272 



in which the coefficient 6=(480/7r^Aa^°) tan (p may be computed from 

 the given value of and the selected increment Aat°. 



In the second term of equation (30A) the negative sign is prefixed 

 to sin (ai+j8) when this function is negative. The factor±sin (at-\-l3) 

 is then positive for all values of at, and w^ll be so distinguished by 

 writing it as sin (ai+jS). The algebraic sum of the last two terms, for 

 values of at at the selected intervals, may be designated as —B. 

 Equation (30A) then becomes: 



b(z-2o)+2zsm (at+^)-R=0 

 whence 



z={2o-B/b)l[l-i-2 sin (at-^^)/b]. (3lA) 



22. The values of z for successive values of a^+/3 may be computed 

 from equation (31A) after an initial determination of Zq has been made. 

 By taking A at° as an integral factor of 180°, these values are repeated 

 after aif+/3 has passed through 360°. Taking then Zq as zero at any 

 value of at-\-^, such as zero, the resulting values of z may be succes- 

 sively computed through 360°, a corrected initial value of Zq derived, 

 and the procedure repeated. Since the divisor of the second term of 

 equation (31 A) is greater than unity, the new values of z successively 

 approach and finally coincide with those previously found. The 

 process is in fact abbreviated, since the values of z repeat themselves, 

 with the sign reversed, after passing 180°. 



23. Second correction. — '\'\Tien the flow is largely frictional, and 4> 

 consequently is a relatively small angle, the values of z derived from 

 the foregoing procedure are so large that their squares are not negli- 

 gible, and are sufficient, also, to reverse the sign of the velocity when 

 the primary current is small. A further correction, 8, is therefore 

 required. Designating the first determination of the correction factor 

 as Zi, the corrected current becomes B [sin {at-\-P)-\-Zi-\-8]. 



Equation (112) then takes the form: 



Scos (at+H°)-{-(aB/g) cos (at+^) + (B/g) {bz^/bt-i-dBldt) 

 ±B'[sm (ai+|3) + 2i+5]7CV=0 



which, by a procedure paralleling that in paragraph 20, may be 

 transformed into: 



(8/37r) tan ^(d^/ad^+dsi/adO- (8/37r) sin (a^^/S) 

 ±[sin (ai+/3) + 2i]'±26[sin {at+ ^)-^Zi\±8^=0 



