70 APPENDIX. 



Again, the resistance in the two cases being as the area into the square of the velocity, 

 we have 



/ c' N g 

 r : r :: (~) s aV* : at,- 



but the power expended and overcome in a given time is as the resistance into the velocity ; 

 therefore the power expended is as 



' 2 



c \2 

 Y av ' : v! ; 



but the force expended by the engine in a given time is a representation of its power ; 

 therefore, 



/ c \i 

 p : p : : c : c : : ( J 3 t> J :t>', 



or, c' 3 : c 3 : : u' ! : t> ! , 



whence, v' = v ^ J ? . 



Then, if we take c' = 2c, or double tonnage, we find the ratio of the areas 1 : 2s, or 1 : 1-587 ; 

 and the ratio of the velocities 1 : 2-^, or 1 : 1'08 : or a decrease of about j^th in the time of 



making a given voyage, and of course a corresponding increase in the capability of the vessel for 

 distance. 



If, instead of increasing the power in proportion to the tonnage, it is only increased in pro- 

 portion to the resistance, or the same speed is given to the vessel, then the power required 

 will be as the section, or as 1 to 1'587, or not much more than half as much again ; and, 

 consequently, a voyage of more than one-and-half times greater length can be performed, 

 while the spare room for cargo will be in the same ratio as the increase of the vessel, or as 

 1 to 2 ; and the vessel will have the same power to contend with adverse weather, as the smaller 

 one with the larger proportion of power. 



These results may be said to be theoretical, but the examination of our present voyages 

 show that they are perfectly consistent with practice. Referring to the performances of Her 

 Majesty's Mediterranean packets, it will be seen, that in the smaller vessels, where the power 

 is large in proportion to the tonnage, and consumption of coals large per ton, the speed of the 

 vessel is generally less than in the larger vessels with a smaller proportion of power, and 

 consequently small consumption of coals per ton. 



It may also be seen to obtain throughout the experiments on Her Majesty's steam vessels 

 at Woolwich, where the larger vessel generally gives a greater speed, although the pro- 

 portion of power is less than in the small ones. 



To exhibit the difference more fully, I have given the following table of the average speed 

 and consumption of coals of nine of the most recently constructed of Her Majesty's steam 

 vessels, and have calculated what speed the largest, viz. the Medea, should make, by the 

 rules given above from each of the other vessels, having Morgan's wheels. I have compared 

 those with the radiating wheels with the largest of that class, viz. the Dee. 



