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11
380 THE RESOLUTION OP PROBLEM*
Ex. Let the curve ABDE be the calenaria, formed by
-^ a flender chain, or perfectly flexible cord, fuf
pended by its two extreme» in the horizontal
line AE: then, fince its centre of gravity muft be the lovveft
poflible, the fluent of yi, when AC rr AE, muft therefore
be a maximum (art. 173) : whence (n being here rr l) our
equation
( A— rr p ± qyn j becomes —- rr p — qy.
But, in order to reduce it to a more convenient form, let
the diftance (DF) of the loweil point of the curve from the
horizontal line AE be put rr b ; then, when y (BC) becomes
zz b, x will be r z; and therefore the equation, in that circumftance, is 1 rr p — qb ; whence p zz\ 4 qb, and con-
£
fequently-— zz \ 4- qb — qy zz \ 4- q x b — y: which,
z
by putting b —y (DH) rr s, and a zz —, is reduced to —- rr
q x
IF —: from whence arix (rr a 4- s\ X xz) zz a 4-
2
ÏÎ X
a
il — sz ; and confequently BD rr *S 2as 4- ss.
For another example (wherein the exponent n will be negative), let the required curve be that along which a body
may defcend, by its own gravity, from one given point A to
another B, in lefs time than through any other line of the
i
fame length. In which cafe, the fluent of iy 2 being a minimum, when x and z become equal to given quantities, our
equation (by writing — \ for «) will here become —- rr p
4- qy 2 : from whence exterminating x or i, by means of
the equation xz F yz = i\ the fluent may alio be determined.
SECTION XI.
The Refolution of Problems of various Kinds.
PROBLEM i.
424. Any hyperbolical logarithm (y) being given, it is
propofed to find the natural number anfwering thereto.
«rt.
| Title | Doctrine and application of fluxions. |
| Alternative Title | The doctrine and application of fluxions containing (besides what is common on the subject) a number of new improvements in the theory, and the solutions of a variety of new and very interesting problems in different branches of the mathematics... To which is prefixed an account of his life. The whole revised and carefully corrected by William Davis. |
| Reference Title | Simpson, Thomas, 1805, The doctrine and application of fluxions. |
| Creator |
Simpson, Thomas, 1710-1761. Davis, William, 1771-1807 . |
| Subject | Calculus. |
| Publisher | London. |
| DateOriginal | 1805 |
| Format | JP2 |
| Extent | 31 cm. |
| Identifier | 138 |
| Call Number | QA303.S45 1805 |
| Language | English |
| Collection | History of Mathematics |
| Rights | http://www.lindahall.org/imagerepro/ |
| Data contributor | Linda Hall Library, LHL Digital Collections. |
| Type | Image |
| Title | Page 380. |
| Format | tiff |
| Identifier | 1138_408 |
| Relation-Is part of | Is part of: The doctrine and application of fluxions containing (besides what is common on the subject) a number of new improvements in the theory, and the solutions of a variety of new and very interesting problems in different branches of the mathematics... To which is prefixed an account of his life. The whole revised and carefully corrected by William Davis. |
| Rights | http://www.lindahall.org/imagerepro/ |
| Type | Image |
| OCR transcript | 11 380 THE RESOLUTION OP PROBLEM* Ex. Let the curve ABDE be the calenaria, formed by -^ a flender chain, or perfectly flexible cord, fuf pended by its two extreme» in the horizontal line AE: then, fince its centre of gravity muft be the lovveft poflible, the fluent of yi, when AC rr AE, muft therefore be a maximum (art. 173) : whence (n being here rr l) our equation ( A— rr p ± qyn j becomes —- rr p — qy. But, in order to reduce it to a more convenient form, let the diftance (DF) of the loweil point of the curve from the horizontal line AE be put rr b ; then, when y (BC) becomes zz b, x will be r z; and therefore the equation, in that circumftance, is 1 rr p — qb ; whence p zz\ 4 qb, and con- £ fequently-— zz \ 4- qb — qy zz \ 4- q x b — y: which, z by putting b —y (DH) rr s, and a zz —, is reduced to —- rr q x IF —: from whence arix (rr a 4- s\ X xz) zz a 4- 2 ÏÎ X a il — sz ; and confequently BD rr *S 2as 4- ss. For another example (wherein the exponent n will be negative), let the required curve be that along which a body may defcend, by its own gravity, from one given point A to another B, in lefs time than through any other line of the i fame length. In which cafe, the fluent of iy 2 being a minimum, when x and z become equal to given quantities, our equation (by writing — \ for «) will here become —- rr p 4- qy 2 : from whence exterminating x or i, by means of the equation xz F yz = i\ the fluent may alio be determined. SECTION XI. The Refolution of Problems of various Kinds. PROBLEM i. 424. Any hyperbolical logarithm (y) being given, it is propofed to find the natural number anfwering thereto. «rt. |
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