Whence 



and 



OF ARTS AND SCIENCES. 327 



xff' :=. — a sin X -{- ^ cos x = q', 

 TVcrcr' = TV (a^ -\- sa sin x — e(3 COS x), 



A = I rTV(u§ -\- sa sin x — e§ cos x). (27) 



Xo 



If the new origin lie anywhere on a or ^, one of the terms of this 

 integration disappears. Suppose the new origin is on a; then £ = 

 ma, where m is a scalar. Hence 



X 



A =z I i TV«^(1 — m cos x) 



= -n- sin ^ a? — m sin a:- . (28) 



IS''. The equation of the hyperbola, 



Q = aChx -\- /3Shx, 



will give results precisely similar to those of the preceding section, 

 with the hyperbolic sine and cosine everywhere substituted for the cir- 

 cular. With the origin of vectors at the centre, the area swept by the 

 radius vector is 



A=^,\nl[x-x,'\; (29) 



or if a _L |3 



This is the area of a portion of the surface exterior to the curve, con- 

 tained between the curve and the limiting positions of the radius vec- 

 tor. To find a portion of the inner area we need only transfer the 

 origin to some point on or within the curve, and proceed as before. 

 Suppose the origin to lie on a, at a distance 7nTa from the centre. 

 The finite area is 



^ = ^ sin f [x- — mShxl ' . (30) 



In formulae (28) and (30), put ?« = cos x^ and m = Chx„, and 

 take T„ and for the limits. Then if a _L ^, the elliptic area (dou- 

 bled) is 



A^ = ai[x„ — cos a?„, sin x^], (31) 



