534 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



having the continuation of OA, and the parallel AH for asymptotes ; the other of 

 like dimensions approaching to the lines OA' and A'H'. 



25. I may conclude this short notice of the properties of recurring derivatives 

 of the second order by considering the case of proportionality. Let it be proposed 

 to investigate the nature of the function cf>t when it is proportional to its second 

 derivative, that is when 2t <pt = c<pt, c being a constant multiplier. 



By taking the successive derivatives of this we obtain — 



3 <£f = c . x (pt ; tft = c 2 .<f>t ; 



5 cpt = c°- . 2 <pt ; 6 (pt = c 3 .<pt ; etc., 



whence, according to Taylor's theorem, 



j- / . \ j. , f 1 cm 2 c 2 u 4 , 1 



<P (t + u) = 0*|l + _ + x 2 3 4 + etc. j 



-* f ^ * 3 I 



10 ( — + TT2T3 +etc - j 



If now we suppose that the value of <po is A, while that of c\<po is B, A and B 

 being two constants, the above expression becomes 



(pu = A . sus (ujc) + B . cat {ujc) , 



which formula includes all functions which are proportional to their second 

 derivatives. 



It is here to be remarked that c cannot have a negative value ; the equation, 



2 0* = - c.(pt , 

 belongs properly to recurring functions of the fourth order, under which head it 

 will afterwards be considered. 



Third Order. 



The functions of the second order of recurrence are only compounds of the 

 exponential function, and might almost have been passed over, if my sole object 

 had been to exhibit what is novel. The functions of the third order, however, 

 cannot be produced by compounding those of the previous orders, and we have 

 now to touch upon ground entirely new. 



26. According to what has already been explained, the fundamental recurring 

 functions of the third order are — 



+ etc. 

 + etc. 

 + etc. 

 For the sake of conciseness in language, it would be advantageous to have distinct 



1 + 



1 



t* 

 .2.3 ' 



* 6 

 ...6 + 



t 9 

 ....9 



t 

 X + 





...4 + 



...7 + 



* 10 

 ...10 



t 2 



1.2 + 





t b 



7778 + 



t 11 



...5 + 



11 



