302 
therefore RO which being to PR 
\vJ h b 
(7) in the condant ratio of a to b, or of PR to RH, the 
triangles ROP and RPH mud therefore be fimilar, and fo 
the angle POH, wliich the ordinate PO makes with the 
evoiute, being every where equal to PRQ, will likewife 
be invariable. Whence it appears that the evoiute is 
alfo a logarithmic fpiral, fimilar to the involute ; and that 
a right line drawn from the center, perpendicular to the 
ordinate, of any logarithmic fpiral, will pal’s through the 
centre of curvature. 
Si. Ex. 2. Let the curve propofed be the fpiral of 
Archimedes; where we have p — ^ 
FLUXIONS. 
On the FLUENTS of QUANTITIES. 
and v — 
fly 2 -flb 2 
V:y 2 -flb 2 
Therefore v— zyy X y 2 -flb 2 - 1 -f yy 
X — \ X 2 yy X y^flb 3 ] 2 = 
w 
y- + t) 2 ]* 
zyyXy 2 -\-b 2 — y 3 y y 3 y-\-Tb 2 yy 
whence the radius of 
Prof. XXXIII .—To find the fluent ofi Z —-~-— F. 
85. Put a’ : —b 2 f z' ! —x 2 , then x* n — x, .\-X-z 2 '‘ i y s 
2 "" 
1 j! - i . 2 . _ • 2 X 2 
=a-, and 2 a z—~xxi hence, - 2 =~X 
n n o 4 -{-x 2 nb 2 
7—— ; confequcntly (Art. 46.) V=~ X cir. arc, 
° 1 no 2 
whofe rad. =Z>, tan. =x. 
i.i—1 £ 
Prop. XXXIV .—To find the fluent ofi —-—p, 
a '•— z" 
86. By the fame fubditution, F=-x—-_—v 
n^b 2 — x 2 ~nb X 
ibx 1 f \ L NT- 1 , , b + X 
,2-- 2 i hence, (Art. 45 -) F =—7 X h.l. -f—- 
— x* nb b—x 
y 2 flb 2 
y 2 —j— 2 12 
y V yy-L-b 2 } 2 b 
curvature — is here -—: which being =-, (Art. 
v y 2 -\-2b- 2 
79.) when y—O, the arch of the evoiute (Art. 74.) 
_ 4 
yy-X- b 2 \ b 
reckoned from the vertex, is therefore — ~ -'-. 
’ y 2 + ib 2 2 
After the very fame manner you may proceed in other 
cafes: but if the value of v (or — J changes, in any cafe, 
v J 
from pofitive to negative, the radius of curvature (RO) 
after becoming infinite, will fall on the other fide of the 
tangent, and the correfponding point of the curve, when 
®=o, will be a point of contrary flexure . Whence it may 
be obferved that the point of inflection, in a curve whofe 
ordinates are referred to a center, may be found by making 
the fluxion of the perpendicular, drawn from the center 
to the tangent, equal to nothing, which cafe is not taken 
notice of in the preceding feCtion. 
On the FLUXIONS of EXPONENTIALS. 
82. A quantity is called an exponential , when its index 
is variable. 
Prop. XXXI.— To find the fluxion of x y . 
83. Put xl—z, and let Xrrzh. 1 . x ; then by the nature 
©f logarithms jyX=zZ, therefore yX-fl'Xy—Z ; but by 
* X • 2 V V* 2 
Art. 45. X— -, and Z=-; hence, -—“» confe- 
X Z X z 
quently J r zXy—yx y ' — 1 xflXx y y. 
It x be condant, then x— o, and zzflXx y y. If y be 
condant, j—o, and z=yx y — , x, as in Art. 11. 
Prop. XXXII.— To find the fluxion of x y . 
84. Put x y czw, and let x y —v, then v z — w; hence, 
it V —h. 1 . V} we have (Art. 83.) zo — zfl > v-\-Vv z z; 
but v—z y , and v=yx y 1 x-\-Xx y j-, hence, by fubditu- 
tion, io—zx y 1 Y.yx y i x -(-X x y j V x y z — zyxl 
Xx y x-\-zXx y Xxyy -flVx y z. If any one cf the 
quantities x, y, z , becomes condant ; its fluxion —o, and 
the term vaniflies where that fluxion enters. In like 
manner we may find the fluxion, whatever be the number 
of quantities. The meaning of this notation is, the z 
power of x y , not the y z power of x. If this latter had 
been the meaning of the notation, we mud have put 
y z ~v ) indead of x y —v. 
Prop. XXXV. Let F= 
i’—i 
2 2 Z 
fl a" - 4 -s” 
to find F. 
87. By the fame fubditution, F—-x- 
n V b 2 flx 2 
hence, 
(Art. 45.) F=-x h. 1 . x-flflb 2 +x a . 
n 
Prop. XXXVI. Let F= 
J.'!- 1 . 
z 2 z 
fl a"—z 
to find F. 
88. By the fame fubditution, F=:-x 
n flb 2 +x 2 ni > 
bx 
7 X 
; hence, (Art. 46.) F=:—x cir. arc,rad. —b, 
t / b 2 —x 2 nl > 
line — x. 
Prop. XXXVII. Let F=- 
fl az 2 flbz-\-c 
89. F=—X 
to find F. 
b 
- : — ■ ; put z -i -= X, then 
J*'+jXz+t. 
\ a a 
b b 2 . b c b 2 
z 2 .+-z-\ -- = .v 2 ; hence, z-+ -z + - = x 2 - U 
a 4 a 2 a a 4 a 2 
~= (by putting^— ~= d 2 ) x 2 +d 2 -, alfo, ia=x; 
• 1 x x 
hence, I-=—rX— --. ; and (Art. 45.) F = —= 
a \/x 2 -\-d 2 y/a 
X h. 1 . x-\-flx 2 -\-d 2 . 
- -1 # 
Prop. XXXVIII. Let F — - J! * to find F. 
fl ax 2 "-fbx"flc 
90. Put fcz, then x K — i x— - X «» alfo, x 2 " = 2 2 j 
hence, F= - x- — ■■ 
n fl az 2 flbzflc 
the lad article. 
Prop. XXXIX.— Let F 
, whofe fluent is given in 
to find F. 
91. Aflurne v: 
fl a 2 x 2 
fla 2 flx 2 
= , then (Art. 43.) v— hi 1 . 
x -fl fl a 2 -b x 2 ; put tv — fl a 2 x 3 -fl x 4 , then a > — 
a 2 xx-fl 
