ELISHA MITCHELL SCIENTIFIC SOCIETY. 73 



Complete the rectangle OABC; draw Ox' and Oy' bisecting 

 the angles between Ox and Oy; draw EBFG perpendicular to 

 AC; draw OK and OH parallel and perpendicular to AC; let 

 CAO=p. 



B is the instantaneous centre, BF is normal to the envelope 

 of AC and its envelope is the required evolute. 



ABF=p, FBC=90°— p, BEO=45°— p, OGK=45° + p 

 OK=HF=a — 2 a siu 2 p=a cos 2 p. 



EG=OG cosec BEO^OK cosec OGK cosec BEO 



=a cos 2 p cosec (45°-|-p) cosec (45° — p). 



Hence, by reduction P]G=2a. 



Since EG is of constant length, and its extremities move on 

 two rectangular axes, its envelope must be a four-cusped hypo- 

 cycloid, which is the required evolute. 



Remark. — If M is the point where EG touches its envelope, 

 BM=BK. For, at the point (x, y) of the curve x%-j-y%=a%, 

 the radius of curvature is 3 (axy)^* and the perpendicular from 

 the origin on the tangent is (axy)^. 



Or, it follows from the formula connecting the segments into 

 which the radius of curvature of the hypocycloid is divided by 

 the instantaneous centre. (See Williamson's Differential Cal- 

 culus, Art. 281). 



*MICA MINING IN NORTH CAROLINA. 



W. B. PHILLIPS. 



Modern mica mining began in North Carolina in 1868— '69. 

 Some little work was done in 1867, but beyond opening two or 

 three pits, and getting out several hundred pounds of fine mica, 

 uot a great deal was accomplished. Reference has already been 

 madef to the fact that some of the mines had been worked by 



This paper has appeared in the Engineering and Mining Journal. 



f\V. C. Kerr, Engineering and Mining Journal, Vol. XXXI, No. 13, p. 211 



