78 APPENDIX. 



is of less magnitude in the proportion of the cosine of the inclination (<f>) of the radius with the vertical ; so that the 

 power employed is less than of the effective power, or, which is the same, the effective power is more than jj of the 

 whole power. 



To state the ease in symbols, we have the pressure of the engine necessary to overcome resistance = cos <f> ( V cos <f> 

 v) 3 , as stated in pages 50 and 51 ; and hence the whole power employed = 5 V cos. $ ( V cos # of = V 2 cos <f> 

 ( V cos <t> )*> where 2 is supposed to include the same expression for each of the paddles immersed at the same 

 time. But the horizontal pressure on the paddles = 2 ( F cos <p u) 2 , which being also the resistance on the vessel, 

 the effective power is v 2 ( Kcos <t> y) 2 . It therefore follows that the proportion of the effective power is, 



effective power v 2 ( V cos <j> ) 2 



whole power V 2 cos. <t> ( V cos <t> )" ' 



which is necessarily greater than 

 Instead of the terms under 2, we may substitute the equivalent mean pressures for one paddle, viz. 



effective power _ v , f d $ (V cos <J> c) a 

 whole power V f~d <J> cos <f> ( F cos <p ) 



If we assume v = V, as stated in the paragraph referred to, the proportions of effective power at the four immer- 

 sions of Table III., page 52, calculated accurately in this way, come out -696, -701, -701, '690 ; Mr. B. gives them 



all -666. When the immersion of the wheel is so much as to cause cos <t> to become less than -=> which occurs in the 



two latter cases, it should be particularly remarked in taking the integrals, or calculating in any way from these 

 formulae, that the factor ( V cos <t> v)* should be accounted negative, as long as Kcos ^> t; is so, since under such 

 circumstances the paddle retards both the wheel and the vessel. EDITOR.] 



IV. TIME AND TRAVERSE TABLE, 



Calculated to show the advantage of sail on large SEA-GOING steam-vessels in 

 extended voyages, masted and rigged on the principle of spreading the 

 greatest quantity of canvas with the least possible resistance from masts 

 and yards when steaming. 



BY CAPTAIN ROBERT OLIVER, R. N. 



In well proportioned steam-vessels, under steam and sail, (being capable of making a 

 4 point course good, and from that to 5J points, acccording to the extent and force of the sea,) 

 the rate of speed is very considerably increased by sail, and great relief is thereby given to 

 the engines and vessel. 



