300 
Proceedings of the Pioyal Society 
-4- 
.^x= + iij.cosic - 3sinx 
_d^x— — - 2cos^i7 
- £C.C0S£I7 + silliC 
^x= +i^.sinii? 
iFx= +x.cosfl? + sinic 
2 Fx= — iij.sina: + 2cosa? 
3 ^^= - ^(7. coscc - 3sinx 
r^= +^.siiiii? - 4cosd7 
etc. 
f'x= + sinii? + 4cosic 
etc. 
etc. 
_4^(j>x= +ic.cosiu- 4sini:i7 
_^<hx— - iP.sinx - 3cos;r 
_^(fiX= - CC.COSCC + 2sincc 
_-^(f)X= + ii;. sin£i? + cosiT- 
<j)X= +^i7.cosa? 
^(f)X= -o;.smfl? + cos^c 
^cf)X= - it?, cosic - 2sinz 
s<f>x= + a?, sins? - 3cosi^J 
4 <^x= + it;, cos a: + 4 sin ^ 
etc. 
eacli term being the derivative of that above it. 
On equating any one of these functions to zero, we get (1st) the 
intersection of its curve with the line of abscissae, (2nd) the cul- 
minations of the curve belonging to the function immediately 
above, and (3rd) the points of reflexure of the curve of two steps 
up ; hence we are mostly concerned with the solution of the general 
equation 
The equation Fx = a;.sinii? = 0 is satisfied in two ways, by making 
x = 0 or by making since = 0. Now since becomes zero for every 
value of x = mr, n being any integer positive or negative number, 
and hence the curve, shown in fig. 5, crosses the line of abscissae at 
equal intervals of tt, counting from the zero point ; and since ce = 0 
coincides with one of the values since = 0, the curve touches the axis 
at the origin. 
The conjugate equation ^cc = ce.cosce, is also satisfied by putting 
= 0 or by cosce = 0, the latter condition giving for x any value of 
the form (?^ + 2)7 t ; hence, besides the crossing at the origin, this 
curve crosses the axis of abscissae at points distant by the interval 
7 T, reckoned, from on either side of the zero. 
The solution of the adfected equations is somewhat more complex. 
Beginning -with the case _ifx= -cccoscc-fsincr, which is that of the 
actual lathe-band, we observe that the function becomes zero when 
the condition 
n- 
Px or „(px = 0 . 
x~ tanx 
