[ 14 ] 
k Is found = 
ay 
and, from thence, x = 
y y , — a a 
a X hyp. log. ^ y y which equation, be- 
CL 
ing impoflible when y is lefs than Ihews that the 
curve (which is here the Catanaria) cannot poffibly 
meet the axis about which the folid is generated ; 
and, confequently, that the cafe will not admit of any 
Minimum, unlefs the firft, or leaft given value of 
y exceeds a certain affignable magnitude. 
When any, or all of the above-fpecified quantities 
.are given, and the contempory fluent of fome other 
n m \—~xn 
expreflion, as -i-yyl xyxy-, is required to be a 
Maximum, or Minimum ; the equation (by taking 
the fluxion of this lafl: expreflion, and joining it to 
n - 
the former) will then be x x -f- y jjj 
ex s: y ^ 
I — 2 n 
y. 2 n xy y 
^ ~ / r i J J \ / . . 
"V y y "V X x-\- y y 
which, when ;« = i, and « = — i, will be that de- 
fining the folid of the leafl: refiftance; and this, 
when the axis only is fuppofed to be given 
(without farther reftridions) will be expreflTed by 
y y\ X — 2 xy y^ d = o, or 2 y y^ 
dxxx^yy ; being the cafe, firfl:, confidered by 
Sir Ifaac Newton. If both the length and the 
folid content be given, the equation will be 
2 xy y^ X X x-{- y y\ d h y^ = o but if, 
befides 
