554 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES, 

 and thus, in order that the base BC may make half a turn, we must have 



■6 = it and 3 T = V | . tt . 



The value of 3T, found in article 37, is 



3T = 3-62759 87283 68435 70118 81568 



while 



tt = 3-14159 26535 89793 23846 26434 



and, on seeking the ratio of these by Brouncker's method of continued fractions, 

 we find the quotients 1 ; 6, 2 ; 6, 2 ; 6,2, the group 6, 2, being repeated ten 

 times. It was this recurrence of the quotients, which I observed in 1850, that 

 led me to seek for a rigid demonstration of this remarkable relation between the 

 intervals of the intersections of the ternary curves, and those of the intersection 

 of the quaternary ones. 



52. The observation that \/f is the sine of 120°, while — \ is its cosine, led at 

 once to the following unexpected generalisation. 



Having assumed a any constant angle, let us put 



( . cos a . s 



t = e . sin (t . sin a) , 



then, by taking the successive derivatives, we obtain 



<f>t = e * ■ cos a 



. sin (a + t sin a) 





. sin (2a + t sin a) 



<pt = e'- C0Sa 



sin (3a + t sin a) 



• , t . cos a 



sin (na + t sin a) 



and in general 



9 _ 



wherefore, if a be taken the nth part of the entire circumference, that is if a = — , 



the rath derivative of <pt comes to be <pt itself, and we have a series of recurring 

 functions of the rath order. It is to be observed, however, that these are not the 

 fundamental functions. 



If we make ra = 1, a = 2 x, cos a = 1, sin a = 0, <pt becomes e +t . sin (02), 

 which is quite useless. If ra = 2, a = tt, cos a = — 1, sin a = 0, and <f>t — e~ 

 . sin (Ot), which also is unavailable. But when we put ra = 3, we have a = %ir 

 = 120°, cos a = — i, sin a = a/|^ and 



• (jit = e-i' . sin (t Vf) 



u (/>t = e~i f . sin (120° + t^l) 



d> = e~f . sin (240° + t J I) 

 •if 



