48 G. H. KNIBBS. 



itself; while for less values of a one index will be less than unity 

 and the other greater than 4*7345 ; inasmuch as this ordinate 

 meets the curve again for a value of p of about that amount. 

 Consequently: — 



Prop. (g). If the symmetrically situated sections be at a distance 

 of 0' 1997 088 from the terminals of the axis, considered as of unit 

 length, only two indices can be satisfied viz., p = 1 and q = 4-'7345* 



10. Two symmetrically situated sections and their conjugate 

 indices. — In Fig. 1, any line drawn parallel to the axis of abscissae 

 at a less distance than 0-2123179 cuts it in two points, and cuts 

 also the heavy vertical line, viz. the ordinate for p = 1 . For any 

 definite value of a let the abscissae of the intersections be called 

 conjugate. Then the limits are as follows : — 



a> 0-1997088 ) Lesser f 1 to 2471; Greater C 2-471 to 47345 

 a<0'1997088J index) to 1 ; index ( 4-7345 to co 



It happens that the indices 2 and 3 are conjugate to one another, 

 a having the value in each case i--^v/3, or 0-2L13249, conse- 

 quently a " two-term formula " applies not only to the prismoid 

 and prismatoid, but to figures and solids whose ordinates or right- 

 sections are cubic functions of the distances along the axes. Or 



Prop. (h). If two symmetrically situated sections be at a dis- 

 tance of not more than 0'2123179 from the terminals of the axis, 

 considered as of unit length, then in general the index 1 together 

 with two conjugate indices, one greater, and one less than 2 % lf71 

 can be satisfied. 



Let u and v denote the conjugate indices and A 1 and B x the 

 corresponding symmetrical positions, then for the function 

 A Z = A+Bz+Cz n + Dz y 



we shall have V=lz(A 1 + B 1 ) (24). 



And further as a special case : — 



Prop. (i). If two sections be taken 0-211324-9 from the terminals 

 of the axis, considered as of unit length, then the indices satisfied 

 will be 1, 2, and 3. That is to say a symmetrical two-term 

 formula applies to a solid whose right-sectional area is a cubic 



