322 



MR. p. W. BARLOW ON THE LAWS WHICH GOVERN 



(V - t; COS (py and the integral of (V — v cos (p)^ d . ^, divided by (p, will be the mean 

 resistance ; or this may be obtained sufficiently near for practical purposes arithmet?- 

 cally. Let this be m, then V m will express the whole power of the engine. Again, 

 the resolved horizontal resistance is expressed generally by (V — v cos <p)2 cos <p, and 

 the mean value of this is found by dividing the integral of (V — v cos (p) cos (pd(p by 9. 

 Or find the same arithmetically : let this be m', then V m' will be the whole resolved 

 horizontal force, which may be supposed to be applied at the circumference, as in the 

 locomotive engine, and the part of this force expressed by m' v, will be the effective 



force exerted on the vessel ; but the whole force is m V, therefore ^ will be the pro- 

 portional part of the power saved, the original engine power being 1. These numbers 

 are computed and arranged within all practical limits in the following little Table. 



In the vertical paddle, the mean resistance applied tangentially is the integral of 

 (V cos (p — vY d<p divided by (p, which may be all supposed to be applied horizontally 

 as in the locomotive carriage ; and the lost power is therefore simply the difference 



V 



of velocity ; that is, the effective horizontal force is y-j the power of the engine being 



1 ; and as in this case the mean velocities are generally as 3 to 2, the part of the 

 whole power which becomes effective is '666. 



In the manner above described, the following Table, exhibiting a comparison of the 

 lost power of the common wheel and that of the new wheel at several states of im- 

 mersion, has been calculated. 



Table III. 



Explanation of the manner in which the Power of the Engine is expended in the two 



IVheels. 



Having obtained an expression for the whole resistance opposed to the engine at 



any angle of the paddle, we may, as before stated, find such a mean resistance, which 



continuing the same throughout the whole arc, would produce the same effect as the 



variable resistances expressed in the above formula; and this multiplied by the num- 



