470 GENERAIi CASE OP THE 



diilum is to be determined. The first will give the time in 

 a vertical direction, and the second the horizontal angle de- 

 scribed about the vertical which passes through the centre of 

 of suspension. 



In order to integrate these equations, let us observe that 

 the oscillating point can neither ascend so high on the sphe- 

 rical surface as to attain a point of which the vertical ordi- 

 nate is equal to 3«, nor descend so low as the horizontal, 

 tangential plane X. The curve described will therefore be 

 contained between two horizontal circles drawn on the sphe- 

 rical surface, and it will evidently touch these circles in two 

 points P and Q, corresponding to the greatest and least va- 

 lues of the vertical ordinate^. Let these values, or the or- 

 dinates of P and U be denoted by p and <?, and let us observe 



Ax 

 that tiie vertical velocity -i- decreases as the oscillating point 



approaches P and Q, and vanishes at the instant of their co- 

 incidence. We shall therefore have Fz= 0, both when x = p 

 and x = q^ and accordingly this equation must be divisible 

 by each of the factors x — p, x — 9, and consequently by their 

 product xf — (p + q)x+pq. Let the third factor of i^ be 

 X — r. Then we shall have 



F or x' — (2a + b)x^+ 2abx — c = ^x'' — (p + q)x + pq\ .(x—r), 



from which, by comparing like powers of x, we get 

 2a + b=p+q + r, 2ab =pq + {p + q)r, c = pqr, 



and thence 



r=2a+ — ^i , c = (2a+ — ^ — )pq. 



2a — p — q ^ 2a — p — q^^ 



The equation F may now be transformed into the product 

 lx'-^(p + q)x +pq] . {x^(2a + -_^---)], 



