52 



G. H. KNIBBS. 



12. Three symmetrical sections: viz. a middle and the terminal 

 sections. — Turning to the case of three sections, obviously the 

 simplest possible condition in regard to their position is, — a = 0, 

 b = J, c = 1 . If further we make their weights symmetrical, a and 

 y may each be unity, and then we can determine /3: this would 

 be the simplest possible solution in regard to the weight-coefficients. 

 Equation (5) thus becomes, keeping the fraction b general, 

 1 \ . p~\ 



j8(jp- 



+ 



1 



0. 



.(36) 



from which if b = J, 



P 



a-p) 



9l> 



9P, 



(P-I) 



.(37) 



(l+^)-2 p 2 p -(p+l) 

 from which values of j3 may be readily computed. The following 

 table, giving the values for a considerable range ; is the basis from 

 which the curve /3 in Fig 2 is plotted. The general coefficient 

 will be 1/(2 + ft), see (38) hereinafter. 



Y. — Weight-coefficients for the middle-section, the weights of 

 initial and terminal sections being unity. 



Index 



P 

 00 

 0-1 

 0-2 



0-3 

 0-4 

 0-5 



06 



0-7 



0-8 



0-9 



0-99 



1-00 



P 



00 



31-884* 

 17-913* 

 12-516* 

 98358 

 8-2426 1 

 7-1934 

 6-4554 

 5-9122 

 5-4993 

 5-2063 

 5-1741 2 



(3+/3) 





 02951 

 05022 

 06889 

 •08449 

 •09763 

 •10877 

 •11827 

 •12639 

 •13334 

 •13877 

 •13939 



Index 



P 



1-01 



M0 



1-2 



1-3 



1-5 



20 



2-40 



2-45 



2-458 



2-5 



2-6 



3-0 



(2 



(2+/3) 



•15714 



•14444 



•13939 



•13134 



•11905 6 



•10815 



•09843 



•09010 



•08292 



•07671 



•07133 



•06661 7 



Since ft may have any value whatever for p=\< any value of /3 

 greater than 3-933647 will satisfy the index unity together with 

 two conjugate indices, as is evident from the /3 curve on Fig. 2. 

 Consequently 



5-1494 



4-9224 



4-7176 



4-5517 



4-3060 



4- exact 



3-9346 



3-9337 



3-9336 4 



3-9341 



3-9387 



4- exact 



73) 



13987 



14446 



14886 



15263 



15858 



16667 3 



16850 



16853 



16853 



16852 



16839 



16667 5 



Index 

 P 



4 



5 

 5-382 



6 



7 



8 



9 

 10 

 11 

 12 

 13 

 14 



P 



4-3636 



4-9231 



5-174 



5-6140 



6 -4 exact 



7-2461 



8-1594 



9-0977 



10-0589 



11-0350 



12-0205 



13-0119 



l 4 + 3^2 

 3 Exactly | 



p - oo, /3 = 



2 Really indeterminate, the curve crosses at this ordinate; 

 -, * More exactly 3 933647; 5 Exactly £ ; 6 Exactly ±£; ^ For 



CO - 1. 



