MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 547 



various ways, we may obtain a vast number of formulae, many of which are 

 singularly interesting. 



In order to fix the mutual relations of these functions easily in the mind, we 

 may construct a regular tetragon, and mark, as in the margin 

 (fig. 5), the numbers 0, 1, 2, 3, at the corners, to stand for the 

 functions 0, 0, 0, respectively. Each one is then the deri- 

 vative of the succeeding, or the primitive of the preceding, taken 

 in the order in which they are written. The six couples which 

 can be made among these four functions may then be represented 

 by the six lines joining the corners of the figure, of which four are lateral, and 

 two diagonal. 



Let now a piece of paper be cut of the size of the square, and let there be 

 written the same figures, 0, 1, 2, 3, at its four corners. If we place this paper upon 

 figure 5, so as that the may agree with the 0, we obtain 

 the appearance shown in figure 6 ; and if we suppose the 

 outer numbers to indicate functions of t, while the inner rig. 6. 

 numbers indicate the corresponding functions of u, this 

 figure 6 will at once picture the value of (t + u) in equa- 

 tion 1. 



If we turn the inner paper until its coincide with the 

 1 of figure 5, we obtain figure 7, which, in the same way, Fig 7. 

 represents the value of (t + u) ; and so similarly of 

 figures 8 and 9, which picture the values of (t + u) and 

 (t + u). 



This same artifice may be applied to recurrences of higher 

 orders, and it might have been used also for the ternary Fig . 8 , 

 functions. 



46. If we make t = u, equations (5) and (7) of the pre- 

 ceding article give, 



00 = 1 = 0*2 - 20* . 0* + \T\f~ (9) 



-0o = = Q]* 2 - 2 0* . 0* + g*2 (io) ^g. 9. 



These are two cases of a general proposition which may 

 be exhibited thus : — 



Let t, 1, 1, 1 be the symbols of any four recurring 

 functions, then I say that the sum of the squares of one- 

 diagonal pair differs from twice the product of the other Fi e- 10 - 

 diagonal pair by a constant quantity. 



For the derivative of the sum 0£ 2 + 0f is 20* . \j>}t + 

 20*. 0*; while that of 20*. 0* is the very same, and, consequently, for 



VOL. XXIV. PART III. 7 : 



