270 



As cos a and sin a are both essentially positive, the factor 



(cos a+sin a) 



has values lying between the comparatively restricted range of a 

 minimum of unity, when a is or 90°, and a maximum of 1.414 

 when a=45°. The value of DHQ+DLQ is the mean value of 



Di (cos a+sin a). 



The value of C, in equation (22A) of this appendix may be shown to 

 be approximately 0.66. The mean value of DHQ+DLQ is then close 

 to but generally less than Ki + Oj. The value of Mn is similarly close 

 to, but a little more than 2M2. It follows therefore, that very roughly: 



(DHQ + DLQ)/Mn= (Ki + Oi)/2M2 



2(DHQ + DLQ)/Mn=(Ki+Oi)/M2 



The ratio 2 (DHQ+DLQ) /Mn generally is somewhat less than that 

 of (Ki + 0i)/M2, but the values of i^(Mn) in table VI change so slowly 

 with this ratio that no large error is introduced by using this approxi- 

 mation in entering the table. 



Correction factor ijB 



20. As stated in paragraph 261, chapter V, the correction to the 

 primary current therein designated as i is such that the corrected 

 velocity: 



5 sin (a^+i3)+i=5[sin (a<+j8)+i/5] 



satisfies the general equation of motion (equation 112) when the sur- 

 face slope has the simple harmonic fluctuation, S cos {at-'rH°), and 

 the velocity head term is dropped. Placing, for convenience, i/B=z, 

 equation (112) therefore becomes: 



>S'cos iat+H°)i-(l/g)bB[sm (at+^) + 2]/dt±B''[sm {at+ l3)-\-z]yC-r=0 



or: 



S cos iat+H°) -f (aB/g) cos {at +13) + {B/g) bzjbt 



±B'[sm (at+l3) + zY/C'r=0. (27A) 



The values of B and /S are such that equation (145), paragraph 243, 



Scos iat+H°)-^(aB/g) cos (ai+i3) + (8/3x)(57CV) sin (at-^l3)=0 



is identically true for all values of t. Equation (27A) therefore may 

 be written: 



(B/g)bz/bt-(8/3Tr) (ByC'r) sin (at+^)±(ByC'r) [sin (at+^) + zY=0. 



