96 Transactions. — Miscellaneous. 



Therefore 

 X = — 2 -y^ 03 — xdl = — 2 dV cos a cos f — r 6- cos^ a cos 



r = — 2^' 03 — 7/0j _ t/0^ = _ 2 0F cos a sin </> — r 0^ sin </> 



'Jt 



Z= 2^ 01 + .T0103 == 2 0T'^ sin a cos ^ + r 0^ sin a cos « cos ^ 



= 2 7 sin Lat. + r 0- sin a cos a cos ^ 

 7 being tlie velocity of tlie body on the eartli's surface. 



The terms in A", Y, Z, not containing V, are tlie forces parallel to the 

 axes, acting on a body at rest on the earth's surface. Their resultant is 

 therefore balanced by centrifugal force, and reaction of the earth's surface. 

 The terms containing 7 in X and Y, being resolved along the tangent at jj 

 to the great chcle A A', cancel, showing that the earth's rotation has no 

 effect in accelerating or retarding a moving body. The term in ^, 2 7 

 sin Lat. represents a force acting in a tangent to the earth's surface, and 

 at right angles to the line of motion of the body, the positive sign showing 

 that the constraining force is du-ected towards the left, the body having an 

 equal tendency towards the right which is the deflecting force. 



Let the mean velocity of a stream be three miles an hour (the stream being 

 assumed to be everywhere of the same depth, and the velocity to be equal 

 at equal depths below the surface), then the deflecting force being propor- 

 tional to the velocity, the mean deflecting force acting on the stream is that 

 due to a velocity of three miles an hour, and it has been shown that the angle 

 of inclination of the surface line of the cross-section to the horizon, is that 

 whose tangent is the mean deflecting force divided by the force of gravita- 

 tion = /i suppose — 



. -./, 2 7 sin Lat. . . ,, , ,. o ,^ ,i 



Anci— L = . where is the angular motion of the earth per 



second which is 0-0000729 



V = 3 miles an hour or 4-4 feet per second 

 Sin Lat. = sin 45° = 0-707 



Theii^- 2 X 0-0000729 x 0-707 X 4-4 _ 0-00001408 

 <j ~ 32-2 



1 

 ~ 71023 

 which multiplied by 63360, the number of inches in a mile, gives -?_ inch 

 nearly, which is the difference of level at the opposite sides of a stream a 

 mile wide. 



