Projection of the Sphere in Crystallography. 



4. Let X, Y, Z be the poles of 

 the sides of the spherical triangle 

 ABC, opposite to A, B, C re- 

 spectively; P any point on the 

 surface of the sphere ; a, b, c any 

 positive constants. Let P be de- 

 noted by the symbol h k I, where 



39 



iC0sPX = ^C0sPY: 



h k 



cosPZ. 



Diameters of the sphere passing through X, Y, Z will be 

 called axes ; the constants a, b, c parameters ; and the quantities 

 h, k, I the indices of the point P. 



It is easily seen that if h, k, I be supposed all positive when P 

 is within the triangle ABC, h will be positive or negative ac- 

 cording as PX. is less or greater than a quadrant, and that h 

 will be zero when Pis in the great circle BC. The signs oik, I 

 are determined by a similar rule. The symbols of the points 

 A, B, C are 1 0, 10, 01 respectively. It is evident that 

 the point P is equally well denoted by the symbol nh nk nl, where 

 n is any positive quantity. The opposite extremities of the dia- 

 meter of a sphere have the same indices with contrary signs. 



5. cos PX= siii PC sin PCB = sin PB sin PBC, 



cos PY -^ sin PA sin PAC = sin PC sin PCA, 

 cos PZ = sin PB sin PBA= sin PA sin PAB. 



Hence 



T sin PAB = - sin PAC, - sin PBC = - sin PBA, 

 b c c a 



-sinPCA=JsinPCB 



a 



(^) 



8inABsinBAP _ sinBH sin BC sin CBP 

 sin CA sin CAP" sin CH' sinABsinABP 



sin CA sin ACP sin AL 



sin CK 



sinAK' 



sin BC sin BCP sin BL" 



Hence 



k smBH _/ sinCH / sin CK_ /t sin AK 

 1) sin AB ~~ c sin CA' c sin BC ~ a sin AB' 



h sin AL k sin BL 



a sin CA b sin BC 



(r) 



