introduction . when considering the design of a jet-flapped aircraft from a stability and control aspect , it is necessary to have fairly accurate information concerning the downwash field behind the jet-flapped wing , particularly in those regions where it is practicable to locate the tailplane . the evaluation of the downwash at the tailplane is dependent upon a knowledge of the strength and position of the vorticity distributions which represent the wing and the jet . in his treatment of the flow past a wing with a jet-flap , of infinite span , Spence assumes that the incidence of the wing and the deflection of the jet are small , and hence the usual assumptions of thin aerofoil theory , in which the wing and jet are replaced by vortex sheets in the direction of the free stream , apply . the results so obtained for the vorticity distributions on the wing and jet are used in part 1 to give the downwash at any position relative to the plane vortex sheet in the form &amp;formula; , where e = downwash angle , t = jet deflection angle , and a = wing incidence . however , in the calculation of the downwash induced at a point ( P ) in the field , it is necessary to allow for its location relative to the actual wing and jet . to the order of accuracy consistent with the previous assumptions , this implies calculating the downwash at a point whose ordinate relative to the plane vortex sheets is equal to the distance of the tailplane from the jet ( as shown in figs 1a and 1b ) . the functions de/dt and de/da depend upon the jet momentum coefficient cj , and on the relative position of the tailplane ; charts for these functions , and for the position of the jet , are given for various specific cj values . the downwash has been evaluated for ranges of the tailplane position , wing incidence , jet deflection and jet momentum coefficient . for the unswept wing of finite span , with a full-span jet-flap , considered in part 2 , Maskell has introduced the concept of an effective wing and jet flap of infinite span , in order to obtain the strength of the bound vorticity , elliptic spanwise loading being assumed . this solution may be used to give the contribution to the downwash from the bound vorticity , in a similar way to that described in part 1 , but it does not account for the effect of the trailing vortices arising from the pressure gradients along the wing and jet spans . in the case of a wing without a jet-flap , it has been found that the downwash is very sensitive to the relative distance between the tailplane and the wake , and that the spanwise loading has more effect on the downwash than the chordwise loading , and so the wing and its wake are replaced by a lifting line and its trailing vortices , the latter being displaced in order to keep the tailplane at the correct height above the wake . the effect of the rolling-up of the wake has also been investigated for a wing without a jet-flap , and it is shown that rolling-up is not important for normal tailplane positions behind wings of large aspect ratio . the distance e behind the wing at which rolling-up may be assumed to be complete is given by &amp;formula; for a wing without a jet-flap , where k&amp;prime; depends upon the plan-form and spanwise loading of the wing . for the jet-flapped wing , the cl will be greater than for the normal wing , but k&amp;prime; may now be a function of cj , and will probably increase with increasing cj ( since the bound vorticity on the jet will tend to resist rolling-up ) , so that e/c will not decrease so quickly with increasing cl and cj , as might have been expected from first considerations . thus , in order to evaluate the contribution to the downwash behind a jet-flapped wing from the trailing vorticity , it is assumed that the majority of the load is carried on the wing , so that the trailing vortices may be considered to arise from one chordwise position on the wing with no rolling-up taking place . the displacement of the jet and trailing vortices is accounted for by taking the position of the tailplane relative to the wake , and a chart is given for the downwash due to the trailing vorticity . calculated values of the downwash are in good agreement with the few experimental results available , especially if the difference between the experimental and theoretical lift coefficients is taken into account . theoretical results for the downwash on the centre-line are also given for a wing of aspect ratio 6.0 , showing variation with tailplane position , wing incidence , and jet parameters . part 1 . 1 . vortex representation of the wing and jet-flap of infinite span . the wing and jet-flap of infinite span may be represented in two dimensions by vorticity distributed on the chordal plane of the wing and the median line of the jet ( assumed to be thin ) . the downwash relations have been solved by Spence , using the assumptions of thin-aerofoil theory , so that the aerofoil incidence and jet deflection are considered to be small . the vorticity distributions and the position of the jet are given in Fourier-series forms , with coefficients as functions of the jet momentum coefficient cj . let U0f(x) be the vorticity distribution on the aerofoil ( at incidence a to the mainstream ) and g(ch) the vorticity distribution on the jet ( emerging at deflection t to the extended chord-line of the aerofoil ) , as shown in fig 1a . the x axis is taken parallel to the main stream , and the z axis vertically downwards , with the origin at the leading edge of the aerofoil . the chord of the aerofoil is taken to be unity , so that x and z are non-dimensional . thus the vortex representation of the flow which is in accordance with the assumptions of thin aerofoil theory is as shown in fig 1b , with U0f(x) located on the x axis , between 0 and 1 , and g(ch) also on the x axis , between 1 and &amp;symbol; . then the expressions for f(x) , g(ch) and zJ(x) , the jet displacement , as obtained from Ref 1 , are : for &amp;formula; &amp;formula; . for &amp;formula; &amp;formula; . 2 . the downwash . the downwash induced by the vortex distributions U0f(x) and g(ch) at the point ( X , Z ) is given by &amp;formula; to the first order in a and t ( see fig 1b ) . in order to apply the results calculated for the simplified configuration ( fig 1b ) to the actual configuration ( fig 1a ) , where the jet is displaced a distance zJ(X) below the x axis , it is assumed that the downwash w(X,z) calculated for the point P&amp;prime;(X,z) in fig 1b is equal to the downwash at the point P(X,z+zJ) in fig 1a . a similar procedure is followed in Ref 3 , where the displacement of the wake of a finite wing has to be considered . in general , the tailplane will be located a distance H above the jet , as indicated in fig 1a , so that to evaluate the downwash at the tailplane , i.e , at the point (X,zJ-H) in fig 1a , we must evaluate the downwash at the point ( X , - H ) in fig 1b . the position of the tailplane is usually given as the distance along and height above the extended chordline . if l is the distance of the aerodynamic centre of the tailplane behind the wing leading edge , measured along the extended wing chord-line , and h the height above the chord-line , when the chord is of length c , as shown in fig 1a , then the non-dimensional co-ordinates ( X , Z ) at which the downwash is to be evaluated are given by &amp;formula; , where zJ may be obtained from fig 3 ( or equation ( 4 ) ) . for the numerical evaluation of the two integrals in equation ( 6 ) , it is necessary to change the variables of integration , in the first integral using equation ( 1 ) in order to avoid the infinite value of f(x) at the leading and trailing edges , and in the second integral using equation ( 3 ) to make the range of integration finite . thus , if we write &amp;formula; , then the downwash at the tailplane is given by &amp;formula; , where &amp;formula; and &amp;formula; remain finite as x and th tend to zero , and as &amp;formula; , &amp;formula; . equation ( 10 ) may be rewritten in the form &amp;formula; , where de/dt and de/da are functions of Cj , X and Z . these have been evaluated for Cj = 0.5 , 1.0 , 2.0 and 4.0 , with &amp;formula; and &amp;formula; , the results being shown as charts in figs 4a to 4d . thus the procedure for the evaluation of the downwash at a given tailplane position , h/c and l/c , and given a , cj and t , is to calculate the functions in the following order : ( 1 ) X from equation ( 8a ) ( 2 ) zJ from fig 3 ( 3 ) Z from equation ( 8b ) ( 4 ) de/dt , de/da from figs 4a to 4d ( 5 ) e from equation ( 11 ) . interpolation will be necessary for cj values other than 0.5 , 1.0 , 2.0 and 4.0 , and it seems better to evaluate e for a range of cj , and then to interpolate the final result , rather than to interpolate for zJ , de/dt and de/da separately . for large X , the downwash is given by &amp;formula; , ( see Ref 1 ) so that &amp;formula; and &amp;formula; . it may be noted that the value of &amp;formula; for the downwash far behind the aerofoil is also obtained when the aerofoil is without a jet-flap . 3 . results . the results for the downwash behind an infinite wing and jet-flap are shown in figs 7 to 11 . it should be remembered that the theory is only strictly valid for small a and t , so that the use of the method to obtain the downwash for the larger values of a and t must wait to be justified or otherwise until experimental data are available . however , the results should indicate the trends in the variation of downwash with the various parameters . in figs 7 and 8 , the variation of the downwash with tailplane position is shown for two values of jet deflection angle , t , and two values of wing incidence , a , for cj = 2.0 . fig 7 shows that on the extended chord-line , h/c = 0 , the downwash decreases quite sharply with increasing distance behind the wing , l/c , but when h = 2c , the downwash is practically constant in each case for &amp;formula; . the results have been replotted in fig 8 to show the downwash field ( i.e , contours of equal downwash ) , in the tailplane region . a comparison between the fields for the various t and a shows that the downwash is more sensitive to tailplane position for the higher t and a values , as might be expected . the results for the variation of e with cj , t and a are given in figs 9 and 10 for a representative tailplane position , l/c = 3.5 , h/c = 1.5 , and also for a position on the extended chord-line , l/c = 3.5 , h/c = 0 . it will be noticed in fig 9a that e does not increase linearly with t for a given cj value ( as might be implied by a glance at equation ( 11 ) ) due to the correction made to the downwash field for the displacement of the jet relative to the tailplane position . fig 9b indicates that de/dCJ decreases with increasing cj . the variation of downwash with wing incidence is more important for stability and control considerations and the results are shown in figs 10a to 10d for t = 30 and 60 deg , and for various cj values . ranges of values of &amp;formula; are also indicated on the diagrams , and are seen to be the same for the two different t values over the same range of cj for a given value of h/c . since &amp;formula; increases with cj , it is not possible to assess a maximum , but for cj = 4.0 , &amp;formula; is well below 1.0 at the tailplane and on the extended chord-line , being 0.20 and 0.35 respectively . it also appears that de/da increases as a increases , but this is only noticeable at the higher values of cj , and for cj = 4.0 , a = 20 deg , de/da is still less than 0.4 at the extended chord-line position . 