776 Dr. J. R. Wilton on the Solution of 



Thus 



Y =-2irp{a*-b 1 2 -2b 2 2 ... -nbj) { f + \ ^" 2 *cos 277 



3a n+l a v I 



and Y t - may be written down by means of the general 

 formula. 



The form taken by V at infinity assumes that the curve 

 does not cross itself. This, of course, restricts the variability 

 of the coefficients 6 l5 b 2 , ... b n . 



The value of V may be similarly written down in the 

 rather more general case of the transformation 



+ 6 ] «-^+'9> + 6,«- 2 C €+l ')+ ... + b m e~ m ^ +ir} \ 



But in this case we have, at infinity, 



log r -^ n% , 



and therefore the coefficient of f in V is — 2nrpn x the area 

 of cross- section, so that a term in f alone occurs in V,-. For 

 example, for the evolute of the ellipse, we have 



os + ty=za cos 3 (77 — ig) + ib sin 3 (rj — 1%) 



= c{3 cosh (X + f+My) + cosh (X— 3f — 3irj)} 9 



on putting 



a = 4c cosh X, 5=4csinhX. 



At the singular points 



Hence we obtain, by integration, 



Y = -^pabtf-ie 2 ^-® cos 2 v + ie-^cos4, V + -}e 2 x ~ 3? ) cos 67?}, 



and therefore 



V»= -7r i o(^ 2 + ?/ 2 ) + f 7rpab% + Y — -feirpab sinh 4f cos 4?7 



+ ^tt/oc 2 (15 cosh 2£ cos 2n + cosh 6£ cos 617) 



+ 37rpc 2 cosh 2X cosh 4f cos 4?;. 



It is not possible to pass to the case of the four-cusped 

 hypocloid by making X -> 00 , 2ce K ->a, for the transformations 

 take different forms at i n finity. 



