traced upon the Surface of the Sphere, 293 



symmetry of the result would be destroyed, as we should find the co-ordi- 

 nates of one of the points involved differently from those of the other points. 

 But by making the substitutions in all, adding the results, and taking one- 

 third of the sum, we shall have a value of cos in the proper form. The 

 work, which is long, but involving no other difficulty, is suppressed here, 

 and the result merely set down. That is 



_ cos a sin a ; sin a,, sin ffi /3 x/ + cos a, sin a sin a,, sin /? /3 + cos a a sin a sin a, sin /3 /3, 

 cos ^^ * ~ ' . * * 1 1*3. i 



'* 



Making these several substitutions in (1), we have a complete solution 

 of the problem. The denominator \f m? + it? + p*, appearing in all the terms 

 of the expression, is omitted altogether. 



cos a sin a, sin a,,sin /3,-/3, ; + cos a, sin a sin a a si n /3,,-/3 + cos a a sin a sin a, sin /3-/3, = 



cos</> { sin a, sin a,, sin j^,-/?,, + sina sin a, sin /?-/? + sina sin a, sin <3-/3,} 

 cos(9{cos a(sin c^sin/^-sin a.sin/3,) + cos a,(sina sin/3-sin a.sin/3,) + cosa^sine^sin^-sin a sin 



, J 1 t \ // I II I 1 ff * f\ I J //I ll\ / ( / .--- J y I 



) 1 

 )) " 



Cor. The condition that must subsist amongst the co-ordinates of three 

 points on the sphere, that they may range in the same great circle, is, 



cosa sin a, sin a,, sin/?,-/?,, + cosa,sina sin a,,sin/3,,-/3 + cos a a sin a sin a, sin /3-/3, = 0. ... (15.) 



XIV. 



To change the Origin and Direction of the Co-ordinates of a Spherical 



Curve. 



1. Let the origin remain the same, but the prime meridian be changed 

 for one whose longitude is /^ ; then the transformation will be effected by 

 writing 6 + ft, instead of 0. 



2. Let the origin be transposed to a point in the prime meridian, whose 

 radius vector is a. 



VOL. XII. PART I. P p 



