DEPARTMENT OF PURE MATHEMATICS. 493 



Itenco we get a simple proof of the theorem that a ' special ' group is the difect 

 product of prime-power groups. 



4. Dr. Miller's results for ' groups of subtraction and division ' bold good for 

 groups generated by any two substitutions of the type .v' = (.r + 6)-4-(c.i"— 1). 



5. The most convenient form to which successive reflexions in an odd number 

 of planes can be reduced is an inversion about a point followed by a rotation 

 about a line through that point. 



2. 071 Two New Symmetric Functions, i^y Major P. A. MacMAhon, F.Ji.S. 



3, A Test for the Convergence of IfuUiple Series, 

 By T. J. I'A. Bromwich, F.R.S. 



Cauchy was the first to show that the convergence of the double series of positivo 

 tetms 



22/(wi, n) 

 lu deducible from a knowledge of the convergence of the double integral 



lind Riemann^ proved that this double integral might be replaced by a single 

 integral in many cases. Riemann's test has been augmented with a test for 

 divergence by Hurwitz ;" and the following test is to some extent connected with 

 those of Riemann and Hurwitz. 



If the jjositive function f (x, y) steadily decreases to zero as X, y increase to 

 infinity, and if f (x, y) has a lower limit g(^) and an upper limit G(^), when 

 X and y take all positive values for ivhich x + j = ^, then 



(i) 22f(m, n) converges if G{$)$d$ converges. 



(ii) 22f(m, n) diverges if g{^)^d^ diverges. 



The proof is immediate ; for the sum of the terms on a diagonal " .r + y = m 

 is then between (n + 1) g{7i) and (w + 1) G{n), because there are (n + 1) terms on 



the diagonal. Consequently the double series converges with 2wG(n), that is, 

 G($)^d^ ; and the double series diverges with ^ng{n), 



/•CO 



that is, with the integral g(i)id^. 



Kvample.—Considev the series 22/(?«'' + n'), where /(.i) steadilj' decreases to 

 iero as .r tends to infinity ; it is then evident that we may write 



because $'^.v- + rf^i$-, if .v + y = ^ and .r, y are restricted to be positive, 

 Hence the series 22f(m'^ + n^) 



(i) converges if /(i^-)^rf^ converges; 



/•» 

 (ii) diverges if f{^')^d^ diverges. 



> Ges. Werhe (187G), p. 452. 



" Math. Annalen, Bd. 44 (1894), p. 83. 



* The terms are supposed arranged in a square, as iisuaU 



