suspended from two points. 425 
evidently the equation of an ellipsis, whose conjugate semidiameters 
are GL, GR. Hence the place of the body may be found upon the sup- 
position that the curve described is an ellipsis, one of whose semidiameters 
GL is found by making SL=b' cosine (a—a' .t4+-c— c’), the other GR being 
taken on the aais of y, and made equal to b' sine (a—a’ .t+c—c’). The 
orresponding point P of the curve being found by taking, on the axis 
of x, GM=b cosine (at+c) and drawing the ordinate MTP, perpendic- 
ular to that axis. 
A second method of computing this ellipsis may be found by put- 
ting a=b. cosine (a’t+c'+h) =6. cosine h. cosine (a’t+c’)—b. sine h. 
sine ('t4c’), andif in this we substitute 6. cosine (@t+c')= 3 and 6 
sine (a't+c’) =b Y" =, it will become a= 4 . cosine h—J. sine h . 
me “2—PM;, in Desie M’T'P parallel to GS. Now if on WK 
we take WL’=4. cosine 4, and draw the line GL’ to cut MP in T’, 
we shall have by the similar triangles GM’T’, GWL, M’'T’= “4 COs 
ae GL?—GT’2 
sine h, T’P=—. sine /.- _ By taking therefore on the 
axis GS thevline GR'=6. ye we sat lave TP oCR. 
/GL?—GT? 
eo Ma 
, which is evidently the equation of an ellipsis whose con- 
jugate semidiameters are GL’, GR’,* consequently the place of 
the body may be found by making WL'=b . cosine (aa .t+o—c’) 
the other GR’ being taken on the axis of x equal to b . sine (s—al «t+ 
—c’). The corresponding point P of the curve being found by taking 
on the axis of y,GM'=b' . cosine (a'tc’ and drawing the ordinate M'T P’ 
perpendicular to that axis. 
* In the present figure the point P is supposed to correspond to a negative 
value of the ordinate T’P. 
712 
