Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 



and which determine the relative velocities — , — , — of the point with respect to the co-or- 



dt dt dt ^ ^ 



dinate axes. 



If the point be fixed relatively to the axes, and a?o» Dm ^o ^^ its co-ordinates, the above 

 equation becomes 



v^ = \i^ . Zq — i2j: . y^, 



one of a set of well known equations, determining the linear velocity of a point in a body 

 revolving with given angular velocities. 



If the point lie in the axis of at, so that y, x both vanish, 



dx 



— - = 7)^, = u„ - wQ^, = v^ + a?Qy. 

 dt 



In these, if iv, y, ss are in the directions of the radius vector, a perpendicular to it in the 



vertical plane, and a perpendicular to this plane respectively, and if r, d, (p denote radius 



vector, altitude and azimuth, then 



w = r, Qj, 



d0 



whence 



cos d, ^z = -rr, 

 dt dt 



dr de ^ d(h 



V. = — , v„ = r — - , v, = r cos —~- , 

 dt " dt ' dt 



the common expressions for the components, relatively to polar co-ordinates, of the velocity of 

 a point. 



b. Accelerations, radial, transversal in the vertical plane, and perpendicular to that plane. 



In our general formulae u will now denote a velocity, and f an acceleration strictly so 



called. And in this case 



dr dd . dd) 



Mi = — , My = »■ ;t- > u,=^rcosO 



dt 



dt 



dt ' 



^ dd) . „ d(b ^ de 



a =--^ sine, ^=--fcos0, Q, = -j-, 

 dt " dt dt 



wherefore, by equations {E) 



d/T f 

 radial acceleration —f, = [r 



-^ df \ 



d9 

 ~dt 



+ r cos'' Q — ^ 

 dt 



)■ 



transversal acceleration in the vertical plane =^ 



d [ d9\ ( d(p 



-p »'-r - -v-sme. cos0-p- 

 dt \ dtj V dt 



1 d f „d9\ . „ „dd> 

 -— ( r' — - + r sin . cos 0-p- 

 rdt\ dt) dt 



dt 



d9\ 

 dt) 



azimuthal acceleration =/. = — ( rcosO — | - f . -^ cos + r sin — - . -^] 



''" dt\ dt) \ dt dt dt dt) 



