544 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



ordinate at T the trigon has made one-sixth part of a turn, the side BA being 

 now horizontal. In this way the rotation continues at the rate of one turn for 

 the distance 6T. For each interval of JT the uniformity of the angular motion 

 is obvious ; it remains to be shown that the same uniformity holds good at inter- 

 mediate positions ; that is, that the angle of inclination is proportional to the dis- 

 tance Ou. This part of the demonstration, however, may be conveniently reserved 

 until we shall have reached the genesis of angular functions. 



43. I shall conclude this slight sketch of the doctrine of ternary recurring 

 derivatives by an investigation into the sum of their cubes. 



Temporarily, for the sake of abbreviation, let us put Ct as the symbol for the 

 sum of the cubes of the ternary functions'; that is, let 



C* = A? + At 3 + At 3 , then 



fit = 3{ At 2 . At + At 2 • At + At 2 • At } , 

 fit = 9{At . At 2 + A* • A* 2 + At • At 2 } , 

 fit = 9{At* + At 3 + At 3 + 6A* . A' • A* } ; 

 now it has been shown (article 28) that 



At 3 + At 3 + At 3 - 3 At . At • At — l 



wherefore fit = 27Ct — 18, and consequently the subsequent derivatives follow 



thus : — 



fit = 27 fit 



fit = 27 fit 



fit = 27 2 Gt - 27 . 18 



fit = 27 2 fit 



fit = 27 2 fit 



fit - 27 8 Gt - 27 2 . 18 , and so on. 



From these we obtain, by Taylor's Theorem, 



G(t + it) = II 1 - A3w } + C* . A3m + fit . | A3« + fit . 9 A3m , 



but when t = o, Ct = 1, fit = 0, fit = wherefore 



2 1 

 Gu — g + g A 3u 



and thus we have the following values of the sums of the triple products, 



A* 8 + A* 3 + A* 3 = l + l A3< , 



A* 2 • A* + A* 2 • A* + A* 2 • A* = I Aot , 



A* • A* 2 + A* • At 2 + At -At 2 = | A3* , 



At. At . A* = - 9 + 9 A3* • 



