another generalization presented in chapter 6 was the application of the technique to large structural assemblies in which we provide also for the so-called interaction or external redundancies . as far as the practical side of the cut-out technique is concerned this was discussed in connexion with windows , doors , wing-fuselage interpenetrations , floors , partial removals of rings , etc . now , we may consider the cut-out process as a special case of the more general modification technique and this was , in fact , usually our approach to the presentation of the relevant theory . however , we did also mention that there is an essential difference between the cut-out and modification techniques in their practical application . this is immediately evident if we have to apply these respective procedures to a large number of elements which may be taken to form a sub-system . thus , when the flexibilities of the elements of the sub-system have to be modified it is obvious that we have to include all stresses specified in the elements to be altered in the matrix b1h and other relevant matrices of the sub-systems . but this is not so if we wish to eliminate the sub-system . here we may achieve its effective removal by detaching it along its boundary in the parent regularized structure , leaving only a statically determinate connexion . hence , in this approach , we actually cut only the redundant members of this connexion without having , at the same time , to break it up internally if it is itself redundant . on the other hand , it is perfectly legitimate to carry out , in addition , these internal eliminations , but this extends inevitably the amount of work involved and the order of magnitude of the matrix to be inverted . but on no account can we cut beyond this stage for we would then create a kinematic mechanism which means mathematically linearly dependent rows in b1h and a consequent singularity of the process . thus , we see that in the case of the elimination of sub-systems there is no unique number of cut-outs and , furthermore , no unique position of these cut-outs . we may achieve a minimum of eliminations by removing only the redundancies along the boundary and we may reach a maximum of eliminations by cutting also the internal redundancies . these subtle considerations are dealt with in great detail in section 36 and illustrated on a wide range of examples showing the alternative ways we can view and solve these problems . prior to this we summarize for the convenience of the reader in section 35 the basic theory of the modification and cut-out procedures as developed in a number of sections of this book and take this opportunity to generalize slightly the presentation . we hope that this joint account of theory and solutions to specimen problems will contribute to a deepening of the understanding of the cut-out technique and of its applications to practical cases . the concluding section of this chapter generalizes the matrix programme for the bending moments in the rings put forward in section 12 . the reader will remember that the method given there ignored any discontinuity of the bending moments at the vertices . now , this may be a too rough approximation when large loads ( e.g at wing fuselage attachments ) are applied at the external vertices or other points of the rings . the necessary simple theory is developed in section 37 . 33 . techniques to improve the conditioning of the D matrix . first a word of apology to our mathematically more knowledgeable readers . we are only too conscious that , in our repeated references to the conditioning of the D matrix , we have been guilty of imprecise language , not having really defined mathematically what we mean by the conditioning of a set of linear equations in the unknown redundancies . indeed , our whole approach to this matter was rather of applying the terminologies well- or ill-conditioned as qualitative terms of praise or abuse to a system of equations . now , ill-conditioning can , in fact , be expressed by various mathematical measures . unfortunately , most of these precise measures involve as long computations as the solution itself of the simultaneous equations and are not , hence , very useful in practice for giving advance warning . we refer the interested reader to the papers of Todd , Turing , and Johannes von Neumann cum Goldstine . even the relatively simple rule that ill conditioning is present when the value of the determinant &amp;formula; is small ( more precisely we should state that &amp;formula; is small compared with the individual terms of expansion of &amp;formula; in the co-factors of the elements of any chosen row or column ) is not of much value in computational work . nevertheless , in structural problems it is usually possible to adopt a simple measure sufficient for practical purposes . to fix ideas , consider a fuselage with two bays where only one set of primary redundancies Y arises at the intermediate frame station . having introduced - for reasons connected exclusively with the application of the digital computer - the inversion technique of Eqs ( 4 , 21 , 44 ) for the direct determination of the complete b1l , b1q matrices , and hence also of b1r , we are necessarily faced with self-equilibrating systems which are spread over the complete cross-section or , at least , over the main ( outer ) periphery . to express then with a high degree of accuracy ( from the practical computational point of view , which is the only one which interests us ) any arbitrary self-equilibrating stress system in terms of the b1l , b1q , b1r distributions , it is mandatory that the columns of these matrices - each of which corresponds to a redundancy - be not even remotely linearly dependent . this is evidently achieved when the flange loads , field forces and ring bending moments due to Y exhibit an increasing waviness with increasing order of redundancy . the search for such distributions brought us , more or less inevitably , to the selection of the trigonometrical matrix Ol as a transformation matrix Al defining the redundancies . as we know from chapter 4 and a large number of other similar computations , it appears that for the cross-sections commonly occurring in practice the flange loads and field forces based on Ol do indeed retain the full waviness of Ol , although naturally they are not any longer orthogonal as in the uniform circular cylinder . if the rings or frames were now rigid this characteristic waviness would ensure the precise determination of the redundancies and we then say that the associated equations are well conditioned . for the associated matrix Dyy - which in the present case where Dyyr is zero reduces to &amp;formula; we observe that the diagonal terms dii must be strongly preponderant , the non-diagonal dij being the smaller the better the waviness of Ol is retained . it is now possible to give the conditioning some measure by the degree of satisfaction of the condition &amp;formula; which is the generalization of the simple requirement usually quoted for 2 x 2 matrices . from the strictly mathematical point of view the inequality ( 2 ) ought to be defined more rigorously to express a sufficient condition theoretically acceptable ; at the same time we know that it is not a necessary prerequisite for good conditioning . nevertheless , for us engineers the relation ( 2 ) yields for structural matrices a sufficient measure for satisfactory conditioning . we must interpolate here in our main argument and refer briefly to the method of establishing systems of redundancies previously advocated by us in Ref ( 30 ) . contrary to what we put forward in the present treatise , we suggested there that it is advantageous to select systems of a distinctly local character . clearly then condition ( 2 ) still holds and is the better satisfied the less overlapping there is between the self-equilibrating systems . for reasons set out in the introduction and subsequently , we preferred here the method of direct inversion for the determination of b1 . considering next the more realistic case of fuselages with rings of finite stiffness , we find that the matrix Dyyr becomes of paramount importance ( this being at least so for the lower order redundancies ) and Eq ( 1 ) must be written as &amp;formula; . we noted in chapter 5 that the internal ring forces b1r are much more prone to lose their full waviness when the cross-section departs significantly from the circular shape . it is inevitable , in such instances , that the off-diagonal terms dijr ( elements of the matrix Dr ) may become of similar order to diir and/or djjr so that the measure of conditioning , Eq ( 2 ) , will consequently deteriorate and yield , in extreme cases of severe loss of waviness , a positive value only slightly above zero ( of course , it can never become negative in structural problems ) . such unfavourable conditions may prevail only in a few isolated spots of the Dyy matrix and we observed in chapter 5 , p 195 , that they do not seem , in our experience , to affect appreciably the accuracy of the solution , as long as these Scho&quot;nheitsfehler are within &amp;formula; . on the other hand , as these unfavourable patches spread , the solution of the equations in the redundancies becomes increasingly inaccurate due to the limited number of digits available and the rapid accumulation of errors . naturally , all methods of inversion or direct solution of equations are not equally sensitive to this danger in each specific case . although such pronounced ill-conditioning should not often occur in practice , it remains a distinct even if remote possibility . we are thinking here of double cell cross-sections with doubly-connected rings of unfavourable shape - for which the conditioning of the symmetrical higher modes deteriorates rapidly - and the rather box-like cross-sections of fuselages specially designed for bulky loads . our unavoidably superficial account leaves many extremely difficult questions unanswered ; in particular , the precise or statistical correlation between order and spread of bad patches , on the one hand , and loss of the accuracy of the solution on the other must unfortunately be ignored . an interesting practical point concerns the acceptable degree of inaccuracy in a solution due to such or other causes of errors . the practising engineer may often , and rightly so , consider a solution as satisfactory , although to us primarily interested in this instance to develop new methods , it may appear unacceptable . we referred to this issue in the introduction to this chapter when we discussed the application of the four-flange systems as redundancies . now , for the reasons stated there and here , we must reject such a narrow utilitarian outlook and seek , in fact , a system of redundancies even better than that based on Ol , if the conditioning of the latter should prove to us unsatisfactory . however , quite apart from the purely technical reasons , which demand such an extension of our original method , it is also perfectionism - a close companion of any intense research activity - which induces us to search for a more appropriate transformation matrix Al . before we proceed to the examination of this question , we must first conclude the bird &apos;s eye view of our theme . the discussion of the previous paragraphs was concerned with the so-called conditioning of the matrix D in the case of a single set of the primary redundancies Ya . when the fuselage extends over more than two bays , there arises at each intermediate frame station i a set of redundancies Yi . it is evident that our previous account is still applicable to the submatrices Dii in the leading diagonal of Dyy . we denote the conditioning of these matrices as peripheral to differentiate from another type presently to be mentioned . now , when a satisfactory conditioning of the leading diagonal submatrices has been achieved this will also apply , in general , to each of the other submatrices ( in the secondary diagonals ) of the five-band supermatrix Dyy . only a very violent change of cross-section could , in extremis , give rise to a significant ill-conditioning in these submatrices . however , from the point of view of the overall conditioning of the complete Dyy matrix , another possible source of ill-conditioning has to be looked for . thus , if the off-diagonal submatrices , say , Di , i+1 ( or Di , i+2 ) arising from the coupling of the sets Yi and Yi+1 ( or Yi+2 ) were proportional to Dii , the proportionality factor being only slightly smaller than unity , then it is evident from what we said previously in the peripheral kind of conditioning , that a new kind of ill-conditioning , conveniently denoted as a longitudinal one , could originate . 