Pallabazzer 



/3p+/3o(l-^) 

 B- - kM ^ - ■ (aligned propulsor) 



= kM^2 _ (l + /3j^j'" [/3^. + /3p + /S„(l-^)] , (S-propulsor) (19.) 



B3= 1^.^/37 • • • . 



Equation (9) furnishes two values of k^, both of which have physical meaning: 



k. = 



B, ± VB,^ f 4B^B3 _ B, ± Ai/2 ^^q) 



2B, 



An analysis of the solutions allows us to observe that ^t is positive every- 

 where; k\ (Mjj) is a monotonic fvinction (Fig. 7), the asymptote of which corre- 

 sponds to the condition Bj = 0, that is, to (The following relationships are de- 

 duced for aligned propulsor.) 



1/2 

 (B^.O) M^' -- (j— ^ (/3/-1)) (21) 



only up to a speed v^ where it happens simultaneously B^ = B2 = 0, that is, 



2k 



k - 1 



(/3/- 1) ■ (22) 



for which v^ can be obtained numerically. Up to the same speed \~ is always 

 negative. Above v^, x.^ is positive above M'J and both solutions merge in a 

 maximum at the condition where A = 0, that is, at 



m;. = 



. 1/2 



/3*- 1 

 2k I /3- ^— 



(21') 



In this range, x.~ shows the same asymptote at M^ = M^'. Therefore, there is 

 a range of m^ where two values of k-^ are possible at v„ > v^^, and M^ shows 

 a maximum, but this does not represent an indetermination because ^i is the 

 physical datum, while M^, which has been selected as a datum in the numerical 

 procedure, is actually a physical effect. In any case, that is, at any speed, there 

 is a maxjmum M^ which cannot be exceeded (it will be M^' at v„ < v„ and M^ 

 at v^ > v„ ). It can be interesting to observe that in any case for X- — » 00 m^^— M'J, 

 which is the Mach number of the gas, as it would have to be. Another significant 

 pseudo- Mach number is the one corresponding to B^ = 0, that is, 



since the speed v„ is obtained at Mj,' = M'J '. All these conditions differently 

 represent physical limits. 



1118 



