190 APPENDIX. 



pendicularly to its plane is V cos. e v cos. (<j> e). Consequently, adopting the same law 

 of resistance as that followed by Mr. Mornay, we have, 



p = I V cos. e v cos. ((f> e) 1 2 . . . . (a). 

 2ff {. 



By putting the effective velocity of the paddle equal to zero, we have V cos. e - 

 v cos. (c e) = o, which gives, 



V v cos. <f> ,, N 



tan. e = : _-r (b) ; 



v sin. <p 



and this determines the position in which the paddle would pass through the water edgewise, 

 without producing any effect. 



The pressure p being resolved tangentially, horizontally, and vertically, gives, 



Tangential pressure = p cos. e, 

 Horizontal pressure = p cos. (< e), 

 Vertical pressure = p sin. (0 e). 

 In an elementary instant dt of time, the distances traversed by the point A are, 



Tangential distance = V . dt "l , .. , ,, T, 



\ relatively to the Engine ; 



Horizontal distance = V cos. <f> . d t i 

 also, 



Horizontal distance = (V cos. <f> v) . dt "l , ,- , ,, m i 



V relatively to the Fluid ; 

 Vertical distance = V sin. < . dt i 



and the horizontal distance traversed by the vessel = v . dt, 



By multiplying the tangential pressure into the tangential distance relatively to the engine, 

 we have, 



Power developed by the engine =pV cos. e . dt. 



By multiplying the horizontal pressure into the horizontal distance traversed relatively to 

 the engine, we have, 



Power developed horizontally = p V cos. <f> cos. (cf> e) . dt. 



By multiplying the vertical pressure into the distance traversed vertically, we have, 

 Power developed vertically = p V sin. <j> sin. (<f> e) . dt. 



By multiplying the horizontal pressure by the horizontal distance traversed with respect 

 to the fluid, we have, 



Power expended horizontally on the water = p (V cos. <f> v) cos. (< e) . dt. 



Lastly, by multiplying the horizontal pressure by the horizontal distance traversed by the 

 vessel, we have, 



Power expended horizontally on the vessel = p v cos. (<f> e) . dt. 

 These results may be conveniently classified thus : 



rby the Engine = p V cos. e . dt ~\ 



I. Entire power 



. < on the Fluid = p ] V cos. e v cos. (6 e) } . dt > . . . . (c). 



expended 



"- on the Vessel = p v cos. (0 e) . dt 



