198 Prof. Young on the Development of certain 



parts of the country, have added some new and very interest- 

 ing minerals to our former catalogue. The following is a 

 list of these : 



Var iolite.— This variety of compact felspar has been lately 

 met with in the hornblende rock of Morne. 



Anthracite. — A compact variety of this mineral, with a 

 highly metallic lustre, occurs frequently in the grauwacke of 

 Down. 



White Carbonate of Lead — accompanies galena and the green 

 phosphate of lead in the Newtonard's lead-mine, Down. 



Colophonile. — This mineral occurs in quartz veins travers- 

 ing siliceous slate near Glassdrummond, Morne. It is of a 

 brownish yellow colour, and is crystallized in rhombic do- 

 decahedrons, with very unequal angles, and having striae pa- 

 rallel to the lesser axis of the rhomboid. The mineral has 

 very much the appearance of cinnamon stone. 



Sulphuret of Molybdena. — This mineral has been lately dis- 

 covered in a siliceous slate, or, I believe, rather a chlorite 

 slate, on the shore near the mountains of Morne. The cry- 

 stalline form is a six-sided table, terminated by a low six-sided 

 pyramid, of which the base angles are truncated. Neither 

 this mineral nor the colophonite have, so far as I know, been 

 before noticed in Ireland. Yours, &c. 



Belfast, Aug. 11, 1834. James Bryce, Jun. 



XXX. On the Development of certain Trigonometrical Func- 

 tions. By J. R. Young, Professor of Mathematics in the 

 Royal College, Belfast,* 



T^HE series given by analytical writers for the development 

 •*• of a circular arc in terms of its sine, cosine, and tangent 

 are as follows : 



si„-i* = ,+ £jjL + x *£^ 6 +, &c. 



-i * & 3x b 



cos l X S3 x 4- — 4-. &c 



—1 X 3 X b X 



tan- 1 * = x — 4- — — - 4-, &c. 



which series, as well as those for the development of the arc 

 in terms of the other trigonometrical lines, are true only for 

 the least of the arcs to which these trigonometrical lines be* 

 long. I am not aware that any explanation has ever been 

 given of this want of generality ; or that any one has made 

 known how it comes to pass, that while the developments of 



* Communicated by the Author. 



