INF 
titles have been kept in heaps without the 
access of fresh air. The oil with which 
it is dressed seems to be the chief cause of 
combustion. Wh eaten flour and Charcoal 
reduced to powder, and heated in large 
quantities have been known to take fire 
spontaneously. 
The cases of the spontaneous human 
combustion have never been satisfactorily 
accounted for ; the facts themselves seem 
to be well authenticated, two are recorded 
in the Philosophical Transactions, and re- 
ferred to under Combustion. They ought 
however to hold out a lesson of warning 
to those habitually given to excess with 
regard to spirituous liquors; for in every 
case, the subjects of this terrible calamity 
were drunkards, whose favourite liquor was 
alcohol, in the shape of brandy, gin, &c. 
INFLECTION, or point of inflection, in 
the higher geometry, is the point where a 
curve begins to bend a contrary way. See 
Fi.exure. 
There are various ways of finding the 
point of inflection ; but the following seems 
to be the most simple. From the nature of 
curvature it is evident, that while a curve is 
concave towards an axis, the fluxion of the 
ordinate decreases, or is in a decreasing ra- 
tio, with regard to the fluxion of the ab- 
sciss ; but, on the contrary, that the said 
fluxion increases, or is in an increasing 
ratio to the fluxion of the absciss, where the 
curve is convex towards the axis ; and 
hence it follows that those two fluxions are 
in a constant ratio at the point of inflection, 
where the curve is neither concave nor 
convex. That is, if .r = the absciss, and 
y — the ordinate, then x is to y in a con- 
x y . 
stant ratio, or -r or A is a constant quantity. 
y x 
But constant quantities have no fluxion, or 
their fluxion is equal to nothing ; so that in 
this case the fluxion of - or of -4 is equal to 
y * 
nothing. And lienee we have this general 
rule : viz. put the given equation of the 
curve into fluxions ; from which equation of 
x y 
the fluxions find either -? or A ; then take 
y x 
the fluxion of this ratio or fraction, and put 
it equal to 0 or nothing ; and from this last 
equation find also the value of the same % 
y 
or J : then put this latter value equal to the 
former, which will be an equation from 
whence, and the first given equation of the 
INF 
curve, x and y will be determined, being 
the absciss or ordinate answering to the 
point of inflection in the curve. Or, putting 
the fluxion of -r equal to 0, that is — — — — 
y. y 
= 0, or x y — xy ~ 0, or xy — xy, or x : 
y::x :y, that is, the second fluxions have the 
same ratio a3 the first fluxions, which is a 
constant ratio ; and therefore If x be con- 
stant, or x = 0, then shall y be = 0 also ; 
which gives another rule, viz. take both the 
first and second fluxions of the given equa- 
tion of the curve, in which make both x 
andjirrO, and the resulting equations will 
determine the values of x and y, or absciss 
and ordinate answering to the point of in- 
flection. 
To determine the point of inflection in 
curves, whose semi-ordinates CM, Cm (Plate 
Miscel. VII. fig. 13 and 14.) are drawn from 
the fixed point C ; suppose C M to be infi- 
nitely near C rn, and make in H — M m ■ let 
T m touch the curve in M. Now the angles 
C m T, C M m, are equal ; and so the angle 
C m II, while the semi-ordinates increase, 
does decrease, if the curVe is concave to- 
wards the centre C, and increases if the con- 
vexity turns towards it. Whence this angle, 
or, which is the same, its measure will be a 
minimum or maxium, if the curve lias a 
point of inflection or retrogression ; and so 
may be found, if the arch TH, or fluxion 
of it, be made equal to 0, or infinity. And 
in order to find the arch T II, draw mL, so 
that the angle TuiL be equal to mCL; 
then if C m = y, mr — x,mT = i, we 
. * tx 
shall have y ; x : : t: — . Again draw the 
y 
arch H O to the radius C H ; then the small 
right lines mr, OH, are parallel; and so 
the triangles O L H, m Lr, are similar; but 
because II I is also perpendicular to m L, 
tlie triangles LHI, m r, are also similar : 
whence i : x : : y : 3 — ; that is, tlie quan- 
tities m T, m L, are equal. But FI L is the 
fluxion of H r, which is tlie distance of 
C m — y, and H L is a negative quantity, 
because while the ordinate C M increases, 
their difference r H decreases ; whence 
x x yy — y y — 0, which is a general 
equation for finding the point of inflection, 
or retrogradation. 
Inflection, in grammar, the variation 
of nouns and verbs, by declension and con- 
jugation. See Grammar. 
INFLORESCENCE, in botany, a term 
