218 Mr Dixon, Geometrical proof of Gonvergency. [May 28, 1894. 



the circumference of the last circle in the ratio ttg : aj. This circle 

 is inside the last. 



Thus, if we go back to the old origin, we find that the point 

 a-iU^ + a^^Un, foi' every value of n, lies within a circle of radius a^R, 

 which is contained in the former circle of radius aiR\ the centre 

 of similitude of the two circles is the point a-^a-i^. 



Let us now move the origin to the point OiWi + a^ih and 

 diminish the lines from this point in the ratio a-^^-.a^. It will 

 follow in the same way that for all values of n the point 



o^u-^ + a2^'2 + a3%3Un 



lies within a circle of radius cisR, contained in the former one of 

 radius ttai?, and the centre of similitude of the two is the point 



aitii + a^ih- 



Carrying on this process we find that the point 



lies within a circle of radius ayR which is contained in every former 

 circle. 



Now by hypothesis a,, decreases without limit, so that this circle 

 can be made as small as we please by making r great enough. 



It follows that the series ^a^Un converges to a definite limit. 



It is clear that the argument will hold just as well if any or all 

 of the points S-m^ lie on the circumference of the circle of radius R. 



In the important special case when 



Un = COS (nd + 0) + t sin (n6 + (f)), 



the successive points Xttn are arranged at equal intervals round a 



1 ^ 



circle of radius ^ cosec ^ , which passes through the origin. 



1 B 



We may take ^ cosec ^ as the value of i^ ; it will be finite un- 



less ^ is a multiple of 27r and therefore, except in that case, the 

 series Xa,i cos {iiiQ + ^) and 2a,i sin {nQ + <^) are convergent if a^ is 

 positive and diminishes without limit as n increases. 



