MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 551 



02n = 2-0n 2 + 2-0n-0n 



!fj2n = 2-[»;n'{0n + 0n} 



02n = 2-0n 2 + Qn 2 + Qn 2 



02n = 2-0n-{Qn + 0n} ; 



which show that the values of 2 II and 2 II are alike ; that is to say, the two 

 curves and cross each other on the ordinate drawn through ?r, if the distance 

 Ot be made double of Oil . 



It is also seen that the value of the difference 



027T — 02tt is — \j~\tt 2 + 20tt . 0tt — 0tt 2 , 



that is 



— {07T — 07r} 2 or — 1 ; 



so that the distance intercepted on the ordinate at 211, between the curves and 

 is unit. 



Fig. n. 



By following a train of reasoning analogous to that which was used for ternary 

 functions, it may be shown that the intersection of the curves and are on 

 ordinates corresponding to the abscissae (2n — 1) II , while those of and are 

 on the alternate ordinates 2nJJ, n being any integer number taken either positive 



VOL. XXIV. PART III. 7 K 



