Moeeils- 
we 
Problems 
—_—\o 
Fig. 45. 
Fig. 4 
464 
Now, because p= 7 = um. RST (art. 67) when 
dew ena te Ry apeirof emery st 
hence » l(a) — #1 (y) = v fetyc +p )f and 
i Sal {reves ripe and and == p+V01 +P") 
a veitng thi epeliialonagtie and putting 
a 
adra 29 re, 
and taking the fluents a second time, 
a yin qayri ‘ 
<=— 7 adi +6; 
but when y=a, + is =0; therefore 0= | 
curve is 
at yi a ti 2na 
88 — Tae aed Ce 
This line is the curve of pursuit remarked by Bou- 
guer and Maupertuis, (Mem. de U Acad. des Sciences, 
1782). 
Pros. 3. If number of straight lines are drawn 
svcmecistentelaa tar; it is required to 
find the nature of a curve to which these are tangents. 
pts poh iri ogo crak r Po- 
sition, (Fig. 45), and K a 3 let 
number of lines KD, ED’, Oe: be Gevtiedewen,: oie 
in D, D’, &c.; and let lars: DC, D’C’, 
&c. be drawn to these lines; it is required to find the 
nature of the curve ACC’, to which perpendicu- 
lars are 
Without to the parti case, we shall re- 
solve the » and AE to be the 
axis of the curve ( re hee a of the co- 
oe cearn Fes ABoam Bios), Poseenie coe 
t in the curve, put AB=a, BC=y ; but again con- 
sieing Ce C as an 4;-~ whatever in the tangent, put 
AB=r’, Then, whatever be the conditions 
of the tangent, the relation 
rt A ms point in it, may be 
= P 4-Q, where P and Q 
disarm functions of con- 
gen may expres the mie which 
che selena anh &c. 
Let us now suppose, that the variable quanti 
anges ite Se eee ieee Be 
the new ion of the tangent corresponding to p+h ; 
then, considering P and-Ges finstions of 9, 0 oth 
lor’s theorem (art. 52.) 
P becomes P+ Sh 5S he. 
Q becomes Q. + ——- anh bt TOT + be. 
FLUXIONS. 
The relation of x’ toy’ in the new position of the tan- 
gut will now Ween eT -% 
¥= Pr 4+Q4(Soe + SEHK + 8. 
whe 4 8 ptr al rng tem 
the series. 
Now, as this holds true of every point 
the tangent C’ Dy, and thoequation # = Pats =P Gs 
at ¢, the intersection of the:to 
aio wy is be true at the same time ; et ath ee 
ve Pre * 
Greta Fo) b+ Ke Be, =05 
and dividing by 4, 92! oe +Kh+ &.=0. 
Colisaiva ape GS to “to coinci- 
dence ; when C’ come to C, then ¢ also fall at C; 
and fh, and all the terms into which it enters,. vanish ; 
also 2’ and y become « and y: and to determine the 
nature of the curve, we have these two equations = 
veer (1) 
= ip tap Ora 
= acterey Lea renredee 
XAMPLE 1. Letus now recur to the 
ses Su SE ae iB pe . K Fig. ts. 
to AE; AF pt Aten By KA KAS oan et 
DB 
ES Cm hh gain 
it will appear BaF a pli $ therefore” p= 
are o=t a? —P, ee SUED» 
a 
The second of tibee egutlncs pine Wane and hence 
the first becomes y= = — es therefore 4 a y= 2® is 
the tion of the curve, which is evidently a ps 
bol, of which AK isthe axis K the focus, and A the 
Bel ‘2. Sappoee w ray of tight RD (Fig. vane * 
from the sun to fall the concave 
po- asphere at D; atid te bestiened-xeteceed in the direc. 
tion DH; it is proposed to find the nature of the curve 
pln) a a all rays reflected in the same man- 
‘a ieee ih the radios of the phere, and AE parallel 
Saecaeped ep theche let C orks = 
the reflected ra the curve ; let DC meet AE 
in BH, and draw CB icular to'AE: Put AD=a, 
aay the print vi be the variable angle DAE. 
~ sae AD bisects the angle 
me whihs'te oceud oo HE, that is to the sum of 
angles DAH, ADH; therefore ‘the angles ADH, 
DAH ‘are: eqaaly aa angle DHE=2p: Now, by 
trigonometry, 
‘ 
S 
= 
¢ 
