488 Mr. S. M. Drach's Supplementary Considerations 

 Ex. p—Si q—% ^=1, gives 



85 cos^ rj/ 4- 4« cos2\[/ + (5 — 3/)2 cos vp — 2a = 2t, 



Wn6IlC6 



^bpx = {Sb+f)af{h'^-bf-r^)-a%{b-f) 



± [a'^b +fr^ - bj- bp) \^^bfr'' + {a^ - ^bf) [b -ff 



for the equation : \^ b^f, 



2/2^ + 2ar2 = + (a V - 2/ V + r^) ; 



ifa = 2/2 = 2i^ 



r^- {a^x + 2aa72)2 = (^2 _,_^2)3, 



But in these quadratics we have a circumstance analogous 

 to the discussion of the equations of the second order. For 

 that cos xp be real, the v^~ must cover a positive quantity or 

 zero; which, when b andyhave different signs, shows that 

 ^bf[r'^-{b-ff] must be ^{ab-afY; whena2=4/^/, 



« cos \p = — 6 — /■+ r. 



Similarly, j9 = 3g— 2/, p-=^q — M would lead to a cubic or 

 biquadratic equation. Even higher powers are thus resolvable, 

 if the intermediate powers disappear through their coefficients 

 vanishing, as 



cos^^ij, -f 2Ai cos^vl/ = A2 

 gives 



cosrI/=-^(-Ai± v^Ai^+Aa). 



The above tricircloidal expansion of 2r^ cos a9 exists what- 

 ever \ be assumed to be, and for cos we may write sin on each 

 side of the equation. 



The straight-lined, curved-cornered bicircloids, cos nQ = 

 funct. (r), are true regular bicircloidal polygons qfn sides, cor- 

 responding to the angular ones of the simple circle (Euc. 

 book iv.). 



For the central bicircloids, 



P — Q « 

 r=2acos^^ — —L 

 p + q 



f^H ,.-. I ,-.-,.-. , ^.J dU\r-^)-dHd{r-^) \df 



becomes 



S{p—q)hi^r~^ + ^pqr~^i 



.'. when q — (excentric circle), as r"^ (Princ. I. vii. 1). 

 The length of these central bicircloids 



= 2afdWy—'^pq{p + q)-^sm^{p-\-q)[p — q)-% 



an elliptic integral, except when §'=0, =2a^ (see fig. Euc. i. 1). 



