MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 537 



the double of the function by the first derivative. Continuing the derivations we 



find 



3 P* = 2Pt + 3 a Pf 3 qt = 2Qt + 3 X Q« 



t Pt = 6Vt + 5^ t Qt = 6Qt + 5 X Q« 



5 p« = lOPf + HjPi 5 q« = ioqt + lljQf 



etc. etc. 



The progression of the numerical coefficients in each of these series is obvious ; 

 one term augmented by the double of the preceding gives the succeeding term of 

 the series, so that the general formula is 



«^- Z 2 -(-1) + 2 -(-1) lift 



and similarly for the function Qt . Hence, by Taylor's theorem, 



P( i + *) = 2P«{ \ + 0-f + 1^- + 13-^-3 + 3-^1+efa, } 

 + ,Pl{lf + lj^ + 3^-3 + 5 J^ + 11-^g + etc. } 



These equations are true for the P and Q of any set of recurring functions of 

 the third order. For those with which we have at present to do, let us put 

 t = o, and in the resulting formulae change u into t, then since 



Qt + 2Pi = e 2i , Qf - Pi = e~ J , we have 

 Qt = l(e* + 2e-') = 1 + o| + 2^ + Sj-J-g + 6-^ + 10-^ + etc., 



R = J(* " e "0 = l| + HTS + "TO + 5 ^i + n ^5 + etc " 



31. When £ = o, the values of the three functions are Ao = 1 , Ao = 0, 

 Ao = , wherefore on attributing a small increment §t to this zero, the function 

 Ao becomes 1 + o8t + 08? + %8t*; it is in a state of conjoined maxi- 

 mum and minimum, and augments very slowly. The function Ao becomes 

 o + l.8t + o8f + o8t 3 , increasing at the same rate with the primary, and the 

 line representing it must cross the line of abscissae at an angle of 50 c (45°). 

 And the function Ao becomes o + o8t + jfo 2 + o8f, it is therefore in a state of 

 minimum, the radius of curvature of the line which represents it being unit. 

 These phases are shown in figure 3, in which OA is the linear unit. 



As t continues to increase, the value of A* approaches and becomes equal to 

 that of A*, that is, the corresponding curves intersect at some point B ; the value 

 of t at that instant is a root of the equation At — At = o ; we may denote this 

 root by T, in other words we may suppose AT to be equal to AT . 



32. If in the equations of article 27 we put t and u each equal to T, we obtain 



