SECTIONAL TRANSACTIONS.— A*. 393 



interval between consecutive tabular entries. Root-extraction, treated 

 otherwise than as inverse interpolation, is a step-by-step determination 

 of a sequence Xi, x^, x^, . . . To obtain Xn + 1 from Xi, X2, . . . ,Xn and 

 functional values at these points by inverse interpolation of the ordinary 

 kind is to ignore the possibility of profiting from the circumstance that 

 Xr + 1 is much nearer to Xr than to Xr-i ; on the other hand, in Newton's 

 formula and in any slight modification of it, the labour of each step is apt 

 to be considerably greater than the labour of a linear interpolation, and since 

 we determine Xn-^i from Xn only, we are continually abandoning information 

 which we have been at pains to acquire. We need not balance disadvan- 

 tages : Prof. Ostrowski's method is one compromise which retains some 

 of the advantages of each extreme ; another, simpler and in the long run 

 more efficient, is the recurrent use of the one formula 



, (jUn, n — 2 ~ ~ inn — i, n — i'V^y'ii — i 



Xn + i = Xn — m,i. « - 1 J" H ~, Z 



yn — yn- I 



where nir, s = (^v — Xs)liyr — ys). 



Mr. D. H. Sadler. — The estimation of computational labour (10.40). 



The difficulties of absolute estimation are summarised, and the care that 

 must be taken in forming relative estimates is stressed. Illustrations are 

 given in the simple case of computing a polynomial expression. 



Symposium on Combinatorial mathematics in the design of experiments 

 (ii.io). 

 Chairman : Prof. R. A. Fisher, F.R.S. 



Dr. C. C. Craig. — Some remarks on randomisation (11. 10). 



The usefulness and validity from the point of view of fiducial probability 

 of significance tests based on the principle of randomisation is well recog- 

 nised. However, it seems of some interest to the author to illustrate how 

 the efl^ectiveness of such a test may depend on the populations from which, 

 in fact, the samples were drawn. In particular, suppose two samples of N 

 are drawn, one from each of two normal populations with equal variances 

 but unequal means. By sampling methods, the probability that the test 

 based on randomisation will indicate that the population means differ is 

 studied. 



Mr. H. W. Norton. — The 7X7 Latin squares (11.30). 



A discussion of 7 X 7 Latin squares leading to Graeco-Latin squares, 

 and of the enumeration of the 7x7 Latin squares. 



Dr. W. J. YouDEN. — Cotnplex square designs in plant physiology and 

 their connection with incomplete randomised blocks (11.50). 



In recent years the use of experimental designs based on the combina- 

 torial properties of numbers has been developed in plant physiology and 

 pathology as in other fields. The natural structure of experimental plants 

 makes it desirable to eliminate causes of variation due both to the indi- 

 viduality of plants, and to leaf order, using a double elimination as in the 

 Latin Square. In addition the principle of balanced incomplete blocks is 



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