SECTIONAL TRANSACTIONS.— A*. 397 



Tuesday, August 23. 



Prof. A. Speiser. — Elliptic functions from an elementary standpoint (lo.o). 



It is a well-known theorem, that a simply connected Riemann surface 

 may be transformed conformally on the Euclidean plane or on the interior 

 of a circle. With the aid of this fact it is proved that the general theory of 

 elliptic integrals consists ultimately in the possibility of paving the plane 

 with congruent quadrilaterals of any shape. 



Dr. B. H. Neumann. — General decompositions of groups (10.30). 



A group G is said to be the general product of its sub-groups A and B 

 whenever (i) A . B = G ; (ii) A/^B = {i}. A and B are called general factors 

 or complementary sub-groups. In the special case, when both factors are 

 self-conjugate in G, we have the well-known direct product. 



The following problems have been attacked, the first with a view to 

 applications in geometry : 



(i) Given G and a sub-group A ; to decide whether A is a general factor 

 of G and to find complementary factors. 



(ii) To characterise the groups in which certain types of sub-groups 

 (e.g. the Sylow sub-groups, or all sub-groups) are general factors (P. Hall). 



(iii) Given A and B, to find their general products. 



However, much.remains to be done, specially as regards the third problem. 



Mr. P. Hall. — The verbal classification of groups (11. 15). 



Let/(xi Xn) be any word, G any group, V = V/(G) that sub-group 



of G which is generated by all elements of the form/(ai, . . ., a»), where 

 the a's are arbitrary elements of G. Let W = W/(G) be the (unique) 

 greatest self-conjugate sub-group of G with the property that 



/(«A, . • •> «n6») =/(«!, . . ., an) 



for every choice of a's in G and b's in W. Then, if V and W are the 

 corresponding sub-groups of another group G', the latter is said to 

 be isological with G (in respect of /) if there exists between G/W and 

 G'/W an isomorphism, aW -*■ a"\N', such that the correspondence 

 fifli, . . ., an) -*f(a\, . . . a'n) determines an isomorphism between 

 V and v. This relation of isologism between groups has the properties 

 of equivalence, and separates all groups into a number of mutually exclusive 

 families in respect of their behaviour as regards the given word /. By 

 choosing a diflferent word in place of/, we obtain (in general) a different 

 classification. The most interesting choice is to take/= Xi~^X2~^XiX2. This 

 gives the commutatorial classification, which is especially appropriate for 

 the discussion of prime-power groups. In this case, V is the derived group, 

 W the central, and isological groups have many numerical invariants in 

 common. 



Dr. Olga Taussky. — Differential equations and hypercomplex systems 



("•45)- 

 It is known that each of a pair of functions satisfying the Cauchy-Riemann 

 equations satisfies the Laplace equation. Similarly for each of a set of four 

 functions satisfying the Dirac equations. It can be shown that the same 



