DOUBLE GAMArA FUNCTION , 
207 
MuUiplkaiion Theory, 
Piige 
§ 62. Expression of r 2 (/H.':) in terms of Donlile Gamma Functions of Argument .s.;j.50 
I 1 /7’CO -f- 
§63. Expre.ssion of U 11' Fjl—^^ in terms of the Double Stirling Function .... 352 
§ 64. The Case of ?» = 2. 354 
§ 65. The Eesults of §§ 62 and 63 obtained from the Contour Integral Expression of the Doul)le 
Gamma Function . '. 354 
§ 66. The Functions 356 
2. = 0 i = 0 " HI / 
Tranufoniuifioii Theory. 
§ 67. Introduction. Transformation of r 2 (s i — , — ).357 
\ ! p <1 / 
§ 68. Transformation Formula for the Doulile Stirling Function. 359 
§ 69. The Transformation of the First and Second Doulde Gamma Modular Forms.361 
Integral Formuhc. 
§ 70. Translation of Alexeieavsky’s Theorem into the Notation of the Paper.363 
§ 71. Alternative Proofs of the Fundamental Eelation.364 
§ 72. The Analogues of Eaabe’s Formula.365 
§ 73. Evaluation of |o' Log r 2 (. 2 | wi, o).,)dz .366 
§ 74. Expression of the Glaisher-Kiidielin Constant A in Terms of p 2 (w) nnd pi(<'j).368 
§ 75. The Eesult of § 73 in the Case of Equal Parameters.369 
§ 76. The Value of jo' Log r 2 (^! + wi, w-I)dz .369 
§ 77. The Ajjplication of Contour Integrals to obtain the Eesult of § 73. Conclusion .... 370 
Part V .—The Asymjgtotic Eo'.pxnsion and Trunscendcntally-transcendental Nature of the Doidde 
Gamma Function. 
§ 78. The Behaviour of Ih,^?) near Infinity.372 
§79. The Asymptotic Expamsion when the Parameters arc equal.373 
§80. Form of the A.symptotic Expansion of r 2 ( 5 :) Avhen .s is Positive with respect to the oj’s. . . 374 
§ 81. „ „ „ „ when z is not Negative with respect to the (o’s . 376 
§ 82. The Negative Quasi-quadrant forms a Barrier-Eegion.377 
§83. The Asymptotic Expansion of Log .377 
J- 2 G) 
§ 84. The Asymptotic Expairsion of Log P^Q + 379 
§ 85. Alternative Method of Proof. 380 
1 f - 
§ 86. The Contour Integral --— —^- F(s a wj oio)ds .381 
2-t J s sin -s' 
§ 87. The Transcendentally-transcendental Nature of r 2 Q).384 
1. Imtboduction. 
The present paper continues my researches in the theory of gamma fimctious. 
Previously to a certain extent I obtained known results by new methods : none of 
2 M 2 
