DOUBLE GAMMA FUXCTIOX. 
349 
Note. —The asymptotic expansions of this jDart were obtained in my original cast of 
this theory, to which reference has already been made in the note which follows the 
“ Theory of the Gamma Function,” by the assumption that they would Involve merely 
powers of n and log n coupled with inductive processes. Such a method, though 
long, is, could the fundamental assumption be justified, probably the most elementary 
way of obtaining these results. 
\^Additional Note added Jvdij 5, 1900.—Dr. Hobson has pointed out that, if we 
admit the validity of the application of the calculus of oj^erators to a parameter in 
the subject of integration of a contour integral, the theory from § 57 onwards may be 
developed in a very elegant manner. 
We take the formula 
J + "■:) — J + <^i) —y + 'y?) + / {*t) — , 
and the known theorem 
1 if (1 - M r 
' / I _ — /r- 
j'e "'( — zy ^dz, 
and deduce 
r(l - -n 
^■{«) = -:u 
-TT 
(g"- du ') • 
^ dpi -.p r 
27r Jfl — e~'^e) n _ '"-I 
Part l\. 
The Midtiidication, Transformation, and Integral Formulce for the. 
Doidjle Gamma Function. 
§ Gl. After the developments of Part IIP, we now return to the pure theory of 
the double gamma function. As regards the multiplication and transformation 
theories, two distinct courses are open to us. We may either proceed entirely alge¬ 
braically, making use of the limit theorems wliich liave been established, and so deduce 
the recpiired results without the intervention of contour integrals at any stage, or vm 
may directly utilise these latter to obtain the formulm in question. The former 
course is, on abstract grounds, preferable we ought to deduce algel)raical results 
by algebraical jwocesses. But it is open to the fatal objection of leading to very 
lengthy algebra. We will employ the two methods, side by side, to deduce the 
multiplication formulae, and it will be observed that the second method is both more 
elegant and more speedy. For the sake of brevity, the results of the transformation 
* In the first sketch of this theory, before the discovery of the contour integral expressions, all the 
results were obtained in this way. 
