110 



Dr. G. H. Bryan and Mr. W. E. Williams. [Jan. 7, 



v v u V 2 - y] Y \f (a) / (a") J ' 



Yu = _ _ # sin ft sin ft 2 J f'(oc') _f'{cc") \ 

 v u V 2 sin(/V-A)V(<> f (*>) } ' 



and an expression for Yq , which is not required. 



The remaining coefficients may for a first approximation be taken 

 the same as in (9). We notice the following points. 



(1) The expressions for X#, Y u , Y VJ G u and G v contain the factor 

 /' (a) /f (a!') - f (a") / / (a"). In the laws of resistance commonly 

 assumed /' (a) //(a) decreases as a increases, and this coefficient, 

 therefore, becomes negative if a! < a", i.e., if the two planes slope towards 

 each other a! referring to the front plane. This is the case in a gliding 

 machine furnished with a rudder, if the rudder is tilted slightly 

 upwards. In such cases rj 2 must be negative, hence X# and G u are 

 positive, Y u and G u are negative. 



(2) If /?i and /3 2 are of the first order of small quantities, ft - /3 2 

 also being of the first order, the expressions G u and G v are of the 

 first order, but Y tl and Y v are of the second order, and hence equations 

 (6) give approximately the five coefficients A, B, C, D, E. 



(3) The fifth stability condition H > O appears at first sight difficult 

 to reconcile with the smallness of the coefficients C, D, E for small 

 inclinations of the planes, as B 2 E is of a lower order of small quantities 

 than BCD. To satisfy this fifth condition, however, the important 

 thing is to make E small compared with C. Now C will be found to 

 consist of two parts, one independent of V and the other negative and 

 proportional to 1/V 2 , while E is proportional to 1/V 2 , hence stability 

 can best be secured by making V 2 sufficiently large.] 



(4) If the planes are not infinitely narrow, it will be found that 

 X«, X«, Y u , Y v are the same as before, and - mG u /v, and mG v /u are 

 both increased by 2KSi«&i (/'( a ') 4* ( a ') + f( a ') 4* ( a '))« This will have 

 the effect of altering C, E, D in the same proportion, and will, there- 

 fore, not alter the stability, except when a = 0, when the additional 

 term will prevent EiD from becoming zero. Ge is increased by the 

 term v^KSia^ (/'(a')<£(a) -f /(a')^'(a )). This term will, in general, 

 be negative, and therefore G<j will be diminished, which will have the 

 effect of diminishing B. 



6. Examples of Two-Plane Gliders. 



We shall now assume certain particular values for the dimensions of 

 gliders of this form, and proceed to calculate the conditions of 

 stability. 



Ex. 4. — Let us first consider the case of the two slats of equal area 

 set at angles such that 



a,\ = 15°, a, = 5°. 



