4S4 
FLUXIONS. 
then, by the compofition of ratios, the quantities gene¬ 
rated will be in the ratio of x : mx, or as i : m, a conftant 
ratio. 
26. Jf BC move parallel to itfelf, and AB and BC in- 
creafe uniformly, the locus of the point C is a ftraight 
line. For let BC 
come into the poii- 
tion Dr ; then as AB 
and BC begin toge¬ 
ther and increafe uni¬ 
formly, they have 
always a conftant ra¬ 
tio to each other, by 
Art. 25. therefore 
AB : BC:: AD: D.f, 
which is the property of fimilar triangles; hence, ACs 
is a ftraight line. Alfo, as BC is parallel to Dr, AB : 
BD :.: AC : Cs, and A B : AC :: BD : Cs ; but AB : AC 
in a conftant ratio; if, therefore, BD the increment of 
the bafe be conftant, the cotemporary increment Cs of 
tiie hypothenufe muft be conftant; or, if the former in¬ 
creafe uniformly, the latter will increafe uniformly. 
Hence, the two uniform motions of C, one in a direclion 
parallel to AB ariling from the motion of BC, and the 
other in the direction BC, generate an uniform motion in 
a right line AC. 
27. The fluxion of the curve line AC, cotemporary 
with CE, Es, (figure to Art. 23.) the fluxions of the ab- 
fcifia and ordinate, is the fpace that would be defcribed 
by the point C with its motion continued uniform for the 
time in which CE, Es, are defcribed. Now the motion of 
C arifes from two motions, one by which it is carried pa¬ 
rallel to AB by the motion of BC, and the other by 
w hich it is carried in the direction BC by the increafe of 
BC ; therefore the uniform motion of C is determined 
by making thefe two motions become uniform ; but when 
thefe two motions become uniform , they are reprefented 
by CE and Es, by Art. 23. and thefe two uniform motions 
produce a cotemporary uniform motion Cs, by'Art. 26. 
hence, by Prop. 1. Cs will reprefent the cotemporary 
fluxion of the curve line at the point C. 
To draw ASYMPTOTES to CURVES. 
28. Definition. —If a right line, interfering the 
axis of a curve at a finite diftance, continually approach 
to the curve, and arrive nearer to it than by any aflign- 
able diftance, but indefinitely produced never meets it, it 
is called an afymptote. 
Prop. XI.— To draw an afmptote to a curve. 
29. Let SDW be an afymptote to the curve AC ; then, 
by the definition, we may confider the afymptote SW as 
.8 T A. is 
the limit to which the tangent approaches, when the ab- 
fcilfa AB is increafed fine limite. Draw AE parallel to 
the ordinate BC produced to D, and let TC be a tangent 
to the curve at C. Put AB=x, BC=y; then, by Art. 
13. BT — hence, ATr=L- Xi From the equation 
y y 
of the curve, find the value of this quantity when .v and 
y are infinite, and if it be then finite, the curve admits of 
an. afymptote SW, and the value of AS is obtained. 
Then having computed the value of BT, find the pro¬ 
portion of TB to BC ; and to get their limit, make x and 
y infinite, and you get the proportion of SB to BD, be- 
caufe the limit of TB to BC is SB to BD; but, by fimi¬ 
lar triangles, SB : BD :: SA :: AE, the ratio therefore 
of SA to AE is known, and as AS is known, AE it 
known ; therefore the point E is determined ; draw SE, 
and produce it indefinitely, and it will be the afymptote. 
Ex. 1. Let AC be the the common hyperbola." Here, 
by Ex. 3. Art. 23. BT—-7——’ therefore AT— 
a-f-x 
2 axf-x* ax 
" a+x fff,' the limit of which, when x is infi. 
. ax 
mte > lS ~ 5 hence, S is the center of the hyper¬ 
bola. Now BC — - x /!«*+**, and BT— l nx + x ~ . 
a a-\-x 
'ZQX— l— Jy __ 
hence, B T : BC :: — ’ ~ X f 2ax-\-x 2 t [] le limit of 
which (when * becomes infinite) is as x : -X#:: a : b :: 
a 
BS : BD :: AS:AE; but AS—a, . •. A E— b ; hence, 
draw AE parallel to BC, and take it —b, join SE, and 
produce it indefinitely, and it will be the afymptote. 
Ex. 2. Let the equation of the curve be y 3 —ax 2 -fx 3 . 
Here 3 \y 2 y — 2 axx -f- 3.v 2 jr, and BT = —-1 - 
y zax-\-2x 3 
alfo, BC=j= V«* 2 +* 3 i hence, BT: BC:; 
3^x 2 -1-3x 3 ,—-- 
2«x-f-3v 2 " v ax ' x3 > *he liniit of which (when * be¬ 
comes infinite) is x : x :: BS : BD :: AS : AE ; AS= 
AE. But AT ^i .^-,-. ax *- - the limi 
2 ax-\-^x 2 - * 
. 2ax + 3 x ' 2 ’ 
of which (when * becomes infinite) is ^=AS; hence, 
AE=-; take, therefore, AS = -, and AE = -, join SE, 
3 3 3 
and produce it indefinitely, and it will be the afymptote. 
To draw TANGENTS to SPIRALS. 
30. Definition.— If an indefinite right line SM re¬ 
volve about S, and a point C move in it continually from 
S, it will defcribe a curve called a fpiral \ S is called the 
center, and SC its ordinate. 
Prop. XII.- —To draw a tangent to any point C of a fpiral. 
31. Let YC s be a tangent to the fpiral at C, and SY 
perpendicular to SC; draw CE perpendicular, and Es 
parallel to SM. Now 
the defcribing point 
C has two motions, 
one in the direction 
SM, and the other 
perpendicular to it, 
arifing from the mo¬ 
tion of SM about 
S. The defcribing 
point C is therefore 
under the very fame ' L 
circumftances as in 
Art. 23. upon fup- 
pofition that CE is there perpendicular to the ordinate 
CB; the fluxions therefore muft be reprefented here in 
like manner as they were there ; for the fluxions at the 
point C in the directions CE, CM, and Cs, depend (Art. 
3.) entirely upon the velocities of the defcribing point C 
in thofe directions, without any regard to what may take 
place afterwards from the further motion of MS about 
M ; the fluxions therefore will be juft the fame as if the 
ordinate were moving parallel to itfelf, and the defcribing 
point C had the fame two motions given to it: hence, 
hy Art. 27. Cs is the fluxion of the curve, and, by Art. 
23. Es is the fluxion of the ordinate, and CE the fluxion 
in 
