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



We take, by way of illustration, the hypotrochoid of n 

 oscillations, 



x — a cos 7) + b cos (n — l)r h | 

 y = a sin rj — b sin (n— V)7].) 



This includes the ellipse when n=2, and when b is not too 

 large — the greatest possible value for b is \\{n — V) — it 

 represents a circle disturbed by an harmonic inequality with 

 n maxima. We have, in fact, 



r 2 = a 2 + b 2 + 2ab cos nrj, 

 and if b is so small that squares of b/a may be neglected, 

 r = a + b cos nrj, 



and 77 differs from 6 by a multiple of bja 9 where r and 6 are 

 polar coordinates. 

 In general, we have 



i(X! 2 + X 2 2 + Y t 2 -I- Y 2 2 ) = a 2 + b 2 + 2ab cosh nf cos tmj, 

 ^ i VxY'-X'Y)^ = {a 2 -^-l)6 2 }f-(l-2/72)a6sinhnf coa nrj. 



And therefore, provided that the boundary does not cross 

 itself, i. e. provided that b >> l/(n — 1), 



•xjri = ^(x 2 +y 2 —a 2 — b 2 — 2ab cosh ?z£ cos 2177) 



+ coab e~ n % cos 7197 + (1 — 2/n)%ab sinh w£ cos nrj 



^lr = coabe~ n ^ cosnrj + {a 2 — {n—l)b 2 }^. 



The singular points of the transformation are given by 



n%= log{(w — l)b/a}, nr) = 2Je7r, 



where h is an integer or zero. Thus a is determined by the 

 condition that c^z/d£=0 at these points. W T e have 



?-l_i (n-1) 2 b 2 

 f n n a 2 ' 



and 



^i = i£0c 2 + y 2 — a 2 -b 2 )-'C(ab/n){e n t + (n-l) 2 (b 2 la 2 ) e- n £} cosnrj, 

 yjr = Z{a 2 - {n-l)b 2 }{% + [(n-l)b/na] e'^cosurj}. 



