Wl 



ith Minimum Partitioned Area. 513 



we find, from (4) with (6), 



(by case = 0) A+ 3a 6 A' = -031578a" 2 . . . (8), 

 (and by case = 30°) A-6/a 6 A' = "031578 ,f. a~ 2 . (9); 

 whence 



A'= - J J . - 9 9 T x -031578 . a~ 8 = - 9 x -0001735 . a~ 8 



= -•001561. a- 8 , 

 A =1(3 -|f) x -031578 . a" 2 =209 x -0001735 . a" 2 



= •036261. a- ; 

 and for required equation of the surface we have (taking 

 a = l for brevity) 



£=•03626 . r 3 cos 30--OO1561 r 9 cos 90 ) (1Q) 



z= -03626. r d (cos 30- '043. r 6 cos 90) J ' ' j ' 



18. To find the equation of the curved edge BEB 7 , take, as 

 in (4), 



a? = l — z v /2 = l — f, where f denotes z^/ 2 . . (11). 



Substituting in this, for s, its value by (10), with for r its 

 approximate value sec 0, we find 



f = -i- (-03626 sec 3 cos 30--OO1561 sec 9 cos 90) . (12) 



a/2 



as the equation of the orthogonal projection of the edge, on 

 the plane BCB', with 



AN=y = tan0; and KP = £ . . . (13). 



The diagram was drawn to represent this projection roughly, 

 as a circular arc, the projection on BCB / of the circular arc of 

 20° in the plane Q, which, before making the mathematical 

 investigation, I had taken as the form of the arc-edges of the 

 plane quadrilaterals. This would give 1/35 of CA, for the 

 sagitta, AE ; which we now see is somewhat too great. The 

 equation (12), with y = 0, gives for the sagitta 



AE = -0245xCA (14), 



or, say, 1/41 of CA. The curvature of the projection at any 

 point is to be found by expressing sec 3 cos 30 and sec 9 cos 90 

 in terms of ?/ = tan and taking a n jdy z of the result. 



By taking x /(3/2) instead of ^(1/2) in (12), we have the 

 equation of the arc itself in the plane Q. 



19. To judge of the accuracy of our approximation, let us 

 find the greatest inclination of the surface to the plane BCB'. 

 For the tangent of the inclination at (r, 0) we have 



/^L + -^V = -lO88.r 2 (l-2x-129.r 6 sec60 + -129V 2 )^ (15), 



Phil Mag. S. 5. Vol. 24. No. 151. Dec. 1887. 2 M 



