Principle of the Screw to the Floats of Paddle-wheels. 257 

 -Jj(4- cob«*) C o,gtan-'(^ 8 + C( ^ g - cogg )^ J. 



To find the value of these expressions for the resistance and 

 moment, the method of Integration by Quadratures has been em- 

 ployed, the values of the functions to be integrated being found 

 at intervals of 5° between the limits and 20°, which limits 

 make the ratio of the breadth to the radius of the wheel the same 

 as that of 7 to 10. 



The value of the circular measure of 5° is '087266. 



Values of the Functions to be integrated, the Integrations being 



taken between 0° and 20°. 

 Let 



(\/8+cos 2 0- cos<9) =2q. 





cos 9. 



cos 2 e. 



2?. 



2§-cos 9. 



o 



- 2| 



•999048 



•998098 



2000635 



1-998731 



+ 21 



•999048 



•998098 



2-000635 



1-998731 



n 



•991445 



•982963 



2005714 



1-988555 



121 



•976296 



•953154 



2-015886 



1-968101 



171 



•953717 



•909576 



2031174 



1-937165 



221 



•923879 



•853553 



2051613 



1-895442 





log e (l + ? 2 ). 



log e (l+? 2 )-cos20. 



tan -1 g . cos 9. 



, 4-cos 2 

 tan- q.cosQ „ ■ 



-h 



+ 21 



71 



121 



171 



221 



•693465 

 •693465 

 •696004 

 •701090 

 •708733 

 •718950 



•692146 

 •692146 

 •684146 

 •668250 

 •644647 

 •613663 



•784809 

 •784809 

 •780093 

 •770643 

 •756422 

 •737407 



1-177960 

 1-177960 

 1-176785 

 1-174015 

 1-168832 

 1-160105 



i 



Hence the 

 ences are: — 



expressions for the resistance with their differ- 



24° 



1-512917 

 1-512917 

 1-495916 

 1-462337 

 1-412980 

 1-349000 

 The expressions for the moment are : — 



12i 

 17* 

 224 



•oooooo 



- -017001 

 - -033579 

 - -049357 

 - -063980 



- 2* 



+ 2J 



n 



124 



174 

 22i 



1-665778 

 1-665778 

 1-657082 

 1-639906 

 1-614657 

 1-581873 



•000000 

 -•008696 

 —•017176 

 -■025249 

 -•032784 



-•017001 

 -•016578 

 -•015778 

 -•014623 



-•008696 

 -•008480 

 -•008073 

 -•007535 



