[ 2G5 ] 
VI. llte Theory of the Double Gamma Function. 
By E. W. Barnes, B.A., Fellow of Trinity College, Cambridge. 
Communicated by Professor A. R. Forsyth, Sc.D., F.R.S. 
Received Feljvuary 26,—Read March 15, 1900. 
Index. 
Page 
§ 1. Introduction.267 
Statement of Notation. 269 
Part I .—The Theonj of Duuhle BernouIUan Functions and Numbers. 
§ 2 . Definition of the Doul)le Bernoiillian Function. 
§ 3. Its Symmetry. 
§ 4. , Initial terms of its Expansion. 
§ 5 . The formula («') = ^ + i;S;f’(^/). 
§ 6 . The value of |o‘’' 2 S,i(n)(/(t. 
§ 7. The Double Bernoullian Numl)ers. 
§ 8 . Reduction for the Case of Eipial Parameters. 
§ 9. Expression for Sn («). 
§ 10. Expression of 2 S„((rji + lo., - a) in Terms of 'PaO and Simple Bernoullian Functions 
§ 11 . The Value of 2 B,j(on, oj.j) . 
§ 12 . The Value of jo 28,1 (a) da. 
§ 13. The Multiplication Formula. 
§14. The Transformation Formula. 
§ 15. The Fundamental Expansion whose Coefficients invoh'e Doul)le Bernoullian Functions 
§16. Derived Expansions. 
§ 17. Relation of 2^ii('0 to the Difterence Equation 
/(a + ioi + (ijo) — /(a + oji) — j b-i + + /(a) = a'* . . . . 
Note on the Alternative Development of the Theory. . . 
271 
271 
272 
273 
274 
276 
276 
277 
277 
278 
279 
280 
280 
281 
283 
Part II .—The Double Gamma Function and its Elementary J'roperties. 
§ 18. Introduction. 285 
§19. Definition of the Double Gamma Function.286 
§20. The Difference Ecpiations for 1 /'.^-’(^;).287 
§21. ,, „ ,, i/'j” (,?■). Introduction of the Numbers//i aiid ?a'.288 
§ 22. Properties of the Numbers m and m! .291 
VOL. CXCVI.— A 279. 2 M 12.4.1901 
