46 



G. H. KNIBBS. 



the equation 



a p + (l-a) p = 



p + 1 



(236). 



Restricting the consideration to real positive values between 

 and i it will be found that for p = l/oo , 2 a = 0-1613782 ; for/? = oo , 

 a = 0; while for p = \,a may have any value from to \ ; but 

 for all other positive values oip, the ordinates a corresponding to 

 the abscissae p are terminated by a continuous curve, the values 

 for p = and p = \ being respectively about 0-1613782 and 

 1997088, and for p = about 2-471 reaching a maximum of about 

 0-2123179. 



II.- 



-Positions o 



f two symn 



letrically situated sections? 



Index. 



a 



Index. 



a 



Index. 



a 



•00 



•1613782 4 



2-45 



•2123160 



6 



•1880587 



•10 



•1674245 



2-46 



•2123172 



7 



•1796675 



•25 



•1752683 



2-47 



•2123178 



8 



•1713937 



•50 



•1857300 



2-471 



•2123179 



9 



•1637491 



•90 



•1974990 



2-48 



•2123176 



10 



•1567357 



100 



•1997088 5 



2-49 



•2123164 



11 



•1503130 



1-00 also 



i to 0-5 



2-50 



•2123147 



12 



•1444266 



1-10 



•2016800 



2-60 



•2122537 



13 



•1390213 



1-50 



•2075308 



3-00 



•2113249 6 



14 



•1340444 



2-00 



•2113249 6 



4 



•2056192 7 



15 



•1294440 



2-40 



•2122970 



5 



•1974029 8 



00 



•0000000 







Note. — 



b=l-a. 







log - 1 x denoting the number of which x is the logarithm. Since a is 

 numerically always less than 0*3, the powers of a are rapidly convergent. 

 For p=9*5, a is about 0'16 ; hence to 7 places of figures, and for p equal 

 10 or more, a? = : therefore b? - 2/ (p+1), from which the above formula 

 is derived. Again, for exceedingly small values of p, 6 p will be much 

 nearer unity than «p ; hence we may commence the approximation by 

 assuming that a p = 2/(j>+1) — 1, and put afterwards the deduced value 

 of 6 p in the place of unity, Other cases may be calculated by (22) or (23), 

 or by such methods as will readily suggest themselves to computers. 



2 Strictly for p = 0, a is indeterminate ; but as p becomes very small a 

 approaches the limiting value given. 



3 The seventh place of figures is generally uncertain. 



4 a = 1 and a may have any value whatsoever : in itself it is therefore 

 indeterminate. In the case considered however it really has a definite 

 limit which may be found as follows : — 



cP + (l-a)P = aP + l-pa[l+— (1-jp) +— (l-jt>+^)+etc] =2(l-p+p 2 -) 

 2 3 3 



