44 G. H. KNIBBS. 



to solve. If, for particular values of p and q, identical values for 

 a and also for b can be derived, they will mark points on the % 

 axis satisfying both equations ; and the mean of the values of the 

 function at these points will be the mean of all values of the 

 function (19) from to I. 



We consider first values symmetrically situated with respect to 

 the middle point on the axis. Put therefore 



a=i_£and b = h + £ (21) 



so that a + b = l; then expanding (20) and dividing by 2, 



$(a* + b>) = L + P(P- l \p+PiP- 1 )(P-2)[P- S )p + etc. = -L(22) 



2\ i 2p '21 2 P ~' 3 4! 2 P "~ 4 p+l 



p of course having any positive value whatever. If p be fractional 

 the series is infinite but convergent, since both the coefficients and 

 the powers of £, £ bein^ a proper fraction, are convergent. If p 

 be integral the series is finite, having pj'2 terms in £, if jo be even; 

 or (p - l)/2 if p be odd. Or again, writing b = 1 - a, expanding, 

 and dividing by — p, we have, when p is odd, 



g-!LZ±a'+&- l )(Pjll)a*-...-a^=_-Pzl- (23), 



2! 3! p{p+l) 



and when /; is even 



p-\ 2 (p-l)(p-2) , n , 2a? p-\ /OQ N 



a- 1 a 2 + x l '11 '-a 3 - ... + a 13 " 1 — — = _l ....(23a); 



2! 3! p p{p+l) J 



that is, the equation for any even integer is of the same degree 



as that for the odd integer next above it, as is obvious also from 



(22). Since also if p be 1, £ or a may have any value from to ^, 



it is at once evident that two equations can be simultaneously 



satisfied as long as the index of one is unity. Therefore 



Prop. (e). // one of the positive indices in the original function 



be unity, then always two points on the axis, symmetrically situated 



with respect to its centre, may be taken, so that the mean of the 



values of the function at those points, will be the mean value of the 



function. In other words a " two-term formula" will always 



apply in such a case. 1 



1 This result was obtained for prismoidal solids, in which the sectional 

 area is a quadric function of the z coordinate (i.e. jp= 1, g = 2) by Echols, 

 Annals of Mathematics 1894; I have not, however, seen his article. See 

 also, « Prismoidal Formula and Earthwork," by T. U. Taylor, 1898— 

 Wiley and Sons, New York. 



