216 Wisconsin Academy of Sciences, Arts and Letters. 



finitesimal angle represents the two opposite wedges in the plane of the 

 principal axes. With C as a center and a a and b b as principal axes de- 

 scribe an ellipse similar to one having axes A A and B B, also describe 

 circles with radii C b, and C B, , On C P as a diameter corresponding to 

 B^ B, as a diameter describe another similar ellipse. From P draw Ph and 

 Pi tangent lines to the ellipse having axes a a and b b. Draw diameter 

 B, B„ or b„ b„ parallel to tangent Ph, also diameter B,„ B„, or b,„ b,„ parallel 

 to tangent Pi. Thi'ough the points of tangency a„ and the center C draw 

 the diameter A„ A„ or a„ a„ , then A„ A„ and B„ B,, or a„ a.„ and b„ b„ are 

 conjugate diameters. A,„ A,„ and B,„ B,„ or a,„ a,„ and b,„ b,„ are likewise 

 conjugate diameters. Through the points of tangency a„ and a,„ draw 

 line m n, and diameter A, A^ parallel to m n, then A, A, and B, B, or a, a, 

 and b, b, are conjugate diameters, and a„ d and a,„ d are equal ordinates. 

 From P draw line Pe tangent to the circle having radius C b,, and from 

 point of tangency g draw line g d 1; then line g 1 is perpendicular to di- 

 ameter B, B^ and a double ordinate.* 



From known demonstrations of Conic Sections, 



g d : a„d :: b, : a, :: B, : A, . 



Draw radius g C and let angle C g d or its equal g P C be represented by 

 $■. Then because g d = b, cos 5, and a„d = a, cos 3-, 



Thickness of elliptic wedge at a, is to that at a„ as a, : a^ cos 3 or 1 : cos 3. 



Likewise thickness of wedge at a, is to that at a„, as 1 : cos 5. 



Through a, and a, draw double ordinates o p and r s. These double ordi- 

 nate are equal because they are equally distant from center C. 



Per Conic Sections, 



Chord o p : chord h t :: B, : B„ . 



Chord r s : chord i f :: B, : B,„ . 



Let a be the difference between a right angle and the angle C a, p or 

 C a^ s, also let b be the difference between a right angle and the angle 

 C a„ t and b, that of a right angle and angle C a,„ f . 



The differential width at a, corresponding to n 1 in Dia. 4 is to differential 

 width at a,, corresponding to s a Dia. 4 as A, (Jos a : A,, cos b. 



Also the differential width at a, is to that at a,„ as A, cos a : A,,, cos b, 



The product then of thickness at a, into length op into width at a, is to 

 that of thickness at a,, into length h t into width at a,, 



As ^ B, 



A, cos a 



cos 3 ) 

 B,, [• :: A,B, cos a : A„B„ cos 3^ cos b :: 1 : cos 2: 



A,, cos b ) 



Because A,B, cos a equals A„B„ cos b the last ratio above is true. 



*These double ordinates and line P C should intersect at d. 



