8 SCIENCE PROGRESS 



measure this deficiency : the paper is occupied only with the 

 formal algorithms, and questions relating to the convergence 

 of infinite processes are therefore not considered. Haripada 

 Datta (Proc. Edinburgh Math. Soc. 191 5-16, 34; Research 

 Paper, 1916, No. 4) shows that for many important known 

 functions the continued-fraction-expression can be obtained 

 directly and practically from the power series by using a known 

 transformation which involves determinants, and 03^ evalu- 

 ating these determinants. By this method a considerable 

 number of known isolated results are connected together and 

 exhibited as parts of a systematic theory, and some new re- 

 sults are obtained. Datta (ibid, 1916-17, 35 ; Research Paper, 

 191 7, No. 4) makes a further contribution towards the same 

 object and shows, amongst other things, that the continued 

 fractions given by Gauss and Heine for the quotient of two 

 hypergeometric or generalised hypergeometric functions may 

 be obtained by a direct use of Heilermann's (1845) transforma- 

 tion. Datta (ibid, and No. 7) examines cases of failure of 

 Heilermann's theorem. 



W. Gibb (ibid. 1915-16, 34; Research Paper, 1916, No. 3) 

 obtains certain integral relations connected with Whittaker's 

 " confluent " hypergeometric function. G. N. Watson (Proa 

 Lond. Math. Soc. 1919, 17, 241-6) gives the integral formula 

 ior generalised Legendre functions in a form different from that 

 given by E. W. Barnes in 1907. 



E. B. Stouffer (ibid. 337-52) completes his discovery (cf. 

 Science Progress, 191 6, 11, 269) of a complete system of 

 seminvariants for a certain system of linear homogeneous 

 differential equations by the calculation of complete systems 

 of invariants, semi-covariants, and covariants. The methods 

 used largely avoid the solution of the complicated systems of 

 partial differential equations which arise by the Lie theory. 

 E. B. Elliott (Proc. Lond. Math. Soc. 1919, 17, 306-15), in con- 

 nection with some papers of 191 2 published in the Proceedings 

 (191 3) of the fifth International Congress of Mathematicians 

 held at Cambridge in 191 2, and in subsequent papers, gives 

 examples of the formal " analysation " of solutions of differ- 

 ential equations of certain classes. A. B. Coble (Amer. Math. 

 Monthly, 191 9, 26, 12-5) determines a particular integral of 

 the linear differential equation with constant coefficients in a 

 certain special case (cf. H. P. Manning, ibid. 113). A. Milne 



