498 
Ex. 8. Let the folid be acbne: 
ofcillation. Pat AB —a, BD —h • 
b> 
FLUXIONS. 
to find the center of 
then a : b x : y = 
52. Ex. j.), the body=: 
x, CG—d-\-%x\ 
x 2 
-—('if m ~~y ,lx 5 hence, pxxd+x z Xy 2J riy^ = ^'X 
^+.v"xw a * 2 +i>K 4 jr 4 , whofe fluent i'sl pd 2 m 2 x 3 A-bpdm 2 x* 
■ alfo, (Art. 52.¥ ‘ 
£/> ffl2 * 3 » aild (Art. 57. Ex. 2 .)AG=|. 
hence, CD-^£ + 3 Q^a- +i 2.v 2 -j- 3w 
zod-\-i$x 
zod 2 -\- 2,0da -f- 12 a 2 + 3b 2 
~~ 2od+ 1 ~Ja. - for l ^ e vv bole cone, when x—a, 
and mx—y—b. 
If the cone be fufpended at the vertex, then d=zo, and 
co = ^ 2 .- 
S a 
Ex. 9. Let tlie body be a fphere ; to find the center of 
ofcillation. Let B be the center ; then if BA=r, y 2 — 
zrx — x 2 . In this cafe, it will be 
mod convenient to apply the rule 
in Art. 65. that is, to get the value 
of CO when C coincides with A, 
and thence to deduce its value in 
any other cafe. Now when C coin¬ 
cides with A, d—o, and the expref- 
fion becomes pxxx 2 y-\-%y A =p x 
r 2 x 2 x-\-rx 2 x — %x A x, whofe fluent is 
Apr 2 x z -\-\ prx* —2 3 oA* 5 ; and when 
x—zr 
it 
i-?- r; confequently 
CB, d :: \r : 
5 d 
becomes || /» 5 for the 
whole fphere. Alio, the body x 
CG (G now coinciding with B) — 
^r 3 Xt=-j/if 4 ; tlierefore AO =2 
BO—|-r. Hence, (Art. 65.) if d— 
-BO when the point of fufpenfion is at 
C ; therefore CO—d-\-- 
bd 
x 2 , is /±xx-\-zxx-\-zxx—6xx-\-zxx. In like manner we 
may find the fucceflive orders of fluxions of any quantity. 
67. If x increafe uniformly, or if x be conftant, x n will 
have n fluxions, and no more, n being an affirmative whole 
number. For the firft fluxion is nx n 1 x ; and x only 
being variable, its fluxion is n.n — i.x n 2 x 2 ; and the 
fluxion of this \sn.n—i.n — z.x"- 3 x 3 , &'c. when there¬ 
fore we have taken the fluxion n times, the index of x 
becomes — o, and a - 0 —1 ; hence, the fluxion then be¬ 
comes n.n —1 . . . 2.1.x' 1 , which being aconflant quantity, 
it has no further fluxion. 
68. The firft fluxion of x 3 ±~ay 2 is 3 x 2 x+zayy and if 
x and j be both variable, its fluxion is 6xx 2 + 3X 2 x + zay 2 
+ zayj>; but if x be conftant, then x—o; therefore the 
fecond fluxion becomes 
1 A-zey 2 -\-zayy ; 
and if y be 
ON SECOND, THIRD, £Sc. FLUXIONS. 
Prop. XXIX. To explain under what circumjlances a 
quantity may have feveral orders of Jluxions. 
66. The fluxion of a quantity being the uniform in¬ 
creafe or decreale of that quantity in a given time, every 
quantity which increafes or decreafes mutt have a fluxion. 
Hence, if the fluxion of any quantity be not conftant, it 
muft have fpme certain rate of increafe or decreafe, 
which rate of increafe or decreafe will therefore be the 
fluxion of that fluxion, or the fecond fluxion of the ori¬ 
ginal flowing quantity. Alfo, if this fecond fluxion be 
not always the fame, it muft have arateof variation,that 
rale therefore will be the fluxion of the fecond fluxion, 
or the third fluxion of the original quantity; and fo on. 
The fluxion of x is denoted thus, x ; the fluxion of x is 
denoted thus, x; and fo on. Thus a quantity will have 
a fucceflive order of fluxions till feme one ’fluxion be¬ 
comes conftant, and then by Art. 3. it will have no more. 
Thus, let x increafe uniformly; then the fluxion of x 2 
is zxx ; now x is conftant, but x itfelf increafes, therefore 
increafes in proportion to the increafe of x ; the 
fluxion therefore of x 2 is not conftant. Hence, confider- 
ing xas the variable part of zxx, its fluxion by Art. 9. is 
zxx—2x 2 , which is the fecond fluxion of x 2 . But if we 
luppofe .v not to increafe uniformly, then zxx will have 
both X' and x variable ; hence, by Art. 14. the fluxion of 
zxx will be zxx-\-2xx, or 2.x 2 -\-zxx, which therefore is 
the fecond fluxion of x 2 . But if we fltould here fuppofe 
neither x nor x to be conftant, then this fecond fluxion 
would be variable. Now the fluxion of zxy- is found by 
Art. 11 confidering here x as the root, and therefore the 
fluxion of the root is x ; hence the fluxion of zx 2 is 
, yxx ; alfo, the fluxion of zxx is found by Art. 9. both x 
and x being variable; hence, its fluxion is zxx + zxX ; 
therefore the fluxion of zx 2 -\-2xx, or the third fluxion of 
conftant, the fecond fluxion is 6 xx 2 + 3x 2 x-\-zay 2 . 
69. The firft fluxion of x”y m , by Art. 15. is ny m x" i x 
+m-xy m 1 y ; and if both x and y be variable, we are to 
confider each of thefe quantities as compofed of three 
variable fadfors, and then the fluxion, by the fame Art., 
will be n.mx n i y" 1 ^yxf-n.n —1. y'"x“ 2 x 2 -ny m x r ‘ t 'x 
■\-m.m —1 ,x”y 2 j 2 +utny”‘ * x " i xy+mx"y m 1 y. 
On the POINT of CONTRARY FLEXURE of a. 
CURVE. 
70. If a curve be concave in one part and convex in 
another, the point where the concave part ends and the 
convex begins, is the point of contrary flexure. 
Prop. XXX. To 'find the point of contrary flexure of a 
curve. 
71. Let PQ^, BC, Dr, be three equidiftant ordinates,, 
and the curve, concave to the axis; and draw QR, CE* 
parallel ter AD, and join QC, 
and produce it to meet Dr in t. 
Then the triangles QRC, CE t, 
being fimilar, and C^R^CE, 
therefore CRzrztE, and hence 
CR is greater than Er; there¬ 
fore if y. reprefent the ordinate, 
moving from A, and x the ab- 
fcilfa, and PB— BD— fa conftant A 
r 
7* 
Pv 
;E 
P B X) 
quantity ; then correfponding to the uniform increafe of 
x, the increment of y, and confequently y, decreafes ; 
now as y increafes, y is pofitive by Art. 16. but as j de¬ 
creafes, its fluxion, or j', is negative by the fame article. 
If the curve be convex to the axis, and the ordinate 
move from A, then the increment of y, and therefore j, 
increafes ; and as y in¬ 
creafes, y is pofitive; 
and, as j increafes, its 
fluxion, cr y , is pofi- 
tive. Therefore when 
the curve is concave to 
the axis, y is negative ; 
when convex, y is pofitive, 
ft-being conftant. Hence, 
at the point of contrary jfi p 73 -q 
flexure y changes its fign ; but a quantity may change its 
fign, either by paftirig through o, or infinity ; hence, at 
the point of contrary flexure, j—o, or infinity. What 
we here mean by infinity is only in refpedt to its value at 
any other time, that term being relative ; and in this cafe 
we are to underhand that j' is indefinitely greater at that 
time than at any other. If we conceive a line to be 
drawn from A parallel to BC, and confider it as an ab- 
lcifl’a to the curve, and draw lines from it to Q^_, C, r, 
parallel to AD; then the former abfciflse AP, AB, 
AD, become equal to the ordinates, and the ordinates 
ECb,, BC, Dr, become equal to the abl'ciflse ; if therefore 
y be made conftant, x—o, or infinity, at the point of con¬ 
trary flexure. Hence we have the following 
Rule : —Put the equation of the curve into fluxions ; 
make x or j conftant and take the fluxion of the equation 
again, and get the value of y or x, and put it —o, or in¬ 
finity ; from which find the value of x, which gives the 
abfeiffa 
