MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 529 



x must be the corresponding function of t. Now, whenever 27b 2 is greater than 

 4<z 3 , the first two forms fail us ; in such event, we must have recourse to the 

 third equation, and when a is positive, we must use the fourth. Thus, by help of 

 tables of angular and catenarian functions, we can resolve all equations of the 

 third order. 



15. As another instance of this close affinity, I may cite, using the notation of 



Leibnitz, the integral / - 7 - . 



° J a + b cos f 



When a is greater than 5, we make the assumption 



b + a cos p 



a + b cos p 



= cos -v|/, which gives 



V(a 2 -W) d<p ,, ... 



— * : — ~ = rf-vL, so that 



a + b cos p 



P d<p 1 _1 b + a cos p 



a + b cos <p s / (a 2 — b 2 ) a + b cos p 



but when a is less than b this method of reduction fails us ; in that case we may 

 assume 





b + a cos p 

 a + b cos p 



SUS -v}/ 



when 



V(ft 2 - 



a + b 



a 2 ) dp 

 cos p 



d^ 



and therefore 



p d<p 

 J a + b cos p 



= 



1 



— 1 

 SUS 



b + a cos p 



V (b 2 - a 2 ) 



« + ft cos p 



Conversely for the analogous integrals, we find 





r d<p 



J a + b sus p 



= 



1 



— 1 

 sus 



b + a sus p 



V {a 2 - b 2 ) 



a + b sus p 



r d 9 



J a x b sus p 



= 



1 



— 1 



COS 



6 + a sus p 



V (& 2 - a 2 ) 



a + b sus p 



to which may be added 











r d? 



J a + b cat p 



= 



-1 



J (a 2 + b 2 



— 1 

 cat 



b — a cat p 

 a + 6 cat p 



16. From these examples, it is apparent that the properties of recurring func- 

 tions of the second order offer a fair field for the exertions of the analyst, inas- 

 much as they promise to give unity to investigations which have hitherto ex- 

 hibited abrupt changes. 



The inquiry into the form of an equilibrated bridge, affords a good instance of 

 their utility in physical researches. Since the whole space between the roadway 

 and the arch stones is, in this case, filled up, the weight must be proportional to 

 the surface of the projection on the side of the bridge. Now, in all arches, this 

 weight is proportional to the tangent of the inclination of the arch line ; this tan- 

 gent is the derivative of the vertical ordinate regarded as a function of the hori- 



