1877.] 



On certain Definite Integrals. 



359 



about 13' nearer the normal of the face (10 0) than that of the blue, the 

 msan direction of the first moan line being about 35° on the normal of 

 10 0, and 8° 40' on that of TO 1. 



Crystals of potassium nitrate, containing less silver nitrate than the 

 double salt, gave angles according somewhat closely with those of ordi- 

 nary nitre ; but the crystals did not give very good reflections. 



The results obtained by E-ose with sodium nitrate containing silver 

 nitrate were confirmed so far as the crystals permitted of measurement. 



Crystals of strontium nitrate containing silver nitrate gave excellent 

 measurements, according with those of a cubo-octahedron. 



VIII. " On certain Definite Integrals/' By W. H. L. Russell, 

 F.R.S. Received June 21, 1877. 



The following paper is a continuation of two papers recently commu- 

 nicated to the Royal Society, and inserted in the ' Proceedings/ 



(40.) j dd log e ( e 2* cos e + 2e x 008 9 cos (cc sin 0) + 1)=2tt log e 2, 



IT 



(41.) f " 2 dd c 2 C08 " 6 cos (sin 26 cos e + sin cos 0) = £ 

 J o 



(42.) j c£0e CO820 + coa *cos(sin20 + sin0)=7r. 

 J o 



(43.) f ' dd el-tee^* COS = l X =7T6 



' 1 1 — 2#cos0 + a? 2 



%} o 



2 



C V 1 — X COB 9 



(44.) 1 dB.e 1 - a " OOB *+« 8 cos- 

 J o 1 



#sin0 



2x cos -f- x 



f J sin0(4+cos 2 0) 7r 

 /^x 0sin 3 0c?0 7r(o? 2 -l) i l+a? 7r 



f ' 8in 3 0cZ0 7T(^ 2 + 1) , 7T 



(47.) 1 — — — = 3 y tan - — . 



Jo cos 2 a? 3 or 



