Pig, 6 
FLUXIONS. 
prop, AB th beam which touches the wall at B, 
pact eth at P, Let C be the centre of 
ity of the beam. P draw the 
DE, and draw CD perpendicular to DE. Put 
PE=a, CB=, these are given quantities ; also, put 
eh ee 
ust . 
ig similar ah peBC : DE:: PC: PD; that is, 
b:a+ax::/(a*+y*): 2; Hence, 
: (ata? (4 y) =P, 
and taking the fluxions, phere Be he 
(a2) (28-49%) + (+2)? (=+y fae. 
Because y is a maximum, dy =o, therefore this last 
equation may be abbreviated to 
(e+2) (2° +9") +2(a+2)=h2. 
When we have elimi z*4y*, by this and the first 
equation, we 
(a+-z)i=a b*, 
Hence -z is determined. 
In this example, as well as in some others, for the 
sake of brevity, we have assumed the possibility of the 
uantity y having a maximum, as it 1s sufficiently in- 
dicated by the nature of the question. For, 
P the of su to be between, C and 
if B, the end of 
FE 
in. esploving this theory, we have only con-« 
sidered functions of a single variable quantity. When 
Ww 
= 
3 
S 
ll 
4 
pet ry plete ag Fonctions Analytiques, No. 160; 
to ix’s Traite du Caleul Differentiel, Vol. 1. 
Method of Tangents 
67. Let CPD be any curve referred to an axis AB, 
(Fig. 8.) and let PQ, P’Q’ be two consecutive ordi- 
hates, and AQ, AQ’ the corresponding abscissee, A be- 
value of y corresponding to x+/ ; therefore, 
d yl? By is 
PQ st i+ Tasty ggt &e- and 
8854 CY BE _PE wy 
Bvnde aah ES SE FE ete 
—pE 8 the trigonometrical expression for 
the tangent of the angle P’PE, or S ; therefore, 
dy h | &y i? 
dee 2 tag t & 
Sup’ now QQ’=A, the increment of x, to be conti- 
nually diminished ; when Q’ comes to Q, the two 
pone of intersection P’, P coincide, and the secant 
P'S becomes PT, a tangent to the curve at P; and as 
theangle S becomes then the angle T, and A=0, we have 
tan. angle T =. (1.) 
The segment TQ of the axis comprehended between 
PT the tangent, and PQ, the ordinate at the point of 
contact, is called the sublangent. By trigonometry, 
_ dy 
(2.) 
Hence, in the right angled triangle TPQ, we have 
tangent PT =v a4tdy*) (3) 
Draw PN i to the tangent at P, the 
point of contact, meeting the axis in N ; the line PN is 
called a Normal to the curve at P; ahd QN, the seg- 
ment of the axis between the ordinate and normal, is 
called the Subnormal. By the elements of geometry, 
the angle QPN is equal to the angle T; now we have 
found tan. T =SY ; therefore, observing that PQ = y, 
we have by trigonometry, ms geometry, . 
Subnormal QN = y. (4) 
_ Normal PN aty(dst+d y) (8.) 
68. We shall now apply these formule to some ex- 
am; 
on . Let the curve be a circle, (Fig. 9.) and let A, Fig. 9. 
igin of the co- 
one extremity of the diameter, be 
ordinates; put the radius OA=a ; the equation of the 
curve isy?=2axr—a2*. Hence, taking the fluxions, 
9 gag A888 & and, by formula (2) (Art. 67.) 
‘x 
F9= a= 10 the subtangent ; 
therefore, OQ: QP :: QP : QT. 
Ex. 2. Suppose the curve a 
that A, the vertex of the axis, is 
the origin SS 
c0-or. 
dinates. Let abe the 
of the axis. The 
Hence it appears that the su 
the vertex; and that the 
meter. 
Ex. 3. Let the curve be an 
let O, the centre, be the origin 
a denote half the , and & half the lesser axis: 
The equation of curve is a®y?-+-b?2?=a*h*; (Co. 
x Sections, Sect. VIII.) hence atydy+0?2dz=0, 
dz 
is bisected at 
is half the. para- 
ellipse 
ay? = z 
‘The general formals for the was investi« 
gated upon the supposition that the abscissa and subtan- 
10.) and Fig. 10. 
e ~ 
Spe; (Bly. aL )'atd p, 
Gr tie es ediaten. to © 
