386 
Proceedings of Royal Society of Edinburgh. [sess. 
x = 7 r mN /H + ( 7 T. tM R + ^t?h— i.P) . 
PmsI H + (p m R + Pm-lD) 
and this after rationalisation of the denominator will be found to 
be simplifiable by the removal of the common factor D from 
numerator and denominator, \/H being thus left in the numerator 
with the coefficient 1. This simplification is the main matter 
deserving of note, but the full result is — 
If the irrational expression which is the equivalent of 
1 a 2 + • • • + a n + • • • 
be denoted by ( JH + R)tD, then the like expression for 
I _1_ J_ 1 
1 a 2 + • ■ * 4- a m + aj + a 2 + • • • + a n + • • • 
* * 
is (VH + R')-D', where 
R (in— 1) Pn + In-1, In $ 27T m p m ) '^‘m—lPm "h ^"mPm— 1 > 
and 
V' = ( $ 2 pm, 2p m p m _i, 2 p^_i), 
7r m /p m and p n /q n being the m <;i and n th conver gents to a x + — — 
a 2 + • • • + a m 
and a n +i — respectively . (VIII) 
1 a 2 + --- + a n 
D', it will be observed, is got from R' by writing p m , p m-1 for 
_ _ * 
7r m , ^m— 1 * 
* It is extremely probable that among the papers of the late Mr C. E. 
Bickmore, Fellow of New College, Oxford, who was an earnest and capable 
student of this branch of algebra, there are to be found quite a number 
of results as yet unpublished and worthy of being made known. In 
a letter dated 29th April 1895, in which some noteworthy theorems are given 
from a paper prepared for the Oxford Mathematical Society, he says: “Our 
Society is too poor to print anything.” It would appear that French 
mathematicians are not similarly hampered (v. Lond. Math. Soc. Proc., 
xxxiv. p. 129). 
( Issued separately February 28, 1903.) 
