526 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



put x for e l our original equation (pt = lt <pt takes the form x = u x, and, observing 

 that lx t is the reciprocal of lt x, we have 



lx t = x 



The continuation of the derivation gives us ix t= — l.x~ 2 ; 3x t = +1-2. x~ z ; 

 ix t — — 1 .2.3.#~ 4 ; etc. ; putting log x for t, and applying Taylor's theorem, the 

 well-known development of a logarithm results, viz. — 



log (w + y) = log x + | - ^ |- 8 + 3 | 3 ~ etc - 



7. Having said enough concerning the recurring function of the first order to 

 show its place in the Theory of Primaries, I proceed to consider those functions 

 which reappear as their own second derivatives. 



According to what has already been explained, the fundamental functions of 

 this order are — 



1 + 172 + 1774 + 1776 + etc - 



t , ^3 tb il 



1 + 1.2.3 + 1....5 + 1....7 + etC - , 



These functions possess very remarkable properties ; and in order to exhibit 

 these clearly, it is convenient to give distinguishing names to the two functions. 



If the values of t be represented by the abscissae, and the corresponding values of 

 the first of the above functions by the ordinates of a series of points, those points 

 indicate a curve which we can show to be the catenaiy ; the length of the curve 

 reckoned from the point corresponding to t = being the second of the two 

 functions. On this account, and in allusion to the construction of a chain bridge, 

 I shall designate the former function the suspensor, the latter function the 

 catena ; that is to say, I shall put 



t 2 t* f' 

 sus t — 1 + y~2 + I — 4 + 1^ 6 + etc ' 



' t i 3 t 5 t~' 



cat ' = 1 + 1X3 + T775 + EZ77 + etc ' 



each being the derivative of the other. 



, 8. The differential coefficient of the square of one of these functions is just 

 equal to that of the square of the other function, 



or u(sus f) 2 = 2 sus t . cat t — i ( (cat t) 2 



and, consequently, the difference between these squares must be constant ; now 

 when t — that difference is unit, wherefore for all values of t 



sus t 2 — cat t 2 = 1 ; 



wherefore if sus t be the absciss and cat t the ordinate of a point, that point lies 

 in an equilateral hyperbola, so that these functions may be called hyperbolic 

 functions. 



