MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 527 



9. By following the course of reasoning given in section 3, we obtain 



sus (t + u) = sus t . sus u + cat t . cat u (1) 



sus (t — u) = sus t . sus u — cat t . cat u (2) 



cat (t + u) = cat t . sus it + sus £ . cat u (3) 



cat (£ — «) = cat t . sus m — sus t . cat u (4) 



which are the counterparts of the four elementary formulse of trigonometry. 



10. By manipulating the four equations of the preceding section, we can 

 obtain theorems exactly analogous to those of the angular calculus ; thus, by 

 additions and subtractions, 



sus (t + u) + sus (t — u) = 2 . sus t . sus u (5) 



sus (t + u) — sus (t — u) = 2 . cat t . cat u (6) 



cat (t + u) + cat (t — u) = 2 . cat £ . sus u (7) 



cat (t + u) — cat (i — m) = 2 . sus £ . cat u (8) 



Hence putting t = nu 



sus (n + 1)m + sus (n — l)w = 2 sus ww . sus u (9) 



cat (n + T)u + cat (n — V)a = 2 cat . «w . sus u (10) 



sus (n — 1)m — 2 sus mm + sus (n + l)w = sus nu . 2(sus u — 1) 

 cat (» — l)w — 2 cat nu + cat (» + \)u = cat n« . 2(sus u — 1) 



and thus a table of the values of catenarian functions may be constructed by 

 help of second differences, just as in the case of the trigonometrical canon; the 

 expression 2(sus u — 1) taking the place of 2(1 — cos u). The same analogy 

 may be extended to differences of the fourth and sixth orders, and so on. 



11. By putting successively t = u, t — 2u, t = 2u, we can form the expressions 

 for the catenarian functions of multiple arguments; thus 



sus 2m = sus u 2 + cat u 2 



= 2 . sus u 2 — 1 = 2 . cat w 2 + 1 

 cat u 

 3 sus u 

 3 cat u 

 8 sus u 2 + 1 

 cat 4m = 8 cat m* + 8 cat u 2 + 1 

 sus 5m = 16 sus m 5 — 20 sus m 3 + 5 sus u 

 cat 5m = 16 cat u 5 + 20 cat m 3 + 5 cat u 

 and so on. 



12. Since the sum of the functions forming a period is a recurring function of 

 the first order, we have 



cat 2m = 



2 . sus u . i 



sus 3m = 



4 SUS M 3 — 



cat 3m = 



4 cat m 3 + 



sus 4m = 



8 SUS M 4 — 





sus t + cat t = e' 



but 



sus t 2 — cat t 2 = 1 



wherefore 



sus t — cat t = e~ l 



and sus t = ~ ( e l + e~ l ) ; cat t = „ ( c' - e~ l ) » as 

 is indeed evident from the ordinary operation for separating the terms containing 



VOL. XXIV. PART III. 7 D 



