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A group of parallel threads of equal length, clamped at each end so that all threads extend equally under tension, is called a bundle, and the maximum load which the bundle can sup port is called its strength: The object of the work is to study the probability distribution of the strength of bundles whose constituent threads are sampled randomly from an infinite population of threads in which the probability distribution of strength is known.The relation between the strength of a bundle and the strengths of its constituent threads is first discussed, and results are stated for bundles so large that the proportions of threads of different strengths approach their expectations. The properties of the probability dis tribution of bundle strength are next developed in detail, attention being confined in the present paper to the case where all threads have the same load-extension curve up to breaking point. Finally, the asymptotic behaviour of the distribution for large numbers of threads is studied, and it is shown that in the commonest cases the distribution tends to assume the normal form. I n t r o d u c t io n 1.Consider an infinite population of threads of equal length whose probability distribution of strength is known. A num ber of threads are sampled random ly from th is population, laid side by side and clamped a t each end to form w hat we term a bundle. W hen a free load is applied to th e bundle all threads extend by an equal am ount, and th e minimum load beyond which all th e threads of th e bundle break is called its strength. The present work is an investigation into th e probability distri-v bution of the strength of such bundles.The problem presents itself naturally in th e theory of strength testing of textile m aterials. F or example, a te st widely used in practice for estim ating th e strength of y am s is known as th e ' hank ' te st for wool yam s, and th e ' lea ' te st for cotton yam s. The m ethod employed is to reel a hank of yarn containing a stated num ber of turns of specified circumference and to apply th e breaking load to the hank by stretching it between two hooks. The w ord ' bundle ' has been used here in preference to ' hank ' or 'le a ', since there is usually a small am ount of slip a t the hooks which makes th e conditions of this te st indeterm inate. The testing of cloth samples for warp and weft strength is another instance of a practical te st in which th e specimens approxim ate to bundles, though in closely woven fabrics th e cross-threads afford a measure of support which introduces complications. Other examples outside the field of textile testing could no doubt also be quoted.2.

1. THE purpose of this paper is to examine the relation between rank correlation and specific features of the population about which information is required. The paper was written after those by P. A. P. Moran and J. W. Whitfield were in draft form, and with their permission I have commented on one or two points arising out of their work.The first question I wish to consider is how far Kendall's 't" and Spearman's p give similar information about a bivariate population of ranks. The population may be ranked from that of an underlying variable, or it may be a population of ranks in its own right. Both measures of rank correlation are usually considered reasonable in that they measure in some sense the "degree of agreement" between two rank orders, and I have given a similar justification of the general coefficient T' (1948). It is sometimes assumed that because the sample estimates* T and R of '" and p are highly correlated in the case of independence, 't" and p describe more or less the same aspect of a separate bivariate population of ranks when the correlation is not zero. The following inequality shows that the assumption may be far from true.2. It is convenient to start with a finite sample of n which is subsequently made indefinitely large.Let Ph Pz, . . . ,Pn and qh q2, . . . , a« be two rankings, and assign scores aij = sgn(pjp.J, b ij =sgn (qj -qi) to the corresponding pairs of ranks, taking aii = b ii = O. Thenand similarly for b ij , so that 1 3 T = n(n _ 1) L Gij bij, R = n(n2 _ 1) L aij bik.all suffixes being summed from 1 to n. Since relations like Pi > Pi > Pk > Pi are impossible, Gij, ajk, ak; cannot be all positive or all negative, hence aij + ajk + au = ± 1, and similarly bij + bj k + b ki = ± 1. Moreover, we can writewhere E'ijk has the following meaning. Let Pi, Pj, Pk and qi, % qk be renamed according to their order of magnitude, becoming P'i, P'j, t/»: q'i, q'j, q'k, where »', q' are now permutations of 123.For example the ranks 472 become 231. Then Eijk = + 1 or -1 according as P' is an even or odd permutation of q', and it is zero if any pair of suffixes is equal. Summing over all values of the suffixes we find 3n L aij bij -6 L ai} bik = L E,jk' * I prefer the symbol T to t, which is standard for Student's ratio. As Whitfield and Moran both use different notations, there seems no harm in introducing a third one.

SynopsisThe paper is concerned with the distributional properties of Markoff chains in two and three dimensions where the transition probability for the length of a step and its orientation relative to that of the previous step is specified.The discrete two-dimensional chain of n steps is first discussed, and by the use of moving axes an equation relating characteristic functions of the end-point distribution for successive values of n is obtained. The corresponding differential equation for the limiting chain with continuous first derivatives is given and asymptotic solutions for long chains are found.The three-dimensional chain is similarly treated in terms of moving axes, and the limiting continuous chain is again discussed. Finally the same methods are applied to the discrete chain of equal steps to obtain the asymptotic form of the end-point distribution for long chains.