214 -Mr Greenhill, On the Rotation of a [March 20, 



A still further reduction is required in order to determine 

 when these values of rf and £ 2 or &> 2 2 and w 2 are positive, and to 

 determine when a is positive. 



From x = to x = l, a 2 > b 2 > c 2 , in the region shaded hori- 

 zontally (region I.) ; from x = 1 to x = 4, b 2 > a 2 > c 2 , the region 

 shaded vertically (region II. A) ; from x=l to x = 9, b 2 > a 2 > c 2 , 

 the region shaded vertically (region II. B) ; from x = 9 to x = oo , 

 6 2 > a 2 > c 2 , the region shaded vertically (region II. B), or b 2 > c 2 > a 2 , 

 in the region shaded with slanting lines (region III.). 



Riemann's condition that b + c < 2a, or sjx + \/y<2, holds 

 therefore within regions I. and II. A, and that b — c>2a, or 

 \Jx—sJy> 2, within regions II. B and III. 



In the reduction of A, B and C to elliptic integrals, lines of 

 constant modulus radiate from the vertex of the parabola in the 

 figure, AO being the line of modular angle 90°, AB of 0", AL of 

 60°, A C of 90°, AS of 0°. 



In order to reduce A, B, and C to elliptic integrals, 

 (1) in region I. we must put 



and 



suppose, where 



Then ^^^fsin 2 ^ 





(1 - a-) V(l - </) 



(F<f>-E<j», 



where sin 2 </> = 1 — '-5 = 1 — y 



a 



and therefore cos <£ = ijy, A</> = .\/x. 



