suspended from two points. 419 
successive operations, we might obtain the co-ordinates to any re- 
quired degree of accuracy. 
7. The equations (B) give +4 for the greatest value of x, and 
+6’ for the greatest value of y. Hence if on GS the axis of x (Plate 
3. Fib. 2.) we take, on each side of the point G, the lines GS=GN= 
6; and in like manner on the axis of y, the lines GE=GW=9, and 
through these points draw lines parallel to the axes, to complete the 
parallelogram ACKD, the projection of the path of the pendulum on 
the plane of xy will always be contained within this figure. 
8. on tangent of the curve must be parallel to the axis of x, 
ab sine (4/0) ig infinite, and perpendicular to that axis 
a’b’ sine (a’t-+-c’) 
when that expression is 0. ‘The first condition takes place when 
b'=o, or sine (a't-+c’}=0 ; the second when 6=0 or sine (at+c)=0. When 
b'=<o, the second equation (B) gives y=o, consequently the pendulum 
must vibrate in the plane of za: when b=o, the first of the equations 
when © a or 
(B) gives a2=o, and the body must then vibrate constantly in the plane 
of zy. When sine (a't+c )=0, y=b' cosine (a t+’) must be +4’, con- 
sequently the line AD or CK must then be tangent to the curve. 
When sine (at+c)=o, x=b cosine (at+c) must be + 6 and the line 
AC or DK must be tangent to the curve. Hence in general we may 
conclude that the sides of the parallelogram ADKC become tangents to 
the curve in every vibration. There is one exception to this, name- 
ly, when sine (at+c) and sine (a't+c’) are both =o, which correspond 
to 2=+h and y=+0', when the body is at one of the corners A, C, 
_ D, of the parallelogram. In this case the preceding expression of 
qa = becomes of the form, and we must, according t to the usual rule, 
ar the fluxions of the numerator and denominator, vishal 
