IN PHYSICAL ASTRONOMY. 
281 
Now, let L S M T be the section of a 
cylinder revolting about an axis, passing 
through O perpendicular to the plane 
LSMT, aod let the cylinder revolve in 
the direction LN. The action of the 
resisting medium will be in the direction 
Z P, perpendicular to O P upon all the 
points P between L S ; and in the con- 
trary direction K P upon all the points, 
P between T M. These remarks show that in what follows, the integrations 
must not be made throughout the whole surface of the body revolving : this 
consideration however does not alfect the nature of the results. 
The equation to a plane perpendicular to the axis of rotation, and passing 
through the centre of gravity of the body, is p x + qy + r z = 0. 
Let the body revolving be a spheroid of which the equation is 
x Ci + y* + z 2 ( l + e 2 ) = a- ( 1 + e 2 ) 
The equation to the tangent plane to the spheroid at the point x, y, z is 
x x' + y y' + z z' ( 1 + e 2 ) = a 2 ( 1 + e-) 
The equations to the planes from whose intersection the line P B results, are 
* z (qz’ — ry') + y (r x' — p z') + z (py' — q »') = 0 
px + qy + rz = D 
D being a constant. The equations to the line P C are 
x^r (qz' — ry') —p(py ' — + y {r (r x' — p z') — q (py' — q a:')} =0 
x { q (q z' — r y') — p (r x' — p z ') } + % { q (p y’ — q x') — r (r x' — p z’) } = 0 
and neglecting p 2 , q 2 , p q, 
x (q z' — ry') = y (p z' — r x') 
x (q y' + p x') = z ( p z' — r x') 
The equations to the direction of motion of the point P are 
x (p s' — r x') = y (r y' — q r ') 
x (q x' — p y') —z(ry' — q z ') 
Cos . angle, which the direction of motion of P makes with the normal to the 
surface or cos A P C 
x 1 ( r y' — qz') + y' (pz f — rx') + z' (1 + e a ) (q —py') 
V { ( r y' ~ ? z ')* + (p z' — r x'P + (qy — pa;') 2 } {* ,2 + y' 2 + z" 3 - (1 + 2e 2 )} 
* The notation is the same as p. 20, except that the accents at foot of x j} y t , z t are omitted. 
2 o 2 
