TRANSACTIONS OP SECTION A. 529 



Retaining, however, an arbitrary value for A, and dividing/(A) by \-x, we infer 

 that if 



& = \* Xi + *(** - ]r/,) + x\ - 1 -ff i x i --g a , 



then 



Tit + « vj, + e- y?7= o, -/{ 4 ~+ e- v'i7- € v'fe^'o, 



for all values of A. With the usual symbolical notation the general value of £ may 

 be written { II f\f x \Qw)- 



The application of these remarks to the trisection of the elliptic functions arises 

 from the fact that, with the usual notation, the four quantities 



/2a>\ /2a>' \ /2a + 2a> 1 \ /4a+2a>'\ 



<T> *(lT> p { 3 > H 3 -) 



satisfy the quartic equation /(#) = 0. 



Also it is easily found that the four quantities 



1 1 1 1 



/■2w\' ,a/2»V ,o/ 2a. + 2a)' \' 2 /4« + 2«V 



satisfy the equation 



, 18 „ 216 27 „ 



H* — -/i" + -— ^ -- ^ = 0, 



A A- A- 



where A = 27«; 2 3 -^ ; to this similar remarks apply, so that, for instance, we have 

 ./2o)\ ,/2a.' 1 \ * ,/2ai + 2a) 1 \ 



+ — £.---£rr=Q. 



where <r= ± 1. 



1 /4a) + 2a)'\ ./^toX ,/2w'\ 



4. On the Convergence of certain Series used in Electron Theory. 

 By Professor A. W. Conway. 



If r(t) denotes the distance of the point x, y, z from the moving point (which 

 for simplicity we may regard as moving in a right line) whose co-ordinates are 

 0, 0, f(t), then it is known that the equation C(£-T) = r(T), where C is the speed of 

 radiation, possesses a real root T between and t, provided that the speed of the 

 moving point is less than C and that Ct > r(0). The series 



C -2 S 2 C -3 g 3 



r~' + r— r- + &c. 



2 ! M* 3 ! St 3 



if convergent, represent the potential of a moving point. Its convergence (and 

 that of other series of this type) may be established by using a complex integral, 



around a circle in the w-plane about the point t as centre, and 



{C(«-«)-r(«)M«0 



of such a radius that C|(# — «)l > | r ( w )l on its boundary. In the case examined the 

 conditions for convergence are the same as those for a real root of C(t -T) = »-(T), 

 where t > T > 0. 



5. Two Notes on Theory of Numbers. 

 By Lieut. -Colonel Allan Cunningham, R.E. 

 Factorisation of N=(2 77 +l). 



N =N t . N 7 . N11 • N 77 ; where 



Ni = (2' + l)=3; N 7 =(2'+l) + N,=43; N n = (2 n +l)-HNi = 683 ; 



N 71 = (2 77 +l)(2- r -l)-r(2 7 + l)(2 11 + l) = 617 . 78233. 35532304099 ; 



[The last 11 -figure factor is prime]. 



