64 Proceedings of the Royal Irish Academy. 



2 1 



two tangents, since ri = r^^-h for tlie ovals, therefore rz----h for 



o o 



the conchoidal branches. Hence (see fig.) the tangential radii O^and 

 OFoi the ovals, when real, are bisected at ^and j^by the conchoidal 

 branches at their sides of the asymptotic axis of the curve. 



The sum of the three products in pairs ro/s + r^r-^ + TiT^ of the three 

 roots ri, r2, r^ of equation (1) being also independent of co, and equal 

 for all directions of r ; therefore, for every three radii of the curve 

 having a common direction, if ri and To, be those to either oval, and rs 

 that to the infinite branch at the same side of the asymptotic axis, 

 ^3 (^1 + ^2) = ^1^2- Hence, for every three radii having a common 

 direction, that to either infinite branch is in magnitude and direction 

 half the harmonic mean between those to the oval at the same side of 

 the asymptotic axis of the curve. 



Differentiating equation (1) with respect to w, we get 



^ = _2/^3.-^.-^— .i- , (2) 



rdia " ' eos'oi' 3r ~2h' r''^ 



dr . 

 which shows that --=- is = only when sin w = ; that is, only for the 



three apsidal points A, B, C of the curve. 



Hence the radius r of the curve has its maxima and minima values 



only at the three apsidal points on the axis for which co = 0, and the 



conchoidal branches have in consequence no dumb-bell depressions 



througliout their whole lengths. 



dr 

 Since, from the same equation (2), —7- = 00 when u --J- tt and when 



rdw 



3r -2h = ; therefore, as already observed, the axis for which co = ^ tt 

 is an asymptotic tangent to the infinite branches, and the two radii 

 OSJ and OF for which 3r - '2h are the two ordinary tangents to the 

 ovals from the origin. 



Putting, in equation (1), x = r cos ut and y - r Bm w, that is trans- 

 forming into rectangular co-ordinates, and solving for y, we get 



y^=h''-x-- ^hWx--- + ¥x-^, (3) 



which shows that, for every finite value of c^, y"^ has but a single finite 

 value, positive or negative ; that, from a; = 00 to a; = OA, y"^ is nega- 

 tive, and therefore y imaginary ; that, from x = OA to a: = OB, y^ is 

 positive and finite, and therefore y real and finite ; that, from x = 0£ 

 to X = OC, y'^ is negative, and therefore y imaginary ; and that, from 

 X = OC to X = y^ is positive and increasing from to 00, and there- 

 fore y real and increasing from to 00. These results obviously verify 

 the form of the curve as obtained above from the equation for ^ (1). 

 Putting y = in equation (3), we obtain for the squares of the three 



