12 



gives r 2 (o=a. constant, namely its value at the outer circumference 



of the screw, denoted hereinafter by R 2 <f>. 



The Table of Horse Power has therefore been calculated on 



the hypothesis that r 2 o)=a. constant, and that v is also constant, 



and therefore the same throughout as at the outer circumference. 



From the fundamental equation we have for this, by transposing 



r 2 o> 2 

 — r-, and K 2 ; also assuming that \p is not large enough to 



COS 2 )/' 



make it necessary to take into account that it is not unity (for 

 instance if \f/ were 20 , cos^ is 0-94 and as R<f> is small, seldom 

 exceeding 0-08 of R£l or 0-32 of K, its exact value to a decimal 

 has little influence) 



v 2 =2R 2 4>Sl + K 2 -R 2 <p 

 and, for the Horse Power 



H.P.=^v& f r ^dr=^-v^r 2 ^ frdr 

 S5°g J 55°£ J 



55°£" V 2 / 



where B stands for the radius of the boss, taken to be one-fourth 

 of the radius of the screw in calculating the Table. 



The calculated results could only exactly be true, therefore 

 for a special screw, designed to fit the vortex assumed ; but, as 

 above shown, they must be approximately correct (exclusive 

 of friction) for any screw having reasonably fair stream lines 

 over the outer parts of its disc ; and the examples mentioned in 

 the Paper show that, on calculation, this is so. The curves of 

 values of rui and v for the case where R§ and K have the 

 values shown are drawn in Fig. 3, the former being denoted 

 by the curve TT. 



A special form of the^ vortex occurs when— (z/tan0)=O through- 

 out, i.e. for a straight cylindrical vortex. In this we find 



0--r 2 o)=— no 2 

 dr 



for the equation of the curve of values of ru. On integration this 



r 2 (o 

 equation gives _ , — =a constant, and the curve is drawn in 

 2li-j-co 



