( 458 ) 



The points of the left-side branch are then given by the values 

 of s between -|- cc and 2, those of the right-side branch by the 

 others. 



For s = '2 we get the two infinite branches belonging to the 

 as3'mptote : 



27'— 3x' = 2 (53) 



19. Nor do we meet with any difliculties in the calculation of 

 the breadth-relations of the regions for very large values of y' men- 

 tioned in § 10. 



For the cubic curve we may put: 



3x'=r2y' + ^l/y' (54) 



through which its equation passes into: 



(_ p _|_ 8I-) i/y' + 10 — 4P = . . . . (55) 



from which appears that for very lai-ge xalues of y' we find 

 — 2 1/2, Ü and +21/2 for /•. We get therefore foV the leftside 

 branch of the cubic curve approximately: 



x' = jr'-j[/2.[/r (56) 



and for that on the right-side: 



x' = ^y'-h|-l/2.i/y' (57) 



while of course the middle branch with asymptote corresponds 

 with /: = 0. For this branch we have: 



"^i^-j <^«' 



In a similar way we find for the parabolic border curve: 



x' = Ay'±|-l/0.|/y' (59) 



Taking this into consideration we may e(iuate the breadth of the 



2 

 yellow regioii at iiifijiity to ~ {S—y/'S)[/'2 .y^^y' , that of the green 



one to — V^G.l/y', that of the blue one to ~—, that of the purple 

 9 3 



2 2 



one again to — |/6.|,/y' and that of the red one to --(3 — \/S)y2Vy' 

 9 "^ 



from which the relations of equation (9) easily follow^ while 

 V 3 — 1=^:0.732. 



