142 The Rev. Samuel Haughton's Account of 



M = (a' cos- a + b^ cos- /3 + r cos- 7) ^ = f^P' = rjjT; (5 j 



(R,!') denoting the radius vector and perpendicular on tangent plane of the 

 ellipsoid (4); and {B'F), the corresponding lines in the ellipsoid (3 ). 



IV. — To find the Magnitude, Position, and Direction of ike Statical Couple produced 



bji the ('entrifugal Forces. 



If from any point (.?■, y, z) of the body, a perpendicular be let fall on the 

 axis of rotation (a, j3, 7), the centrifugal force will be represented by the pro- 

 duct of the square of the angular velocity and tliis perpendicular ; the corres- 

 ponding elementary statical couple will be found by multiplying the centrifugal 

 force by the distance from the loot of the perpendicular to the origin, which is 

 represented by the quantity (.r cos a + y cos ^ -\- z cos 7). The components of 

 the elementary couple will be proportional to the projections of the triangle 

 formed by the lines before mentioned. The components of the elementary 

 couples must be integrated for the entire extent of the body, and the integrals 

 thus found will be the components of the couple produced by centrifugal force : 

 the expressions are as follows : 



or (,r cos a- + y cos fi + z cos 7) (z cos § — y cos 7) dm ; 

 o>^ {x cos a + y cos ji + z cos 7) (.r cos 7 — - cos a) dm ; 

 or (.!• cos a + y cos (i + z cos 7) (y cos a — ./■ cos /3) dm. 



If the axes of co-ordinates be principal axes, these expressions, when integrated, 

 will become 



or cos ji cos 7 (B - C) = qr (B - C) ; 



or cos a cos 7 ( C — A)= pr ( C — A) ; (6) 



w' cos a cos /3 (.4 —B) = pq (A — B) : 



p, q, r being the components of the angular velocity m. The position of the 

 resultant couple may be expressed by means of the ellipsoid (4 ). If a tangent 

 plane be drawn to this ellipsoid at the point (x, y, z), and perpendicular to the 

 line (a, /3, 7), it may be easily shown that the projections of the triangle formed 

 by the radius vector and perpendicular are represented by the quantities 



