454 Mr. W. Galloway. On the Influence [June 16,. 



5a cos JSaCos 



Since, in the equation of differences 



U n+2 +py<n+i + qu n = 0, 



the ratio can always be expressed by continued fraction, and 



since if 



ps x n 



where ft and a. are the roots a + bx + cx 2 =0, 



2n — l b n — la _ n 



2n <y i n g \ . \ V 



... . f/s x r+l fjs x r 



it is evident that the ratio of , — to I / — - — — can 



J « v a-\-bx + cx° }a V a + bx -\-cx 2 



be expressed by a continued fraction. Similar treatment will apply to 



the integrals — 



f dx . x u P x r dx 



J V^uW> ■ • (214) ' )VJY-W^X=¥) ■ ' (215) ' 



de 



. . . . (217), 



\</0-kW)(l-x°~) (216) ' J(a + ftcos0)» 



[ . . (218), f M , „ . . . (219),. 



and many other integrals. 



XXV. " On the Influence of Goal-dust in Colliery Explosions. 

 No. III." By W. Galloway. Communicated by R. H. 

 Scott, F.R.S. Received May 30, 1881. 



(Abstract.) 



The information that can be gleaned from a study of the three great 

 colliery explosions of the past year, namely, Risca, Sean am, and 

 Penygraig, appears to throw more light upon the question than any- 

 thing that has been elicited experimentally either by myself or others 

 since I last had the honour of addressing the Society on the same 

 subject. These explosions took place on the 15th of July, the 8th of 



