394 SECTIONAL TRANSACTIONS.— A*. 



needed owing to the limited number of leaves. A group of designs com- 

 bining the two qualities, and which have proved useful in practice, is 

 exhibited. The remaining unsolved combinatorial problems are indicated. 



Mr. F. Yates. — The use of lattice squares in plant improvement (12.10). 

 Efficient methods of comparing, under field conditions, large numbers 

 of new varieties produced by genetical segregation are a vital need in prac- 

 tical plant improvement. The new quasi-factorial and allied designs give 

 methods of considerably increasing the accuracy of the field comparisons, 

 using the same amount of experimental material, and supersede the older 

 methods of arrangement, such as the use of ' control ' varieties. Many of 

 these designs depend on the existence of Graeco-Latin and higher order 

 orthogonal squares, and the question of the existence of such squares, first 

 investigated by Euler, has therefore become of practical importance. In 

 particular the existence of complete sets of orthogonal squares is necessary 

 for the construction of designs in lattice squares (which enable p^ varieties 

 to be arranged in squares of side p, eliminating fertility differences between 

 rows and between columns) and for the construction of certain types of 

 incomplete randomised blocks (i.e. randomised blocks each of which con- 

 tains only a proportion of all the varieties to be compared). 



Mr. W. L. Stevens. — Completely orthogonalised squares (12.30). 

 It is known that for certain values oi p, (p — i) Latin squares may be 

 formed such that any two of the squares are mutually orthogonal. Solutions 

 are now known for p = 2, 3, 4, 5, 7, 8, 9, and any prime number. It is 

 believed that a solution exists when p is any power of a prime. The case 

 proved is for the square of any prime, and the theory has been applied to 

 develop a completely orthogonalised square of side 25. 



Afternoon. 



Visit to the Mathematical Laboratory. Symposium and Demonstration 

 on Mechanical methods of computation (3.0). 



Prof. J. E. Lennard-Jones, F.R.S. — Bush differential analyser. 



Mr. M. V. WiLKS. — Mallock machine (3.20). 



Demonstration of Bush and Mallock machines (3.30). 



Dr. J. WiSHARTand Mr. D. H. Sadler. — Description and demonstration 

 of Hollerith and National machines (4.45). 



Friday, August 19. 



Symposium on From function to printed table : some aspects of the work 

 of preparing a table of a mathematical function (11.30). 

 Chairman : Prof. E. H. Neville. 



Dr. W. G. BiCKLEY .—Computation from series and by recurrence 

 formula:. 



Some elementary considerations concerning computation by power series, 

 especially with regard to labour saving and checking, are discussed. 



For greater values of the argument, convergence of the power series is 



