MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 541 

 the inverse ratio being 



e-&= 1-83051 94665 55609 67148 01998 . 



40. If we take any absciss u less than T , and draw an ordinate to cut the 

 three lines ; and if beyond T at the same distance u we draw another ordinate, 

 the distances intercepted on these ordinates, between the curves, are reciprocally 

 proportional. For since AT = AT, the expressions for the functions of T + u 

 may be written 



^(T + u) = AT {A« + A^} + AT • Au , 



A(T + u) = AT { A« + Aw} + AT . A« , 

 A(T + «) = AT { A" + A"} + AT • Aw ; 



wherefore, 



A(T + u) - A(T + u) = (AT - AT) {Aw - Aw} 

 A(T + u) - A(T + u) = (AT - AT) {A« - A«} ; 



that is to say, the intervals intercepted on the ordinates between T and 2T are 

 proportional to those intercepted on the ordinates between and T, these latter 

 being reduced in the ratio of e~ iT to 1, that is, of -54629, etc., to 1. 



The same law extends for every interval between nT and [n + 1) T, and that 

 on both sides of the zero-point ; so that if the complete details of the intersections 

 on the ordinates between and T were once computed, those for every other 

 interval could thence be easily obtained. 



41. If, on the other side of the origin, we measure off Ou' equal to Ou, and 

 draw an ordinate at u', the distances intercepted thereon are again proportional 

 to those intercepted on the ordinate at u, only in a different order. For, accord- 

 ing to article 29, we have 



A(— u) = Aw 2 — A M • A w 

 A(— w) = Aw 2 — A« • A M 

 A(— w) = Aw 2 — Aw • A w 

 and, consequently, 



A(— w) - A(— «) = {Aw — A«} et ; 

 A(-w) - A(-w) = {Aw - A^} e l . 



Hence, if we measure the distance Tu backwards from T and equal to Ou, the 

 distances intercepted on the ordinate drawn through u" are again proportional to 

 those intercepted at u ; and thus we may expect that, for the ordinate which 

 bisects the distance OT, the intercepted distances should be alike. 



For the purpose of examining into this matter, let us assume some absciss ?;, 

 such that Av — Av = Au — Av, and let us compute the distances intercepted 

 on the ordinate at 2v. 



