to the second change 
x =p’ + u', and so on. 
As the instant ¢/ is at once the last of the first interval of time, 
(a sin (nt + $) —b cos (nt + 9)). 
and the first of the succeeding interval, x as well as oe must 
preserve the same value for ¢ = ¢’; and, in the above general 
expression, the values p®, a°, 5°, or p', a’, ', may be substituted 
for p, a,b. Thus 
0 = p'—p® + (a — a°) e—*" cosnt! + (b'—6°) e~*" sinn?’, 
0 = (a'—a°) sin (nt’ + ¢) — ‘0’ — BD) cos (né! + 9), 
