56 



G. H. KNIBBS. 



which it is sometimes convenient to put in the form 



6 P (1 + p) + py = 1 + a (426/ 



Thus the solution for b is 



log b = - j log (1 + a - py) - log (1 + p) ^ 



= JLi log (l+a-py)-log (1+2) 1 = etc.. .(43) 



In (42) and (43) the only solutions of utility are those which give 

 values of b lying between and 1 : it is moreover convenient to 

 employ only positive values of a and y. Hence the conditions of 

 limitation are, f3 being unity, 



— 1 < (a - py) < p 

 and the equation obviously can be made to satisfy any two positive 

 values of p. The degree of the resultant equation for b will n 

 general, be the same as the index. If however the value of b be 

 assigned the solution for a and y is in all cases merely linear. 

 Table VII. indicates a few formulae, such as may readily be 

 deduced, the section B being at the distance b from the initial end 

 and the other two sections, viz. A and C, being terminal sections. 



VII. — Position and coefficients of intermediate section, with 

 coefficients of terminal sections. 2 





K 



= A + 



Bz* + Cz*; 



V=^ 



-* Mo 



+ PB h + yC ) 















(T 











Indices 

 P 



a 



b 





1 

 0" 



a 









7 



i 





1 



2 5 

 6"* 





l 

 4 6 



3 



32 





10 



J? 





55 



~2~5 





1 

 3 6 



2 



25 





9 



55 





55 



i-S. 

 3 6 





1 

 I 2 



1 



9 





2 



15 





55 



2.6. 



4 9 





W 



5 



49 





6 



1 





2 



ft" 





3 6 



3 



25 





8 



5' 





55 



.3. 

 6 





1 

 3 6 



8 



25 





3 



5) 





51 



i 

 2 





1 

 6 



1 



4 





1 



55 





,, 



4 

 9 





4 



5 



27 





8 



55 





5' 



5_ 





1 

 4 0" 



8 



25 





5 



2 





4 



i 

 2 



etc.. 



_1_ 

 4 5 



, etc., 



6 

 etc. 



32 





7 



i Thus 



we may write 















Hrt 



> + iy 



= 2b + y 



= SI 



» 2 + 2y 



= 46 3 + 



3y — etc. = 



: 1 + 



a 



and solve by inspection. 1'or example, for p — %, q = 1 the first and second 

 expressions give V6 = i [3 ± </(9 - 16y)] ; and similarly for p = l, q=2 the 

 second and third give b =£ [1 ± V(l - 3y)]. 

 2 The table is merely illustrative. 



