32 Dr. Booth on the Rectification a7id 



Now if the integrals in the last two equations be taken be- 

 tween the same limits of A and <p, their values will be equal, 

 hence, subtracting the former from the latter, 

 (t' — 0" = — y. 



Now as sin u = -g- and u = -7^, sin u = j> -r^ , and as neither 



P nor p pass through infinity or zero, they always retain the 

 same sign +, hence the sign of sin u will depend upon that of 



~-i but v^ s= a^ cos^ X + 6^ sin' X : hence 

 dK ^ 



p-^zn — (a^ — b^) sin A cos A, 



therefore sin is negative, and as u is always less than le, v is 

 negative, and may be written — u ; making this change in the 

 last formula for rectification, 



(t' — (T = V, (55.) 



a formula precisely analogous to (35.). 



Thus as the difference of two elliptic arcs may be exhibited 

 by a right line, so may the difference of two arcs of a spheri- 

 cal ellipse be represented by an arc of a great circle. 



XXVIII. To show the geometrical interpretation of the 

 assumption made in (53.). 



In the first place we may observe, 

 that if O A, OB are arcs of great cir- 

 cles at right angles, a point P on the 

 surface of the sphere may be referred 

 to those axes either by the arcs P m, 

 P w, which are secondaries to the arcs 

 O A, O B, or by the arcs Om,On; 

 let P 7» = ^, Pn = £, O w = ^, 

 O w = >j, O P = p, and the angle 

 PO?»= co; we shall then have by the 

 common rules for right-angled spherical triangles, 



sin ^ = sin p sin CO, sinj; = sinpcosc«~\ /cc ^ 



tan )3 = tan p sin ctf, tan^ = tanpcoswj * 

 hence sin^_y + sin^^ = sin^p, tan^^ + tan^)j = tan^p. 

 We may easily establish a relation between x and ^, ^ and >;, 

 for in the preceding equations, eliminating the functions of 00, 

 we find 



sina; = tan^cosp, sin3/ = tan>;cosp; 

 by the help of these equations we may pass from the one 

 system of spherical coordinates to the other. 



If now, between equations (4.), (5.) and {56.), we eliminate 



