MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 535 



names for these. The task of finding appropriate appellations is, however, more 

 difficult than that of finding a designation for a new asteroid or a new metal ; 

 without attempting it, I shall content myself with a simple piece of notation. 



Placing the trigon A as the general symbol for recurring functions of the third 

 order, we may indicate the separate cases by writing the index of the first term 

 within it ; in this way we have 



At= x + irks + ^e + etc - • 



A* = 4- + 



...4 



+ — ~ + etc. , 



t 2 



t 5 

 ....5 



H s + etc. , 



and we shall afterwards use a similar notation for functions of the fourth order, 

 these being represented by a tetragon having the appropriate numbers inscribed, 

 thus— 0^0^0^0« • 



27. Every recurring function of the third order may be represented by the 

 formula 



AA* + BA« + CA* 



in which A , B , C are constant multipliers, which may be positive, zero, or 

 negative. In general, if cpt represent such a compound function, x $t and n (pt 

 being its derivatives, we have 



(p(t + u) = <pt. A^ + A>t • A u + 2<pt • A u t 



and, in the case of the fundamental functions themselves, 



A(* + u) = At • A« + A* • Au + At • A** > (l) 

 &(t + u) = At-Au + At.Au + At-Au . (2) 

 A(* + u) = At- Au + At ■ Au + At ■ A u ; (3) 

 these three equations give by addition 



A(t + u) + A(t + u) + A(* + ») = (A' + A* + AO (Aw + Au + Au) , or 



they are analogous to the values of sus (t + u) and cat (t + u) given in article 9 ; 

 but they cannot, like those, be converted into functions of the difference t — u by 

 a change of sign. 



28. The sum of the cubes of three recurring functions exceeds three times the 

 continued product of those functions by a constant quantity. 



The first derivative of the sum of the cubes 



At 9 + At 3 + At* is 



3 A* 2 . A* + 3A* • A* 2 + 3A* • A* 2 



VOL. XXIV. PART III. 7 F 



