(50) 



VOLUMES OF SOLIDS AS BELATED TO TRANSVERSE SECTIONS. 61 



^P+dp = x?(l +\ og Kdp) (49); 



the logarithm being of course Napierian, we have also 

 2 2 2 dp 



p + dp + 1 p + 1 (p + l) 2 



For brevity let the expressions of the type (48), but not multiplied 



k 

 by weight-coefficients (k), be denoted by F (p 2): then remem- 

 bering that 



\og(l -A) = _ JL -... - _A_ -etc (51) 



\ n / n rn T 



we have from equations (48) to (51), 



F(p + dp.±.-2)=F(p.^.-2) + {(-Ylog* + (l- *f 



n n [ \n I n \ n I 



A __A_ _ etc.) + - 2 \dp (52) 



v n 2n* ' (p + l) 2 J ^ v y 



which is quite general. If now in passing from p to p + dp the 



function F (p . &/w . 2) is continually zero, we must have 



(-VW- + (l--j P (-— - _*1- -_*!_- etc.) = -— -? ...(53) 



\n) *n V n K n 2n 2 3ri 3 ; (p+^ 



that is, the quantity in the larger brackets in (52) must be zero. 



This last equation determines the values of k/n in terms of p ; 



when p = 0, it becomes ; — 



log — - — - A_ _..._ JL -etc. = -2 (53a) 



n n 2n 2 rn L 



and when p - 1 ; — 



k k k k 2 k v 



- log— - — + — — — + ... + — + etc. = —A (536) 



n * n n 1 .2n 2 (r-l).rw 1 2 v 7 



expressions which can readily be transformed so as to suit the 



exigencies of computation with respect to convergency etc., and 



which are the basis of values already found for such limits. It is 



now evident that the graph of the function F (p . k/n. 2) = is 



of the type shewn by heavy lines on Fig. 1, viz., the two lines 



whose abscissae are and 1, and the curve numbered 2. 



When however k/n is constant, and only p is variable, the curve 

 is of very different form, as may be seen in Fig. 3. In this, 

 curve 6 is the graph of 2/(p + I); curves 7, 8, and 2 of Uk/n) p 

 + (1 -k/n) v \ in which the fraction has the values J, ^, and ^ 



