382 Mr DAVIES on the Equations of Loci 



The 6 in this is measured on a circle whose polar distance is (p ; but if we 

 wish to substitute for it the 6 of the equator, the latter 6 and its differen- 

 tials must be multiplied by sin $ and its powers corresponding with the 

 powers of the several differentials themselves in the denominator. Then (1.) 

 will be changed into 



sin 3 (> d& ' 1 .2.3 



2. To ymd fAe inclination of a given chord of a spherical curve to 

 the radii-vectores of its extremities. 



Let MN be the chord; PM = ; PN = 0'; EPM = 6, and EPN = 6'. 



Then, by spherics, 



/\f /\ 



tan PMN = 



cot <f>' sin <p cos & Q cos (f) 



sin Q' 6 sin <p' 



cos d>' sin <b cos (b sin d)' cos & 6 



,_ . 



But cos & 6 = 1 2 sin 2 ^ # 6, which inserted in (3.) converts it into 



tan PMN = ain <fr an fr-0 



sin 9' 2 sin (f)' cos sin* % & 6 



3. From this we can find the inclination of the radius-vector to the 

 tangent. 



Let & 6 be put = ?. Then, by TAYLOR'S Theorem, (xliii. 1.), (4.) 

 becomes 



