suspended from two points. 415 
the integrals of which, combined with the equation of the curve sur- 
"face, will give the values of x, y, z, corresponding to the time ¢. 
FP ih axdt 
4 eee 
; aydt 
O=d. yt (A) 
at VW2zz+yy 
O=d. om: == edt. 
dt Wzztyy ° 
4. These equations may be investigated by the same method that 
La Place has used fora single pendulum in vol. i. page " of his “ Me- 
canique Celeste,” md substituting 3 in the equations o=d. < > adt & =) 
ond. 7a (S “) 5 O=d. Sad (3) see the value of u=o re- 
presentiug the equation of the curve surface on which the pendulous 
body moves. This equation is easily investigated by observing that 
MI=MS+SI>r'4yr7—22 ; ML=/2zz+yy, and MI=ML gives 7+ 
V pena EEF yy 5 5 OF O=r'4+/ rr—az—V zz4yy=u. This value of 
u, substituted in the preceding equations of La Place, gives the sys- 
tem (A) of the preceding article. 
If AB were inclined to the horizon, and GD to the vertical, by the 
angle I, the force of gravity in direction a would be gsine I, and m 
direction z would be g cosine 1; which would produce, in the first 
equation, the term — gdf sine I, and in the last — gd¢ cosine I, instead 
of —gdt, which when I is very small and a, y, small in comparison of 
r, (which, asin § 5 following, give A=g nearly) may be reduced to the 
same form as the equations (A), by writing a+ sine I for 2, which 
is the same as changing the origin of the co-ordinates, consequently 
the values of x, y, z, given in the following articles for the case of AB 
