330 G. H. KNIBBS. 



conform to the fundamental equation, where the generatrix is 

 one-dimensional. 



When the generatrix is two-dimensional, it is quite otherwise. 

 The value of Act no longer affects the generated elements, the 

 volume thereof depending upon Ap alone, that is to say, 

 8A t = 8x.8y.8t (l+A P /p) (16) 



in all cases, hence 



2(8A t .A P y=Q (17) 



is the only condition to be satisfied; i.e. the centre of inertia of 

 the generatrix must lie continually in the binomial axial surface. 



8. Theory of equivalent generatrix of unit value. — It has been 

 shewn herein, that when the £-axis is curved or tortuous, the mean 

 value of the factor (l+Ap/p) must, for every position of the 

 generatrix, be continually unity; otherwise some correction to 

 the integral will be required. Let us suppose however, this con- 

 dition to be abandoned, and the mean value (depending upon the 

 place of the centre of inertia of the generatrix elements or area) 

 to vary in such a manner, that this variation is also a function of 

 the position of the generatrix on the axis, such as may be repre- 

 sented by the expression 



k' = k +ft + gt h + hl c + etc (18) 



We shall then have, for the value of what may be called the 

 equivalent generatrix, the factor of which will be unity, for every 

 position on the axis, 

 A t = kA t + A(ft* + gt h +...) + fi(ft*+» + gt h +v+...) 



+ C(ft*+*+...)+Z>(ft»+*+...) (19) 



which, when a, b, c and p, q, r are actually given can be arranged 

 according to ascending values of the indices, and written in a form 

 identical with the original expression (1), viz. 



A' t = A' + B'V>' + C'P' + D'f + etc (20) 



the integral of which will be therefore the same as (2) in form. 



9. Necessary number of equidistant values of equivalent gener- 

 atrix. — I have shewn elsewhere 1 that the prismoidal formula 



1 Some applications and developments of the prismoidal formula. — 

 Journ. Eoy. Soc, N. S. Wales, Vol. xxxm., pp. 129- 145, 1899. See §§ 

 15, 16. 



