MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 555 



And if we make n = 4, we have a = — , cos a = 0, sin a = 1, whence 



<pt = e ot sin £ or </)£ = sin £ 



These results are remarkable in this respect, that for binary functions they 

 give the multiplier e~\ which agrees with the difference between sus t and cat t ; 

 for ternary functions e~ ht , which corresponds with the diminution of the side 

 BC of the equilateral trigon; and for quaternary functions e° , in accordance with 

 the fact that the intervals intercepted between the two curves and are the 

 same for t as for rm ± t, whatever may be the value of the integer number n. 



The functions obtained from the formula <pt = e l C0Sa . sin (t . sin a) are not 

 fundamental, but compound functions, and do not indicate a general solution of 

 the equation nt x = x . Thus the absolutely general solution of x — 3t x is 



in which A, B, C, may be any coefficients, positive or negative ; but this generality 

 could not be obtained from the above three functions, <pt, ^>t, 2 <pt . 



In order to render this matter, which is of importance in physical investi- 

 gations, quite clear, we may express the above ternary functions in terms of the 

 fundamental ones, and, contrariwise, seek to deduce the fundamental functions 

 from them. For this purpose we shall suppose that the first of them is the x of 

 the preceding equation ; that is, 



« ' . sin (t \/y = A A* + B At + At , 



sin (l20 J + l-\f^\ =BA« + CAUA^ , 

 sin (240° + t \/?\ =GA' + AA< + BA< . 

 Hiving in these formula to t the value zero, we obtain 



A = O , B = sin 120° = \/| ; C = sin 240° = - \/| ; 



whence the three values 



<pt = <T* . sin (/ v^) = *J\ { At - At) 



rft = <T ,( . sin (l20° + t\/'f) = \/| { A* - At) 



3 <pt = <T' . sin (240° + f x/ 3 ^ = V^ { A' - At } 

 which give <j)t, ^t, <pt, in terms of the fundamental ternary functions. But if 



VOL. XXIV. PART III. 7 L 



it 



it 



. sin 



