AJO) DETEEMINAJ^TS OF LINES. 
491 
But those resulting equations can be found far more simply and directly in the following 
manner. 
For this purpose let us revert to the fundamental formula 
^’■=©+(i')-2'iet(Q,V) (I.) 
Now section 30 of Chapter II. shows us how to find the moment-axis with respect to 
the origin of det (H, V). If, namely, ns and -y be the magnitudes of O and V, and ^ be 
the angle between O and the radius vector, then the moment-axis of det (O, V) is 
(ix 
—vsm cos Therefore the moment about the axis of x of —2 det (O, V) is l-csr cos <p 
and similarly, its moments about the axis of y and z are respectively 2ro-r cos (p and 
2'mr cos <p Moreover, since <p is the angle between the radius vector and O, it is evident 
that 
Tsrr cos (p=mjJc-{-7iy^y-{-i!y^=m(x COS X+2 sin a), 
X being the latitude of the origin. 
Let us now take the moments of Fj about the axes of coordinates, and equate them to 
the moments of those components of Fj which are given in the formula (I.), 
p 
The moments of — about the axes are zero in this case of the pendulum. The 
moments of g about the axes of x, ?/, and z are respectively gy, — gx, and 0 ; and the 
moments of —2 det(0, V) we have just found. Hence we obtain at once the following 
three equations : — 
d^z 
dt^ 
^ df — 
gy-\-2vy 
dx 
dt 
{X COSX-1-2 
sin X), 
d^x 
d^z 
—gx-\-27s 
dv 
(^cosX-|-2 
sin X), 
dt^ 
dt^— 
dt 
d^y 
d^x 
2t«3' 
dz 
(a:" cos X 4 - 2 ; 
sin X). 
1?' 
~yi?— 
dt 
• (H.) 
These are the three equations given in Hansen’s elaborate ‘ Theorie der Pendel- 
Bewegung, and which are generally obtained by means of very complicated analysis. 
One simple equation can be deduced by means of the proposition contained in section 
45 ; for the principle of vis viva gives 
= 2 ^( 2 ) -f-a constant. 
Moreover oif' -\-y‘-\-z^ is a constant, and these two equations, combined with any one of 
the equations (H.), determine the particle’s motion. 
