38 G. H. KNIBBS. 



coefficients be denoted by <r : i.e. let 



a<6<cetc. <1; and a + /3 + y + etc. = o~; 

 then since the coefficients may have any value whatever we may 

 evidently write 



V = -L | aA>+pA h + etc. | = 



a + p + etc. ( ) 



^ + - B(aa? + pb*+ etc.)+ - C(aa q + /?6 q + etc.) + etc (3) 



O" or 



A & being the value of the original function (1) for z — a, A h that 

 for z — b etc. Remembering that z is to be considered unity, we 

 have by equating (3) with (2) 

 B (aa p + /56 p + etc.) + C (aa* + /3b q + etc.) + etc. = 



B-?— + C—^-~ + etc...(4> 



The disposable terms, included within the brackets on the left- 

 hand side of this last equation are aa, f3b, etc., viz., the fractions, 

 and the weights assigned to the corresponding functions repre- 

 senting say sections or ordinates thereat : we propose to investigate 

 the general relations which subsist among these — when so deter- 

 mined as to satisfy this last equation, viz. (4). Or to restate this 

 in the light of an application in solid 'geometry of the results of 

 such investigation, the inquiry may be thus expressed : — 



In a solid, whose right section is the function (1) of the distance 

 along its z axis, what weight should be assigned to the sections at 

 different distances on that axis so as to give the mean value of 

 the section ; and conversely at what points thereon may the xy 

 planes be taken, so that with equal, or with any other definite 

 system of weights, their mean shall be the mean cross-sectional 

 area of the solid % x A similar statement will apply to the case of 

 ordinates and areas, since mathematically it is identical in form. 



2. General relation between indices, number and positions of 

 sections, and weight-coefficients. — In order that a relation of the 

 character indicated may obtain, it must be wholly independent of 

 the values of the constants, viz., A, B, C etc. The necessary and 



1 Determined by the expression A m = V z /z. 



