194 Mr Jolliffe, On Certain Trigonometrical Series which have 



Hence for all values of 6 



I Un+^ + ... + ?i^ i < (tt + 2/7r) M, 



and therefore the condition X,ia„ -*- is sufficient for uniform con-"" 

 vergence and a fortiori for continuity in any interval. 



il 



4. If 'Xn tends to infinity more rapidly than n, the series does 

 not seem to be capable of any modification. If A,^ = ^?2 + B, where 

 A and B are fixed, we obtain practically the series 2a„ sin nO and 

 nothing more. But when X^ tends to infinity more slowly than n, 

 and with a certain measure of regularity, the theorem can be 

 transformed in an interesting manner. We have, in fact, the 

 following theorem : 



If \n tends steadily to infinity and \n+i — ^n tends steadily to 

 zero, then the necessary and sufficient condition for the uniform 

 convergence of 



Zft^i (Xji+i ~ Xji) sm A.,jC7 

 is Xndn -* 0. 



As before, I prove rather more, viz. that the condition is suffi- 

 cient for uniform convergence and necessary for continuity. 



This theorem will follow at once from the theorem just proved, 

 if we can show that the series 



Sa« {(cos \nd - cos Xn+i0)/6 - (Xn+i - Xn) sin XnO] 



is uniformly convergent throughout any interval. Here the con- 

 dition Xn+ittn ■-* is equivalent to X^an -^ 0, since Xn+i — X^ -^ 0. 

 We can verify immediately that 



cos y — cos X — sin y sin (sc — y) 



= sin^ h{x — y) (cos y — cos x) + ^ sin {x — y) (sin x — sin y). 



It follows by Abel's lemma that, if Xn+i — X^ decreases steadily, 

 so that sm{Xn+i — Xn)d and sin ^(Xn+i — Xn) decrease steadily, 

 then 



S {cos Xn — cos X,i+i 6 — sin {Xn+^ — Xn) sin Xnd] 



-rt+1 



< 2 sin^ 1 {Xn+i ~Xn)6 + sin {Xn+i - X«) 0- 



Also, given any e, we can choose v so that Xn+i — X^ < e for n ^ v. 

 Hence, for n'^v, we have 



p 



2 {cos XnO — cos Xn+i6 — sin (Xn+i — Xn) 6 sin XnO] 



w+1 



< 2e2^2+66'<3e^, 

 for any interval of values of 0, if e is sufficiently small. 



