264 ANIMAL MECHANICS, 



Hence we have 



p 1 = A 2 + A' 2 - 2 A A' cos x. 

 In the triangle bSa\ let 



bet = p' bS = A (by hypothesis) a 8 = A' bSa = y. 

 Hence we have 



p" 1 = A 1 + A' 2 - 2 A A' cos y. 



If we imagine a sphere described round 8, intersecting 

 the several lines, we shall have (Fig. 70), the line SK, which 

 is the axis of the cone, described by the line 8a. meeting the 

 sphere in the centre of a small 

 circle, in which the cone pier- 

 ces it. 



The spherical radius of this 

 circle will be the angle 



KSa = <t. 

 Let S, a, a be the original 

 positions of all the lines, and 

 let b denote the new position 

 taken up by a, after a rotation Fig. 7c 



through an angle 



aSb = to 



Draw the arc b a\ which will be equal to y, while aa will be 

 equal to x. 



In the spherical triangle ba'S, we have 



bS = ba' = y, Sa' = a - x, bSa' = w. 



Therefore 



cos y = cos w sin <r sin (d - x) + cos cr cos (cr - x), 

 and when w is a moderately small angle 



OJ 2 



