in the higher Geometry . 
395 
analysis ; and in itself, perhaps, no inelegant specimen of alge- 
braic demonstration. 
PROP. XXI. PROBLEM. Fig. 11. 
To find the Curve whose Tangent is always of the same Mag- 
nitude. 
Analysis . Let MN be the curve required, AB the given axis, 
SM a tangent at any point M, and let d be the given magni- 
tude ; then, SM . q . = SP . q . + PM . q . = or, y*-j- 
= d z 3 and ~ = - -- iJ ..; therefore, x — fiy.s/d z —y z . In order 
to integrate this equation, divide W d 2 — y* into its two parts, 
d% y A —yy 
y Vd 1 — f ana Vd % —~f' 
To find the fluent of the former, 
A 
1 1 + — = 
*y 
y V&- ~f 
£y 
y 
x 
V d. % — y z 
d -f Vd^—y l 
— dxl - ^ - <Pj 
y V d? — y x J 
d - f- v' d z — y a 
d x fluxion of _ 
— — ; therefore, the fluent of —L 
d + d* . 
y Vd z —y % 
is — d x hyp. log. fil . fi (l% — and the fluent of the other part. 
’ y y • r — - - 
=== is T v d z — y 2 ; therefore, the fluent of the aggregate 
V.d 7 
7 Set —f is h. 1. d + or s/d'-f -f 
^ x h. L j— ; a, final equation to the curve required. 
Q . E. I. 
^ E a 
