536 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES, 

 which is also the first derivative of 



3 At • At - At 

 wherefore the difference 



A* 3 + A* 3 + A* 3 - 3A* -At-At 



must be constant. This proof applies to all recurring functions of the third 

 order ; in the case of the fundamental functions we have for the value t = o , 



Ao 3 + Ao 3 + A<> 3 - 3A<> • A<> • A<> = l , 



and, therefore, the sum of the cubes exceeds three times the continued product 

 of those functions by unit. 



29. If we substitute — t for u in the equations of article 27, we have 



Ao = l = At - A(- + A* • A(- + A* • A(- 

 Ao = = At • A(- + At - A(- t) + At- A(- 

 Ao = = At ■ A(- + A* • A(- t) + At- A(- 



from which we can obtain the functions of — t in terms of those of + t . 

 On eliminating A(— t) , A(— t) from these equations, there results this 



At 2 - At • At = { A* 3 + A' 3 + A* 3 - 3A* ■ At . At] • A(- 

 wherefore, according to article 28, 



A(- t) = At 2 — At-At ; and similarly, (4) 



A(- = At 2 - At- At , (5) 



A(- = A* 2 -At-At ■ (6) 



The sum of these three functions must be e~\ now on dividing unit under the 



form 



A* 3 + A* 3 + A* 3 - 3A* . A* • A* = 1 



by e* under the form At + At + At we obtain the very sum in question, viz., 



A* 2 + A* 2 + A* 2 - At - At - At • At - At- At - e~* . (7) 



30. If, for the sake of conciseness, we put Qt (quadratics) for the sum of the 

 squares, and Tt for the sum of the products of these functions, that is, if 



qt = At 2 + At 2 + At 2 

 Ft = At- At + At- At + At- At 

 we obtain, on taking the successive derivatives, 



X P* = Ft + qt jQ« = 2P* 



2 Pi = 2Pi + jVt a Q« = 2Q« + x qt 



and hence each of these new functions is such that its second derivative exceeds 



