104 



TRANSFORMATION OF SURFACES BY BKNDIXG. 



and therefore the difference between the angle of a and b and that of c and d is 



*, dl * d I 1 Y\ c , 

 S( = -,- 5u = j- ( : j j ,) 85u ; 

 au au \pH\ntp an J 



whence the differential of p with respect to M may be found. 



We must therefore find 8/, and this is done by means of the quadrilateral 

 on the sphere described in Art. (12). 



15. To find tJie values of 81 and 81'. 



In the annexed figure let abed repre- 

 sent the small quadrilateral on the surface 

 of the sphere. The exterior angles a, b, 

 c, d are equal to those of the four facets 

 which meet at the point P of the surface, 

 and the sides represent the angles which 

 the planes of those facets make with each 

 other ; so that 



and the problem is to determine 81 and 81' in terms of the sides I and I' and 

 the angles o, b, c, d. 



On the sides ba, be complete the parallelogram abed. 



Produce ad to p, so tiiat op = a8. Join S^>. 

 Make cq = cd and join dq. 

 then. 8l = cdab, 

 = cq c8, 

 = - (qo + 08). 



Now qo = qd tan qdo 



= cd sin qcd cot qod, 

 but cd = l nearly, sin qcd = qcd = (c + 6 - tr) and 



