12 SCIENCE PROGRESS 



The researches of T. J. I 'A. Bromwich (Proc. Lond. Math. 

 Soc. 191 7, 15, 401-48) on the solution of dynamical problems 

 by means of complex integrals, which is more general than the 

 method (Routh, Heaviside) of normal co-ordinates, are of 

 interest to mathematicians in connection with the work of 

 Cauchy, Dini, and W. B. Ford [Studies on Divergent Series and 

 Summability , New York, 191 6) on the application of complex 

 integrals to the summation of series of normal functions. 



W. V. Lovitt (Amer. Journ. Math. 1917, 39, 27-40) discusses 

 some of the singularities of a contact transformation. 



Tomlinson Fort (ibid. 1-26) generalises the fact that, if 

 all the coefficients of a linear difference or differential equation 

 have the same period co, then, if y(x) is a solution of the equa- 

 tion, y(x-\-(o) is also a solution. He treats equations of the 

 second order, and the fundamental facts developed are so 

 primarily with a view to their application, that he also gives, 

 to self-adjoint boundary -value problems in one dimension. 



W. E. Milne (Bull. Amer. Math. Soc. 191 7, 23, 166-9) estab- 

 lishes some asymptotic expressions in the theory of linear 

 differential equations. 



J. F. Ritt (Trans. Amer. Math. Soc. 191 7, 18, 27-49) studies 

 a general class of linear homogeneous differential equations of 

 infinite order with constant coefficients. The first part of his 

 paper is devoted to the theory of the " entire differential 

 operator of genus zero," and, in the second part, on the homo- 

 geneous equation of genus zero, Ritt finds it desirable to have 

 information as to the possibility of resolving into partial frac- 

 tions the reciprocal of a whole function of genus zero, and this 

 forms the subject of another paper (ibid. 21-6). 



T. H. Hildebrandt (ibid. 73-96) applies some of the concep- 

 tions of E. H. Moore's " general analysis " and some of the 

 results of the general theory of integral equations to the theory 

 of linear differential equations in general analysis. 



E. Delassus obtained a canonical form useful for the study 

 of systems of partial differential equations, but L. B. Robinson 

 and Gunter found independently of one another in 191 3 that 

 the form is not absolutely general. Were the integration of the 

 given differential system the only question of interest, there 

 would be no need to improve the above form, in consequence of 

 a theorem due to Riquier ; but since canonical forms are 

 often useful in the study of comitants of either differential or 



