620 SCIENCE PROGRESS 



that of Clebsch and Gordan (1869) than to subsequently 

 developed methods. Glenn also (ibid. 545-56) studies the 

 functions of the a's and x's in a binary quantic of order m 

 whose coefficients are variables which are invariantive under 

 the operation of transforming this quantic by the group repre- 

 sented by the general binary linear transformation in which 

 the coefficients are parameters representing residues of a 

 prime number. 



Analysis. — P. Nalli (Giorn. di Mat. 191 5, 53, 169-77), i n 

 connection with some researches of Denjoy, proves a theorem 

 on the generalised second derivatives of a function. The 

 theorem of course throws light on the question of the develop- 

 ment of a function in a trigonometric series. 



A. Denjoy (Bull. Soc. Math, de France, 191 5, 43, 161-248) 

 gives a paper on summable derived functions in which the con- 

 ception of " approximate continuity " of a function at a point 

 is introduced and plays an important part. 



In 191 1, C. N. Moore gave a method for establishing the 

 convergence or uniform convergence in series of Bessel's func- 

 tions, and determined the values to which they converge by 

 long and complicated considerations. Moore now gives (Bull. 

 Amer. Math. Soc. 191 6, 23, 18-27) a fairly simple method of 

 determining the value of a development when we know it 

 converges, " and consequently this paper combined with the 

 previous one gives the first complete discussion of the conver- 

 gence and value of the developments in Bessel's functions 

 under conditions that are usually satisfied in the applications." 



G. A. Bliss (ibid. 35-44) gives a very careful analysis of 

 W. F. Osgood's Madison Colloquium Lectures entitled Topics 

 in the Theory of Functions of Several Complex Variables (New 

 York, 1 91 4). 



The general definition of " adjoint systems " of boundary 

 conditions associated with ordinary linear differential equations 

 was given by Birkhoff in 1908, and the idea was further 

 developed by Bocher in 191 3. Bocher obtained a condition 

 that a system of the second order be self-adjoint. Dunham 

 Jackson (Trans. Amer. Math. Soc. 191 6, 17, 418-24) extends the 

 discussion of this problem to the case of systems of any order. 

 He expresses a condition for self-ad jointness of the boundary 

 conditions in matrix form without any requirement that a corre- 

 sponding property be possessed by the differential equations. 



