49i Transactions of section A. 



Both results may ba summed up iu tho sLatomenl : — 



The double aeries 22/(?)i'^ + n'^) converges or diveryts loith the integral 

 r f{.v)dx. (Hurwitz, I.e., p. 87.) 



This result can be at once extended to the series t2f{am~ + '2hmn + cn'^), in 

 which am^ + 2bmn + cn"^ is restricted to be always positive. Familiar special cases 

 are given by taking /(.r) = .r-ii (the double series for Weierstrass's elliptic 

 function ^'!<) or f{.r)=€-' (the double theta-functiono). 



It is easy to extend the test given above to a ^-fold series :^- 



If the 2')ositive function f(Xj, Xo, ..., Xp steadily decreases to zero as 

 X,, X,,, ..., Xp increases to infinity, and if f(x,, x,, ..., Xp) has n lower limit 

 g(^) and an upper limit G(|) ivhen x,, x.,. ..., Xp takes all 2>o.ntivp values such 

 that 



,ri + .v., + . . . + .!■;, = ^ 



then (i) 2f(mj, xa.,, ..., nip) converges if G(^)^i'^^d^ converr/es j 



J 00 

 g{i)^^-^d^ diverges. 



In particular, if the general term takes the form 



f(aiim,'' + 2aj„mim2 + ... ) 



where the quadratic form Sors x, Xg is always positive, and f(x) decreases as x 

 increases, the convergence or divergence of the series depends on that of the integral 



rf(x) x5P-'dx. 



4. Many - Valued Fitnctions of Jieal Variables. T>y A. R. UlClIARDSON. 



Some of the general properties of functions of real variables which are many- 

 valued are investigated. 



Let F(^)) denote the set of values of the function at the point p, and ^(^^) 

 denote the set obtained as the limits of the values of the function at points near^. 



Let 7;(^;) denote those values of (^(/') which belong to F(^)) ; A(^) denote 

 those values of <^(^-') which do not belong to F(?^)' 



The difference between the upper and lower limits of 7;(^)) is denoted by 

 /^rj{p), and the upper limit to the distance between a point of X(^) from all points 

 of F(p) or X(;j) is denoted by oi{p). 



Let Q{p) be the greater of (oi^p) and ^r){p). 



Then many-valued functions may be classified in a similar manner to one- 

 valued functions, and analogous theorems are found to hold. 



A more natural classification, however, from one point of view, is to divide the 

 functions into two broad classes, according as fl(^') is or is not continuous. 



Interpreting this in the language of one-valued functions, these will be 

 classified according as their degree of discontinuity is or is not continuous. It is 

 then found that a large number of the properties of continuous functions are 

 possessed by functions whose degree of discontinuity is continuous. 



5. Note on the Semi-convergent Series for 3„x. 

 By Professor Alfred Lodge, M.A, 



The semi-convergent series for the nth Bessel function, viz., 



