..«!'.») 



fn.iu which, after reduction, we obtain 

 I _ 2m+p-l*v[ 4mp+tp-l) a 

 1 I 2»p 



-rj denoting the value of the abscissa for a point of inflexion. 

 The following values for the abscissa3 of the maximum 

 or minimum points and of the points of inflexion, with 

 values of p from± \ to ± 3, will indicate some of the charac- 

 teristics of the curve. 



Positions of maxima or minima and inflexions.— 



- 3 



(3*/m)i 



{12n/[4m-8± ,(64- I8m)]}i 



2* 



(2*»/m)* 



j 10n/[4m - 7 ± 449 - 40m)] | ! 



2 



(2 »/m)* 



; 8ti/[4m-6 ± •(36-32m)]}* 



n 



(l|»/m)f 



| 6»/[4m-5 * •(25-24m)]}f 



i 



(1 »/m) 1 



j 4n/[4m-4± ^(16- 16m)]} * 



i 







+ i ( 



( H/»)' 



{ 2n/[4m-3± •( 9- 8m)]}' 



-2m/*)' i 



:-[4m-l ± •( 1+ 8m)]/ 2»}' 



1 ( 



-lm/n) 1 j 



[-[4m w( 0+16m)]/ 4»; x 



H ( 



-W)f : 



:-[4m+l ± v(l + 24m)]/6»:f 



2 ( 



-M*)± ; 



1 -[4m + 2 * •( 4 + 32m)]/ 8«}* 



2* ( 



-Hn/«)* 1 



-[4m + 3± •( »+40m)]/l(m}* 



3 ( 



-Jjii/n)* 



! -[4m + 4± /(16+48m)]/12«}* 



The above expressions are left in their unreduced form 



in order to 



shew the progression of the quantities : that 



for p = + 



1 reduces t< 



d — (m ± ^m-)/" indicating that the 



abscissa? of points of inflexion are equidistant from the 



abscissa of 



the maximi 



nm or minimum ordinate -m/li by 



the distance Sm/n, and 



that this condition is independent 



of the value 



! of either n 



i or n. 



When p q 



as any other value, we have, from (15) and (19), 



and after m 



ultiplyingby (2^ +1 np) p 



