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COMMENTARY
NEWTON'S PRINCIPIA.
A SUPPLEMENTARY VOLUME.
DESIGNED FOR THE USE OF STUDENTS AT THE UNIVERSITIES.
BY
J. M. F. WRIGHT, A. B.
> V
LATK SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE, AUTHOR. OF SOLUTIONi
OF THE CAMBRIDGE PROBLESIS, &c. &C.
IN TWO VOLUMES.
VOL. I. ^ ^'
LONDON:
PRINTED FOR T. T. & J. TEGG, 73, CHEAPSIDE;
AND RICHARD GRIFFIN & CO., GLASGOW.
MDCCCXXXIlf.
GLASGOW:
OEOBGK BROOKHAN, PRUNTKR, VILLAPIKLD.
TO THE TUTORS
OF THE SEVERAL COLLEGES AT CAMBRIDGE,
THESE PAGES,
WHICH WERE COMPOSED WITH THE VIEW
OF PROMOTING THE STUDIES
OVER WHICH THEY SO ABLY PRESIDE,
ARE RESPECTFULLY INSCRIBED
BY THEIR DEVOTED SERVANT,
THE AUTHOR.
•^i^^i^ •^■. ^^^i>
PREFACE.
The flattering manner in which the Glasgffjo Edition of Kewton's Prin
cipia has been received, a second impression being already on the verge
of^ publication, has induced the projectors and editop of that work, to
render, as they humbly conceive, their labours still liiore acceptable, by
presenting these addition^ vohmres to thd public. From amongst the
several testimonies of the esteem in which their former endeavours have
been held, it may suffice, to avoid the charge of selfeulogy, to select the
following, which, coming from the high authority of French mathematical
criticism, must be considered at once as the more decisive and impartial.
It fias been said by one of the first geometers of France, that *^ L'edition
de Glasgow fait honneur aux pi'esses de cette ville iftdtusfrieuse. Gn peut
affirmer que jamais Fart typographique ne rendit un plus bel hommag*
a la memoire de Newton. Le merite ^e I'impression, quoique tresremar
quable, n'est pas ce que les editeurs ont recherche avec le plus de soin,
pour tant le materiel de leur travail, ils pouvaient s'en rapporter k ['habi*
lite de leur artistes : mais le choix des meilleures editions, la revision la
plus Scrupuleuse du texte et des epreuves, la recherche attentive des fautes
qui pourraient ^chapper meme au lecteur studieujt, et passer inaper^ues
ce travail consciencieux de rintelligence et du savoir, voilJi ce qui ^leve
cette edition audessus de toutes celles qui I'ont prec^i^e.
" Les editeurs de Glasgow ne s*etaient charges que d'un travail de re
vision. S'iU avaient conpu le projet dtamelioj'er et completer Voeimre des
a3
VI PREFACE.
commentateurSf ih auraient sans doute employe, comme eux, les travaux des
successews de Nekton sur les questions iraitees dans le livre des Prmcipes.
" Les descendans de Newton sont nombreux, et leur genealogie est
prouvee par des titres incontestibles ; ceux qui vivent aujourd'hui verraient
sans doute avec satisfaction que Ton format un tableau de leur famille, en
reunissant les productions les plus remarquables dont I'ouvrage de Newton
a fourni le germe: que ce livre immortel soit entoure de tout ce Ton peut
regarder comme ses developpemens : voila son meilleur commentaire.
U edition de Glasgow pourrait done etre continuee, et prodigieusement
enrichie'*
The same philosopher takes occasion again to remark, that ** Le plus
beau monument que Ton puisse elever a la gloire de Newton, c'est une
bonne edition de ses ouvrages : et il est etonnant que les Anglais en aient
laisse ce soin aux nations etrangeres. Les presses de Glasgow viennent
de reparer, en partie, le tort de la nation Anglaise : la nouvelle edition
des Principes est efFectivement la plus belle, la plus correcte et la plus com
mode qui ait parujusqi^ici. La collation des anciennes editions, la revi
sion des calculs, &c. ont ete confiees a un habile mathematicien et rien
n'a ete neglige pour eviter toutes les erreurs et toutes les omissions.
*' II faut esperer que les editeurs continueront leur belle entreprise, et
qiCils y seront assez encourages pour nous donner, non seulement torn les
ouvrages de Newton, mais ceux des savans qui ont complete ses travaux."
The encouragement here anticipated has not been withheld, nor has
the idea of improving and completing the comments of "The Jesuits",
contained in the Glasgow Newton, escaped us, inasmuch as long before
these hints were promulgated, had the following work, which is composed
principally as a succedaneum to the former, been planned, and partly writ
ten. It is at least, however, a pleasing confirmation of the justness of our
own conceptions, to have encountered even at any time with these after
suggestions. The plan of the work is, nevertheless, in several respects,
a deviation from that here so forcibly recommended.
The object of the first volume is, to make the text of the Principia, by
PREFACE. Vll
supplying numerous steps in the very concise demonstrations of the pro
positions, and illustrating them by every conceivable device, as easy as
can be desired by students even of but moderate capacities. It is univei*
sally known, that Newton composed this wonderful work in a very hasty
manner, merely selecting from a huge mass of papers such discoveries as
would succeed each other as the connecting links of one vast chain, but
without giving himself the trouble of explaining to the world the mode of
fabricating those links. His comprehensive mind could, by the feeblest
exertion of its powers, condense into one view many syllogisms of a pro
position even heretofore uncontemplated. What difficulties, then, to him
would seem his own discoveries ? Surely none ; and the modesty for
which he is proverbially remarkable, gave him in his own estimation so
little the advantage of the rest of created beings that he deemed these
difficulties as easy to others as to himself: the lamentable consequence of
which humility has been, that he himself is scarcely comprehended at this
day — a century from the birth of the Principia.
We have had, in the first place, the Lectures of Whiston, who des
cants not even respectably in his lectures delivered at Cambridge, upon
the discoveries of his master. Then there follow even lower and less
competent interpreters of this great prophet of science — for such Newton
must have been held in those dark days of knowledge — whom it would be
time misspent to dwell upon. But the first, it would seem, who properly
estimated the Principia, was Glairaut. After a lapse of nearly half a cen
tury, this distinguished geometer not only acknowledged the truths of the
Principia, but even extended the domain of Newton and of Mathematical
Science. But even Clairaut did not condescend to explain his views and
perceptions to the rest of mankind, farther than by publishing his own
discoveries. For these we owe a vast debt of gratitude, but should have
been still more highly benefited, had he bestowed upon us a sort of run
ning Commentary on the Principia. It is generally supposed, indeed,
that the greater portion of the Commentary called Madame Chastellet's,
was due to Clairaut. The best things, however, of that work are alto
a 1
Vlll PREFACE
gether unworthy of so great a master ; at the most, showing the perforra
pnce WAS not one of his own seeking. At any rate, this work does not
ileserve tjie name of a Commentary on the Principia. The same may
safely be affirmed of many other productions intended to facilitate New
ton. Pemberton's View, although a bulky tome, is little more than
a eulogy. Maclaurin's speculations also do but little, elucidate the
dark passages of the Principia, although written more immediately for
that purpose. This is also a heavy unreadabje performance, and not
worthy a place on the same shelf with the Qth^v works of that great
geometer. Another great rnathematician, scarcely inferior to Maclaurin,
has also laboured unprofitably in the same field. Emerson's Comment^
is a book as small in value as it is in bulk, affording no helps worth th^
perusal to the student. Thorpe's notes to the First Book of the Princi
pia, however, are of a higher character, and in many instances do really
facilitate the reading of Newton. Jebb's notes upon certain sections deserve
the same commendation ; and praise ought not to be withheld from several
other commentators, who have more or less succeeded in making small
portions of the Principia more accessible to the student — such as the Rev.
Mr. Newton's work, Mr. Carr's, Mr. Wilkinson's, Mr. Lardner's, &c.
It must be confessed, however, that all these fall far short in value of the
very learned labours, contained in the Glasgow Newton, of the Jesuits
Le Seur and Jacquier, and their great coadjutor. Much remained, how
ever, to be added even to this erudite production, and subsequently to its
first appearance much has been excogitated, principally by the mathema
ticians of Cambridge, that focus of science, and native land of the Princi
pia, of which, in the composition of the following pages, the author has
liberally availed himself. The most valuable matter thus afforded are the
Tutorial MSS. in circulation at Cambridge. Of these, which are used in
explaining Newton to the students by the Private Tutors there, the author
confesses to have had abundance, and also to have used them so far as seem
ed auxiliary to his own resources. But at the same time it roust be remark
ed, that litde has been tlie assistance hence derived, or, indeed, from all
PUEF ACK.  nC
Other known sources, which from the first have been constantly at com
mand.
Tlje plan of the work being to make those parts of Newton easy which
are required to be read at Cambridge and Dublin, that portion of the
Principia which is better read in tlie elementary works on Meclianics,
viz. the preliminary Definitions, Laws of Motion, and their Corollaries,
has been disregarded. For like reasons the fourth and fifth sections have
been but little dwelt upon. The eleventh section and third book have
not met with the attention their importance and intricacy would seem to
demand, partly from the circumstance of an excellent Treatise on Physics,
by Mr. Airey, having superseded the necessity of such labours; and
partly because in the second volume the reader will find the same subjects
treated after the easier and more comprehensive methods of Laplace.
The first section of the first book has been explained at great length,
and it is presumed that, for the first time, the true principles of what has
been so long a subject of contention in the scientific world, have there
been fully established. It is humbly thought (for in these intricate specu
lations it is folly to be proudly confident), that what has been considered
in so many lights and so variously denominated Fluxions, Ultimate Ratios,
Differential Calculus, Calculus of Derivations, &c. &c. is here laid down
on a basis too firm to be shaken by future controversy. It is also hoped
that the text of this section, hitherto held almost impenetrably obscure, is
now laid open to the view of most students. The same merit it is with some
confidence anticipated will be awarded to the illustrations of the 2nd, 3rd,
6th, 7th, 8th, and 9th sections, which, although not so recondite, require
much explanation, and many of the steps to be supplied in the demon
stration of almost every proposition. Many of the things in the first
volume arc new to the author, but very probably not original in reality —
so vast and various are the results of science already accumulated. SuflSce
it to observe, that if they prove useful in unlocking the treasures of the
Principia, the author will rest satisfied with the meed of approbation,
which he will to that extent have earned from a discriminating and im
partial public
X PREFACE.
The second volume is designed to form a sort of Appendix or Supple
ment to the Principia. It gives the principal discoveries of Laplace, and,
indeed, will be found of great service, as an introduction to the entire
perusal of the immortal work of that author — the Mecanique Celeste.
This volume is prefaced by much useful matter relative to the Integra
tion of Partial Differences and other difficult branches of Abstract Ma
thematics, those powerful auxiliaries in the higher departments of Physical
Astronomy, and which appear in almost every page of the Mecanique
Celeste. These and other preparations, designed to facilitate the com
prehension of the Newton of these days, will, it is presumed, be found
fully acceptable to the more advanced readers, who may be prosecuting
researches even in the remotest and most hidden receptacles of science ;
and, indeed, the author trusts he is by no means unreasonably exorbitant
in his expectations, when he predicates of himself that throughout the
undertaking he has proved himself a labourer not unworthy of reward,
THE AUTHOR.
A COMMENTARY
ON
NEWTON'S PRINCIPIA,
SECTION I. BOOK I.
1. This section is introductoiy to the succeeding part of the work. It
comprehends the substance of the metliod of Exhaustions of the Ancients,
and also of the Modejn Theories, vai'iously denominated Fluxions^ Dif
ferential CalculuSi Calculus of Derivations, Functions, &c. &c. Like
them it treats of the relations which Indefinite quantities bear to one ano
ther, and conducts in general by a nearer route to precisely the same
results.
2. In what precedes this section, fnite quantities only are considered,
such as the spaces described by bodies moving uniformly infinite times
with finite velocities ; or at most, those described by bodies whose mo
tions are uniformly accelerated. But what follows relates to the motions
of bodies accelerated according to various hypotheses, and requires the
consideration of quantities indefinitely small or great, or of such whose
Ratios, by their decrease or increase, continually approximate to certain
Limiting Values, but which they cannot reach be the quantities ever so
much diminished or augmented. These Limiting Ratios are called by
Newton, " Prime and Ultimate Ratios," Prime Ratio meaning the Limit
from which the Ratio of two quantities diverges, and Ultimate Ratio that
towards which the Ratio converges. To prevent ambiguity, the term Li
miting Ratio will subsequently be used throughout this Commentary.
A COMMENTARY ON [Sect. I.
LEMMA I.
3. Quantities and the Ratios of Quantities.] Hereby Newton
would infer the truth of the Lemma not only for quantities mensurable
by Integers, but also for such as may be denoted by Vulgar Fractions.
The necessity or use of the distinction is none ; there being just as much
reason for specifying all other sorts of quantities. The truth of the Lemma
does not depend upon the species of quantities, but upon their confor
mity with the following conditions, viz.
4. That they tend continually to equality, and approach nearer to each
other than by any given difference. They must tend continually to equa
lity, that is, every Ratio of their successive corresponding values must be
nearer and nearer a Ratio of Equahty, the number of these convergen
cies being without end. By given difference is merely meant any that can
be assigned or proposed.
5. Finite Time.] Newton obviously introduces the idea of time in this
enunciation, to show illustratively that he supposes the quantities to con
verge continually to equality, without ever actually reaching or passing that
state ; and since to fix such an idea, he says, " before the end of that
time," it was moreover necessaiy to consider the time Finite. Hence
our author would avoid the charge of " Fallacia Suppositionis," or of
" shifting the hypothesis." For it is contended that if you frame certain
relations between actual quantities, and afterwards deduce conclusions
from such relations on the supposition of the quantities having vanished,
such conclusions are illogically deduced, and ought no more to subsist
than the quantities themselves.
In the Scholium at the end of this Section he is more explicit. He
says, The ultimate Ratios, in "which quantities vanish, are not in reality the
Ratios of Ultimate quantities ,• but the Limits to nsohich the Ratios of quan
tities continually decreasing always approach ; "which they never can pass
beyond or arrive at, unless the quantities are continually and indefinitely
diminished. After all, however, neither our Author himself nor any of
his Commentators, though much has been advanced upon the subject, has
obviated this objection. Bishop Berkeley's ingenious criticisms in the
Analyst remain to this day unanswered. He therein facetiously denomi
nates the results, obtained from the supposition that the quantities, before
Book I.] NEWTON'S PRINCIPIA. 3
considered finite and real, have vanished, the " Ghosts of Departed
Quantities " and it must be admitted there is reason as well as wit in the
appellation. The fact is, Newton himself, if we may judge from his own
words in the above cited Scholium, where he says, " If two quantities,
whose DIFFERENCE IS GIVEN are augmented continually, their Ultimate
Ratio will be a Ratio of Equality," had no knowledge of the true nature
of his Method of Prime and Ultimate Ratios. If there be meaning in
words, he plainly supposes in this passage, a mere Approximation to be
the same with an Ultimate Ratio. He loses sight of the condition ex
pressed in Lemma I. namely, that the quantities tend to equality nearer
than by any assignable difference, by supposing the difference of the quan
tities continually augmented to be given, or always the same. In this
sense the whole Earth, compared with the whole Earth minus a grain of
sand, would constitute an Ultimate Ratio of equality ; whereas so long as
any, the minutest difference exists between two quantities, they cannot be
said to be more than nearly equal. But it is now to be shown, that ,
6. If two quantities tend continually to equality, and approach to one
another nearer than by any assignable difference, their Ratio is ULTIMATE
LY a Matio of ABSOLUTE equality. This may be demonstrated as fol
lows, even without supposing the quantities ultimately evanescent.
It is acknowledged by all writers on Algebra, and indeed selfevident, that
if in any equation put = 0, there be quantities absolutely different in kind,
the aggregate of each species is separately equal to 0. For example, if
A + a + B V~2 + b V~2 + C V~^^^ = 0,
since A + a is rational, (B + b) V^2 surd and C V — 1 imaginary,
they cannot in any way destroy one another by the opposition of signs,
and therefore
A + a = 0, B + b = 0, C = 0.
In the same manner, if logarithms, exponentials, or any other quantities
differing essentially from one another constitute an equation like the above,
they must separately be equal to 0. This being premised, let L, L' de
note the Limits, whatever they are, towards which the quantities L + I,
L' + 1' continually converge, and suppose their difference, in any state of
the convergence, to be D. Then
L + 1_L'— 1' = D,
or L — L' + 1 — r — D = 0,
and since L, U are fixed and definite, and 1, 1', D always variable, the
former are independent of the latter, and we have
A2
4 A COMMENTARY ON [Sect. I.
L
L — L' = 0, or j> = 1, accurately. Q. e. d.
This way of considering the question, it is presumed, will be deemed
free from every objection. The principle upon which it rests depending
upon the nature of the variable quantities, and not upon their evanescence,
(as it is equally true even for constant quantities provided they be of dif
ferent natures), it is hoped we have at length hit upon the true and lo
gical method of expounding the doctrine of Prime and Ultimate Ratios,
or of Fluxions, or of the Differential Calculus, &c.
It may be here remarked, in passing, that the Method of Indeterminate
Coefficients, which is at bottom the same as that of Prime and Ultimate
Ratios, is treated illogically in most books of Algebra. Instead of
" shifting the hypothesis," as is done in Wood, Bonnycastle and others,
by making x r= 0, in the equation
a + bx + cx2+dx3+ = 0,
it is sufficient to know that each term x being indefinitely variable, is he
terogeneous compared with the rest, and consequently that each term
must equal 0.
T. Having established the truth of Lemma I. on incontestable princi
ples, we proceed to make such applications as may produce results useful
to our subsequesnt comments. As these applications relate to the Limits
of the Ratios of the Differences of Quantities, we shall term, after Leib
nitz, the Method of Prime and Ultimate Ratios,
THE DIFFERENTIAL CALCULUS.
8. According to the estabhshed notation, let a, b, c, &c , denote con
stant quantities, and z, y, x, &c., variable ones. Also let A z, A y, A x,
&c., represent the difference between any two values of z, y, x, &c., re
spectively,
9. Required the Limiting or Ultimate Ratio of A (a x) and A x, i. e.
the Limit of the Difference of a Rectangle having one side (a) constant, and
the other (x) variable, and of the Difference of the variable side.
Let L be the Limit sought, and L + 1 any value whatever of the va
rying Ratio. Then
A (a x) a (x + A x) — ax , xt r.
L = a.
Book I.] NEWTON'S PRINCIPIA. 5
In this instance the Ratio is the same for all values of x. But if in the
Limit we change the characteristic A into d, we have
d (a x)
= a
"" : (b)
or " ^
d (a x) = a d X
d (a x), d X being called the Differentials of a x and x respectively.
A{x2)
10. Required the Limit of ' ^.v '
Let L be the Limit required, and L + 1 the value of the Ratio gene
rally. Then
A (x *) (x + A x) * — x *
Ax ■" Ax
2 X A X + ^ X
2
AX =2X + AX. .
.. L — 2x + l — Ax=0
and since L — 2 x and 1 — Ax are heterogeneous
L — 2 X = 0,
or
L = 2x.
and .*.
or
d(x2) = 2xdx (c)
A (x")
1 L Generally i required the Limit of ■ ^ .
Let L and L + 1 be the Limit of the Ratio and the Ratio itself re
spectively. Then
T J_ 1 _ ^(^°) _ (X + AX)°— X°
^ + ^  Ax  AX
n. (n — 1)
= n x"» + — ^^"2 . X "' A X + &c.
and L — n x ° — * being essentially different from the other terms of
the series and from 1, we have
d(x°)
jj X =L = nx°'ord(x") = nx''*dx (d)
or in words,
AS
6 A COMMENTARY ON [Sect. I.
The Differential of any power or root of a variable quantity is equal to
the product of the Differential of the quantity itself, the same powei' or
root MINUS one of the quantity, and the index of the po'wer or root.
We have here supposed the Binomial Theorem as fully established by
Algebra. It may, however, easily be demonstrated by the general prin
ciple explained in (7).
12. From 9 and 11 we get
d (a X °) = n a X "  * d X (e)
„ . ,, ^ . ,A (a + bx° + cx°> + exP + &c.)
13. Required the Limit of —
Let L be the Limit sought, and L + 1 the variable Ratio of the finite
differences; then
A(a + bx" + cx'° + &c.)
^ + ^ = AX
a + b(x + Ax)° + c(x + Ax)*" + &c. — a— bx" — ex"— &c.
— Ax
= nbx''^ +mcx'"» +&c. + Pax + Q(Ax)2 + &c.
P, Q, &c. being the coefficients of A x, A x * + &c. And equating the
homogeneous determinate quantities, we have
dfa + bx^^+cx'^ + Scc.) ^ , , , , , « ,«
^—^ ^ ^ = L = nbx'*^ + racx"'»+pexP» + &c...(f)
A(a + bx'' + cx" + &c.) '
14. Required the Limit of ^^ *
By 11 we have
d. (a + bx" + cx"" + &c.) '
d(a + bx + &c.) =r(a + bx'»+cx + &c.)'»
and by 13
d(a+bx" + cx'" + &c.) = (nbx"» + mcx""* + &c.) dx
dfa+bx'* + cx™+&c )''
.. ^^ ^ ^ = r(nbx'»'+mcx'°' + &c.)(a+bx'^+&c.)'>..(g)
the Limiting Ratio of the Finite Differences A(a + bx'^4cx™ + &c.),
A X, that is the Ratio of the Differentials ofa + bx'^4cx'" + &c.,
and X.
A+Bx"+Cx°' + &c.
15. Required the Ratio of the Differentials ^ a4bx' + Cxi^4&c
and X, or the Limiting Ratio of their Finite Differences.
Let L be the Limit required, and L + 1 the varying Ratio. Then "^
__ A + B (x 4 A x)*^ + C (x + A x)" + &c. A + B X ° + &c.
^ "*■ ^  a + b(x + Ax)' + c(x + Ax)/* + &c. ~ a+ bx' +&C.
Book I.] NEWTON'S PRINCIPIA. 7
which being expanded by the Binomial Theorem, and properly reduced
gives
L X ( a 4 b X' + &c.)' + L X JP. Ax + Q (A x)* +&c. + 1 X fa+bx' + &c.
+ P. A X + Q (A x)^ + &C.J =(a+bx' + cx^ + &c.) X (nBx"'
+ m C X «» + &c.) — (A + Bx"+Cx'"+ &c.) X (v b x   '
+ /t c X A* 1 + &c) + P'. A X + Q' (A x) 2 + &c.
P, Q, F, Q' &c. being coefficients of a x, (a x) "^ &c. and independent of
them.
Now equating those homogeneous terms which are independent of the
powers of a x, we get
L(a + bx' + &c.)^ = (a + bx' + &c.)(nBx''»4mCx'"'+&c.)
— (A + Bx" + Cx'^ + &c.) — (cbx'' + /icx/*' + &c.)
J .. ^ + B x^+Cx^ + Sc c. , ^ „
and puttmg u = a"+ b x ' + cxm + &cr ^« ^^^^ ^^^^^
d u d u
g~^ = L, and therefore g^ =
(a+bx'+&c.)(nBx+"'mCx^'+&c.)(A+Bx"+&c.)(vbx''+AtcxM^+&c.)
(a + bx' + cx'' + &c.) *
the Ratio required.
16. Hence and from 1 1 we have the Ratio of the Differentials of
(A + Bx+Cx"' + &c.) P
(a4bx'+ cx/^ + &c.) 1 ^^ ^ » ^"^ "* short, from what has al
ready been delivered it is easy to obtain the Ratio of the Differentials of
any Algebraic Function "whatever of one variable and of that variable.
N. B. By Function of a variable is meant a quantity anyhow involving
that variable. The term was first used to denote the Powers of a quan
tity, as X % x ^, &c. But it is now used in the general sense.
The quantities next to Algebraical ones, in point of simplicity, are Ex
ponential Functions; and we therefore • proceed to the investigation of
their Differentials.
17. Required the Ratio of the Differentials of ^^ and x ; or the Limit
ing Ratio of their Differences.
Let L be the required Limit and L + 1 the varying Ratio ; then
A(a^) 3^ + ^* — a*
L + 1 =
AX AX
a^'^— 1
= a'^ X
A X
8 A COMMENTARY ON [Sect. 1.
But since
ay = (l+a — i)y
y. (y — 1)
= 1 + y (a — 1) + •'^•^2 \ (a  1) 2 +
y.(y_l)(y2)
273 (a — 1) + &c.,
it is easily seen that the coefficient of y in the expansion is
, (aJip (al)3
a — 1 — 2 + g — &c.
Hence
a* (a— 1)^ (a^l)3
^ + ^ = Z^ {(a— 1— —2 + 3 — ^^•) A X + P (^x)2 + &c.}
and equating homogeneous quantities, we have
d. (a^) ^ (a_l)2 (a_l)3
= A a^ (h)
or the Ratio of the Differentials of any Exponential and its exponent is
equal to the product of the Exp07iential and a constant Quantity.
Hence and from the preceding articles, the Ratio of the Differentials of
any Algebraic Function of Exponentials having the same variable index,
may be found. The Student may find abundance of practice in the Col
lection of Examples of the Differential and Integral Calculus, by Messrs.
Peacock, Herschel and Babbage.
Before we proceed farther in Diiferentiation of quantities, let us inves
tigate the nature of the constant A which enters the equation (h).
For that purpose, let (the two first terms have been already found)
a^= 1 +Ax + Px2 + Qx3+&c.
Then, by 13,
d (a ^)
^^ = A + 2Px + 3Qx'* + 4Rx' + &c.
But by equation (h)
d (a^)
1 also = A a *
= A + A2x + APx2 + A.Qx3 + &c.
.. A42Px + 3Qx24.4Rx3 + &c. rrA + A'^x + APx^ + ficc.
and equating homogeneous quantities^ we get
2 P = A % 3 Q = A P, 4 R = A Q, &c. = &c.
Book I.] NEWTON'S PRINCIPIA. 9
whence
P= 2»Q 3 2. 3'^ = ~i~ = 27374 ^^' ^^'
Therefore,
A' A' A*
a^=l + Ax + 2x'' + 273x' + 37374 x ^ + &c.
Again, put A x = 1, then
X 111
a = 1 + 1 + 2 + 2T3 + 27171 + &^
= 2.718281828459 as is easily calculated
= e
by supposition. Hence
loff. a
A = 41 (k)
(a^l)'' (a 1)3 ^ log. a
.. a  1 2 + 3 &c. = 13^3 = 1. a
for the system whose base is e, 1 being the characteristic of that Bystem,
This system being that which gives
(e1)* (el)3
€ 1 2 + 3 &C* — ^
is called Natural from being the most simple.
Hence the equation (h) becomes
d(a^)
17 a. JRequired the Ratio of the Differentials of 1 (x) and x.
Let 1 X = u. Then e " = x
.. d X = d (e '») = 1 e X e '» d u = e " d u, by 16
d(lx) 11
.. "dlT = iT =........ (m)
Ix
In any other system whose base is a, we have log. (x) = j^.
d loff. X 1 1
••• "dV = U ^ X (")
We are now prepared to differentiate any Algebraic, or Exponential
Functions of Logarithmic Functions, provided there be involved but
one variable.
Before we differentiate circular functions, viz. the sines, cosines, tan
gents, &c., of circular arcs, we shall proceed with our comments on the
text as far as Lemma VIII.
10 A COMMENTARY ON [Sect. I.
LEMMA II.
18. In No. 6, calling L and L' Limits of the circumscribed and inscribed
rectilinear figures, and L + 1, L,' + V any other values of them, whose
variable difference is D, the absolute equality of L and L' is clearly de
monstrated, without the supposition of the bases A B, B C, C D, D E,
being infinitely diminished in number and augmented in magnitude. In
the view there taken of the subject, it is necessary merely to suppose them
variable.
LEMMA IIL
19. This Lemma is also demonstrable by the same process in No. 6,
as Lemma II.
Cor. 1. The rectilinear figures cannot possibly coincide with the curvi
linear figure, because the rectilinear boundaries albmcndoE,
aKbLcMdDE cut the curve a b E in the points a, b, c, d, E in
finite angles. The learned Jesuits, Jacquier and Le Seur, in endeavour
ing to remove this difficulty, suppose the four points a, 1, b, K to coincide,
and thus to form a small element of the curve. But this is the language
of Indivisibles, and quite inadmissible. It is plain that no straight line,
or combination of straight lines, can form a curve line, so long as we un
derstand by a straight line " that which lies evenly between its extreme
points," and by a curve line, " that which does not lie evenly between its
extreme points ;" for otherwise it would be possible for a line to be
straight and not straight at the same time. The truth is manifestly this.
The Limiting Ratio of the inscribed and circumscribed figures is that of
equality, because they continually tend to a fixed area, viz, that of the
given intermediate curve. But although this intermediate curvilinear
area, is the Limit towards which the rectilinear areas continually tend and
approach nearer than by any difierence ; yet it does not follow that the
rectilinear boundaries also tend to the curvihnear one as a limit. The
rectilinear boundaries are, in fact, entirely heterogeneous with the interme
diate one, and consequently cannot be equal to it, nor coincide therewith.
We will now clear up the above, and at the same time introduce a strik
ing illustration of the necessity there exists, of taking into consideration
the nature of quantities, rather than their evanescence or infinitesimaUty.
Book L]
NEWTON'S PRINCIPIA.
11
Take the simplest example of Lemma II., in the case of the right
angled triangle a E A, having its two legs A a, A E equal.
The figure being constructed as in the text of Lemma II, it fol
lows jfrom that Lemma, that the Ultimate Ratio of the inscribed and cir
cumscribed figures is a ratio of equality ; and moreover it would also
follow from Car. 1. that either of these
coincided ultimately with the triangle
a E A. Hence then the exterior boundary
albmcndoE coincides exactly with
a E ultimately, and they are consequently
equal in the Limit. As we have only
straight lines to deal with in this example,
let us try to ascertain the exact ratio of
a E to the exterior boundary.
If n be the indefinite number of equal
bases A B, B C, &c., it is evident, since
A a = A E, that the whole length of
albmcndoE = 2nxAB. Also since
K
\
n
1
b
\
L
\
n
c
\
M
\
o
d
^
B
C
D
E
= &c.
b = b c
= V a P + b 1 * = V 2. A B, we have a E = n V 2. A B.
Consequently,
albmcndoE:aE: ; 2: V2: : V~2 : 1.
Hence it is plain the exterior boimdary cannot possibly coincide with
a E. Other examples might be adduced, but it must now be sufficiently
clear, that Newton confounded the ultimate equality of the inscribed and
circumscribed figures, to the intermediate one, with their actual coinci
dence, merely from deducing their Ratios on principles of approximation
or rather of Exhaustion, instead of those, as explained in No. 6 ; which
relate to the homogeneity of the quantities. In the above example the
boundaries being heterogeneous inasmuch as they are incommensurable,
cannot be compared as to magnitude, and unless lines are absolutely equal,
it is not easy to believe in their coincidence.
Profound as our veneration is, and ought to be, for the Great Father
of Mathematical Science, we must occasionally perhaps find fault with
his obscurities. But it shall be done with great caution, and only with
the view of removing them, in oi'der to render accessible to students in
general, the comprehension of " This greatest monument of human ge
nius."
20. Car. 2. 3. and 4. will be explained under Lemma VII, which re
lates to the Limits of the Ratios of the chord, tangent and the arc.
12
A COMMENTARY ON
[Sect. L
LEMMA IV.
21. Let the areas of the parallelograms inscribed in the two figures be
denoted by
P, Q, R, &c.
p, q, r, &c.
respectively ; and let them be such that
P : p : : Q : q : : R : r, &c. : : m : n.
Then by compounding these equal ratios, we get
P + Q+R + :p + q4.r + ::m:n
But P + Q + R . . . . and p + q + r + . . . . have with the curvili
near areas an ultimate ratio of equality. Consequently these curvilinear
areas are in the given ratio of m : n.
Hence may be found the areas of certain curves, by comparing their
incremental rectangles with those of a known area.
Ex. 1. Required the area of the common Apollonian parabola comprised
between its vertex and a given ordinate.
Let a c E be the parabola,
whose vertex is E, axis E A and
LatusRectum = a. Then A A'
being its circumscribing rectan
gle, let any number of rectan
gles vertically opposite to one
another be inscribed in the areas
a E A, a E A', viz. A b, b A' ;
B c, c B', &c.
K
^^^^^^
1
b
^\^
m
c
n
d
\
A'
B'
C
And since
A B
A b = A K. A B
A'b = A' 1. A' B' = ^^^^. A' B'
D
E
from the equation to the parabola.
A b g. AB
•'•A'b  AK. A'B'
Also
or
(Aa)*— •Bb'^rzaxAE — axBE = aXAB
(A a + B b) X A' B' = a X A B
NEWTON'S PRINCIPIA.
Book I.]
a X AB ^ „ ,
•*• A^ B' = A a + B b
A b _ Aa + Bb 2Bb + Ka
*• A' b 
13
Ka
= 2 + Bb
Bb  B b
. . A b
Hence, since in the Limit ~r~^ becomes fixed or of the same nature with
the first term, we have
A b
A'b
= 2
ultimately.
And the same may be shown of all other corresponding pairs of rec
tangles ; consequently by Lemma IV.
a E A : a E A' : : 2 : 1
.*. a E A : rectangle A A' : : 2 : 3.
or the area of a 'parabola is equal to trvo thirds of its circumscribing rec
tangle.
Ex. 2. To compare the area of a semielUpse "joith that of a semicircle
described on the same diameter.
^.^^^
r
Ql
■^<
y
^^^^
X
/p'
r'
■"^X
r
^
M sr
B
Taking any two corresponding inscribed rectangles P N, P' N ; we
have
P N : F N : : P M : P' M : : a : b
a and b being the semiaxes major and minor of the ellipse ; and all other
corresponding pairs of inscribed rectangles have the same constant ratio ;
consequently by Lemma IV, the semicircle has to the semielUpse the ratio
of the major to the minor axis.
As another example, the student may compare the area of a cycloid
with that of its circumscribing rectangle, in a manner very similar to
Ex. 1.
This method of squaring curves is very limited in its application. In
the progress of our remarks upon this section, we shall have to exhibit a
general way of attaining that object.
■^_
14 A COMMENTARY ON [Sect. I.
LEMMA V.
22. For the definition of similar rectilinear figures, and the truth of this
Lemma as it applies to them, see Euclid's Elements B. VI, Prop. 4, 19
and 20.
The farther consideration of this Lemma must be deferred to the ex
planation of Lemma VII.
LEMMA VL
23. In the demonstration of this Lemma, " Continued Curvature" at
any point, is tacitly defined to be such, that the arc does not make nsoith the
tangent at that point, an angle equal to ajinite rectilinear angle.
In a Commentary on this Lemma if the demonstration be admitted,
any other definition than this is plainly inadmissible, and yet several of
the Annotators have stretched their ingenuity to substitute notions of
continued curvature, wholly inconsistent with the above. The fact is,
this Lemma is so exceedingly obscure, that it is difficult to make any
thing of it. In the enunciation, Newton speaks of the angle betiaeen the
chord and tangent ultimately vanishing, and in the demonstration, it is
the angle between the arc and tangent that must vanish ultimately. So
that in the Limit, it would seem, the arc and chord actually coincide.
This has not yet been established. In Lemma III, Cor. 2, the cointi
dence ultimately of a chord and its arc is implied ; but this conclusion by
no means follows from the Lemma itself, as may easily be gathered from
No. 19. The very thing to be proved by aid of this Lemma is, that the
Ultimate Ratio of the chord to the arc is a ratio of equality, it being
merely subsidiary to Lemma VII. But if it be already considered that
they coincide, of course they are equal, and Lemma VII becomes nothing
less than " argumentum in circulo."
Newton introduces the idea of curves of " continued curvature," or
such as make no angle with the tangent, to intimate that this Lemma does
not apply to curves of noncontinued curvature, or to such as do make a
Jinite angle isoith the tangent. At least this is the plain meanmg of his
words. But it may be asked, are there any curves whose tangents are
inclined to them ? The question can only be resolved, by again admitting
Book I.]
NEWTON'S PRINCIPIA.
15
the arc to be ultimately coincident with the chord ; and by then showing,
that curves may be imagined whose chord and tangent ultimately shall be
inclined at a finite angle. The Ellipse, for instance, whose minor axis
is indefinitely less than its major axis, is a curve of that kind ; for taking
the tangent at the vertex, and putting a, b, for the semiaxes, and y, x, for
the ordinate and abscissa, we have
b^
y2 = — , X (2ax — x'')
and
b /2 a
= a\/T"
X 1 =
V 2a
a s/ X
.*. since b is indefinitely smaller than a V x, x is indefinitely greater than
y, and supposing y to be the tangent cut off by the secant x parallel to
the axis, x and y are sides of a right angled a, whose hypothenuse is the
chord. Hence it is plain the Z opposite x is ultimately indefinitely
greater than the z_ opposite to y. But they are together equal to a right
angle. Consequently the angle opposite x, or that between the chord and
tangent, is ultimately finite. Other cases might be adduced, but enough
has been said upon what it appears impossible to explain and establish as
logical and dhect demonstration. We confess our inabihty to do this,
and feel pretty confident the critics will not accompUsh it
24. Having exposed the fallacy of Newton's reasoning in the proof of
this Lemma, we shall now attempt something by way of substitute.
Let AD be the tangent to the curve at the
'point A, and A B its chord. Then if ^ be
supposed to move indefinitely near to A, the
angle BAD shall indefinitely decrease, pro
vided the curvature be not indefinitely great.
Draw R D passing through B at right an
gles to AB, and meeting the tangent AD and
normal A R in the points D and R respective
ly. Then since the angle BAD equals the
angle A R B, if A R B decrease indefinitely
when B approaches A ; that is, if A R be
come indefinitely greater than, A B; or
which is the same thing, if the curvatiue at A, be not indefinitely great ;
the angle BAD also decreases indefinitely. Q. e. d.
We have already explained, by an example in the last article, what is
16 A COMMENTARY ON [Sect. I.
meant by curvature indefinitely great. It is the same with Newton's ex
pression " continued curvature." The subject will be discussed at length
under Lemma XI.
As vanishing quantities are objectionable on account of their nothing
ness as it has already been hinted, and it being sufficient to consider va
riable quantities, to get their limiting ratios, as capable of indefinite diminu
tion, the above enunciation has been somewhat modified to suit those
views.
LEMMA VII.
25. This Lemma, supposing the two preceding ones to have been fully esta
blished, would have been a masterpiece of ingenuity and elegance. By
the aid of the proportionality of the homologous sides of similar curves,
our author has exhibited quantities evanescent by others of any finite
magnitude whatever, apparently a most ingenious device, and calculated
to obviate all objections. But in the course of our remarks, it will be
shown that Lemma V cannot be demonstrated without the aid of this
Lemma.
First, by supposing A d, A b always finite, the angles at d and b and
therefore those at D and B which are equal to the former are virtually
considered finite, or R D cuts the chord and tangent at finite angles.
Hence the elaborate note upon this subject of Le Seur and Jacquier is
rendered valueless as a direct comment.
Secondly. In the construction of the figure in this Lemma, the de
scription of a figure similar to any given one, is taken for granted. But
the student would perhaps like to know how this can be effected.
Lemma V, which is only enunciated, from being supposed to be a mere
corollary to Lemma III and Lemma IV, would afford the means immedi
ately, were it thence legitimately deduced. But we have clearly shown
(Art. 19.) that rectilinear boundaries, consisting of lines cutting the inter
mediate curve ultimately atjinite angles, cannot be equal ultimately to the
curvilinear one, and thence we show that the boundaries formed by the
chords or tangents, as stated in Lemma III, Cor. 2 and 3, are not ulti
mately equal, by consequence of that Lemma, to the curvilinear one.
Newton in Cor. 1, Lemma III, asserts the ultimate coincidence, and
therefore equality of the rectilinear boundary whose component lines cut
the curve at finite angles, and thence would establish the succeeding cor
Book I.l NEWTON'S PRINCIPIA. 17
ollaries a fortiori. But the truth is that the curvilinear boundary is the
limit, as to magnitude, or length, of the tangential and chordal bounda
ries ; although in the other case, it is a limit merely in respect of area.
Yet, we repeat it, that Lemma V cannot be made to follow from the
Lemmas preceding it. According to Newton's implied definition of simi
lar curves, as explained in the note of Le Seur and Jacquier, they are the
curvilinear limits of similar rectilinear fgures. So they might be consi
dered, if it were already demonstrated that the limiting ratio of the chord
and arc is a ratio of equality ; but this belongs to Lemma VII. Newton
himself and all the commentators whom we have perused, have thus
committed a solecism. Even the best Cambridge MSS. and we have
seen many belonging to the most celebrated private as well as college tu
tors in that learned university, have the same error. Nay most of them
are still more inconsistent. They give definitions of similar curves wholly
diiFerent from Newton's notion of them, and yet endeavour to prove
Lemma V, by aid of Lemma VII. For the verification of these asser
tions, which may else appear presumptuously gratuitous, let the Cantabs
peruse their MSS. The origin of all this may be traced to the falsely
deduced ultimate coincidence of the curvilinear and rectilinear boundaries,
in the corollaries of Lemma III. See Art. 19.
We now give a demonstration of the Lemma without the assistance of
similar curves, and yet independently of quantities actually evanescent. '
By hypothesis the secant R D cuts the chord and tangent at finite an
gles. Hence, since
A + B + D = 180°
.. B + D = 180° — A
or L h 1 lL'l 1' = 180° — A
L and \J being the limits of B and D and 1, V their variable parts as in
Art. 6 ; and since by Lemma VI, or rather by Art. 24, A is indefinitely
diminutive, we have, by collecting homogeneous quantities
L + L' = 180°
But A B, A D being ultimately not indefinitely great, it might easily
be shown from Euclid that L = L', and ••. A B = A D ultimately, (see
Art. 6 ) and the intermediate arc is equal to either of them.
18
A COMMENTARY ON
[Sect. I.
OTHERWISE,
If we refer the curve to its axis,
A a, B b being ordinates, &c. as
in the annexed diagram. Then, _
by Euclid, we have
AD« = AB^ + BD^ + 2BD.Bd
A D
= 1 + B D.
B D + 2 B d
"AB^"^ AB^
Now, since by Art. 24 or Lemma VI, the z. B A D is indefinitely less
than either of the angles B or D, .. B D is indefinite compared with A B
AD
or A D. Hence L being the limit of . — p, and 1 its variable part, if we
extract the root of both sides of the equation and compare homogeneous
terms, we get,
L = 1 or &c. &c.
26. Having thus demonstrated that the limiting Ratio of the chord, arc
and tangent, is a ratio of equality, "when the secant cuts the chord and tangent
at FINITE angles, we must again digress from the main object of this work,
to take up the subject of Article 17. By thus deriving the limits of the rati
os of the finite differences of functions and their variables, directly from the
Lemmas of this Section, and giving to such limits a convenient algorithm
or notation, we shall not only clear up the doctrine of limits by nume
rous examples, but also prepare the way for understanding the abstiniser
parts of the Principia. This has been before observed.
Required to find the Limit of the Finite Differences of the sine of a cir
cidar arc and of the arc itself, ^or the Ratio of their Differentials.
Let X be the arc, and a x its finite variable increment. Then L being
the limit required and L  1 the variable ratio, we have
L + l =
A sin. X _ sin. (x  a x) — sin. x
AX AX
_ sin. X. cos. (a x) + cos. x. sin. (a x) — sin. x
A X
sin. X
A X
sin. (a x) sin.x. cos. ax
= cos. x. ^ A
A X AX
Now by Lemma VII, as demonstrated in the preceding Article, the li
mit of
sm. A X
A X
.  J cos. (a x)
is 1, and i ,
A X
SUl. X
A X
have no definite limits.
Book I.] NEWTON'S PRINCIPIA. 19
Consequently putting
sin. (a x)
COS. X. ^ = COS. X + r,
AX '
we have
sin. X. COS. A X sin. x
L + 1 = COS. X + 1' +
AX AX
and equating homogeneous terms
L = COS. X
or adopting the differential symbols
d. sin. X
— T = cos X
d X
or
d sin. X = d X. cos. x
27. Hence and from the rules for the differentiation of algebraic, expo
nential, &c. functions, we can differentiate all other circular functions of
one variable, viz. cosines, tangents, cotangents, secants, &c.
Thus,
}
(a)
or
or
dsin.(x)
d. cos. X
— dx
d. cos. X
= ^°^ G^) =
<2 ■■ ^^"^^
= sin. X
= — sm. X
d ^ (b)
d. COS. X = — d X. sin. x
Again, since for radius 1, which is genei'ally used as being the most simple,
1
1 + tan. ^ X = sec. * x =
2 tan. X. d. tan. x = d.
cos. ^ X
1 — 2 cos. X. d. COS. X
cos. " X COS. X
See 12 (d). Hence and from (b) immediately above, we have
J d X. sin. X
tan. X. d. tan. x = , —
COS. ^ X
.'. d. tan. X = d X. 3— (c)
COS. ^ X
Again,
cot. X =
tan. X
B2
20 A COMMENTARY ON [Sect. I.
Therefore,
1 ^ J 1 — d. tan. X ,„ ,,
d. cot. X = d. — = r (12. d)
tan. X tan. ^ x ^ '
— d X — d X
tan. * X. COS. * x sin. * x
Again,
(d)
sec X =
COS. X
dj 1 — d COS. X /,« j\
. sec. X = d. = 5 (12. d)
COS. X COS. '^ X ^
d X. sin. X
COS.
(e)
and lastly since cosec. x = sec. ( — x)
we have
d. (~ — x) Sin. ( — x^
J ji f^ \ V2 / V2 /
d. cosec. X = d. sec f — — xj =
COS.
(i'')
— d X. COS. x
(0
sin. '■ X
Any function of sines, cosines, &c. may hence be differentiated.
28. In articles 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 26 and 27, are to
be found forms for the differentiation of any function of one variable,
whether it be algebraic, exponential, logarithmic, or circular.
In those Articles we have found in short, the limit of the ratio of the
first difference of a function, and of the first difference of its variable.
Kow suppose in this first difference of the function, the variable x should
be increased again by a x, then taking the difference between the first
difference and what it becomes when x is thus increased, we have the dif
ference of the first difference of a function, or the second difference of a
function, and so on through all the orders of differences, making a x al
ways the same, merely for the sake of simplicity. Thus,
A (x ^) = (x f A x) ^ — X ^
= 3x'^AX + 3xAX^ + AX^
and A* (x)' = 3 (x f Ax) = AX H 3 (x + ax) AX^ + AX^ — Sx^Ax
3 X A X* A X^
= 3. 2xax= + 3ax'
Book L] NEWTON'S PRINCIPIA. 21
denoting by a * the second difference.
Hence,
— ^P = 3. 2. X + 3 A X
A X^ '
and if the limiting ratio of A * (x ^) and Ax*, or the ratio of the second
differential of x % and the square of the differential of its variable x, be
required, we should have
L + l = 3. 2. X + 3AX
and equating homogeneous terms
.\^ = L = 3. 2. X
d x^
In a word, without considering the difference, we may obtain the se
cond, third, &c. differentials d ^ u, d ^ u, &c. of any function u of x im
mediately, if we observe that ^ — is always a function itself of x, and
make d x constant. For example, let
u = ax" + bx™ + &c.
Then, from Art. 13. we have
3 — = nax°' + mbx"* + &c.
d X
^'(dH) d(du) d«u., ^ . .
= n. (n— l)ax'»« + m (m— l)bx "» ^ j &c.
Similarly,
T— ^ = n. (n — 1). (n — 2) a x"  ' + &c.
&c. = &c.
Having thus explained the method of ascertaining the limits of the ra
tios of all orders of finite differences of a function, and the corresponding
powers of the invariable first difference of the variable, or the ratios of the
differentials of all orders of a function, and of the corresponding power
of the first differential of its variable, we proceed to explain the use of
these limiting ratios, or ratios of differentials, by the following
B3
22
A COMMENTARY ON
[Sect. I.
APPLICATIONS
OF THE
DIFFERENTIAL CALCULUS.
29. Let it be required to draw a tangent to a given curve at any given
point of it.
Let P be the given point, and A M'
being the axis of the curve, let P M
= y, A M = x,be the ordinate and
abscissa. Also let P' be any other
point; draw P N meeting the ordi
nate P' M' in N, and join P P^ Now
let T P R meeting M' P' and M A in
R and T be the tangent required.
Then since by similar triangles
F N : P N : : P M
MT'
.. M T' = M T + T T' = y.
A X
Now y being supposed, as it always is in curves, a fimction of x, we have
seen that whether that function be algebraic, exponential, &c.
A X . . . d x .
in the limit, or 5 — is always a definite function of x. Hence putting
Ay
we have
^ = ^ + 1
A X
a7
dy
M
T + TT— y(^^ + l)
(e)
the point T will be
and equating homogeneous terms,
MT = ydiS
dy
which being found from the equation to the curve,
known, and therefore the position of the tangent P T. M T is called
the subtangent.
Ex. 1. In the common parabola,
y* = a X
Book I.]
NEWTON'S PRINCIPIA.
Therefore,
and
dx
dy
2y
a
MT :
2y2
~ a
= 2x
or the subtangent M T
is equal to twice the abscissa.
Ex. 2. In the
ellipse,
y^ =
b^
:^(a^x^)
23
and" it will be found by differentiating, &c. that
— (a^ — x^)
M T = ^^
Ex. 3. In the logarithmic curve,
y = a *
.♦. M T = A
1 a
which is therefore the same for all points.
The above method of deducing the expression for the subtangent is
strictly logical, and obviates at once the objections of Bishop Berkeley
relative to the compensation of errors in the denominator. The fact is,
these supposed errors being different in their very essence or nature from
the other quantities with which they are connected, must in their aggre
gate be equal to nothing, as it "has been shown in Art. 6. This ingenious
critic calls P' R = z ; then, says he, (see fig. above)
^^ = dy + z accurately ;
whereas it ought to have been
y A X ^y
MT =
Ay + z Ay
+
AX ~ A X
A y
the finite differences being here considered. Now in the limit, 7— becomes a
A X
d y
definite function of x represented by 7—' Consequently if 1 be put for
'^ y
the variable part, of ~~, we have
24 A COMMENTARY ON [Sect. I.
dx + ^ + A X
and it is evident from Lemma VII and Art. 25, that z is indefinite com
z d y
pared with ax. .*. t— is indefinite compared with M T, j— , and y ;
and 1 is also so ; hence
MT.  + (l + ^)MT = y
gives
which proves generally for all curves, what Berkeley established in the
case of the common parabola ; and at the same time demonstrates, as had
been already done by using T T' instead of P' R, incontestably the ac
curacy of the equation for the subtangent.
30. If it were required to draw a tangent to any point of a curve, re
ferred to a center by a radiusvector g and the £. 6 which g describes by
revolving round the fixed point, instead of the rectangular coordinates
X, y ; then the mode of getting the subtangent will be somewhat different.
Supposing X to originate in this center, it is plain that
X = I COS. 6 )
y = g sin. 6 J
and substituting for x, y, d x, d y, hence derived in the expression (29.
e.) we have
. d f cos. tf — f d ^ sin. 6
M T = g sm. 6 X dTihTT+TdT^^tf . . . . (f)
Ex. In the parabola
_ 2a
^ ~ 1 — cos. 6 '
where a is the distance between the focus and vertex, or the value of g at
the vertex. Then substituting we get, after proper reductions
f ■« „, 1 + COS. 4
and the distance from the focus to the extremity of the subtangent is
/I + COS. 6 cos. 6 \
MTs cos. ^ = 2 a [i^Z^^^^ " 1  cos. d)
Booic L] NEWTON'S PRINCIPIA. 85
 ^^ __
— 1 COS. d "^ ^'
as is well known.
30. a. The expression (f) being too complicated in practice, the following
one may be substituted for it.
Let P T be a tangent to the
curve, refei'red to the center S,
at the point P, meeting S T
drawn at right angles to S P,
in T ; and let P' be any other
point. Join P P' and produce
it to T', and let T P be pro
duced to meet S P' produced in
R, &c. Then drawing P N parallel to S T, we have
PN
ST' = ST + TT' = j^, X S F
But
and
P N = § tan. A ^, S P' = ^ + A f
NF = SPSN = , + A,,,3.(^,).
Therefore, substituting and equating homogeneous terms, after having
applied Lemma VII to ascertain their limits, we get
(g)
O A
 dg ' • ' '
•
• • •
F.x. L
In the
spiral of Archimedes
f = a <?;
.•.ST = ^
we
have
Ex.2.
In the
hyperbolic spiral
a
.. S T = — a
31. It is sometimes useful to know the angle between the tangent and
axis.
_ _ PM dy
TT=MT = dx (^)
See fig. to Art. 29.
26
A COMMENTARY ON
[Sect. I.
Again, in fig. Art. 30 a.
SP dg
T^T = ST = g"dM (^)
32. It is frequently of great use, in the theory of curves and in many
other collateral subjects, to be able to expand or develope any given func
tion of a variable into an infinite series, proceeding according to the
powers of that variable. We have already seen one use of such develop
ments in Art. 17. This may be effected in a general manner by aid of
successive differentiations, as follows.
If u = f (x) where f (x) means any function of x, or any expression
involving x and constants ; then, as it has been seen,
d u =r u' d x
(u' being a new function of x)
Similarly
d u' = u'' d X
But
d u''  u'" d X
&c. = &c.
X dx — d^x X du
(6 k)
and (d x) 2 by d X %
^d x' d X '
&c. = &c.
denoting d. (d u), d. (d x) by d " u, d ^ x,
according to the received notation ;
Or, (to abridge these expressions) supposing dx constant, and .*. d^ x = 0,
d^u
dx
du' =
du
••• dl^ = "'
(Pu
dx^ = ^'
d^u
dx^  "
&c. = &c.
which give the various orders of fluxions required.
Ex. 1. Let u = X "
Then
du
d^rrnx^
(a;
d. = n. (nl)x«'
Book I.] NEWTON'S PRINCIPIA. 27
j^3 = n. (n— 1). (n_2)x — '
&c. = &c.
d° u
j^ = n. (n — 1). (n — 2) 3. 2. 1.
Ex. 2. Let u = A + B X + C x* + D X 3 + E X * + &c.
Then,
j^ = B + 2Cx + 3Dx* + 4Ex3 + &c.
jY2 = 2 C + 2. 3 D X + 3. 4 E x2 + &c.
J^3 = 2. 3 D + 2. 3. 4 E X + &c.
&c. = &c.
Hence, if z^ be known, and ^e coefficients A, B, C, D, &c. be un
known, the latter may be found ; for if U, U', U'', U'", &c. denote the
dud'^ud^u
values of u, j — , j — „ , t — , , &c.
' d x» d x^'d x^'
when X = 0, then
A = U, B = U, C = ^ U", D = 2;^; U "', E = ■^^^ W",
&c. = &c.
and by substitution,
u = U + U' X + U" Y + U"' O + ^"^ (^^
This method of discovering the coefficients is named (after its inventor),
MACLAURIN'S THEOREM.
The uses of this Theorem in the expansion of functions into series are
many and obvious.
For instance, let it be required to develope sin. x, or cos. x, or tan. x,
or 1. (1 + x) into series according to the powers of x. Here
u = sin. X, or = cos. x, or = tan. x, or = 1. (1 + x),
du II
'• a~^ = ^os. x, or = — sm. x, or = ^^2—' or = j^ ^
d'u ^ 2 sin. X 1
d^2 = — sm. X, or = — cos. x, or = ^^^3' or = — jr+^'
28 A COMMENTARY ON [Sect. I.
d^u 2 + 4sin.*x 2
5^3 = — COS. X, or = sin. x, or = ^^^;t^^ » or = (T+I53
&c. = &c.
U =0, or = 1, or = 0, or =
U' = 1» or = 0, or = 1, or = 1
U" =0, or = — 1, or = 0, or = — I
U"' = — 1, or = 0, or = 2, or = 2
&c ;= &c.
Hence
sin. X = X ^27s + 2. 3. 4. 5 — ^^•
x* x^
COS. X = 1 — g + iTsTi" ~" ^^•
x^ 2x^ 17 x^
tan. X = X + 3 + 37^ + 3T5I7 + &c.
x^ x^
L (1 + x) = X — 2 + 3 — &c.
Hence may also be derived
TAYLOR'S THEOREM.
For let
f(x) = A + Bx + Cx* + Dx^ + Ex* + &c.
Then
f (x + h) = A + B. (x + h) + C. (x + h) ' + D . (x + t) ' + &c.
= A + Bx + Cx2 + Dx3 + &c.
+ (B + 2 Cx + 3Dx)h
+ (C + 3Dx + 6 Ex«) h*
+ (D + 4 Ex + 10 Fx*) h'
+ &c.
d. f(x) d.^f(x) h*
d»f(x) h^ ^ '
+ dir^2:3 + &^ <^)
the theorem in question, which is also of use in the expansion of series.
For the extension of these theorems to functions of two or more varia
bles, and for the still more effective theorems of Lagrange and Laplace,
the reader is referred to the elaborate work of Lacroix. 4to.
Having shown the method of finding the differentials of any quanti
Book I.] NEWTON'S PRINCIPIA. 29
ties, and moreover, entered iii a small degree upon the practical applica
tion of such differentials, we shall continue for a short space to explain
their farther utility.
33. Tojind the MAXIMA and Minima of quantities.
If a quantity increase to a certain magnitude and then decrease, the
state between its increase and decrease is its maximum. If it decrease
to a certain limit, and then increase, the intermediate state is its mi
nimum. Now it is evident that in the change from increasing to decreas
ing, or vice versa, which the quantity undergoes, its differential must have
changed signs from positive to negative, or vice versa, and therefore (since
moreover this change is continued) have passed through zero. Hence
W/ien a quantity is a MAXIMUM or MINIMUM, its differential z= 0. . . (a)
Since a quantity may have several different maxima and minima, (as for
instance the ordinate of an undulating kind of curve) it is useful to have
some means of distinguishing between them.
34. To distinguish betisoeen Maxima and Minima.
Lemma. To show that in Taylor's Theorem (32. c.) any one term can
be rendered greater than the sum of the succeeding ones, supposing the
coefficients of the powers of h to be finite.
Let Q h " ~ ' be any term of the theorem, and P the greatest coefficient
of the succeeding terms. Then, supposing h less than unity,
P h" (1 + h + h + . . . .minfin.) = Ph" X ■ ^.
is greater than the sum ( S) of the succeeding terms. But supposing h to
decrease in infin.
1
Ph." I ^ = P h " ultimately. Hence ultimately
Ph°> S
Now
Q h °  ' : P h ° : : Q : P b,
and since Q and P are finite, and h infinitely small ; therefore Q is > P h,
Hence Q h "  ' is > P h >», and a fortiori > S.
Having established this point, let
u = f(x)
be the function whose maxima and minima are to be determined ; also
when u = max. or min. let x = a. Then by Taylor's Theorem
., u\ c, ^ du , d^u h^ d'u h^
f(ah) = f(a)_p^h + g^. __g^. — + &0.
30 A COMMENTARY ON [Sect. I.
and
and since by the Lemma, the sign of each term is the sign of the sum of
that and the subsequent terms,
.. f (a — h) = f (a) — i^. M
^ ^ d a
f(a + h) =f(a) + ^. N
Now since f (a) = max. or min. f (a) is > or < than both f (a — h)
and f (a + h), which cannot be unless
d u ^
T = 0.
d a
Hence
d^u
f(ah)=f(a)+^. MO
f(a+h) = f(a)+^. W)
d a'
and f (a) is max. or min. or neilher, according as f (a) is >, •< or = to
both f (a — h) and f (a + h), or according as
d^u .
3 —  IS negative^ positive, or zero
If it be zero as well as ; — , we have
d a
f(ah) = f(a)^. MM
f(a + h)=f(a) + i^" N 3
and f (a) cannot = max. or min. unless
d^u „
d7^ = ^'
which being the case we have
d*u
f(a — h) = fa + ^. M''0
f(a + h) = fa + il^. N)
da'
and as before,
BookL] NEWTON'S PRINCIPIA. 31.
f (a) IS max. or min. or neither^ according as ^ —  is negative, positive, or
zero, and so on continually.
Hence the following criterion.
If in u = f (x), y— = 0, the resulting value of x shall give u = MAX.
or MIN. or NEITHER, according as ^ — „ is negative, positive, or zero.
If  — = 0, ; — „ = 0, and :; — , = 0, then the resulting value of u
•^dx dx^ dx^
d*u .
shall be a MAX., min. or neither according as ^ — ^ is negative, po
sitive, or ZERO ; and so on continually.
Ex. 1. To find the MAX. and MIN. of the ordinate of a common para
bola.
y = V a X
d y _ 1 V a
* * d X " 2 ' ^"^
which cannot = 0, unless x = a .
Hence the parabola has no maxima or minima ordinates.
Ex. 2. To find the maxima and minima of y in the equation
y^ — 2axy + x^ = b^
Here
2 a ?^  f^f  .
dy_ay — xd'^y_ dx ^dx/
a=^
dx y — ax'dx^ y — ax '
• dy «
and putting t=^ = 0, we get
 +ab _ + b d _ + 1
^  V (1 — a^)' y ~ a/ (1 — a«)' d x^ ~ b V (1 
which indicate and determine both a maximum and a minimum.
Ex. 3. To divide a in such a manner that the product of the m^^ power
of the one part, and the 7i^^ power of the other shall be a maximum.
Let X be one part, then a — x = the other, and by the question
u = x*". (a — x) ° = max.
d u ^ .
.*. r— = X "»  ^ (a — x) "  ^ X (ma — x. m + n)
88
and
d*u
: X •" ■
(a
T» d u
Put^—
d X
= 0;
then
A COMMENTARY ON [Sect. I.
X (m 4 n — 1. m + n. x' — &c.)
ma
X = 0, or X = a, or X =
m + n'
the two former of which when m and n are even numbers give minima^
and the last the required maximum.
}_
Ex. 4. Let u = X ^
Here
d u 1 — 1. X
'T~ = u. "jTz — = 0, .*. 1. X = 1, and x = e the hyperbolic base
= 2.71828, &c.
Innumerable other examples occur in researches in the doctrine of
curves, optics, astronomy, and in short, every branch of both abstract and
applied mathematics. Enough has been said, however, fully to demon
strate the general principle, when applied to functions of one independent
variable only.
For the maxima and minima of functions of two or more variables, see
LacroiXf 4to.
35. If in the expression (30 a. g) ST should be finite when g is infinite,
then the corresponding tangent is called an Asymptote to the curve, and
since g and this Asymptote are both infinite they are parallel. Hence
To Jlnd the Asymptotes to a curve,
In S T = §^ i — , make ^ = a , then each Jinite value of S T gives an
s
Asymptote ; which may be drawn, by finding from the equation to the
curve the values of ^ for f = a, (which will determine the positions of g),
then by drawing through S at right angles to g, S T, S T', S T", &c. the
several values of the subtangent of the asymptotes, and finally through
T, T', T", &c. perpendiculars to S T, S T', S T'', &c. These perpen
diculars will be the asymptotes required.
Ex. In the hyperbola
_ b'
^ "" a ( 1 — e cos. 6)'
Here f = a , gives 1 — e cos. ^ = 0, .*. cos. 6 =
'. + 6 = £. whose cos. is —
e
6
1
Book I.]
Also S T
NEWTON'S PRINCIPIA.
33
= b ; whence it will be seen that
a e sm. 6 a V e '^ 1
the asymptotes are equally inclined (viz. by c 6) to the axis, and pass
through the center.
The expression (29. e) will also lead to the discovery and construction
of asymptotes.
Since the tangent is the nearest straight line that can be drawn to the
curve at the point of contact, it affords the means of ascertaining the in
clination of the curve to any line given in position ; also whether at any
point the curve be injlectedi or from concave become convex and vice ver
sa ; also whether at any point two or more branches of the curve meet,
i. e. whether that •point be double, triple, &c.
36. To Jind the inclination of a curve at any point of it to a given line .•
fnd that of the tangent at that given point, which will be the inclination
required.
Hence if the inclination of the tangent to the axis of a curve be zero,
the ordinate will then be a maximum or minimum ; for then
tan. T
_ dy __ ^
dx
(31. h)
37. To f.nd the points of Inflexion of a curve.
A B A B
Let y = f (x) be the equation to the curve a b ; then A a, B b being
any two ordinates, and ana tangentat the point a, if we put A a = y,
and A B = h, we get
A a = f X
Bb = f(x + h)=y4^^h + i^,
dy
1. 2
+ &c. (32. c)
But Bn = y + mn = y. 4 r^. h. Consequently B b is < or > B n
dx
d^y
according as 5—^ is negative or positive, i. e, the curve is concave or con
84 A COMMENTARY ON [Sect. I.
d * V
vea; ionaards its axis according as , — \ is negative or positive.
Hence also, since a quantity in passing from positive to negative, and
vice versa, must become zero or infinity, at a point of inflexion
T— ^ = or a
d X ^
Ex. In the Conchoid of Nicomedes
X y = (a + y ) V (b ^ — y »)
which gives, by making d y constant,
d'x _ 2 b *^ — b' ys — 3b'ay'
d y «  (b"« y ^ iirjr^)"^^!^^ _ y «)
and putting this = 0, and reducing, there results
y' + 3ay'' = 2b2a
which will give y and then x.
These points of inflexion are those which the Theory of (34) indicates
as belonging to neither maxima nor minima ; and pursuing this subject
still farther, it will be found, in like manner, that in some curves
d* v d^y
T — ^ = or a , j — ^ = or a , &c. = &c.
d x* d x^
also determine Points of Inflexion.
38. Tojind DOUBLE, triple, S^c. points of a curve.
If the branches of the curve cut one another, there will evidently be as
many tangents as branches, and consequently either of the expressions.
Tan. T = ^' (31. h)
d x ^
M T = ^^ (29. e)
d y '
as derived from the equation of the curve, will have as many values as
there are branches, and thus the nature and position of the point will be
ascertained.
If the branches of the curve touch, then the tangents coincide, and the
method of discovering such multiple points becomes too intricate to be in
troduced in a brief sketch like the present. For the entire Theory of
Curves the reader is referred to Cramer's express treatise on that subject,
or to Lacroix's Different, and Integ. Calculus, 4to. edit.
39. We once more return to the text, and resume our comments. We
pass by Lemma VIII as containing no difficulty which has not been al
ready explained.
As similar figures and their properties are required for the demonstra
Book I.] NEWTON'S PRINCIPIA. 85
tion of Lemma IX, we shall now use Lemma VII in establishing Lemma
V, and shall thence proceed to show what figures are similar and how to
construct them.
According to Newton's notion of similar curvilinear figures, we may
define two curoilinear Jigures to be similar when any rectilinear polygon
being inscribed in one qfthem^ a rectilinear polygon similar to the former ^
may always be inscribed in the other.
Hence, increasing the number of the sides of the polygons, and dimi
nishing their lengths indefinitely, the lengths and areas of the curvilinear
figures will be the limits by Lemmas VII and III, of those of the recti
linear polygons, and we shall, therefore, have by Euclid these lengths
and areas in direct and duplicate proportions of the homologous sides
respectively.
40. To construct curves similar to given ones.
If y, X be the ordinate and abscissa, and x' the corresponding abscissa
of the required curve, we have
X : y : : x' : ^ X x' = y' (a)
the ordinate of the required curve, which gives that point in it which
corresponds to the point in the given curve whose coordinates are x, y ;
and in the same manner may as many other points as we please be de
termined.
In such curves, however, as admit a practical or mechanical construc
tion, it will firequently be sufficient to determine but one or two values of y'.
Ex. 1. In the circle let x, measured along the diameter from its extre
mity, be r (the radius) ; then y = r, and we have
y' = ^ X x' = x'
•' X
where x' may be of any magnitude whatever. Hence, all semicircles^ and
therefore circles, are similar Jigures.
Ex. 2. In a circular arc (2 a) let x be measured along the chord (2 b),
and suppose x = f sin. a ; then y = r . vers, a
vers, a
y = — X X
•' sm. a
which gives the greatest ordinate to any semichord as an abscissa, of the
required arc, and thence since
y = r' — V r' * — x' «
it will be easy to find the radius r' and centre, and to describe the arc
required.
36 A COMMENTARY ON [Sect. I.
But since
y' _ r' vers, a! __ vers, a vers, a
x' T* sin. a' sin. a
sm. a
therefore
 2 sin. ' ^ , , 2 sin. ^J
1 — COS. a 2 1 — COS. a 2
sin. a _ a . a sin. «' „ a' . a'
2 COS. — sin. — 2 COS. — sm. —
or
a a
tan. — = tan. — ,
and
which accords with Euclid, and shows that similar arcs of circles subtend
equal angles.
Ex. 3. Given an arc of a parabola, ishose latusrectum is p, to find a
similar one, whose latusrectum shall be p'.
In the first place, since the arc is given, the coordinates at its extremi
ties are ; whence may be determined its axis and vertex ; and by the usual
mode of describing the parabola it may be completed to the vertex.
Now, since
y ' = p X
X, x' being measured along the axis, and when
P P
.'. y = ^ . X = — . X = 2 X
^ X y
which shows that all semiparabolas, and therefore parabolas, aje similar
figures. Hence, having described upon the axis of the given parabola,
any other having the same vertex, the arc of this latter intercepted be
tv/een the points whose coordinates correspond to those of the extremi
ties of the given arc will be the arc required.
Ex. 4. In the ellipse whose semidiameters are a, b, if x be measured
along the axis, when x = a, y = b. Hence
b ,
y = — . X
^ a
and x' or the semiaxis major being assumed any whatever, this value of
y' will give the semiaxis minor, whence the ellipse may be described.
This being accomplished, let (a, jS) (a', /S") be the coordinates at the
Book I.j NEWTON'S PRINCIPIA. SI
extremities of apy given arc of the given ellipse, then the similar one of
the ellipse described will be that intercepted between the points whose
coordinates, (x', y') (x'', y") are given by
y' = ^ V (2 a' x' — x «)
":l::';/:^and' i
■' y' = ^ V (2 a X ' — X *)
In hke manner it may be found, that
All cycloids are similar.
Epicycloids are so, "dihen the radii of their isoheels a radii of the spheres.
Catenaries are similar when the bases a tensions, S^c. S^c.
40. If it were required to describe the curve A c b (fig. to Lemma
VII) not only similar to A C B, but also such that its chord should be of
the given length (c) ; then having found, as in the last example, the co
ordinates (x', y') (x", y") in terms of the assumed value of the absciss^
(as a' in Ex. 4), and (a, /3), (a', /3') the coordinates at the extremities of
the given arc, we have
c = vT^^=:i?r + (y' — y'r = f K)
a function of a' : whence 2! may be found.
Ex. In the case of a parabola whose equation is y ^ = a x, it will be
found that (y'^ = ol yJ being the equation of the iequired parabola)
a'
c = ,. (S — 3') V(/3 + /3')^ + a^
whence (a') is known, or the latusrectum of the required parabola is so
determined, that the arc similar to the given one shall have a chord = c.
41. It is also assumed in the construction both to Lemma VII and
Lemma IX, that. If in similar figures, originating in the same point, the
chords or axes coincide, the tangents at that origin 'will coincide also.
Since the chords A B, A b (fig. to Lemma VII), the parallel secants
B D, b d, and the tangents A D, A d are corresponding sides, each to
each, to the similar figures, we have (by Lemma V)
A B : B D : : A b : b d
and z. B = z. b. Consequently, by Euclid the z. B A D = Z. b A d,
or the tangents coincide.
«8
A COMMENTARY ON
[Sect. I.
To make this still clearer. Let
M B, M B' be two similar curves,
and A B, A' B' similar parts of them.
Let fall from A, B, A', B', the or
dinates A a, B b, A' a', B' b' cut
ting off the corresponding abscissae
M a, M b, M a', M b', and draw
the chords A B, A' B' ; also draw
A C, A' C at right angles to B b, B' C
Then, since (by Lemma V)
Ma
Ma'
..Ma
Ma'
• ..AC
A'C
But
Ma
..AC;
and the Z. C = z C
.. the triangles A B C, A' B' C are similar, and the ^ B A C =
z. B' A' C, i. e. A B is parallel to A' B'.
Hence if B, B' move up to A, the chords A B, A' B' sha]l ultimately/
be parallel, i. e. the tangents (see Lemma III, Cor. 2 and 3, or Lemma
VI,) at A, A' are parallel.
Hence, if the chords coincide, as in fig. to Lemma VII, the tangents
coincide also.
The student is now prepared for the demonstration of the Lemma.
He will perceive that as B approaches A, new curves, or parts of curves,
A c b similar to the parts A C B are supposed continually to be described,
the point b also approaching d, which may not only be at ajinite distance
from A, but absolutely fixed. It is also apparent, that as the ratio be
tween A B and A b decreases, the curve A c b approaches to the straight
line A b as its limit
42. Lemma XI. The construction will be better understood when
thus effected.
Take A e of any given magnitude and draw the ordinate e c meeting
A C produced in c, and upon A c describe the curve Abe (see 39)
Mb :
Mb'
: A a : B b "»
: A'a' : B'b'i
ab : :
a'b' :
Aa : B C 1
: A' a' : B' C /
BC :
B'C
: Ma : Aa ■»
: : Ma' : A'a'/
A a :
BC :
. M a' : A' a'
: A' C : B' C
Book L] NEWTON'S PRINCIPIA. 89
A D
similar to A B C. Take A d = A e X r—^ and erect the ordinate d b
A hj
meeting A b c in b. Then, since A d, A e are the abscissae corre
sponding to A D, A E, the ordinates d b, e c also correspond to the
ordinates D B, E C, and by Lemma V we have
d b : D B : : e c : E C : : A e : A E
: : A d : A C (by construction)
and the z. D = A d. Hence
b is in the straight line A B produced, &c. &c.
43. This Lemma may be proved, without the aid of similar curves, as
follows :
A B D = ^ . (D F + F B)
. ^, tan. a , A D . B F
= A1J*. —  — H
and
ACE = AE^^^^" + ^^;^^
where a = z. D A F.
. AJ^  AD^tan. g + AD.BF
•'ACE ~ A E« . tan. a + A E . C G
Now by Lemma VII, since ^ B A F is indefinite compared with F or B ;
therefore B F, C G are indefinite compared with A D or A E. Hence
if L be the limit of „ and L + 1 its varying value, we have
A v^ iLd
 AD', tan, a + A D . B F
+ A E ^ tan. a + A E . C G
and multiplying by the denominator and equating homogeneous terms
we get
L . A E * . tan. a = A D ^ tan. a
^. . ^ABD AD^
., Limit of ^^ = ^^,.
44. Lemma X. " Contimially increased or diminished." The woi*d
" continually" is here introduced for the same reason as '' continued
curvature" in Lemma VI.
If the force, moreover, be not ^^Jinite^'' neither will its effects be ; or
the velocity, space described, and time will not admit of comparison.
n
K
b
r
m
B'
^^^^^
40 A COMMENTARY ON [Sect. I.
45. Let the time A D be divided into several portions, such as D d,
A b B being the lociis of the extremities of the ordinates which D repre
sent, the velocities acquired D B, d b, b
&c. Then upon these lines D d, &c.
as bases, there being inscribed rect
angles in the figure A D B, and when
their number is increased and bases
diminished indefinitely, their ultimate
sum shall = the curvilinear area D d D' A
A B D (Lemma IIL) But each of these rectangles represents the space
described in the time denoted by its base ; for during an instant the ve
locity may be considered constant, and by mechanics we have for constant
velocities S = T X V. Hence the area A B D represents the whole
space described in the time A D.
In the same manner, ACE (see fig. Lemma X) represents the time
A E. But by Lemma IX these areas are " ipso motus initio," as A D *
and A E '^ Hence, in the very beginning of the motion, the spaces de
scribed are also in the duplicate ratio of the times.
46. Hence may be derived the differential expressions for the space
described^ velocity acquired^ &c.
Let the velocity B D acquired in the time t (AD) be denoted by v,
and the space described, by s.
Then, ultimately, we have
Dd = dt,Bn = dv,
and
Hence
Dnbd = ds=rDdxdb = dtXv.
d s   , d s , .
v=T,ds = vdt,dt = — (a)
d t V
Again, if D d = d D', the spaces described in these successive instants,
are D b, D' m, and therefore ultimately the fluxion of the space repre
sented by the ultimate state of D' m is b n r m or 2 b m B'. Hence
d (d s) = 2 X b m B' ultimately,
and supposing B' to move up to A, since in the limit at A, B' coincides
with A, and B' m with A D, and therefore b m B' or d (d s) represents
the space described " in the very beginning of the motion."
Hence by the Lemma,
d (ds) a 2dt« a dt*
or with the same accelerating force
d^ s a d t^ (b)
Book I.]
NEWTON'S PRINCIPIA.
41
With different accelerating forces d ^ s must be proportionably increased
or diminished, and .*. (see Wood's Mechanics)
d^s a Fd t^
Hence we have, after properly adjusting the units of force, &c.
d*s = Fdt'
and .*.
(c)
F
= F d t^
_ d's Y
~ dt^ ^
Hence also and by means of (a) considering d t constant,
F = ^, vdv = Fds . , . .
d t'
(d)
all of which expressions will be of the utmost use in our subsequent
comments.
47. Lemma X. Cor. I. To make this corollary intelligible it will be
useful to prove the general principle, that
Jf a body, moving i7i a curves he acted upon by any new accelerating
force, the distance between the points at "which it wotdd arrive WITHOUT
and WITH the new force in the same time, or " error" is equal to the space
that the new force, acting solely, "would cause it to describe in that same
time.
e c
Let a body move in the curve ABC, and when at B, let an additional
force act upon it in the direction B b. Also let B D, D E, E C ;
B F, F G, G b be spaces that would be described in equal times by the
body moving in the curve, and when moved by the sole action of the new
force. Then draw tangents at the points B, D, E meeting D d, E e, C c,
each parallel to B b, in P, Q, R. Also drawF M, G R, b d parallel to
B P; MS, R N, d e parallel to D Q; and S V, N T, e c parallel to
ER.
42 A COMMENTARY ON [Sect. I.
Now since the body at B is acted upon by forces which separately
would cause it to move through B D, B F, or, when the number of
the spaces is increased and their magnitude diminished in infinitum,
through B P, B F in same time, therefore by Law III, Cor. 1, when
these forces act together, the body will move in that time through the
diagonal up to M. In the same manner it may be shown to move from
M to N, and from N to C in the succeeding times. Hence, if the num
ber of the times be increased and their duration indefinitely diminished,
{he. body will have moved through an indefinite number of points M, N,
&c. up to C, describing a curve B C. Also since b d, d e, e c are each
parallel to the tangents at B, D, E, or ultimately to the curve B D E C ;
.'. b d e c ultimately assimilates itself to a curve equal and parallel to
B D E C ; moreover C c is parallel to B b. Hence C c is also equal
to Bb.
Hence, then, The E7'ror caused by any disturbing force acting upon a
body moving in a curve, is equal to the space that laould be described by
means of the sole action of that force, and moreover it is parallel to the
direction of thai force. Wherefore, if the disturbing force be constant, it is
easily inferred from Lemmas X and IX, and indeed is shown in all books
on Mechanics, that the errors are as the squares of the times in isohich they
are generated. Also, if the disturbing forces be nearly constant, then the
eiTors areas the squares of the times quam proxime. But these conclusions,
the same as those which Note 118 of the Jesuits, Le Seur and Jacquier,
(see Glasgow edit. 1822.) leads to, do not prove the assertion of Newton
in the corollary under consideration, inasmuch as they are general for all
curves, and apply not to similar curves in particular.
48. Now let a curve similar to the above be constructed, and completing
the figure, let the points corresponding to A, B, &c. be denoted by
A', B', &c. and let the times in v/hich the similar parts of these cui^ves,
viz. B D, B' D' ; D E, D' E' ; E C, E' Of are described, be in the ratio
t : t'. Then the times in which, by the same disturbing force, the spaces
B F, B' F'; F G, F' G'; G b, G' b' are described, are in the ratio of
t : t'. Hence, " in ipso motus initio" (by Lemma X) we have
B F : B'F : : t^ : t'^
F G : F'G' : : t^ : \!^
&c. &c.
and therefore,
B F + F G + &c. : B' F' + F' G^ + &c. :
: t
Book I.] NEWTON'S PRINCIPIA. 43
But, (by 15,)
B F + F G + &c. = the error C c,
and
B' F 4 F G' + &c. = the error C c',
and the times in which B C, B' C are described, are in the ratio t : t'.
Hence then
Cc : C'c' : : t* : t'°
or The Errors arising from equal farces, applied at corresponding points,
disturbing the motions of bodies in si?nilar curves, "johich describe similar
parts of those curves in proportional times, are as the squares of the times
in "which they are generated EXACTLY, and not " quam proxime."
Hence Newton appears to have neglected to investigate this corollary.
The corollary indeed did not merit any great attention, being limited by
several restrictions to very particular cases.
It would seem from this and the last No. that Newton's meaning in
the forces being " similarly applied," is merely that they are to be applied
at corresponding points, and do not necessarily act in directions similarly
situated with respect to the curves.
For explanation with regard to the other corollaries, see 46.
49. Lemma XI. " Finite Curvature." Before we can form any precise
notion as to the curvature at any point of a curve's being Finite, Infinite or
Infinitesimal, some method of measuring curvature in general must be de
vised. This measure evidently depends on the ultimate angle contained by
the chord and tangent (A B, AD) or on the angle of contact. Now, although
this angle can have no finite value when singly considered, yet when two
such angles are compared, their ratio may be finite, and if any known
curvature be assumed of a standard magnitude, we shall have, by the
equality between the ratios of the angles of contact and the curvatures, the
curvature at any point in any curve whatever. In practice, however, it
is more commodious to compare the subtenses of the angles of contact
(which may be considered circular arcs, see Lemma VII, having radii in
a ratio of equality, and therefore are accurate measures of them), than the
angles themselves.
50. Ex. I. Let the circumference of a circle be divided into any num
ber of equal parts and the points of division being joined, let there be f
tangent drawn at every such point meeting a perpendicular let fall from
the next point ; then it may easily be shown that these perpendiculars or
subtenses are all equal, and if the number of parts be increased, and their
44
A COMMENTARY ON
[Sect. I.
magnitude diminished, m hifinitum, they will have a ratio of equality.
Hence, the circle has the same curvature at every poi7it, or it is a airve
of uniform curvature.
51. Ex. 2. Let two circles touch one
another in the point A, having the
common tangent A D. Also let B D
be perpendicular to A D and cut the
circle A D in B'. Join A B, A B\
Then since A B, A B' are ultimately
equal to A D (Lemma VII) they are
equal to one another, and consequently
the limiting ratio of B D and B' D, is
that of the curvatures of the respective
circles A C, A D (by 17.)
But, by the nature of the circle,
AD" = 2 R X D B' — D B'2 = 2r X D B — D B«
R and r being the x*adii of the circles.
Therefore
T a. 1  2L?L _ 2 R — D B^
■^DB'"'2r— DB
and equating homogeneous terms we have
^ ^ >
i. e. The curvatures of circles are inversely as their radii.
52. Hence, if the curvature of the circle whose radius is 1, (inch, foot,
or any other measure,) be denoted by C, that of any other circle whose
radius is r, is
C
63. Hence, if the radius r of a circle compared with 1, he ^nite, its
curvature compared with C, \sjinite , if r be irifinite the curvature is
infinitesimal ; if r be infinitesimal the curvature is iiifinite, and so on through
all the higher orders of ijifnites and infinitesimals. By infinites and in
finitesimals are understood quantities indefinitely great or small.
The above sufficiently explains why curvature, compared with a given
standard (as C), can be said to hejinite or indefinite. We are yet to show
the reason of the restriction to curves o^ finite curvature^ in the enuncia
tion of the Lemma.
64. The circles which pass through A, B, G; a, b, g, (fig. Lemma XI)
Book L] NEWTON'S PRINCIPIA. 46
have the same tangent A D with the curve and the same subtenses. Hence
(49. and 52.) these circles idtimately have the same curvature as the curve,
i. e. A I is the diameter of that circle which has the same curvature as the
curve at A. Hence, according as A I is finite or indefinite, the curvature
at A is so likewise, compared with that of circles of finite radius.
Now A G ultimately, or
AB2
BD
whether A I be finite or not. If finite, B D a A B % as we also learn
AI =
)e finite or i
from the text.
A B*
55. If the curvature be infinitesimal or A I infinite ; then since
r> JJ
is infinite, B D must be infinitely less than A B ^, or, A B being
always considered in its ultimate state an infinitesimal of the first order,
B D is that of the third order, i. e. B D cc A B ^. The converse is
also true.
Ex. In the cubical parabola, the abscissa a as the cube of the or
dinate; hence at its vertex the curvature is infinitely small. At other
points, however, of this curve, as we shall see hereafter, the curvature is
finite.
To show at once the different proportions between the subtenses of the
angles of contact and the conterminous arcs, corresponding to the differ
ent orders of infinitesimal or infinite curvatures, and to make intelligible
this intricate subject, let A B ultimately considered be indefinitely small
A B^
compared with I ; then since . ^ = A B, A B ^ is infinitesimal com
A B°
pared with A B ; and generally . p n — i = A B, shows that A B " is
infinitely small compared with A B "^ ~ ^ so that the different orders of in
Jinitesimals may be correctly denoted by
AB, AB^ AB^ AB*, &c.
Also since 1 is infinite compared with the infinitesimal A B, and A B
compared with A B ^, &c. the different orders of infinites may be repre
sented by
^ J ^ i &c
AB' AB^' AB^' AB^'
56. Hence if the curvature at any point of a curve be infinitesimal in
the second degree
46 A COMMENTARY ON [Sect. 1
A B" 1
T „ a T— TToj and B D a A B*. and conversely.
BD A B* •'
And generally, if the curvature be infinitesimal in the n'** degree,
A B*^ 1
rjrrp a ■ ^ , and BDcx AB°+% and conversely.
BD AB" •'
Again, if the curvature be infinite in the n'"" degree,
A B^
g^ a A B ", and B D a A B *  ", and conversely.
The parabolas of the different orders will afford examples to the above
conclusions.
57. The above is sufficient to explain the first case of the Lemma.
Case 2. presents no difficulty ; for b d, B D being inclined at any equal
angles to A D, they will be parallel and form, with the perpendiculars let
fall from b, B upon A D, similar triangles, whose sides being propor
tional, the ratio between B D, b d will be the same as in Case 1.
Case 3. If B D converge, i. e. pass through when produced to a given
point, b d will also, and ultimately when d and D move up to A, the
difference between the angles A d b, A D B will be less than any
that can be assigned, i. e. B D and b d will be ultimately parallel ;
which reduces this case to Case 2. (See Note 125. of PP. Le Seur and
Jacquier.)
Instead of passing through a given point, B D, b d may be supposed
to touch perpetually any given curve, as a circle for instance, and B D
will still a A D ^ ; for the angles D, d are ultimately equal, inasmuch as
from the same point A there can evidently be but one line drawn touch
ing the circle or curve.
Many other laws determining B D might be devised, but the above
will be suflScient to illustrate Newton's expression, " or let B D be deter
mined by any other law whatever." It may, however, be farther observed
that this law must be definite or such as viiWJix B D. For instance, the
Lemma would not be true if this law were that B D should cut instead of
touch the given circle.
58. Lemma XI. Con. II. It may be thus explained. Let P be
the given point towards which the sagittae S G, s g, bisecting the chords
A B, A b, converge. S G, s g shall ultimately be as the squares of
A B, A b, &c.
Book I.]
NEWTON'S PRINCIPIA.
47
For join P B, P b and produce
them, as also P G, P g, to meet the
tangent in D, d, T, t. Then if B
and b move up to A, the angles
T P D, t P d, or the difFerences be
tween the angles ATP and A D P,
and between A t P and A d P, may
be diminished without limit; that is,
(Lemma I), the angles at T, D and
at t, d are ultimately equal. Hence
the triangles ATS, A D B are
similar, as likewise are A t s, A d b.
Consequently
S T : D B
and
s t
d b
A S : A B
Ab
and
.. S T : s t :
: DB
db
Also by Lemma VII,
ST : St :
: S G :
sg
and by Lemma XI, Case 3,
D B : d b :
: AB^
: Ab*
.•• S G : s g :
: AB«
Ab^
Q. e. d.
Moreover, it hence appears, that the sagittcE which cut the chords, in
ANY GIVEN RATIO WHATEVEBi and tend to a given pointy have ultimately
the same ratio as the subtenses of the angles of contact, and are as the squares
of the corresponding arcs, chords, or tangents.
59. Lemma XI. Cor. III. If the velocity of a body be constant or
"given," the space described is proportional to the time t Hence
A B a t, and .. S G a A B 2 « t «.
60. Lemma XI. Cor. IV. Supposing B D, b d at right angles to
A D (and they have the same proportion when inclined at a given angle
to A D, and also when tending to a given point, &c.) we have
48
A COMMENTARY ON [Sect. I.
A
ADB
: A A d b :
. AD X DB Ad X db
22
:??XAD: Ad
A D* , _
:: ^,.xAD:Ad
♦
: AD^ : Ad^
Also
A
ADB
: A Adb :
:f ^ xD"B:db
A d
•• Vdb ^  " • ^"
: : (DB)^ : (db)^.
It may be observed here, that the tyro, on reverting to Lemma IX,
usually infers from it that
A A D B a A D 2 and does not ol AD =",
but then he does not consider that A D, in Lemma IX, cuts or makes a
j^7iite angle with the curve, whereas in Lemma XI it touches the curve.
61. Lemma XL Cor. V. Since in the common parabola the ab
scissa a square of the ordinate, and likewise BDorACcx AD^or
CD", it is evident that the curve may ultimately be considered a
parabola.
This being admitted, we learn from Ex. 1, No. 4, that the curvilinear
area A C B = f of the rectangle C D. Whence the curvilinear area
A B D = ^ of C D = f of the triangle A B D, or the area A B D a
triangle A B D a A D^ &c. (by Cor. 4.) So far B D, b d have been
considered at right angles to A D. Let them now be inclined to it at a
given angle, or let them tend to a given point, or " be determined by any
other law ;" then (Lemma, Case 3, and No. 25) B D, b d will ultimately
be parallel. Hence, B D', b d' (fig. No. 26) being the corresponding
subtenses perpendicular to A D, it is plain enough that the ultimate dif
ferences between the curvilinear areas A B D, A B D' and between
A b d, A b d' are the similar triangles B D D', b d d', which
differences are therefore as B D % b d % or as A B *, A b *, i. e.
BDD'a AB\
But we have shown that A B D a A B '.
Book I.] NEWTON'S PRINCIPIA.
49
Consequently
ABD' = ABD+BDD' = axAB3 + bxAB*=AB5(aq:bxAB)
and b X A B being indefinite compared with a, (see Art. 6,)
ABD' = a X A B^* a A B^.
Q. e. d.
SCHOLIUM TO SECTION I.
62. Wliat Newton asserts in the Scholium, and his commentators Le
Seur and Jacquier endeavour [unsiiccessfiilly) to elucidate, with regard to
the different orders of the angles of contact or curvatures, may be briefly
explained, thus.
Let D B ex A D '". Then the diameter of curvature, which equals
A D^
^g (see No. 22 and 24), a A D ^  "». Similarly if D B ot. AD", the
diameter of curvature cc A D '^ ~ ". Hence D and D' represents these
diameters, we have
T^ = —, . Two — .. = — jAD^^fa and a' being finite)
D a' X A D^" a' ^ o /
and if n = 2 or D' he^tiite, then D will hejinitei infinitesimal, or infinite,
according as m = 2, or is any number, (whole, fractional, or even transcen
dental) less than 2, or any number greater than 2. Again, if m = n
then D compared with ly is finite, since D : D' : : a : a'. If m be less
than n in any finite degree, then n — m is positive, and D is always in
finitely less than D'. If m be greater than n, then
Dal
X
D' ~ a' A D "> °
and m — n being positive, D is always infinite compared with D'.
Hence then, there is no limit to the orders of diameters of curvature,
with regard to infinite and infinitesimal, and consequently not to the
curvatures. ,
63. In this Scholium Newton says, that " Those things which have
been demonstrated of curve lines and the surfaces which they comprehend
are easily applied to the curve surfaces and contents of solids." Let us
attempt this application, or rather to show,
1st, That if any number of parallelopipeds of equal bases be inscribed in
any solid, atid the same Jiumher having the same bases be also circumscribed
Vol. I. D
50
A COMMENTARY ON
[Seci. I.
about it ; then the number of these jparallelopipeds being increased and their
magnitude diminished IN INFINITUM, the ultimate ratios "^hich the aggre
gates of the inscribed and circumscribed parallelepipeds have to one another
and to the solid, are ratios of equality.
Let A S T U V Z Y X W A be any portion of a solid cut ofF by three
planes A A' V, A A' Z and Z A' V, passing through the same point A',
and perpendicular to one another. Also let the intersections of these
planes with one another be A A', Af V, A' Z, and with the surface of the
solid be A U V, A Y Z and Z 1 V. Moreover let A' V, A' Z be each
divided into any number of equal parts in the points B', T', U'; D', X', Y',
and through them let planes, parallel to A A' Z and A A' V respectively,
be supposed to pass, whose intersections with the planes A A' V, A A' Z
Book I.] NEWTON'S PRINCIPIA. 51
shaU be S B', T T', U U'; W D', X X', Y Y^ and with the plane
A' Z V, 1 B', m T', n U' ; t D', s X', o ¥', respectively. Again, let the
intersections of these planes with the curve surface be S P 1, T Q m,
URn; WPtjXQs, YRo respectively. Also suppose their several
mutual intersections to be P C, F E', P" x, P"' G', Q F', Q' H', Q'' K',
&c. ; those of these planes taken in pairs and of the plane A' Z V, being
the points C'', E', x, G', F', H', K', I', &c. and those of these pairs of
planes and of the curve surface, the points P, P', P", P"', Q, Q', Q'', R, &c.
Now the planes, passing through B^ T', U', being all parallel to
A A' Z, are parallel to one another and perpendicular to A A' V. Also
because the planes passing through D', X', Y' are parallel to A A' V,
they are parallel to one another, and perpendicular to A A' Z. Hence
(Euc. B. XL) S B', T T', U U', W D', X X', Y Y', as also P C, F E',
P'' X, F"' G/, Q F', Qt H', Q'' K', &c. &c. are paraUel to A A' and to
one another. It is also evident, for the same reasons, that B' 1, T'm, U' n,
ai'e parallel to A' Z and to one another, as also are D' t, X' s, Y' o to
A' V and to one another. Hence also it follows that A' B' C D',
B' C E' T', &c. are rectangles, which rectangles, having their sides equal,
are themselves equal.
Again, from the points A, P, Q, R in the curve surface, draw A B,
A D; P E, P G; Q H, Q K; R L, R N parallel to A' B', A' D';
C E', a G'; F' H', T' K', T o, I' n and meeting B' S, D' W; E' P',
G' F''; H' Q', K' Q" produced in the points B, D; E, G; H, K, re
spectively. Then complete the rectangles A C, P F, Q I which, being
equal and parallel to A' C, C F, F' I', will evidently, when C P, F' Q,
1' R are produced to C, F, I, complete the rectangular parallelopipeds
A C, P F', Q V. Moreover, supposing F' I' the last rectangle wholly
within the curve Z V produce K' F, H' F and make V L', I' N' equal
K' I', H' F, and complete the rectangle I' M'. Also complete the
parallelopiped R M'.
Again, produce E P, G P, H Q, K Q ; L R, N R to the points d, b ;
g, e ; k, h, and complete the rectangles Pa, Q p, R q thereby dividing
the parallelopipeds A C\ P F', Q I', each into two others, viz. A P,
aC; PQ, pF; Q R, q F.
Now the difference between the sum of the inscribed parallelopipeds
a C^ p F', q F, and that of the circumscribed ones A C, P F', Q I', R M',
is evidently the sum of the parallelopipeds A P, P Q, Q R, R M'; that
is, since theii bases are equal and the altitudes F R', R I, Q F, PC
are together equal to A A', this difference is equal to the parallelopiped
A C. In the same manner if a series of inscribed and circumscribed
D2
52 A COMMENTARY ON [Sect. I.
rectangular parallelopipeds, having the bases B' E', E' H', H' U, be
constructed, the difference between their aggregates will equal the paral
lelepiped whose base is B' E' and altitude S B', and so on with every
series that can be constructed on bases succeeding each other diagonally.
Hence then the difference between the sums of all the parallelopipeds
that can be inscribed in the curve surface A Z V and circumscribed about
it, is the sum of the parallelopipeds whose bases are each equal to A' C
and altitudes are A A', S B', T T', U U', W D', X X', Y Y'. Let
now the number of the parts A' B', B' T^ T' U', U' V, and of the parts
A D', D' X', X' Y', Y' Z be increased, and their magnitude diminished
in infinitum, and it is evident the aforesaid sum of the parallelopipeds,
which are comprised between the planes A A' Z, S B' 1 and between the
planes A A' V, W D' t, will also be diminished without limit ; that is, the
difference between the inscribed and circumscribed whole solid is ulti
mately less than any that can be assigned, and these solids are ultimately
equal, and a fortiori is the intermediate curvesurfaced solid equal to either
of them (see Lemma I and Art. 6.) Q. e. d.
Hitherto only such portions of solids as are bounded by three planes
peipendicular to one another, and passing through the same point, have
been considered. But since a com'plete curvesurfaced solid will consist of
four such portions, it is evident that what has been demonstrated of any
one portion must hold with regard to the whole. Moreover, if the solid
should not be curvesurfaced throughout, but have one, two, or three plane
faces, there will be no difficulty in modifying the above to suit any parti
cular case.
2dly, If in two curvesurfaced solids there be inscribed two series of paraU
lelopipeds, each of the same number ; and ultimately these parallelopipeds
have to each other a given ratio, the solids themselves have to one another
that san£ ratio.
This follows at once from the above and the composition of ratios.
3dly, All the corresponding edges or sides, rectilinear or airvilinear, of
similar solids are proportionals , also the corresponding surfaces, plane o)'
curved, are in the duplicate ratio of the sides ; and the volumes or contents
are in the t?iplicate ratio of the sides.
When the solids have plane surfaces only, the above is shown to be
true by Euclid.
When, however, the solids are curvesurfaced, wholly or in part, we
must define them to be similar when any plane surfaced solid whatever
being inscribed in any one of them, similar ones may also be insaibed in the
Book L] NEWTON'S PRINCIPIA. 53
others. Hence it is evident that the corresponding plane surfaces are
similar, and consequently, by Lemma V, the corresponding edges are
proportional, and the corresponding plane surfaces are in the duplicate
ratio of these edges or sides. Moreover, if the same number of similar
parallelopipeds be inscribed in the solids, and that number be indefinitely
increased, it follows from 63. 1 and the composition of ratios, that the
curved surfaces are proportional to the corresponding plane surfaces, and
therefore in the duplicate ratio of the corresponding edges ; and also that
the contents are proportional to the corresponding inscribed parallelopi
peds, or (by Euclid) in the triplicate ratio of the edges.
These three cases will enable the student of himself to pursue the ana
logy as far as he may wish. We shall " leave him to his own devices,"
after cautioning him against supposing that a curvedsurface, at any point
of it, has a certain fixed degree of curvature or deflection from the tangent
plane, and therefore that there is a sphere, touching the tangentplane at
that point, whose diameter shall be the limit of the diameters of all the
spheres that can be made to touch the tangentplane or curvedsurface
— analogously to A I in Lemma XI. Every curvilinear section of a curved
surface, made by a plane passing through a given point, has at that point
a difierent curvature, the curvedsurface being taken in the general sense;
and it is a problem of Maxima and Minima To determine those sections
'which "present the greatest and least degrees of curvature.
The other points of this Scholium require no particular remarks. If
the student be desirous of knowing in what consists the distinction be
tween the obsolete methods of Exhaustions, Indivisibles, &c. and that of
prime and ultimate ratios, let him go to the original sources — to the
works of Archimedes, Cavalerius, &c.
64. Before we close our comments upon this very important part of the
Principia, we may be excused, perhaps, if we enter into the detail of the
Principle delivered in Art. 6, which has already afforded us so much
illustration of the text, and, as we shall see hereafter, so many valuable
results. We have thence obtained a number of the ordinary rules for
deducing indefinite forms from given definite functions of one variable ;
and it will be confessed, by competent and candid judges, that these ap
plications of the principle strongly confirm it. Enough has indeed been
already developed of the principle, to prove it clearly divested of all the
metaphysical obscurities and inconsistencies, which render the methods of
Fluxions, Differential Calculus, &c. &c. so objectionable as to their logic,
and which have given rise to so many theories, all tending to establish
D3
54 A COMMENTARY ON [Sect. I.
the same rules. It is incredible that the great men, who successively in
troduced their several theories, should have been satisfied with the
reasonings by which they attempted to establish them. So many con
flicting opinions, as to the principles of the science, go only to show that
all were founded in error. Although it is generally difficult, and often
impossible, for even the most sharpsighted of men, to discern truth
through the clouds of error in which she is usually enveloped, yet, when
she does break through, it is with such distinct beauty and simplicity that
she is instantly recognized by all. In the murkiness around her there are
indeed false lights innumerable, and each passing meteor is in turn, by
many observers, mistaken for the real presence ; but these instantly vanish
when exposed to the refulgent brightness of truth herself. Thus we have
seen the various systems of the world, as devised by Ptolemy, Tycho
Brahe, and Descartes, give way, by the unanimous consent of philoso
phers, to the demonstrative one of Newton. It is true, the principle of
gravitation was received at first with caution, from its nonaccordance
with astronomical observations ; but the moment the cause of this discre
pancy, viz. the erroneous admeasurement of an arc of the meridian, was
removed, it was hailed universally as truth, and will doubtless be coeval
with time itself. The Theories relative to quantities indefinitely variable,
present an argument from which may be drawn conclusions directly op
posite to the above. Newton himself, dissatisfied with his Fluxions, pro
duces PRIME AND ULTIMATE RATIOS, and again, dissatisfied with these, he
introduces the idea of Moments in the second volume of the Principia.
He is every where constrained to apologize for his obscurities, first in his
Fluxions for the use of time and velocities, and then again in the Scholium,
at the end of Sect. I of the Principia, (and in this instance we have shown
how little it avails him) for reasoning upon nothings. After Newton comes
Leibnitz, his great though dishonest rival, (we may so designate him, such
being evidently the sentiments of Newton himself), who, bent upon oblite
rating all traces of his spoil, melts it down into another form, but yet falls
into greater errors, as to the true nature of the thing, than the discoverer
himself. From his Infinitesimals, considered as absolute nothings of the dif
ferent orders, nothing can be logically deduced, unless by Him (we speak
with reverence) who made all things from nothing. Sxxch Jiats we mortals
cannot issue with the same effect, nor do we therefore admit in science, finite
and tangible consequences deduced from the arithmetic of absolute no
things, be they ever so many. Then we have a number of theories pro
mulgated by D'Alembert, Euler, Simpson, Marquise L'Hopital, &c. &c.
Book I.] NEWTON'S PRINCIPIA. 66
all more or less modifications of the others — all struggling to establish
and illustrate what the great inventor, with all his almost supernatural
genius, failed to accomplish. All these diversities in the views of philo
sophers make, as it has been already observed, a strong argument that
truth had not then unveiled herself to any of them. Newton strove most
of any to have a full view, but he caught only a glimpse, as we may per
ceive by his remaining dissatisfied with it. Hence then it appears, to us
at least, that the true metaphysics of the doctrine of quantities indefinitely
variable, remain to this day undiscovered. But it may be asked, after
this sweeping conclusion, how comes it that the results and rules thence
obtained all agree in form, and in their application to physics produce
consequences exactly in conformity with experience and observation?
The answer is easy. These forms and results are accurately true, al
though illogically deduced, from a mere compensation of errors. This has
been clearly shown in the general expression for the subtangent (Art. 29),
and all the methods, not even Lagrange's Calcul des Fonctions excepted,
are liable to the paralogism. Innumerable other instances might be
adduced, but this one we deem amply sufficient to warrant the above
assertion.
After these preliminary observations upon the state of darkness and
error, which prevails to this day over the scientific horizon, it may per
haps be expected of us to shine forth to dispel the fog. But we arrogate
to ourselves no such extraordinary powers. All we pretend to is self
satisfaction as to the removal of the difficulties of the science. Having
engaged to write a Commentary upon the Principia, jve naturally sought
to be satisfied as to the correctness of the method of Prime and Ultimate
Ratios. The more we endeavoured to remove objections, the more they
continually presented themselves ; so that after spending many months in
the fruitless attempt, we had nearly abandoned the work altogether;
when suddenly, in examining the method of Indeterminate Coefficients in
Dr. Wood's Algebra, it occurred that the aggregates of the coefficients of
the like powers of the indefinite variable, must be separately equal to zero,
not because the variable might be assumed equal to zero, (which it never
is, although it is capable of indefinite diminution,) but because of the
diffijrent powers being essentially different from, and forming no part of
one another.
From this a train of reflections followed, relative to the treatment of
homogeneous definite quantities in other branches of Algebra. It was
soon perceptible that any equation put = 0, consisting of an aggregate of
D4
56 A COMMENTARY ON [Sect. I.
different quantities incapable of amalgamation by the opposition of plus
and mtJiuSj must give each of these quantities equal to zero. Reverting to
indefinites, it then appeared that their whole theory might be developed
on the same principles, and making trial as in Art 6, and the subsequent
parts of the preceding commentary, we have satisfied oui'selves most fully
of having thus hit upon a method of clearing up all the difficulties of
what we shall henceforth, contrary to the intention expressed in Art. 7,
entitle
THE CALCULUS
OF
INDEFINITE DIFFERENCES.
65. A constant quantity is such, that from its very nature it cannot be
made less or greater.
Constants, as such quantities may briefly be called, are denoted generally
by the first letters of the alphabet,
a, b, c, d, &c.
A definite quantity is a GIVEN value of a quantity essentially variable.
Definite quantities are denoted by the last letters of the alphabet, as
z, y, X, w, &c.
An INDEFINITE quantity is a quantity essentially variable through all
degrees of diminution or of augmentation short of absolute NOTHINGNESS w
INFINITUDE.
Thus the ordinate of a curve, considered generally, is an indefinite,
being capable of every degree of diminution. But if any particular value,
as that which to a given abscissa, for instance, be fixed upon, this value is
definite. All abstract numbers, as 1, 2, 3, &c. and quantities absolutely
fixed, are constants.
66. The difference between two definite values of the same quantity (y) is
a definite quantity, and may be represented by
Ay (a)
adopting the notation of the Calculus of Finite (or dejinite) Differences.
In the same manner the difference between two definite values of a y is
a definite quantity, and is denoted by
A (a y)
Book I.] NEWTON'S PRINCIPIA.
or more simply by
and so on to
67
A'^y
(b)
A" y
67. The difference between a Definite value and the Indefinite value of
any quantity y is Indefinite, and we call it the Indefinite Difference of y, and
denote it, agreeably to the received algorithm, by
dy (c)
In the same manner
d (d y)
or
d«y
tlie Indefinite Difference of the Indefinite Difference of y, or the second in
definite difference of y.
Proceeding thus we arrive at
d"y (d)
which means the n* indefinite difference of y.
68. Definite and Indefinite Differences admit of being also represented
by lines, as follows :
t"
Let P P' =: y be any fixed or definite ordinate of the curve A U, and
taking P' Q' = Q' R' = R' S' = &c let ordinates be erected meeting
the curve in Q, R, S, T, &c. Join P Q, Q R, R S, &c. and produce
them to meet the ordinates produced in r, s, t, &c. Also draw r s', s t',
58 A COMMENTARY ON [Sect. I.
&c. parallel to R S, S T, &c. and draw s t", &c. parallel to s t', &c. ; and
finally draw P m, Q n, R o, &c. perpendicular to the ordinates.
Now supposing not only P P' but also Q Q', R R', &c. fixed or defi
nite; then
Qm = QQ'— PF = APF = Ay
Rr =nr — nR = Qm — Rn = AQm
= a(aPP') = a'^PF = A^y
ss' =Ss — Ss'=Ss — Rr =ARr
= ^'y
t t" = t t' t' t" = t t' — S S' = A S S'
= A (A^y) = A*y
and so on to any extent.
But if the equal parts P' Q', Q' R', &c. be arbitrary or indefinite, then
Q m, R r, s s', 1 1", &c. become so, and they represent the several Inde
Jinite Differences of y, viz.
dy, d^y, d^y, d * y, &c.
69. The reader will henceforth know the distinction between Definite
and Indefinite Differences. We now proceed to establish, of Indefinite
Differences, the
FUNDAMENTAL PRINCIPLE.
It is evidently a truth perfectly axiomatic, that No aggregate of indefi
nite quantities can be a definite quantity, or aggregate of definite quanti
ties, unless these aggregates are equal to zero.
It may be said that (a — x) + (a + x) = 2 a, in which (x) is indefinite,
and (a) constant or definite, is an instance to the contrary ; but then the
reply is, a — x and a + x are not indefinites in the sense of Art. 65.
70. Hence if in any equation
A + B X + C x« + D x' + &c. =
A, B, C, &c. be definite qtumtities and x an indefinite quantity ; then we
have
A = 0, B = 0, C = 0, &c.
For B x + C X * + D x' + &c. cannot equal — A unless A = 0.
But by transposing A to the other side of the equation, it does = — A.
Therefore A = and consequently
Bx + Cx^ + Dx' + &c. =
or
X (B + C X + D X ^ + &c.) =
Book L] NEWTON'S PRINCIPIA. 69
But X being indefinite cannot be equal to ; .•.
B + Cx + Dx^ + &c. =
Hence, as before, it may be shown that B = 0, and therefore
X (C + D X + &c.) =
Hence C = 0, and so on throughout.
71. Again, if in the equation
A + Bx + B'y+Cx'=+C'xy + C'y+DxHD'x^y4.D"xy^+D'''y3+&c.
A, B, B', C, C, C", D, &c. be definite quantities^ and x, y INDEFINITES i
then
A = 0^
B X + B' y = Vwhen y is a Junction qfx.
Cx^ + C'xy + C"y2 = o)
&c. =
For, let y = z x, then substituting
A + X (B + B' z) + X* (C + C z + C z^)
+ x^" (D + D' z + D" z^ + D'"z') + &c. =
Hence by 70,
A = 0, B + B' z = 0, C + C z + C" z ^ = 0, &c.
and substituting — for z and reducing we get
A = 0, B X + B' y = 0, &c.
In the same manner, if we have an equation involving three or more
indefinites, it may be shown that the aggregates of the homogeneous terms
must each equal zero.
This general principle, which is that of Indeterminate Coefficients
legitimately established and generalized, (the ordinary proofs divide
B X + C X  + &c. = by X, which gives B + C x + D x " + &c. = —
and not ; x is then put = 0, and thence truly results B = — , which
instead of being 0, may be any quantity whatever, as we know from alge
bra ; whereas in 70, by considering the nature of x, and the absurdity of
making it = we avoid the paralogism) conducts us by a near route to
the Indefinite Differences of functions of one or MORE variables.
72. Tojind the Indefinite Difference of any junction o/'x.
Let u = f x denote the function.
Then d u and d x being the indefinite diiFerences of the function and
of X itself, we have
u + du = f(x + dx)
Assume
f (X + d x) = A + B d x + C d x ' + &c.
«0 A COMMENTARY ON [Sect. I.
A, B, &C. being independent of d x or definite quantities involving x and
constants ; tlien
u + d u = A + B d X + C d X  + &c.
and by 71, we have
u = A, du = B.dx
Hence dien this general rule,
The INDEFINITE DIFFERENCE of any function of s.^ f x, is the second
term in the developement of^{x. + d x) according to the increasing powers
ofdx.
Ex. Let u = X ". Then it may easily be shown independently of the
Binomial Theorem that
(x + dx)'* = x''+n.x«idx + Pdx2
.. d (x»).= n.x "' d X
The student may deduce the results also of Art 9, ] 0, &c. from this general
rule.
73. To find the indefinite difference of the product of two variables.
Let u = X y. Then
u + du=(x + dx).(y + dy) = xy+x dy + y dx + dx dy
.*. du = x dy+y dx + dx dy
and by 71, or directly from the homogeneity of the quantities, we have
du = xdy + ydx (a)
Hence
d (x y z) = « d (y z) + y z d x
=:xzdy + xydz + yzdx . . . (b)
and so on for any number of variables.
X
Again, required d . — .
Let — = u. Then
y
X = y u, and dx = udy + ydu
, X  d X u J
.'. d — =du= dy
y y y
__y dx — X dy
y'
(c)
Hence, and from rules already delivered, may be found the Indefinite
Differences of any functions whatever of two or more variables. We
refer the student to Peacock's Examples of the Differential Calculus for
practice.
The result (a) may be deduced geometrically from the fig. in Art. 21.
The sum of the indefinite rectangles A b, b A' makes the Indefinite
Difference.
Book I.]
NEWTON'S PRINCIPIA.
61
We might, in this place, investigate the second, third, &c. Indefinite
Differences, and give rules for the maxima and minima of functions of two
or more variables, and extend the Theorems of Maclaurin and Taylor to
such cases. Much might also be said upon various other applications,
but the complete discussion of the science we reserve for an express
Treatise on the subject. We shall hasten to deduce such results as we
shall obviously want in the course of our subsequent remarks ; beginning
with the research of a general expression for the radius of curvature of a
given curve, or for the radius of that circle whose deflection from the
tangent is the same as that of the curve at the point of contact.
74. Required the radius of curvature for any point of a given curve.
Let A P Q R be the given
curve, referred to the axis A O
by the ordinate and abscissa
P M, A M or y and x. P M
being fixed let Q N, O R be
any other ordinates taken at
equal indefinite intervals M N,
N O. Join P Q and produce
it to meet O R in r ; and let
P t be the tangent at P drawn
by Art. 29, meeting Q N, O R
in q and t respectively. Again
draw a circle (as in construc
tion of Lemma XI, or other
wise) passing through P and Q and touching the tangent P t, and there
fore touching the curve ; and let B D be its diameter parallel to A O.
Now
Qn = dy, Pn = dx, Pq=PQ (Lemma VII) =
V (d x^ + d y'^) or d s, if s = arc A P.
Moreover let
P M' = y' ;
then it readily appears (see Art. 27) that d s = , R being the ra
dius of the circle.
Again
Pq« = Qq X (Qq + 2QN0
= Qq(Qq + 2dy + 2yO
62 A COMMENTARY ON [Sect. I.
or
2 Rdx>
But since
and
(ds)«=Qq(Qq + 2dy + ^^°^)
R t : Q q : : P r^ : P Q2 : : 4 : 1 (Lemma XI)
Q q : t r : : 1 : 2
.♦, R t = 2 t r, or R r = t r = 2 Q q
t r d^ V
••. Q q = V^ = V ^^y ^''' ^^^
Consequently
, , d « y d 2 y , 2 R d XV
(d*y)' , , ,, ^ R dx d^y
and equating Homogeneous Indefinites
d s
. R _ ds' __ (dx« + dy^)^
"" dx d'y ~" dx d^y
dy\2
i}^m
~ S^j
dx*
the general expression for the radius of curvature.
Ex. 1. In the parabola y * = a x.
. dy _ a
" d X 2y
and since when the curve is concave to the axis d * y is negative,
d*y a dy_ a*__
~ dx* ~ ~~ 2y~* * dx ~ ""* irp ""
a*x^ 4y'
(d)
2a
= (4y* + a*)tx5L
Hence at the vertex R = — , and at the extremity of the latus rectum.
3
2^
R = — r a = a V 2.
Book L] NEWTON'S PRINCIPIA. 63
Ex. 2. If p be the parameter or the double ordinate passing through
the focus and 2 a the axismajor of any conic section, its equation is
y*^ = p X + ^x*
•^ ^ — 2 a
Hence
2ydy = pdx+. — xdx
and
2dy« + 2yd2y = + ^ d x *
3,
"dx 2y
and
d^y _
dx* ~
.. R =
p*(l
^ir^
3p ,
{4y
4y'
which reduces to
pe + ?^(2a + p)x + ^(p+2 a)x^}
^= 2p^
Ex. 3. In the cycloid it is easy to show that
d y _ / 2r — y
dx ^ y
r being the radius of the generating circle, and x, y referred to the base
or path of the circle.
d* y _ r
•'• dx"2 ~ "~ y*
.. R = 2 V 2 r y = 2 the normal.
Hence it is an easy problem iojind the equation to the locus of the centres
of curvature for the several points of a given curve.
If y and x be the coordinates of the given curve, and Y and X those of
the required locus, all referred to the same origin and axis, then the stu
dent will easily prove that
64 A COMMENTaR\ ON [Sect. I.
and
1 + izl
Y = y + d!7"'
dx^
which will give the equation required, by substituting by means of the
equation to the given curve.
In tlie cycloid for instance
X = X + V (2ry — y^)
Y = y
whence it easily appears that the locus required is the same cycloid, only
differing in position from the given one.
75. Required to express the radius of curvature in terms of the polar co
ordinates of a curve, viz. in terms of the radius vector f and traced
angle 6,
X = e COS.
and
y = g sm.
.*. taking the indefinite differences, and substituting in equation (d) of Art.
74, we get
G^ + ^Y
dO' ^dtf*^^
which by means of the equation to the curve will give the radius of curva
ture required.
Ex. 1. In the logarithmic spiral
i = a;
dp, e
••• J; = 1 a X a (Art. 17.)
s. ^ "\
1. 6)
••• — T7i = — (la)'^Xa'' = — (1 a) ^ g
d6'
. p  (g'+ (la)'e^)t g3^i +(la)n'
2(la)V— (ia)V+g^  p (1 + (1 a)')
= ?n + (la)2}^
Book I.} NEWTON'S PRINCIPIA. 66.
Ex. 2. In the spiral of Archimedes
g = ad
and
2a^ + f* •
3
2 J o2^?
Ex. 3. Jh the hyperbolic spiral
? = 
., R = s±L+i
a
Ex. 4. Jm Me Lituus
_ j_ (4a^ + g^)^
■ Ba'' * 4 a* — e*
Ex. 5. In the Epicycloid
g = (r + r') 2 ._ 2 r (r + r') cos. <>
r and r' being the radius of the wheel and globe respectively.
Here
_ (r + r^) (3 r ' — 2 r r^ — r^ ' + 2 g)^
~ 2 (3r^ — 2rr' — r'*) + 3 g
Having already given those results of the Calculus of Indefinite Differ
ences which are most useful, we proceed to the reverse of the calculus,
which consists in the investigation of the Indefinites themselves from their
indefinite differences. In the direct method we seek the Indefinite Differ
ence of a given function. In the inverse method we have given the Inde
finite Difference to find the function whose Indefinite Difference it is. This
inverse method we call
THE INTEGRAL CALCULUS
INDEFINITE DIFFERENCES.
76. The integral of d x is evidently x + C, since the indefinite differ
ence of x + C is d X.
77. Required the integral o/^a d x ?
By Art. 9, we have
d (a x) = a d x.
Vol. I. E
«6. A COMMENTARY ON [Sect. I.
Hence reversely the integral of a d x is a x. This is only one of the in
numerable integrals which there are of a d x. We have not only d (a x)
= a d X but also
d(ax + C) = adx
in which C is any constant whatever.
.. ax + C =/adx = a/dx . . . (a) (see 76)
generally, y being the characteristic of an integral.
78. Required the integral of
a X P d X.
By Art. 12
d(ax°fC)= nax°»dx
..ax" 4 C =/n a x^'d x
= nx/ax'''dx (77)
/~ 1 1 ax \j
ax«»dx= 1 .
n n
C
But since C is any constant whatever — may be written C.
.•./ax«'dx = i^ + C
n
Hence it is plain that
Or To find the integral of the prodiict of a constant the p*** pffwer of the
variable and the Indefinite Difference of that variable, let the index of the
paaoer be increased by 1, suppress the Indefinite Difference, multiply by the
constant, divide by the increased index, and add an arbitrary constant.
79. Hence
/(a xPdx + bx«»dx + &c) =
a X P+i , b X 1+1 , a , ^,
pHTf + <nrf + & + c
80. Hence also
/a X" d X = 7 rr r + C
•^ (n — 1) x"*^
81. Required the integral of
ax'°idx(b + ex"')P.
Let'
u = b + e x"^
.*. d u = m e X ° ~ * d X
.*. a x" ~* d X = . d u
m e
.'.fa. x">*dx(b + e x"')P —f — u p d u
Book L] NEWTON'S PRINCIPIA. 67
a
m e . (p + 1)
a
m e (p + 1)
_ .... , d X
82. Required the integral of — .
By 80 it would seem that
and if when
. uP+» + C (78)
. (b X ex«)P+' + C.
rdx'^ /dx 1 1
/_=0,C=:C,,v/— =5 = .
But by Art. 17 a. we know that
d. 1 X =
X
Therefore
/^=:1X + C.
•' X ,
Here it may be convenient to make the arbitrary constant of the form 1 C
Therefore
r^  Ix + IC = ICx
'^ x
Hence the integral of a fraction ivhose numerator is the Indefinite Differ
ence of the denominator f is the hyperbolic logarithm of the denominator PLUS
an arbitrary constant.
83. Hence
x™*dx a r mx^ — 'dx
/»a x™'dx __ __a_ / m x""' d :
'bx"4e~bm/ „. e
^ / x™ + J
and so on for more complicated forms.
84. Required the integral o/'a* d x.
By Art. 17
d.a'^ = la.a'^dx
.•./a''dx = ri./da*
^ la*'
E 2
«)8 A COMMENTARY ON [Sect. I.
85. If y, X, t, s denote the sine, cosine, tangent, and secant of an angle
d; then we have, Art. 26, 27.
dy _ — <ix _ ^*_ ds
dtf =
VJi — y') ~ ^ (1 — X*)  1 + t'' s V 2 s — s«
■•♦ /v(fiy) = ^ + ^ = "°"y+^
/jq:^ = ^+ C = tan.'t+C
r — =£i_ = ^ + C = sec's + C
•^ s V 2 s — s^
sin.~'y, cos. ~' x, &c. being symbols for the arc whose sine is y, cosine is
X, &c. respectively.
86. Hence, more generally,
f du _ _i_ f ^T''"
^ a '
1 ' / K
= —pr X angle whose sine is u ^ — to rad. 1 + C.
/■^:ii_= 1 .COS.U J^ + C . . (b)
•/ V (a— bu^) / b V a . ^
A ^1 / V — d u
/ d u __ 1 / a
'a + bu'^ ~ VTh' J i^iu2
« a
= — L=tan. »u^— + C . . (c)
or
Also
Again
Vab
and
f da _ j_ / 7r^°
Book I] NEWTON'S PRINCIPIA. 69
Moreover, if u be the versed sine of an angb 6, then the sine
= V (2u — u*) and'
d u = d (1 — cos. 6) = d^ . sin. tf (Art 27.)
= d^. V(2u— u')
du
.. dO =
v'(2u— u*)
Hence
du
(2u — u')
= vers. — ' u + C
and generally
/v(2u — u')  ^ + C
2b ,
du f T
^^vers.>.u + C . . . (e)
87. Required the integrals of
dx dx d X
a + bx' a — bx' a — bx**
/ * dx __ 2 r d. (a + bx)
•'a + bx b'' a + bx
= i. 1. (a + bx) + C , .... (f)
and
/ ' d X 1_ / »d(a — bx)
a — bx~ b'' a — bx
= — i..l.(a — bx) + C ... (g)
see Art. 17 a.
Hence,
/• , f 1 , 1 \ _ r 2ad x
^ la + bx"*"a — bxj ~^a» —
b«x^
= 1. ]. (a + bx)l. 1. (abx) + C
= l.l.i+bf + c.
b a — bx
E3
70 A COMMENTARY ON [Sect. I.
Hence we easily get by analogy
/. d_3c 1 J ^^ a + V b. X
J a — bx^^Vab Va — bx^ » ,,.
(n)
1 J V a
2 Vab' ' V a — Vb. X
88. Required the integral of
dx
ax'^ + bx + c*
In the first place
Hence, putting
^ b
we have
d X = d u
and
d X d u
b'^ — 4ac>
ax* + bx + c afu* ^ \
which presents the following cases.
Case 1. Let a he negative and c be positive ; then
d X d II
— ax^ + bx + c — a ( ^ + u ^j
. r dx V^ tan'u / ^^ t c
* '•'ax= + bx+c~' — V^'\/(b''+4ac) ^* " V b*+4ac
(see Art 86) = — / — — ? . tan.»rx+— ") / rv^4— +C •••(»)
^ ^ V a(bH4ac). V ^2a/'\ b + 4ac^ ^'
Case 2. Ze^ c be negative and a positive ; then
dx __ r d u
/ a (u
ax' + bx — c / /, b* + 4ac>
/ b * + 4 a c\
(" 2l )
\f du
b '^ + 4 a c
2a
/b' 44ac b_
=  / 1 ^^/~2a +^^2a^r! ,u
>V 2a(b*+4ac) • /b^+4lj_ __b
V 2a ^ 2a
see Art 87.
Book I.] NEWTON'S PRINCIPIA. 71
Case 3. Let b * i^ > 4 a c and a, c he both positive ; then
r dx /' d u
•'ax^ + bx + c" / ~7~; b * — 4 a c>
/ a (u
/ „ 0=' — 4a c\
(" " 2ir)
/
du
b' — 4 a c
2a
b^— 4ac . . b
b'' — 4ac
X
1 , V 2 a "^^"^2a
b^ — 4ac_ b^'
S 2"a ^ 2 a
/ 1 , ^ 2 a • 2 a
• V 2a(b2— 4ac) * b'^— 4ac b '*"^ ' ^'^
Case 4. ii?/ b '^ 6e < 4 a c and a, c ie 6o/^ positive ,•
/• dx _1a du
•^ ax'^+bx+c ~~a / 4ac — b^ ,
/ ~2a ■*■"
= V a(4aLb) ^"""^ (^^^a) V 4^1^ + ^' ' ^"^^
/
Case 5. i)^b* 6e > 4 a c and a, c 6o/A negative ;
Then
d X 1 A d u
: — c — a /
/
— ax^+bx — c — a / b^ — 4ac
2 a ^
Case Q.Ifh^be<i 4 a c an^Z a and c &o^A negative;
Then
d X 1 /■ d u
ax'^+bx — c
""2a
1 /" du
a / 4ac — b*
. .V
4ac — b*
■X + ZT7
 / 1 1 ^ ^^ 2a . ^
~ V 2a(4ac— bn • , 4ac— b^ ^+^....(o;
« / X
N 2 a 2 a
89. Required the integral of any rational Junction 'whatever of one
variable, multiplied by. the indefinite difference of that variable.
Every rational function of x is comprised under the general form
Ax" + Bx'°7' + Cx "' + &c. K X + L
ax" + bx°» + cx'»«+ &c. kx +1
E4
7fi A COMMENTARY ON [Sect. L
in which A, B, C, &c. a, b, c, &c. and m, n are any constants whatever.
If
n = 0,
then we have (Art. 77)
/(A X «. + B K " . + &c.) — = (^^^ + 5ji^
+ 5 — f &c. 1 \ constant.
m — 1 /a
Again, if m be > n the above can always be reduced by actual division
to the form
A'x— + B'x'""» + &c. + ^'Ti^t^'lT.'t^'''
ax°+bx"'f&c.
and if the whole be multiplied by d x its integral will consist of two parts,
one of which is found to be (by 77)
A / B' . X "" ~ "
:^=— : . X ""+' + ^^ + &C.
m — n+1 m — n
and the other
A''x°' + B''x"2+ &c.
a x« + b x"~
dx.
'+ &c.
Hence then it is necessary to consider only functions of the general
form
x"^ + Ax°'=+ Bx°^+&c. _ U
x"+ax"i + bx'''+&c. "V
in order to integrate an indefinite difference, whose definite part is any
rational function whatever.
Case 1. Lei the denominator V consist ofn unequal real factors^ x — a,
X — /3, &c. according to the theory of algebraic equations. Assume
V X — a X — 8 X — y
and reducing to a common denominator we shall have
U = P.x — /S.x — 7...to(n — 1) terms
+ Q.X — a.x — y
+ R.X — a.x — ^
= (P + Q + R + &C.)x°»
— JP.(S— a) + Q.(S— /3) + &c.x»»
+ fP.(S — a.S3i)4Q(S — ^.S^^)+&c.x''^
1. J I i.» 1
— &c.
where S, S &c. xlej^ote the sum of a, /S, y &c the sum of the products of
1 I.J
every two of them and so on.
Book L] NEWTON'S PRINCIPIA. 7»
But by the theory of equations
S= — a
S= b
1.2
&c. = &c.
.. U = (P + Q + R + &c.)x«»
+ {a (P + Q + R + &c.) + Pa+Q^ + Ry +&c.} X x"*
+ Jb (P + Q + R + &c.) + a(Pa + Q8+ &c.) +
(Pa«+ Qi82+ Ry2 + &c.)} x"^ + &c.
Hence equating like quantities (6)
P + Q + R + &c. = 1
a + Pa + Q/3+R7 + &c. = A
b + a(A — a) + Pa2+QiS2+Ry2 + &c. = B
&c. = &c.
givuig n independent equations to determine P, Q, R, &c.
Ex.l. LetH= ' + 6=' + 3
V~ x^ + 6x2+ llx + 6
Here
P+ Q + R = 1v
6 + P + 2Q + 3R = 6 Vwhence
11 + P + 4Q + 9R = 3J
P = — 1, Q = 5 and R = 3
Hence
3dx
/• U d X _ r — ax r o ax p
J V ~/x+l'*"'x + 2 'x + S
= C — L (X + 1) + 5 ]. (X + 2) — 3 1. (X + 3).
P, Q, R, &c. may be more easily found as follows :
Since
x"* + Ax'»''&c. = P (x — /S). (x — y). &c.
+ Q (X _ «). (X — y). &c.
+ R (x — a), (x — ^). &c.
+ &c.
let X = a, jS, y, &c. successively ; we shall then have
a «  1 + A a «» 2 + &c. = P . (a — /3) . (a — y) &C.V
i8''» + Ajg^'^ + ac. = Q.(/3 — a). (^— y)&c. v. . .(A)
y»»4 Ayn^' + Stc. = R.(y — a) . (y — i8)&cJ
&c. = &c.
In the above example we have
a — — 1, iS = — 2, y = — 3 and n = 3
A = 6 and B = 3.
74 A COMMENTARY ON [Sect. I.
p
=
1
— 6 + 3
1. 2. ~
—
■ 1
Q
=
4
— 6. 2 + 3
=
5
— 1. 1
R
^^
9
— 6. 3 + 3
_
3
— 2 X — 1
as before.
Hence then the factors of V being supposed all unequal, either of the
above metliods will give the coefficients P, Q, R, &c. and therefore
enable us to analyze the general expression ^ into the partial fractions
as expressed by
V X — ax — p
and we then have
f^~^ = p.l(x_a) + Ql. (x^) + 8cc. + C.
F 2 /'*' + bx^_ y«a dxab/» dx a + b /• d x
* J a^x — x^ ~ J ~x~ "*" ^2 J a — x 2""' a f x
= alx^±^l(ax)_^+^l.(a + x) + C
= a 1 x —(a + b) 1 V a* — x« + C
by the nature of logarithms.
= .l(x_4)— il.(x — 2) + C.
Ex. 4./ , ^/^ = /^AiL + /Ql2L = P 1 (X + «)
'x* + 4ax — b= ^x + a^^x + ^ ^ ^ '
+ Q 1 . (X + i8) + C
where
a = 2a+ V(4a2 + b^), j8 = 2a'v/(4a' + b')
and
P = " _ 2a+ V(4a + b')
a — /3~ 2V(4a* + b*)
Q = —^ _ ^^ (4a' + b') — 2a
Case 2. Z>/ a// Me factors qf\be real and equal, or mppose a = ^
= 7 = &c.
Then
U __ X ■  ' + A X "  ' + &c.
V (x — «)"
Book I.] NEWTON'S PRINCIPIA. 75
and since
a — /3 = 0, a — 7 = &c.
the forms marked (A) will not give us P, Q, R, &c. In this case we
must assiune
V ~ (X — a)°^ (X — «)ni^ (X_a) ""^ *^ '
to n — 1 terras, and reducing to a common denominator, we get
U = P + Q . (X — a) + R (x — a) ^ + &c.
now let X = a, and we have
„nl^ Aa''2+ &c. =P.
Also
^ = Q + 2 R . (x — a) + 3 S . (X — a) 2 + &c.
dx
d'U
dx^
d^U
= 2 R + 3 . 2 . S . (x — a) + 4 . 3 . T (x — a) ' + &c.
^ = 2 . 3 . S + 4 . 3 . 2 T (x — a) + &c.
d x^
&c. = &c.
and if in each of these x be put = a, we have by Maclaurin's theorem
the values of Q, R, S, &c.
U X'— 3x+ 2
h,x. 1. Let ^ = — 7 Tv3 — •
V (x — 4) ^
Then
U = x« — 3x + 2
J— = 2 X — 3
dx
llI?2
dx^ ~
.. P = 6
Q = 8 — 3 = 5
R = ^. 2 = 1
.*. /• U d X _ V* 6 d X /» 5dx ^dx
/ ~V~ ~ J (x — 4) ' "^ J (X — 4) ^ + V ^E^=l
= CS^^j,^^ + l(K4)
= C + iE? + l(K4).
Ex.2. Let K = '''+^'
V ~ (x — 3) 6 •
19 A COMMENTARY ON [Sect. I.
Here
U =
x^ +
x^
dU
dx "■
5x*
+ 3x''
d*U
dx« "
20x3
+ 6x
d^U
dx^ "
60 x'
^+6
d*U
dx* ~
120 X
d = U
1 — k" —
120
ax"
.. P = 3 ^ + 3^ = 2*7 X 10 = 270
Q r= 27 X 16 = 432
' R = 20 X 2'y + 6 X 3 _ ^^^
9 X 60 + 6 __
^  2X3  ^^ ,
rr 360 _
^ = 2:1:4. = ^^
w = l^ = i.
2.3.4.5
Hence
 »»»• (7^  »»• or^=  ¥ • (^'  >«• K^ + '• c ^)
which admits of farther reduction. ♦
Ex. 3. Let . — ^ = Tf •
(x — 1)* V
Here
U = x*+x
dx ^
and
dx« "■ "^^
Book I.] NEWTON'S PRINCIPIA. 77
Hence
P = 2
Q = 3
R = f = 1
. ril±± d X  2 A i^— + 3 r ^^ + r ^^
= c
(X_l)* (X_l)3 ^(XI)
X^
2(x— 1)*
B appears from this example, and indeed is otherwise evident, that the
number of partial fractions into vohich it is necessary to split the function
exceeds the dimension qfyi in U, by unity.
This is the first time, unless we mistake, that Maclaurin's Theorem
has been used to analyze rational fractions into partial rational fractions.
It produces them with less labour than any other method that has fallen
under our notice.
Case 3. JLet the factors of the denominator V be all imaginary and un
equal.
We know then if in V, ' which is real, there is an imaginary factor of
the form x + h^kV — 1, then there is also another of the form
x + h — k V — 1. Hence V must be of an even number of dimensions,
and must consist of quadratic real factors of the form arising from
(X + h + k V^=T) (X + h — k V~^^^)
or of the form
(x + h)^ + k^
Hence, assuming
U _ P + Qx F + Q^x
V ~ (X + a) 2 + /3* "^ (X + a')' + ^''
and reducing to a common denominator, we have
U = (P + Qx) f{x + aO^ + ^''l {(x + a'O' + ^"'] X &c.
+ (F+ Q'x)f(x + a)2 + /3=^] J(x +a'0^ + /3"^J X &c.
+ (F'+Q"x) J(x+a)2 + /3^J J(x+ «')'+ ^"1 X&c.
+ &c.
Now for X substitute successively I
et + /3 v/l^ri, a' + j3' V^^, a" + ^" V"^^, &C
then U will become for each partly real and partly imaginary, and we
have as many equations containing respectively P, Q ; P', Q' ; V", Q'\ &c.
as there are pairs of these coefficients ; whence by equating homogeneous
quantities, viz. real and imaginary ones, we shall obtain P, Q ; P'j Q'. &c.
TB A COMMENTARY ON [Sect. 1.
Ex. 1. Required the integral of
x^ d X
x* + 3x* + 2*
Here the quadratic factors of V are x * + 1, x ' + 2
.. a = 0, a' = 0, jS = 1, and /S' = V~2 .
Consequently
x' = (P + Qx)(x2 + 2)
+ (F + Q'x)(x« + 1)
Letx = V — \. Then
— • — 1 = (P + Q V — \) . (— 1 + 2)
= p + Q v"^:ri
__• • ^=^' Q = — 1
Again, let x = V 2. V — 1, and we have
— 2^ V — 1 = (P' + Q' •2. V_l)(_2+ 1)
= — F— Q' V'2 . V"=rT
.. F = 0, and Q' = 2
Hence
r x^ d x _ /. — xdx / »2 x d x
»'x*+3x24.2 ~^ x2+ 1 ^xM^S
= C— ^l(x« + l) + l.(x* + 2)
Ex. 2. Required the integral of
' d
X
To find the quadratic factors of
1 + X * "
we assume
X 2 « + 1 = 0,
and then we have
x«'' = — 1 =cos.(2p+ 1)^+ V — lsin.(2p+ 1)*
T being 180** of the circle whose diameter is 1, and p any integer what
ever.
Hence by Demoivre's Theorem
2 p + 1 . , 7 . 2 p + 1
X = cos. —^ ff + V — 1 . sm. V. — ^r
2 n 2 n
But since imaginary roots of an equation enter it by pairs of the form
A i ^ — 1 . B, we have also
2p+ 1 — . 2p+ 1
X = cos. ^^ — s — V — 1 . sni. '— — T
2 n 2 n
Book L] NEWTON'S PRINCIPIA. 79
and
/ 2 p + 1 , . 2 p + 1 \
.. (x — COS. — %^5 — ^ — ^ — ] , sin. \^^ T ) X
V 2 n 2 n /
/ 2p+ 1 , , . 2p+ 1 X
(x — COS. ^ — ff + V — 1 sm. \1 «•) =
V 2 n ' 2 n /
X' — 2XCOS. ^P"^ ^ cr+ 1
2 n
which is the general quadratic factor of x * ** + 1. Hence putting
p = 0, 1, 2 n — 1 successivelj',
x» +1 = (x^ — 2xcos. JL + 1 ) . fx='2xcos. ^+ 1 ) X
<s n / ^ <& n J
(x* — 2xcos. ~+ 1 ) X (x^ — 2xcos. °~ + l) .
Hence to get the values of P and Q coiTesponding to the general factor,
assume
1 P+Qx ,N
Then
But
1 + x'" , ^ 2p+l , ,^ M
X* — 2xcos. ^ i t{ 1
2n.
1=(P + Qx).M + N(x« — 2xcos.i§±^^+l).
1+X*"
M =
X* — 2xCOS.5^ir+l
2 n
and becomes of the form — when for x we put cos. " «■ + V — 1
sin. — ^ ff ; its value however may thus be found
2 p + 1 , . 2 p + 1
Let cos. —^ — ff + V — 1 sm. — ^ ?r = r
2 n 2 n
then
2p+l , r . 2 p+I 1
cos. — ^ It — V — 1 . sm. — ^ AT = — 
2 n 2 n r
and
1 + x*''
M =
(xr. (xi.)
Again let x — r = y ; then
M  ^ + y"'+^"y'°~'r+&c 2nyr'^°' + r'
(1)
80 A COMMENTARY ON [Sect. I.
But
r«" z= COS. 2p + l.»+ V — Isin. 2pf l.w = — I
yi'ni I 2n y*"~*.r+ . . . . 2n r^"'
.*. M = ^ ■ j •
X
r
Hence when for x we put r, y = 0, and
,_ 2n r^"^
= ^J
r
and from the above equation we have
r
2^T=lsin.ig±^^==?nP.cos.?JBjiLLilllZlJ.^+2nP V~="l X
2 n 2 n
. 2p + 1.2n — 1 _, r\ / • en i\
sin. ^ ^ cr — 2 n Q (since r ^ ° = — 1)
2 n ^
.*. equating homogeneous quantities we get
. 2p+l _ . 2p+1.2n — 1
sm. —^ — gzrnP.sm. ^
2 n 2 n
and
But
^ 2p+1.2n— 1 ^
P . COS. ^ ^ cr = Q.
2 n
2p+1.2n— l.ff tr — r 2p+l
^^ t, = 2p+ l.T % It
2 n ^ 2 n
Hence the above equations become
• 2p+ 1 ^ . 2p + 1
.*. sm. — £_ «r = n P sin. ~
2n 2 n
T, 2p+ 1 ^
— P COS. — ^r^ — «• = Q
2 n
.pi .rk 1 2p+ ]
•*. i^ = , andQ= . cos. —^ b".
n ^ n 2 n
Hence the general partial integral of
dx .
Book I.] NEWTON'S PRINCIPIA. 81
, / ( 1 — X COS. — ^ w) d X
1 / \ 2 n /
2p+ 1 . ,
X » — 2 X COS. ^r^ — w + 1
2 n
COS. — ^^^ It /2xdx — 2 COS. — ^r cr . d x
2 n / 2 n
/ax d X — a COS. — ^r ff . d X
?i^ +
X* — 2 X COS. ^P*^ ^ T + 1
2n
2n
2p+ 1
sm.= ^ — « / J
d X
~^i
2 „ 2 p + 1 , 
X 2 — 2 X COS. V. 9r + 1
2 n
2p+ 1
COS. ^ T o , 1
C ^ l(x^2xcos.^^+i.+ 1)
2 n V 2 n ^ /
,.2p+l / 2p+l.
sm. — ^r ^ / X — COS. —^ — ^\
+ IJ^ Xtan.'^ 2u_\
n 'V . 2p+ 1 y
\ sm. ^^ «•/
2 n
see Art, 88. Case 4.
d X
Hence then the integral of yr — ^ , which is the aggregate of the results
obtained from the above general form by substituting for p = 0, 1, 2 . . .
n — 1, may readily be ascertained.
As a jparticular instance let f y i — i *^ required.
" Here
n = 3
and the general term is
2p+ 1
cos. — *^ T o , 1
^ . 1 . (x ^  2 X COS. ?^^ ^ + 1)
.2p+l 2p+l
sm. ^  — * X — COS. ^  —
+ ^ . tan.
^ w A. ^;u». ^
O t. 1 O
. 2p+ 1
Let p = 0, 1, 2, collect the terms, and reduce them ; and it will appear that
f dx ..irVj 1 x^+xV3 + l 3x(l x^))
By proceeding according to the above method it will be found, that the
general partial fractions to be integrated in the integrals of
Vol. I. F
82
are respectively
and
A COMMENTARY ON [Sect. I.
.^___. and ^H:::^!
COS. 2 p ^ ^ _ J
1. 2 ^ .dx
" x^2xcos.i£^+l
n
(r + l).2p* 2 r p ff
— X ■ gZ— d X .
n
and when these partial integrals are obtained, the entire ones will be
n n —
— or
2 2
n n ^_ 1
found by putting p = 0, 1 or — — according as p is even or
odd.
Ex. 3. Required the integral of
x'dx
x««» — 2ax"+l
•where a is < 1.
First let us find the quadratic factors of x * ° — 2 a x " + 1. For that
purpose put <
x2n — 2ax'^= —1
Then
x° = a+ Va'' — I
= a+ V — 1 . V 1 — a'
since a is < 1.
Now put a = cos. i\ then
X ° = COS. 6 + V — 1 sin. 3
= COS. (2 p AT + 3) + V^^l sin. (2 p t + a)
2p^+3 , ^ i „• 2p^ + g
.*. X = COS. — • n^ V — 1 sin. :;:
n
and the general quadratic factor of
X
2 D
2ax''+ 1
^s 2 p ^ + « ,
x * — 2 X COS. — i — =^— + 1
n
where p may be any number from 0, 1, &c. to n — 1.
Hence to find the general partial integral of the given indefinite differ
ence, we assume
X' P + Qx N
■^ M
x«»— Sax'+l" , ^ 2pcr + a
X* — 2cos. — ^: + 1
n
Book L] NEWTON'S PRINCIPIA. 83
and proceeding as in the last example, we get
Q = sin. Cl+l) (2p^+a) ^ _1_
n n sin. 8
and
P = sin. '"'•>• (^P + '>Xl^ ■
n n sin. 3
whence the remainder of the process is easy.
Case 4. Let thejactors of the dejiominator be all imaginary and equal in
pairs.
In this Case, we have the form
u _ u
and assuming as in Case 2.
H  P + Qx F + Q^ X ^ . ^
V  (x + a« + /3^)" +(rHr^'' + /3^)"' + ^''
K 4 L X , K^ + L^ X
and reducing to a common denominator,
U=P+Qx + (F + Q'x)(r+i;]«+/32) + &c.
and substituting for x one of its imaginary values, and equating homoge
neous terms, in the iesult we get P and Q. Deriving from hence the
values of —, — , —, —  , &c. and in each of these values substitutiufj for x
d X d X* *=•
one of the quantities which makes x + al ^ + jS '^ = 0, and equating ho
mogeneous terms we shall successively obtain
P', Q'; P", Q", &c.
This method, however, not being very commodious in practice, for the
present case, we shall recommend either the actual developement of the
alaove expression according to the powers of x, and the comparison of the
coefficients of the like powers (by art. 6), or the following method.
Having determined P and Q as above, make
_ U — (P + Q x)
 r+^' + ^'
^ U(F + Qx)
_ u//_ (F^ + Q^^x)
 (x + a)« + ^«
&c. = &c.
Then since U', U", U'", &c. have the same form as U, or have an
F2
84 A COMMENTARY ON * [Sect. I.
integer form, if we put for x that value which makes (x + a) * + j8'^ =r
0, and afterwards in the several results, equate homogeneous quantities
we shall obtain the several coefficients.
P', Q'; P", Q",&c.
Case 5. If the denominator V consist of one set of Factoi'S simple and
unequal ofthefmm
X — ax — a', &c. ;
of several sets of equal simple Factors, as
(x — e) P, (x — eO S &c.
and of equal and unequal sets of quadratic fcLctors of the forms
X 2 + a X + b, x« + a' X + b', &c.
(x* + 1 X + r) /*, (x« + 1' X + r') ', &c.
then the general assumption for obtaining the partial fractions must be
U M M' ,
V X — ax — a!
E F YJ F
+ (X — e)P + (X— e)P' + *^'^ (X — e')'' "•" (x — e')«Ji + ^*''
P + Qx F + Q^x . .
"^ X* + a X + b ■•■ x^ + a X + b' "*■ ^
R + Sx R^ + Sx . ,. G + Hx G^+H^x
■^(xHlx + r)^'*"(x^+k + r)A*i"*'^*^(xHl'x+r')'"^(x^+rx+r')'''*"
and the several coefficients may be found by applying the foregoing rules
for each corresponding set. They may also be had at once by reducing
to a common denominator both sides of the equation, and arranging the
numerators according to the powers of x, and then equating homogeneous
quantities.
We have thus shown that every rational fraction, whose denominator
can be decomposed into simple or quadratic factors, may be itself analyzed
into as many partial fractions as there are factors, and hence it is clear
that the integral of the general function
Ax^ + Bx^' + Scc. Kx + L ^^^
a X " + b X •'^ + &c. k X + I
may, under these restrictions, always be obtained. It is always reducible,
in short, to one or other or a combination of the forms
Having disposed of rational forms we next consider irrational ones.
Already (see Art. 86, &c.)
/ +dx /» d X r d X
V(a — bx^)' ^xV(bx« — a)' ^ V (ax — bx«)
Book I.J NEWTON'S PRINCIPIA. 85
have been found in terms of circular arcs. We now proceed to treat of
Irrationals generally ; and the most natural and obvious way of so doing
is to investigate such forms as admit of being rationalized.
90. Required the integral of
^ < i 1 i i. J
dxXF^x, x% x**, xP, xS&c. s
isohere F denotes any rational function of the quantities betweeti the brackets.
Let
X = U ""^ P 1 , &c.
Then
X'" — U°P^''...
J.
X** rs U" P^' . . . .
J.
&c. = &c.
and
dx = mnpq.... xu"""p« *Xdu
and substituting for these quantities in the above expression, it becomes
rational, and consequently integrable by the preceding article.
„ x^ + 2ax^ + x^ ,
Ex. — — 7^ d X
b + cx*
Here
X = u*
i x'=u«»
x^ = u"*
x* = u*
x^ = u'^
and
dx = 6u*9du.
Hence the expression is transformed to
u^ + 2au*+ 1
60u'='du Z, 15^^
b 4 c u '*
whose integral may be found by Art 89, Case 3, Ex. 2.
91. Required the integral of
dx X F Jx, (a + b x) °, (a + bx)^, SccJ
where F, as before, means any rational function.
Put a + bx = u""P — then substitute, and we get
"""P^ . uP•••^duXF(^?^^^^^^=^^uP•••,uP •,&c.)
which is rational.
86 A COMMENTARY ON [Sect. I.
Examples to this general result are
x*dx ,xMx(a + bx)^
3 and ^^ g,
cx= + (a+bx)^ x+c(a + bx)7
which are easily resolved.
92. Required the integral of
f /a + b x\ ::; /a + b x\ E . \
dx F SX, ( — ) "'(jr^ )<1, &C. > ■
I \f + g x/ Vf + g x/ J
Assume
a I bx
L — u n qs
f+gx"
and then by substituting, the expression becomes rational and integrabie.
93. Required the integral of
d X F Jx, V (a + b X + c X *)]
Case 1. When c is positive, let
a + bx + cx^ = c(x + u)''
Then
a — cu* ,, 2c (cu* — bu + a)du
• X = and d X = ^^^r .—^
2cu — b (2cu — b)*
/ / . 1 . o^ cu^ — bu+a ,
V (a + bx+ ex') = X ^. Vc
^ 2 cu — b
and substituting, the expression becomes rational.
Case 2. When c is negative, if r, t' be the roots of the equation
a + bx — ex* =
Then assume
V c (x — r) (r' — x) = (x — r) c u
and we have
__cru*"fr', _(r — r')2cudu
""" cu*4 1 ''*'' (cu*+ 1)^
V(a + bxcx')= ^"""T^^
^ cy*+ 1
and by substitution, the expression becomes rational.
94. Required the integral of
dx F 5x, (a + b x) K (a' + b'x) ^ •
Make
a + bx = (a' 4.b'x)u«;
Then
_a — a^u^ _ (a'b — b'a)2u dn
^b'u« — b' (b'u'^ — b)*
./f LK X u^(ab' — a'b) ,/,,,, V V(ab^ — a^b)
V(a + bx)= ^(Vu'b) ' ^("+^^^= vVu^b) .
Book I.] NEWTON'S PRINCIPIA. m.
Hence, substituting, the above expression becomes of the form
duFJu, V(b'u« — b)l
F' denoting a rational function different from that represented by F.
But this form may be rationalized by 93 ; whence the expression becomes
integrable.
95. Required the integral of
^mi dx(a + b x" )q.
This form may be rationalized when either — , or 1 — is aa integer.
p ijq a
Case 1. Leta+bx"=u'; then(a+bx")q = u p, x" = — r — , x"::
/u*! — a\^ ^ ,j qui^du /u^ — a\Z2z^
(b)°''' ''^= nb (— b) ° •
Hence the expression becomes
q «j.n ij f^'* — aN*""
V uP + 1^ du V \ ) ^
nb b '^
which is rational and integrable when — is an integer.
Case 2. Let a + bx" = x"u'i; then substituting as before, we get the
transformed expression
q a"'*'? uP + qidu
n (u" — b)^ + T + '
which is rational and integrable when — + — is an integer.
Examples are
x'dx x±^"dx
(a^ + x*)^' (a* + x«)^*
x*°'dx(a2 + x2)— T,
(a'+x^)*
96. Required the integral of
x'"idx(a + bx")i X F(x '').
This expression becomes rational in the same cases, and by the same sub
stitutions, as that of 95. To this form belongs
x"'+°' dx(a + bx''^?
and the more general one
P p
^ X "  • d X X (a + b x) 1
F4
88 A COMMENTARY ON [Sect. I.
where
P = A + Bx° + Cx*° + &c.
and
Q = A' + B'x» + C'x'*'' + &c.
97. Required the integral of
x"^dx X fJx", x°, (a + bx°)~^?
Make a + bx°=u'J; then
dx = y— .(b)'^ du
m
and in the cases where — is an integer, the whole expression becomes ra
n
tional and integrable.
98. Required the integral of
Xdx
X' + X" + V (a + b X + c x*)
ichere X, X', X" denote any rational functions of's..
Multiply and divide by
X' + X"— V(a + bx + x«)
and the result is, after reduction,
XXMx XX^Mx V(a + bx + cx«)
X'« — X"^(a + bx + cx*) X'^— X''*(a +bx + cx^j
consisting of a rational and an irrational part. The irrational part, in
many cases, may also be rationalized, and thus the whole made integrable.
99. Required the integral of
x"'dxF{x°, \^(a + bx'^ + cx'^")}
Let x ° = u ; then the expression may be transformed into
1 '°+^ ,
— u n 'duF{u, V (a + bu + cu*)]
which may be rationalized by Art. 93, when — '^^— is an integer.
100. Required the integral of
x'^dxFJxS ^(a + b^x*''), bx»+V(a + b^x^")}.
Let
bx"+ ^(a + b«x«°) = u;
then
n(2b)=±^ " ^ " *
and the whole expression evidently becomes rational when is an
integer.
Many other general expressions may be rationalized, and much might
Book L] NEWTON'S PRINCIPIA. 89
be said further upon the subject ; but the foregoing cases will exhibit the
general method of such reductions. If the reader be not satisfied let him
consult a paper in the Philosophical Transactions for 1816, by E. Ffrench
Bromhead, Esq. which is decidedly the best production upon the Integrals
of Irrational Functions, which has ever appeared.
Perfect as is the theory of Rational Functions, yet the like has not been
attained with regard to Irrational Functions. The above and similar arti
fices will lead to the integration of a vast number of forms, and to that of
many which really occur in the resolution of philosophical and other
problems ; but a method universally applicable has not yet been discover
ed, and probably never will be.
Hitherto the integrals of algebraic forms have been investigated. We
now proceed to Transcendental Functions.
101. Required the integral of
a^'dx.
By Art 17,
d.a'' = l.a X a^dx
.•./a«dx = j/da«
= u=''^
ht;
Hence
/"a ™ * d X = — T— a "•
•^ mla
'' + C
102.
Required the integral of
Xa^dx
ixihere X is an algeb
By the form (see
we have
raic function qfyi.
73)
d (u v) = ij d V f
vdu
fvi d V = u V — fy d u.
Hence
/XaMx = X.,— 
1 S
f la
(a)
(b)
/ »dX a'dx _dX a^ > » a' d'X
^ dx * la ~dx*(la)'' *'(la)* dx
/ »d'X aMx _ d«X j^ __ r a^ d^
•^ dx^" (la)"~ dx2 *(la)3 ^'(la)^ dx^
&c. = &c.
the law of continuation being manifest.
90 A COMMENTARY ON [Sect. I,
Hence, by substitution,
/Xadx = Xj— — ^.^p^, + j^.pp&c.
which* will terminate when X is of the form
A + Bx+Cx2 + &c.
^ ^ , , a^x» 3a^x* , 3.2a^x 3.2a'' , ^
Ex./x'a''dx = j^^^j^+ jYaj^ lUr +^'
OTHERWISE
/a^Xdx = ayXdx— /la.a^x/Xdx
= a''X' — la/a^X'dx
putting
X'=/Xdx.
Hence
/a^X'dx = a^X" — la/a^X^'dx
&c. = &c.
and substituting, we get
/a^ Xdx= a^ X' — la.a'' X" + (la)^a'' X"' — &c.
X', X'', X% &c. being equal to/X d x, /X' d x, /X'' d x, &c. re
spectively.
which does not terminate.
By this last example we see how an Indefinite Difference may be in
tegrated in an infinite series. If in that example x be supposed less
than 1, the terms of the integral become less and less or the series is con
vergent Hence then by taking a few of the first terms we get an ap
proximate value of the integral, which in the absence of an exact one, will
frequently suffice in practice.
The general formula for obtaining the integral in an infinite or finite
series, corresponding to that of Taylor in the Calculus of Indefinite
Differences, is the following one, ascribed to John Bernoulli, and usually
termed
JOHN BERNOULLI'S THEOREM.
/XdxrrXx— /xdX
rd^X , _ dX x^_ /» xMx d^X
^ dx ^"x dx • 2^2* dx^
/* d'X x^dx_d^X x^ /»x^dx d^X
J diX^' 2 ~dx^'2.3 ^ 2.3 * dx^
&c. = &c.
Book I.] NEWTON'S PRINCIPIA. 91
Hence
the theorem in question.
Ex.l./x»dx = x + i— ?x+'+^^^^^^x™ + »— &c. + C
II /, Ml m . m — 1 , o \ , r^
= x» + 'x (l_ + — ^3— + &c.)+C
But since
(1 — !)» + '= 1 — I^T+I + "" +J •'" — &c. =
2
../x™ dx= r + C
•^ m+1 ^
as in Art. 78.
102. Required the integral of
Xdx(lx)»
'where X is any Algebraic Function ©/"x, 1 x the Hyperbolic logarithm of x,
andn a positive integer.
By the formula
f\x d V = u V — y* V d u
we have
/Xdx(lx)° = (lx)"/Xdx — n/(lx)'»^— /Xdx
= (lx)«>X'n/(lx)— »^X'
•^'dx(lx)'' = (lx)°»X(nl)/(lx)"^^
&c. = &c
.X' , rX"
where X', X'', X"', &c. are put for/Xdx,/— dx,/— d x, &c. re
spectively.
Hence
/Xdx(lx)° = X'(lx)°— nX"(lx)'>J + n.(n— l)X"'(lx)^2— &C. + C.
Ex.l./x«'dx(lx)"=^{(lx)'>^^(lx)"»&c.}
1 93. Required the integral of
X
where U is any function qfl's..
92 A COMMENTARY ON [Sect. L
Let u = 1 X.
Then
J d X
d u =r .
X '
and substituting, the expression becomes algebraic, and therefore integra
ble in many cases.
104. Required the integral of
Xdx(lx)«
lahere n is negative.
Integrating by Parts, as it is termed, or by the formula
fvi d V = u V — f\ d u
we get, since
/Xdx ^ dx„ . „
r— =/X X. (1 X)",
(1 X) " "^ X ^ ' '
/. X_dx X X 1 / . dx d(Xx);
•/(lx)"~ (n— 1) (lx)"''^n— IV (lx)°'* dx
and pursuing the method, and writing
^, ^dJXx)
d X
^„ ^ d (Xx)
d X
&c. = &c.
we have
/»Xdx_ Xx X^x ^^ f X^")'dx
^(Ix)" (nl)(lxr> „_i.n— 2.(lx)^ *''' ^ (n1).. .2. l(k)
or
Xx p X("' + ^) dx
(n— l)(lx)"' *^ ^(n— l).(n — 2)....(n — m)(lx)"'"
according as n is or is not an integer, m being in the latter case the
greatest integer in n.
P, /. x_^dx __ x°' + ^ f 1 m 4 1 ,0,1
^"^'J (Ix)" ir^=rrt(lx)"' "*■ (n — 2) (lx)«=^'*"^''j
(m + 1)"^ /' X^dx
(n— 1) (n — 2) . ... 1 •/ Ix
when m is an integer.
105. Required the integrals of
d ^ d ^ d ^
d 6 . cos. 6. d. 6 . sin. ^, d ^ . tan. ^, d ^ . sec. 6, ;, , —. — . , .
' ' ' cos. 6 sm. 6 tan. 6
By Art. 26, &c.
d sin. ^ = d ^ . cos. 6, and d cos. ^ = — d ^ sin. 6
.'.fd 6 COS. 6 = sin. ^ + C . . • (a)
and
/d 6 sin. = C — cos. 6 (b)
Book I.] NEWTON'S PRINCIPIA. 93
Again let tan. ^ = t ; then
A A dt
and
1 + t^
t d t
/dtftan.^==/.jiAl_==il(l +t«) + C
= C — 1 . COS. 6 (c)
since
Again
1 + t== = sec.'^^ =
d 6 sec. =
COS. "= d
d 6 d 6 COS. 6
COS. 6 1 — sin. 2 6
d (sin. 6)
1 — sin.
1
— ¥
d (sin. 6) d sin. 6
' 1 — sin. ^ + * * 1 + sin. 6
.'./d 6 sec. 6 =^l.(l+sin.^)— ^1(1— sin.^) + C
= l.tan. (450+1) + C. . . (d)
d^
which is the same as / .
*f cos. d
Again
/: = fd 6 cosec. 6
sin. 6 '^
= /d^sec.(^)=_/d.(_^)sec.(_^)
=l.tan.(45" + ^i) + C
= l.(tan.) + C (e)
Again
= lcos. (^) + C(byc)
= 1 . sin. ^ + C (f)
106. Required the integral of
sin. ™ 6 cos. ** ^ . d ^.
m a7id n being positive or negative integers.
94 A COMMENTARY ON [Sect. I.
Let sin. ^ = u ; then d 6 cos. ^ = d u and the above expression becomes
n — 1
u*"du(l — u'^) ~ir
which IS mtegrable when either — ^ — or — ^ 1 ^ — = — ^ —
is an integer (see 95.) If n be odd, the radical disappears ; if n be even
and m even also, then — ^~ = an integer j if n be even and m odd, then
— ^ — is an integer. Whence
u^d u (1 — u«)""2
is integrable by 95.
OTHERWISE,
Integrating by Parts, we have
Sin ni *■■ 1 4 Tn ^"^ 1
/do sin." 6 cos."* 6= — '. , cos."+ ^ 6+ — — t/cos." + 2 $. sin.^^ 6xd6
'' n + 1 m + I''
sin.™*0 „a.i/,."i — Izj • «. 9.
= cos. " + M H ; — /dx sin. ""^d COS. "0
m + n m + n*'
and continuing the process m is diminished by 2 each time.
In the same way we find
/ 1 X • ms n A sin."* + M COS. "' 6 n — 1 ,,
fd6 sm. " 6 COS. " = i / d x sin. "» 6 cos. " ^ ^
•^ m + n ' m + n*^
and so on.
107. Required the integrals of
d u = d sin. (a + b) cos. (a' 6 + b')
d V = d sin. (a + b) sin. (a' 6 + b')
and
d w = d cos. (a + b) cos. (a' + b')
By the known' forms of Trigonometry we have
du = do {sin. (a + a'.O+b + b') + sin. (a — a'.O+b — h')\
d v= do Jcos. (a+a'.O + b + bO — cos. (a — a'. 0+b — b')}
d w = do {cos. (a + a'. 0+b + b') + cos. (a^T' . 0+b — b')}
Hence by 105 we have
^ , f cos.{a + a'.0+ b + b') , cos.(a — a\ + b— bO \
" = ^~H ^r+ij "^ '^^^^^' i"
— C I i f sin. (a + a^ + b + bQ _ sin, (a — a^O + b — bQ \
~ \ a + a' ~ a — a' j
^_ Q ■ I / s^" (a + a\ + b + bQ sin. (^^I^'^. + b — bQ 1
*\ a + a' "*" a — a' /
These integrals are very useful.
Book I.] NEWTON'S PRINCIPIA. 95
108. Required the integrals of
^ " d ^ sin. tf, and tf " d ^ cos. d.
Integrating by Parts we get
/■^"Xd^sin. ^=C — ^"cos. ^+n ^"^ sin. ^4n . (n — 1) ^"^cos.^ — &c.
and
/^ " X d ^ COS. ^= C + ^ ° sin. ^ + n ^ °  * cos. 6 — n.(n — l)^"^ sin. 6 + &c.
109. Required the integrals of
X d X sin. — 1 X
X d X tan,  ' x
X d X sec. ~ ' X
&c.
Integrating by Parts we have
/Xdxsin.ix=sin.ix/Xdx — ^4^^^^^
/Xdxtan.^x = tan.^x/Xdx— /'^f'^^.f'^
yXdxsec.^x = sec. ^ x/Xdx — f — fro 7\
&c. = &c.
see Art 86.
1 10. Required the integral of ■
(f + g cos. 6) d d
(a + b cos. 6Y '
Integrating by Parts and reducing, we have
" ~ (n— 1) (a^^b^) (a + b cos. ""^ (n — 1) (a^ — b^) ^
^(n— l)(af— bg)+ (n — 2)(ag — bf)cos.^ ^ ,
J (a + bcos.d)»i
which repeated, will finally produce, when n is an integer, the integral
required.
V /  d^ _ 2 , _^ (ab)tan.  ^ ^
»/a + bcos.^~ V(a2 — b^) "* ^(3=^ — b^) ■*"
or
1 , b+acos. ^+ sin. 6 V {h^ — a^) p
V (b« — a^) • a + b cos. 6 " + ^*
Notwithstanding the numerous forms which are integrable by the pre
ceding methods, there are innumerable others which have hitherto resisted
all the ingenuity that has been employed to resolve them. If any such
appear in the resolution of problems, they must be expanded into con
96 A COMMENTARY ON [Sect. I
verging series, by some such method as that already delivered in Art. 101 ;
or with greater certainty of attaining the requisite degree of convei'gency,
by the following
METHOD OF APPROXIMATION.
111. Required to integrate between x = b, x = a, any given Indefinite
Difference, in a convergent series.
Let f (x) denote the exact integral of y X d x; then by Taylor's
Theorem
f.(x + h)~fx = Xh + ^ ]^ + &c.
and making
h = b — a
f(x + ba)~fx = X.(ba) + ^^^.i^^^ + &c.
Again, make
X = a
then
A, A', &c.
become constants
and we obtain
f(b)f(a) = A(ba) + ^. (ba)« + ^3 (ba)»
which, when b — a is small compared with unity, is sufficiently conver
gent for all practical purposes.
If b — a be not smaU, assume
b — a = p.^
p being the number of equal parts ^, into which the interval b — a is sup
posed to be divided, in order to make jS small compared with unity. Then
taking the integral between the several limits
a, a + /3
a, a + 2 j8
&c.
a, a + p /3
Book L] NEWTON'S PRINCIPIA. 9T
we get
f. (a + ^) f (a) = A/3 + ^. /S» + ^3 . i8» + &c.
f(a+2iff)— f (a+^) = B^+f . /3* + ~^^ + &c.
&c. = &c.
f(a+p^) — f (a+J=n./3) = P/3 + "2 /S' + 2;^iS' + &c.
A, A', &c. B, B', &c P, P^ &c.
being the values of
when for x we put
a, a + /3, a + 2 /3, &c.
Hence
f(b)f(a) = (A + B + ....P)^
+ (A' + B'+....F)^
+ (A + B" + ....FOi:^
+ &c.
the integral required, the convergency of the series being of any degree
that may be demanded.
If /3 be taken very small, then
f (b) — f (a) = (A + B + P) /3 nearly.
Ex. Required the approximate value of
/X'idx X (l_x")f
m m p
between the limits of x = and x = 1, when neither —i wor ~ + ~
is an integer.
Here
X = x»i (i — x")?
and
dX p , .^ np i_i
j^=:(m + n^— ])x«2(i_x'')i _^x»2(l— x")"
b_a= 1—0= 1.
Assume 1 = 10 X jS, and we have for limits
' 10 ' ' 10 ' *'*
Vol. I. G
08 A COMMENTARY ON [SEcr. I.
Hence m being > 1,
A =
C=(fo)""{>(.^)"}^
°=(^r{'(fon'
&c. = &c.
r=(for'{'(fo)"}^
Hence, between the limits x = 1 and x =
yXdx = ! — X {(IC — 1)1"+ (10« — S")!"
10 »" + "T
+ (10" —S*")^ + &c. + (10«— 9")f jnearly.
We shall meet with more particular instances in the course of our
comments upon the text.
Hitherto the use of the Integral Calculus of Indefinite Differences has
not been very apparent. We have contented ourselves so far with
making as rapid a sketch as possible of the leading principles on which
the Inverse Method depends ; but we now come to its
APPLICATIONS.
1 12. Required to Jind the area of any curve, comprised between two
given values of its ordinate.
Let E c C (fig. to Lemma II of the text) be a given or definite area
comprised between and C c, or and y. Then C c being fixed or De
finite, let B b be considered Indefinite, or let L b = d y. Hence the
Indefinite Difference of the area E c C is the Indefinite area
B C c b.
Hence if E C = x, and S denote the area E c C ; then
dS = BCcb=CL + Lcb
= ydx+ Lcb.
But L c b is heterogeneous (see Art. 60) compared with C L or y d x.
.*. d S i= V d X
Book I.] NEWTON'S PRINCIPIA. 99
Hence
S=/ydx,
the area required.
Ex. 1. Required the area of the common parabola.
Here
y * = a X.
_ 2ydy
and
•. d X = y— J
a
and between the limits of y = r and y = r' becomes
If m and m' be the corresponding values of x, we have
2
S = 5 (r m — r' m')
Let r' = 0, then
= ~ of the circiunscribing rectangle.
2
S = — r ra (see Art. 21.)
3
Ex. 2. Take the general Parabola whose equation is
y " = a X °.
Here it will be found in like manner that
s=:HLiy + c
mfn
. a p
m + n
between the limits of n = y = 0, and x = a, y = /3.
Hence all Parabolas may be squaredy as it is termed ; or a square may
be found "whose area shall be equal to that of any Parabola.
Ex. 3. Required the area of an HYPERBOLA comjprised by its asympic^Cf
and one infinite branch.
If X, y be parallel to the asymptotes, and originate in the center
X y = ab
is the equation to the curve.
Hence
y*
G2
100 A COMMENTARY ON [Sect. I.
and
S=/iMy = C_ably.
Let at the vertex y = /S, and x = ; then the area is and
C = a b . 1 a
Hence
S = ab.l.'^.
y
1 13. If the curve be referred to ajixed center by the radiusDector § and
tracedangle 6; then
ds = ll^^
2
For d S = the Indefinite Area contained by f , and f+df=(f+df)  — ^
= „ + ^ — I (Art. 26) and equathig homogeneous quantities we
have
Ex. 1. In the Spiral of Archimedes
^ = a^
Ex. 2. In the Trisectrix
g = 2 COS. tf + 1
.. dS = i/(2cos. <J± lydd
which may easily be integrated.
Hence then the area of every curve could be found, if all integrations
were possible. By such as are possible, and the general method of ap
proximation (Art. Ill) the quadrature of a curve may be effected either
exactly or to any required degree of accuracy. In Section VII and many
other parts of the Principia our author integrates Functions by means of
curves ; that is, he reduces them to areas, and takes it for granted that
such areas can be investigated.
114. To find the length of any curve comprised within given values of the
ordinate ; or To RECTIFY any curve.
Let s be the length required. Then d s = its Indefinite Chord, by
Art. 25 and Lemma VII.
.. ds = • (dx^ + dy«)
imd
s =:/V(dx' + dyO (a)
Book L] NEWTON'S PRINCIPIA. 101
Ex. 1. In the general parabola
y " = ax".
Hence
m^ 2m_2
dx^ = gy n . dy2
n* a n
and
ds = dy. V(l +J^/T^)
n ' a n
which is integrable by Art. 95 when either
1 1
ihat is, when either
n n
In 1 m
or
2 m — n 2 m — n
is an integer ; that is when either m or n is even.
The common parabola is Rectifiable, because then m = 2. In this case
ds=dy V(I+^,y^) (r)
Hence assuming according to Case 2 of Art. 95,
we get the Rational Form
I'^du
ds =
Hence by Art. 89, Case 2,
a
4 . o\ 4 ,4
But u = V ^ . Hence by substituting and making the ne
y
cessary reductions
G3
4
. — + V u
ll. ^ + C.
>f a*
102
A COMMENTARY ON
[Sect. I.
s =
y^(y'+i) y + ^(y' + ir)
+ al.
+ c.
Let y = ; then s = and we get C =
and .*. between the Limits of y = and y = jS
s =
+ al.
In the Second Cubical Parabola
y ^ = a X*
and
"•jyVo + H)
which gives at once (Art. 91)
Ex. 2. In the circle (Art. 26)
ds =
dy
which admits of Integration in a series only. Expanding (1 — y *)~»
by the Binomial Theorem, we have
Hence,
and
^^y + fa + Ar^y' + ""'■ + ^
and between the limits of y = and y =  or for an arc of 30° we have
1
2. 3. 2^ "*■ 2.4.5.2
Tb + &c.
1m _J_ 4. _L a. J J ^il oR
~ 2 "^ 3. 2* "•" 5. 2« "^ 7. 2'i "^ 9. 2'« '^^^"
r.5
I .0208333333
= i .0023437500
I .0003487720
L.0000593390^
&c.
> = .5235851943 nearly.
Book L] NEWTON'S PRINCIPIA. 103
Hence ISC of the circle whose iadius is 1 or the whole circumference
sr of the circle whose diameter is 1 is
T = . 5235851943 ... X 6 nearly
= 3.1415111658
which is true to the fourth decimal place : or the defect is less than .
^ 10000
By taking more terms any required approximation to the value of v may
be obtained.
Ex. 3. In the Ellipse
where x is the abscissa referred to the center, a the semiaxis major and
a e the eccentricity (see Solutions to Cambridge Problems, Vol. II. p. 144.)
115. If the curve be referred to polar coordinates, g and 6; then
s =/^/(gM<J^+dg^) (b)
For
y = g sin. 6
X = m + g COS. 6
and if d X % d y ^ be thence found and substituted in the expression
(114. a) the result will be as above.
Ex. 1. In the Spiral of Archimedes
g = ad
•••^^ = ^^(1^ +
«& a a
see the value for s in the common parabola, Art. 114.
Ex. 2. In the logarithmic Spiral
f = e
or
6 = l.g
and we find
s = V~2fd g = g V 2 + C.
116. Required the Volume or solid Content of any solid formed by the
revolution of a curve round its axis.
Let V be the volume between the values and y of the ordinate of this
generating curve. Then d V = a cylinder whose base is t y ^ and alti
tude d X + a quantity Indefinite or heterogeneous compared with either
d V or the cylinder.
G4
104 A COMMENTARY ON [Sect. I.
But the cylinder = a y M x. Hence equating homogeneous terms, we
have
d V = cry«dx
and
V = ff/y^dx (c)
Ex. 1. In the sphere (rad. = r)
y ^ = r*^ — X*
.. V = ^/r ^ d X — ^/x ° d X
and between the limits x = and r
which gives the Hemisphere.
Hence for the whole sphere
4
Ex. 2. In the Paraboloid.
y2 = ax
.'. V = ^fsL X d X
•jt a
and between the limits x = and a
V _ g .a .
Ex. 3. In the Ellipsoid.
.. V = ^^'./(a^dx — x^dx)
(a.x_) + C;
a^ \ 3
and between the limits x = and a
V=_ a3 = _.ab«.
Hence for the whole Ellipsoid
V = ic^ab^
o
The formula (c) may be transformed to
Vrr^yS — c/Sdy (d)
Book L] NEWTON'S PRINCIPIA. 106
where S =ry"y d x or the area of the generating curve, which is a singular
expression, yS d y being also an area.
In philosophical inquiries solids of revolution are the only ones almost
that we meet with. Thus the Sun, Planets and Secondaries are Ellip
soids of diiFerent eccentricities, or approximately such. Hence then in
preparation for such inquiry it would not be of gieat use to investigate
the Volumes of Bodies in general.
If X, y, z, denote the rectangular coordinates, or the perpendiculars let
fall from any point of a curved surface upon three planes passing through
a point given in position at right angles to one another, then it may easily
be shown by the principles upon which we have all along proceeded,
that
d V = d yyz d x"
or
= d z/y d X L (e)
or
= d x/z d y^
according as we take the base of d V in the planes to which z, j', or x is
respectively perpendicular
For let the Volume V be cut off by a plane passing through the point
in the surface and parallel to any of the coordinate planes ; then the area
of the plane section thus made will be
/z d x"
or
yy d X ^see Art. 112.
or I
/zdyj
Then another section, parallel toyz d x, oryy d x, oryz d y and at
the Indefinite distance d y, or d z, or d x from the former being made,
the Indefinite Difference of the Volume will be the portion comprised by
these two sections ; and the only thing then to be proved is that this por
tion is = d yyz d X or d zy y d x, or d x y z d y. But this is easily to
be proved by Lemma VII.
This, which is an easier and more comprehensible method of deducing
d V than the one usually given by means of Taylor's Theorem, we have
merely sketched ; it being incompatible with our limits to enter into de
tail. To conclude we may remark that in Integrating both y z d x, and
y d y y z d X must be taken within the prescribed limits, first considering
y Definite and then x.
106 A COMMENTARY ON [Sect. I.
117. To find the curved surface of a Solid of Revolution.
Let the curved surface taken as far as the value y of the ordmate re
ferred to the axis of revolution be (t, and s the length of the generating
curve to that point; then d o = the surface of a cylinder the radius of
whose base is y and circumference 2 s y, and altitude d s, by Lemma VII.
and like considerations. Hence
d(j = 2'Tyds
and
or
6 = 2 vfy d s (a)
= 2^ys — 2^/sdy (b)
which latter form may be used when s is known in terms of y ; this will
not often be the case however.
Ex. In the common Paraboloid.
and
y'' = ax
. = ^/ydy ^/(y'= + a^)
= H (r + a') U c.
Let y = and /3, then a between these limits is expressed by
If the surface of any solid whatever were required, by considerations
similar to those by which (116. e) is established, we shall have
d ff = V (dy^ + dz2)/>/ (dx^ + dz^) . . . . (c)
and substituting for d z in V d x^ + d z'' its value deduced from z = f.
(x, y) on the supposition that y is Definite ; and in V (d y '^ + d z *) its
value supposing X Definite. Integrate first V (d x^ + d z^) between the
prescribed limits supposing y Definite and then Integrate V (d y ^ + d z ^)
/V(dx'ldz^) between its limits making x Definite. This last result
will be the surface required.
We must now close our Introduction as it relates to the Integiation of
Functions of one Independent variable.
It remains for us to give a brief notice of the artifices by which Func
tions of two Independent Variables may be Integrated.
118. Required the Integral of
X d x + Y d y = 0,
•where X is ant/ function ofx, and Y a function ofy the same or different.
Book L] NEWTON'S PRINCIPIA. 107
When each of the terms can be Integrated separately by the preceding
methods for functions of one variable, the above form may be Integrated,
and we have
/Xdx+/Ydy = C.
This is so plain as to need no illustration from examples. We shall,
nowever, give some to show how Integrals apparently Transcendental
may in particular cases, be rendered algebraic.
Ex. 1. ^ + AZ_ = 0.
X y
.. 1 X + 1 y = C = 1 . C
.•.l(xy) = l.C
and
.. X y = C or = C.
Ex. 2. ^ (l_x^) "^ V (l — y^) = ^•
Here
sin.  ' X + sin. ~ ^ y — C = sin.  ' C
.*. C = sin. {sin. ~ ' x = sin. ~ ' y]
= sin. (sin. ~ ^ x) . cos. (sin. ~ ' y) + cos. (sin. ~ ' x) sin. (sin. ~^ y)
= X. V (1— y^) + y V (1 — x^)
which is algebraic.
Generally if the Integral be of the fofni
f^x) + f.My) = C
Then assume
C = f.'(C)
and take the inverse function of f ~' (C) and we have
C = f{f'(x)4f'(y)l
which when expanded will be algebraic.
119. Required the Integral of
" Ydx + Xdy = 0.
Dividing by X Y we get
X t Y
which is Integrable by art. 118.
120. Required the Integral of
Pdx + Qdy = 0;
uohere P and Q are each mch functions qfs. and y that the sum of ike expo
nents of Si and y in every term of the equation is the same.
108 A COMMENTARY ON [Sect. I.'
Let X = u y. Then if m be the constant siim of the exponents, P and
Q will be of the forms
U X y« — U'y™
U and U' being functions of u.
Hence, since dxr=udy + ydu, we have
U.(udy + ydu) + U'dy =
and
(Uu + U0dy + Uydu = O
• • y + U U + U'  " ^""^
which is Integrable by art. 118.
Ex. 1. (a x + b y) d y + (f X + g y) d X = 0.
Here
P = fx+gy, Q = ax + by
U= fu+ g, U' = au + b
. ^_y , (fu + g)du _
•• y "*fu^4 (g + a)u + b"
which being rational is Integrable by art. (88, 89)
Ex. 2. X d y — y d X = d X V (x* + y*)
Here
Q = x, P = ~y— V (x'^+y*)
U' = u, U = — 1 — V (1 + u»)
. dy 1 + V(l +u^)
' ••T+ uV(l+u^) ^"^
or
y ^ u ^ u V (1 + u«)
which is Integrable by art. (82, 85.)
These Forms are called Homogeneous.
121. To Integrate
(ax + by + c)dy + (mx+ny+p)dx = 0.
By assuming
ax + by + c = u
and
m X
we get
, mdu — adv , , bdv — ndu
d V = i , and a x = r r — 
^ mb — na' mb — na
and therefore
(mu — nv)du + (bv — au)du =
which being Homogeneous is Integrable by Art. 120.
+ by + c = uj
+ n y + p = V j
Book I.] NEWTON'S PRINCIPIA. 109
We now come to that class of Integrals which is of the greatest use in
Natural Philosophy — to
LINEAR EQUATIONS.
122. Required to Integrate
dy + yXdx = X'dx,
where X, X' are functions of X.
Let
y = u V.
Then
udv + vdu + Xuvdx = X'dx
Hence assuming
dvivXdx = . (a)
we have also
Hence
V d u = X' d X (b)
— + Xdx =
V
.. Iv +/Xdx = C
or
V — g C— /Xdx
= e^ X e'"^"*^
= C X ef^^\
Substituting for v in (b) we therefore get
1 /Xdx
du=^.e X'dx
which may be Integrated in many cases by Art 118.
Ex. dy + ydx = ax^dx.
Here
X = 1, X' = a x»
yx d X = X
and
/X'dxe^xdx _ a/x^e^dx
= a e '^ (x' — 3 X* + 6 X — 6)
see Art. (102)
Hence
y= Ce'' + a(x' — 3x« + 6x6)
no A COMMENTARY ON [Sect. I.
122. Required to Integrate the Linear Equation of the second order
dx* d X •'
tshere X, X' are Junctions qfx.
d V
Lety = e^"""^*; then 3^ = ue^"
•^ d X
dx
dll=e/«("+u')
dx* Vd X /
and .*. by substitution,
^+u*+Xu+X' =
d X
which is an equation of the first order and in certain cases may be Integ
rable by some one of the preceding methods. When for instance X and
X' are constants and a, b roots of the equation
u«+ Xu+ X' =
then it will be found that
y=Ce''''+C'e»'*.
123. Required the Integral of
d x'^ d X •'
•cohere X" is a new function of&..
Let y = t z ; then Differencing, and substituting, we may assume the
result
dx*^ "^ '"■ dx
^'% X^+ X'z = (a)
and
••■<'(d:)+(K)(^+iE)ov' •••(")
Hence (by 122) deriving z from (a) and substituting in (b) we have a
Linear Equation of the first order in terms of T j — ^; whence fv — j may
be found ; and we shall thus finally obtain
dx« "*" dx* X "■^'^•y ~ x«— r
Here
XL. ^ "Vf ' "v//
x' X*'  x^— 1*
Book I.] NEWTON'S PRINCIPIA. Ill
Equat. (a) becomes
d^ z . d z 1 z
whence
d X 2 "^ d X ' X x''
du+(u= + ^l)dx =
wherein z = e^"^^; which becomes homogeneous when for u we put y~\
Next the variables are separated by putting (see 120)
X = V s
and we have
and
Hence
and
Again
and
and
d V s^ + s — 1 ,
V s (s^ 1)
 1 ilAJ
 s Vs — r
^'+^ ,/udx = l.^
X (x^— 1)
X2 1
z = e/"'''^ = .
X
g/Xdx __ glx _ X
/X" e^X'i'' z d X =/a d x = a x + C
_ x''— 1 /»(ax + C) xdx^
y ~ X ^ (x«— 1)2 '
which being Rational may be farther integrated, and it is found that
finally
^ _ ax+C xJ1 (c\ ^i^) .
Here we shall terminate our long digression. We have exposed both
the Direct and Inverse Calculus sufficiently to make it easy for the
reader to comprehend the uses we may hereafter make of them, which
was the main object we had in view. Without the Integral Calculus, in
some shape or other, it is impossible to prosecute researches in the higher
branches of philosophy with any chance of success ; and we accordingly
see Newton, partial as he seems to have been of Geometrical Synthesis,
frequently have recourse to its assistance. His Commentators, especially
112 A COMMENTARY ON [Sect. II.
the Jesuits Le Seur and Jacquier, and Madame Chastellet (or rather
Clairaut), have availed themselves on all occasions of its powers. The
reader may anticipate, from the trouble we have given ourselves in establish
ing its rules and formulas, that we also shall not be very scrupulous in that
respect. Our design is, however, not perhaps exactly as he may suspect.
As far as the Geometrical Methods will suffice for the comments we may
have to offer, so far shall we use them. But if by the use of the Algo
rithmic Formulas any additional truths can be elicited, or any illustrations
given to the text, we shall adopt them without hesitation.
SECTION II. PROP. I.
124. This Proposition is a generalization of the Law discovered by Kepler
from the observations of Tycho Brahe upon the motions of the planets
and the satellites.
" When the body has arrived at B," says Newton, ^Het a centripetal
force act at once with a strong impulse, Sfc"~\ But were the force acting
incessantly the body will arrive in the next instant at the same point C.
For supposing the centripetal force
incessant, the path of the body will
evidently be a curve such as A B C.
Again, if the body move in the chord
A B, and A B, B C be chords de
scribed in equal times, the deflection
from A B, produced by an impulsive
force acting only at B and communi
cating a velocity which would h ave been
generated by the incessant force in the time through A B, is C c. But
if the force had been incessant instead of impulsive, the body would have
been moving in the tangent B T at B, and in this case the deflection at the
end of the time through B C would have been half the space describ
ed with the whole velocity generated through B C (Wood's Mech.)
But
CT = ^ Cc
.*. the body would still be at C.
Book I.] NEWTON'S PRINCIPIA. 113
AN ANALYTICAL PROOF.
Let F denote the central force tending constantly to S (see Newton »
figure), which take as the origin of the rectangular coordinates (x, y)
which determine the place the body is in at the end of the time t. Also
let f be the distance of the body at that time from S, and d the angular
distance of g fi'om the axis of x. Then F being resolved parallel to the
axis of x, y, its components are
F.andF.^
and (see Art. 46) we .*. have
Hence
d'x _ __ T^ X d^j _ _ p 2
dt^ ~ P ' dt^ ~ P
y d'x _ T^ X y _ xd*y
dt^ ~ e ~ dt*
y d* X — X d* y .
d t
But
yd'^x — xd"y = dydx + yd'^x — dxdy — xd^y
= d.(ydx — xdy)
.*. integrating
ydx — xdv ^ ^
^ . = constant = c.
d t
Again,
X = f cos. ^, y = g sin tf, x ^ + y ' =r ^ *
.'. d X = — f d ^ sin. ^ + d g cos. d
d y = f d ^ COS. ^ + d f sin. 6;
whence by substitution we get
ydx — xdy = f*d^
p2d^
dt
= C
But (see Art. 1 13)
^— ^ — = d . (Area of the curve) = d . A
.'. d t = ' := — . d A.
c c
Vol. I. H
114 A COMMENTARY ON [Sect. II.
Now since the time and area commence together in the integration
there is no constant to be added.
.. t = — X A a A.
c
Q. e. d.
125. CoR. 1. Pkop. II. By the comment upon Lemma X, it appears
that generally
ds
" = dl
and here, since the times of describing A B, B C, &c. are the same by
hypothesis, d t is given. Consequently '
V a d s
that is the velocities at the points A, B, C, &c. are as the elemental spaces
described A B, B C, C D, &c. respectively. But since the area of a a
generally = semibase X perpendicular, we have, in symbols,
d . A = p X d s
d.A
.'. V a d s « ;
P
and since the a A B S, B C S, C D S, &c. are all equal, d A is constant,
and we finally get
1 c
V a — or = 
P P
the constant being determinable, as will be shown presently, from the
nature of the curve described and the absolute attracting force of S.
1 26. Cor. 2. The parallelogram C A being constructed, C V is equal and
parallel to A B. But A B = B c by construction and they are in the
same line. Therefore C V is equal and parallel to B c. Hence B V is
parallel to C c. But S B is also parallel to C c by construction, and
B V, B S have one point in common, viz. B. They therefore coincide.
That is B V, when produced passes through S.
127. CoR. 3. The body when at B is acted on by two forces ; one in
the direction B c, the momentum which is measured by the product of its
mass and velocity, and the other the attracting single impulse in the di
rection B S. These acting for an instant produce by composition the
momentum in the direction B C measurable by the actual velocity X mass.
Now these component and compound momentums being each propor
tional to the product of the mass and the initial velocity of the body in
the directions B c, B V, and B C respectively, will be also proportional
to their initial velocities simply, and therefore by (125) to B V, B c, B C.
Book I.]
NEWTON'S PRINCIPIA.
llo
Hence B V measures the force which attracts the body towards S when
the body is at B — and so on for every other position of the body.
128. CoR. 1. Prop. II. In the annexed
figure B c = A B, C c is parallel to
S B, and C c is parallel to S' B. Now
A S C B = S c B = S A B, and if the
body by an impulse of S have deflected
from its rectilinear course so as to be
in C, by the proposition the direction
in which the centripetal force acts is that
of C c or S B. But if, the body having
arrived at C, the a S B C be > S A B
(the times of description are equal by
hypothesis) and .*. > S B C, the vertex
C falls without the a S B C, and the
direction of the force along c C or B S',
has clearly declined from the course
B S in consequentia.
The other case is readily understood
fiom this other diagram.
129. To prove that a body cannot de
scribe areas proportional to the times round
two centers.
If possible let
aS'AB = aS'BC
and
S A B = S B C.
Then
aS'BC(= S' AB)= S'Bc
and C c is parallel to S' B. But it is
also parallel to S B by construction.
Therefore S B and S' B coincide, which
is contrary to hypothesis.
130. Prop. III. The demonstration of this proposition, although strictly
rigorous, is rather puzzling to those who read it for the first time. At least
so I have found it in instruction. It will perhaps be clearer when stated
symbolically thus :
Let the central body be called T and the revolving one L. Also lef
the whole force on L be F, its centripetal force be f, and the force ac
H2
116 A COMMENTARY ON [Sect. II.
celerating T be f . Then supposing a force equal to f to be applied to
L and T in a direction opposite to that of f , by Cor. 6. of the Laws,
the force f will cause the body L to revolve as before, and we have
remaining
f = F — f '
or
F = f + f .
Q. e. d.
ILLUSTRATION.
Suppose on the deck of a vessel in motion, you whirl a body round in a
vertical or other plane by means of a string, it is evident the centrifugal
force or tension of the string or the power of the hand which counteracts
that centrifugal force — i. e. the centripetal force will not be altered by the
force which impels the vesseL Now the motion of the vessel gives an
equal one to the hand and body and in the same direction ; therefore the
force on the body = force on the hand + centripetal power of the hand.
131. Prop. IV. Since the motion of the body in a circle is uniform by
supposition, the arcs described are proportional to the times. Hence
, ., 1 arc X radius
t a arc described <x
it
(X area of the sector.
Consequently by Prop. II. the force tends to the center of the circle.
Again the motion being equable and the body always at the same dis
tance from the center of attraction, the centripetal force (F) will clearly
be every where the same in the same circle (see Cor. 3. Prop. I.) But
the absolute value of the force is thus obtained.
Let the arc A B (fig. in the Glasgow edit.) be described in the tune T.
Then by the centripetal force F, (which supposing A B indefinitely small,
may be considered constant,) the sagitta D B (S) will be described in
that time, and (Wood's Mechanics) comparing this force with gravity as
the imit of force put = 1, we have
S = fFT^'
g being = 32 ig feet.
But by similar triangles A B D, A B G
Book I.]
(Lemma VII.)
If T be given
NEWTON'S PRINCIPIA.
^ _ 2 S _ (a rc AB)'
117
(arcAB)'
r a ^ .
If T = arc second
(arcAB)^
gR •
132. Cor. 1. Since the motion is uniform, the velocity is
arc
V =
V V
•••^ = iR°^R
133. CoR. 2. The Periodic Time is
circumference 2 * R
P =
velocity
^ __ 4g«R' _ 4ff'R R^
"• " gRP* ~ gP^ * P^'
134. CoR. 3, 4, 5, 6, 7. Generally let
P = k X R%
k being a constant.
Then
and
v =
1
2flr R
2 9
a
k R"' R
4^* R
4w'
g
k« R2n  gk* R«ni ^ R^"*
Conversely. If F a „ gp_^ ; P will a R ■».
For (133)
Pa^^a V R'°a R".
135. CoR. 8. A B, a b are similar
arcs, and A B, a h contemporaneous
ly described and indefinitely small.
Now ultimately
a n : a m : : a h * : a b *
and
a m : A M : ; a b : A B
(Lemma V)
..an: A M : : a h' : a b . A B
118
A COMMENTARY ON
[Sect. II.
or
f : F
or
ah» A B«
ah''
AB«
ab • AB •
a s
AS
y2 V*
a s ' A S
''''as
(Lemma V)
And if the whole similar curves A D, a d be divided into an equal
number of indefinitely small equal areas A B S, B C S, &c. ; a b s, b c s,
&c. these will be similar, and, by composition of ratios, (P and p being
the whole times)
P:p
Hence
time through A B
AB . ab . A S
V • V • • V
.. P cc A S
V •
„ V« AS
F oc a ST— .
AS P«
time through a b
a s
136. CoR. 9. Let A C be uniformly described,
and with the force considered constant, suppose
the body would fall to L in the same time in
which it would revolve to C. Then A B being
indefinitely small, the force down R B may be
considered constant, and we have (131)
A C^: AB'
nr 2 •
AB
T* :
AL
AL
AL
AC
rjig
RB
: RB(131)
AB«
AD
Hence
AB« = AL X AD.
Peop. VI. Sagitta ex F when time is given.
Lemma XI, « t ^ when F is given
.'. when neither force nor time is given
sag. a F X t * ;
Also sag. a (arc) * by
Fa
Book I.] NEWTON'S PRINCIPIA. 119
OTHERWISE.
By Lemma X, Cor. 4,
j^ space ipso motus initio
t"
To generalize this expression, let ^ be the space described in I" at
the surface of the Earth by Gravity. Also let the unit of force be Gravi
ty. Then
F • 1 . . !^ . —S
t« • 2X1''*
T< 2 sag. 2 s , .
.. F = — f = — X  . (a)
gt' g t^
by hypothesis.
137. Cor. 1. F a g# a QJ^
t (area S F Q) *
"^ S P* X QT»*
To generalize this, let a be the area described in 1". Then the area
A u ^ • .// s^ ^ SP X QT
described in t" = a X t = .
. . _ SP X QT
• 21 '
and substituting in (a) we get
x,_ 8a* QR ,,.
^ ~ "^ SP« X QT* ^°'
Again, if the Trajectories turn into themselves, tiiere must be
a : I" : : A (whole Area) : T (Period. Tune)
A
.*. a = «^.
Hence by (b) we have
F  1^' V QR (c\
gT* SP* X QT* ^ '
which, in practice, is the most convenient expression.
138.COB.2. F = A!x g y^Qp, .(d)
139.Cor.3. F = gA! X gy/^ p^ (e)
120 A COMMENTARY ON [Sect. U.
Hence is got a di£Perential expression for the force. Since
P V = ?PiJ
dp
„ 8 A* 1
.. F = —r^ X
g T« ^ 2p'pdg
dp
= *;5;:x fp (f)
gT* p'dg ^'
Another is
the
'. following in terms of the reciprocal of the Radius Vector
g and the traced
angle 6.
Because
P'd6
P  V(dg^+gM^*)'
1 _ dg^ + g^d 0^
**P*~ g*d^2
~ g*d^* "^ g*'
Let
1
■— = u.
f
Then
J d u
d g = r
also
1 du* ,
p* ~ d()« "*■ "
2dp 2dud2u „ ^
3^ = jTz — + 2 u d u
p3 d^
dp _ d«u
•p»dg d^'' "" + "
and substituting
in f we have
^ = t«x('3^'*+»') te)
140. Cor.
4.
v
«PV
This is generalized thus. Since
V — ^^^^ _ P Q
~ Time ~ t
and
F2
Book I.] NEWTON'S PRINCIPIA. 121
A
T
A
aXt(=FfXt) = area described
Hence
_ P Qx S Y
~ 2
■ , PQ 2 A 1
.•. V = — — = — X .
t T ^ SY
1 T*
X V^
SY« ~ 4 A'
and by Cor. 3.
F = x^ . (h)
From this formula we get
Y' =x F X P V
P V
But by Mechanics, if s denote the space moved thi'ough by a body
urged by a constant force F
V^ = 2gF X s
P V ...
•••^ = 4 ^'^
that is, the space through which a body must fall 'when acted on by the force
continued constant to acquire the velocity it has at any point oj the Trajec
tory^ is \ of the chord of curvature at that point.
Also
V«=2gFx^ = gFx^ • • • • W
The next four propositions are merely examples to the preceding formulae.
141. Prop. VII.
R P^ (= Q R X R L) : Q T* : : A V^ : P V*
QR X R L X PV'_ ^T.,
••• AV^  ^ ^
S P*^
and multiplying both sides by gp .and putting P V for R L, we have
S P2 X p V3 __ SP^ X QT^
AV« ~ QR
Also by (IST c.)
V A V^ 1
** SP'' X PV3°^ SP* X P V^'
V  ^J^ V AV _ 3 2^r^ 1
gT«^SP^xPV^~ gT^ ""sp^xpy
122 A COMMENTARY ON [Sect. II.
OTHERWISE.
From similar triangles we get
AV: PV:: SP: SY
SP X PV
.. SY =
AV
SP^ y P V^
SY« X P V = ^ A V^ ^ ^^
F«7T.fvv^ — ,s^, a
S p 2 X PV^
AV«
1
as before.
SY'xPV SP*xPV»
OTHERWISE.
P = — 2T^
is the equation to the circle ; whence
dp _j_
df ~ r
••'^gT^''p^dfgT^''rp'
_ 4gr 8r^
gT^ ^ ^ ^ (r^ — a=^ + ?'')'
_ ^2'!tT^ f
 gT« ^ (r«_a« + f^)^*
OTHERWISE.
The polar equation to the circle is
__ 2 a cos. ^
^ — 1 + COS. * 6
/ 1 \ _ 1 COS. tf
•*•■" V" yj ~ 2 a COS. "^ 2 a
d u 1 / sin. 6 .\
__ 1 sin. ^ tf
"" 2 a COS. * ^
" d tf * "~ 2 a v cos. tf COS. ' ^/
1 sin. ^6 ,_ . » .»
= 5— X zA X (3 — sm. « 6),
2 a COS. ' ^ ^ '
Book I.] NEWTON'S PRINCIPIA. 123
Hence
d^u _ sin. ^ 6 • 2m . J" , cos. ^
d7^+ "  2acos.=^^ . (^ — sm 6) + 3 a"cos. ^■*' "sT
X (3 sin 2 ^ — sin. * 6 + cos. ^ d + cos. * 6)
2 a cos. ^ &
1
2 a COS. ^ ^
1
X (2sin.2 ^— sin.* ^+ 1 + 1 — 2 ^in. ^ ^+sin.*tf)
a COS. ^ 5 *
which by (139) gives
T, 4A2 u
F = ^„ X
g T '^ a COS. ^ ^
+ CO!
a^ CO
(1 + cos.^d)
_ 4 A' (1 +cos.^^)
"ffT^^ 4a^cos.*d
X
ga^T^ cos.'O
142. CoR. 1. F a spT^pys
But in this case
S P = P V.
1 32^r* 1
.. F « orTS , or = —Ff^ X ^
SP5> gX2  sps
CoE. 2. F: F:: RP^ X PT^: SP*^ X PV^
SP3 X pv^
SP X R P^
PT^
:: SP X RP^ : SG^
by similar triangles.
This is true when the periodic times are the same. When they are
different we have
T
F: F:: SP X RP^^fr X SG^
S R A
R
where the notation explains itself.
143. Prop. VIII.
CP^: PM'':: PR^: QT^
and
PR^ = QRx(RN + QN) = QRx2PM
.. CP^: PM^:: QR X 2PM:QT2
QTj _ 2PM^
•'• Q R ~ C P 2
124 A COMMENTARY ON [Sect. II.
and
QT' X SP ' __ 2PM^ X SP'
QR ~ CP«
J, CP* 1
2PM=' X SP« PM»
Also by 137,
4a' CP'
' ~ g "^ SP^'x PM^'
But
_ S P X velocity _ SP X V
a g _ g
V* CP^
.. F = — X
g PM^*
OTHERWISE.
By Prop. VII,
Fa '
SP* X P V
But S P is infinite and P V = 2 P M.
1
.'. F a
PM'"
OTHERWISE.
The equation to the circle from any point without it is
c' — r' — g«
P = 2T— ^
where c is the distance of the point from ^e center, and r the radius.
. if = —1
•• dg r
Moreover in this case
g=c+PM=c+y
c'' — r* — c* — 2cy — y*
••• P = 27 —
_ _iy
r
" p' d g r c' y'
 c«y»'
Book L] NEWTON'S PRINCIPIA. 126
Hence (139)
„ _ 4a^r 1 _ V^r' 1_
eg y' g y
SCHOLiyM.
144. Generally we have
P R^ : QT^ : : PC* : P M'
But
and
P R' , Q T' . . PCS. p M '
P R*
— — — P V
P C : P M : : 2 R (R = rad. of curvature) : P V
But
and
QT^
=
PV X
PM*
PC* ~
2R
X PM
QR
PC
2R X
PM^
pQS
R
=
AC*
BC^ ^
PC^
QT*
QR
nr
2 AC*
BC*
1
X PM'
PM^*
From the expression (g. 139) we get
4a* d*u
e d^*
But
Also
axt=^^ =
a
dx
X V
_ 4.a* _ V*g*
• dO^ " d X* *
1
u =  .
.•.du= J
126
and
A COMMENTARY ON
= — ^ (see 69)
r
[Sect. II.
Hence
F =
V
i r
gdx^
g dx«
V« d'y
^ X —
g d x^
This is moreover to be obtained at once from (see 48)
(1).
d«y
F=x ,
g dt
For
dt =
If
V
T? V« d»y
.. F = — X — ^, .
g d x^
145. Prop. IX. Another demonstration is the following ;
Let Z.PSQ = ^pSq. Then from the nature of the spiral the
angles at P, Q, j), q being all equal, the triangles S P Q, S p q are simi
lar. Also we have the triangles R P Q, r p q similar, as likewise Q P T,
qpt.
Hence
QR
and by Lemma IX.
q'r:
. Ill
qr
q r : ; p r'
S P : S p
pr* : : q't'*' : qt*
Book L] NEWTON'S PRINCIPIA. 127
Hence
and
q' t'_2 _ q_tf
q' r' " q r '
iil! . QZ: . . s D • s p
q'r'QR^P'^^
O T^
.. ^ a S P
QR
QT" X S P
QR
1
ocSP*
.. F a
SP'
OTHERWISE.
The equation to the logarithmic spiral is
b
d^ _ b
* ' d g ~ a
and by (f. 139) we have
^ 4>a^ tip 4a2 b
F = X ^T = X— X
p^dg g a h^ g^
4 a ^ a^ 1_
Using the polar equation, viz.
b
X lojT. i
 V (a^ — b^) °*a
the force may also be found by the formula (g).
146. Prop. X.
P V X vG : Qv^ : : P C« : CD*
Qv*: QT* : : P C* : P F
.. Pv X vG : QT* : : P C*: CD* X PF*
...vG:^l':: PC ^^'"^ ^^'
:}
Pv • •  ^ • PC2
But
P V = Q R, and C D X P F = (by Conies) B C X C A
also
ult. v G = 2 P C.
. 3 p p . Q T\ BC X CA'
••^^^' "OR • • ^^ • PC^
128 A COMMENTARY ON , [Sect. II.
* QT^xCP^ 2BC^XCA^ •
Also by expression (c. 137) we get
^ 8A* PC
But
gT* 2B C^ X C A«
•A = «rxBCxCA
.. F = i^, X P C.
The additional figure represents an Hyberbola. The same reasonino
shows that the force, being in the center and repulsive, also in this curve,
a CP.
ALITER.
Take
T u = T V
and
u V : vG : : D C« :
PC»
Then since
Q V* : Pv X V G : : D C^ :
P C*
.. u V : V G : : Q V 2 :
P V X V G
.. Q V* = P V X u V
.•.Qv» + uPxPv= Pvx (uV
+ uP)
= P V X V P.
But
Qv^ = QT^ + T»^ = QT' + Tu2
= PQ^— PT^ + Tu''
= P Q'^— (PT^ — Tu2)
= PQ2_PuxPv
(chord PQ)* = Pv X VP.
Now suppose a circle touching P R in P and passing through Q to
cut P G in some point V. Then if Q V be joined we have
z.PQv = /.QPR = ^QV'P
and in the AQ P v, Q V P the z. Q P V is common. They are there
fore similar, and we have
P V : P Q : : P Q : P V
.. PQ2 = PvxV'P = Pvx VP
.. V P = V P
or the circle in question passes through V ;
.*. P V is the chord of curvature passing through C.
Book I.] NEWTON'S PRINCIPIA. 129
Again, since
' D C^
u V = V G X ppi = C X V G
or
p V — P u = C (P G — P v)
and
P V, P G
being homogeneous
2DC^ 2CD^
.. (Cor. 3, Prop. VI.)
PC
"2 PF2 X CD^'
But since by Conies the parallelogram described about an Ellipse is
equal to the rectangle under its principal axes, it is constant. .*. P F x
C D is.
and •
F ot p C.
OTHERWISE.
By (f. 139) we have
T. 4. A^ dp
F = — 7^„ X ^
g T^ P'df
But in the ellipse referred to its center
#g. ^ a^ + b^^'
1 _ a^ + b' — g
•'• p* ~ ab*
and differentiating, and dividing by — 2, there results
dp _. i
p^ d f ab"
which gives
„ _ 4 A^ I _ 4 ?r
 ^T^ ^ ^TmT^  '^'^ ^ ^'
In like manner may the force be found from the polar equation to the
ellipse, viz.
b^
' 1 — e ^ COS. ^ &
by means of substituting in equat. (g. 139.)
Vol. I. I
130 A COMMENTARY ON [Sect. II.
147. Cor. 1. For a geometrical proof of this converse, see the Jesuits'
notes, or Thorpe's Commentary. An analytical one is the following.
Let the body at the distance R from the center be projected with the
velocity V in a direction whose distance from the center of attraction is P.
Also let
F = fi s
fi being the force at the distance 1. Then (by f )
r? 'I'A^ dp
which gives by integration, and reduction
p^ 4 A^ ^ ^ ^ P^ 4 A« ^ ^
R and P being corresponding values of § and p.
But in the ellipse referred to its center we have
1 _ ag + bg g^
p«~ a^b^ a^b^
which shows that the orbit is also an ellipse with the force tending to its
center, and equating homogeneous quantities, we get
and
a« + b^ _ ^gT' r>, . JL"
a*b« ~ 4 A« ^ ■*" P«
f^g^''
But
b«~ 4A
A r= ff a b
T = Sl= (1)
V fi g
which gives the value of the periodic time, and also shows it to be con
stant. (See Cor. 2 to this Proposition.)
Having discovered that the orbit is an ellipse with the force tending to
tne center, from the data, we can find the actual orbit by determining its
semiaxes a and b.
By 140, we have
and
,. 2 A 1
» — T P
+ b^ _ __?L J. 1
1_ _ 1
a»'o«  ''g ^ v^ P«
Book L] NEWTON'S PRINCIPIA. 131
and
2 a b =
V/Ug
and
g'
V2 2 V P^
/ V* 2 V P\
^ /^ g V ^ c'
which, by addition and subtraction, give a and b.
OTHERWISE.
By formula (g. 139,) we have
p. 4. A^ „ /d^ u , \ fL
d^u, gfiT ^ A_n
•*• d^2 + " ^X2~ ^ u^"
and multiplying by 2 d u, integrating and putting ^ . „ = M, we have
(2)
d^2t"+ ^.t^
= u
To deteimine C, we have
du^ 1 d^2
d ^2  g4 d ^2
and in all curves it is easily found that
d6 p^ ^^ ^ '
du^ ?2_p2 2
" dd^ ~ g2p2  p2
1
Hence, when f = R, and p = P,
^+MR2+ C = . (3)
P
which gives the constant C.
Again from (2) we get
u d u
V(— M — Cu'^ — u^)
which being integrated (see Hersch's Tables, p. 160. — Englished edit,
published by Baynes & Son, Paternoster Row) and the constants properly
determined will finally give g in terms of 6 ; whence from the equation to
the ellipse will be recognised the orbit and its dimensions.
12
132
A COMMENTARY ON
[Sect. II.
_ A\ cab b
ot — ) a a —
a / a a
148. Cor. 2. This Cor. has already been demonstrated — see (1).
Newton's Proof may thus be rendered a little easier.
By Cor. 3 and 8 of Prop. IV, in similar ellipses
T is constant.
Again for Ellipses having the same axismajor\ we have
T(.
But since the forces are the same at the principal vertexes, the sagittae
are equal, and ultimately the arcs, which measure the velocities, are equal
to the ordinates, and these are as the axesminores. Hence, a (which
v X S Yx .
= — ^— )ccb.
.*. T X T a 1 or is constant,
b
Again, generally if A and B be any two ellipses whatever, and C a third
one similar to A, and having the same axismajor as B ; then, by what
has just been shown,
T in B = T in C
and
T in C = T in A
.. T in B = T in A.
149. ScHOL. See the Jesuits' Notes. Also take this proof of, " If one
curve be related to another on the same axis by having its ordinates in a
given ratio, and inclined at a given angle, the forces by which bodies are
made to describe these curves in the same time about the same center in
the axis are, in corresponding points, as the distances from the center."
8 R
The construction being intelligible from the figure, we have
P N : Q N : : p O : q O
.. PN: pO : : QN q O
: : N T : O T ultimately.
Book I.]
NEWTON'S PRINCIPIA.
133
.'. Tangents meet in T,
the triangles C P T, C Q T are in the ratio of P N : Q h or of parallelo
grams P N O p, Q N O q ultimately, i. e. in the given ratio, and
CpP:CPT::pP:PT ultimately.
: : NO: NT
: : qQ:QT
: :CQq: CQT
.*. C p P : C q Q in a given ratio.
.*. bodies describing equal areas in equal times, are in corresponding
points at the same times.
.*. P p, Q q are described in tlie same time, and m p and k q are as the
forces.
Draw C R, C S parallel to P T, Q T; then
nO: lO
pO
: qO : : PN
QN:
: n(
.. nO
: p O : : 1 O
: qO
and
n p
n O : : 1 q :
IQ)
but
y
nO
nR : : IQ :
isj
(since n O
O R : : T O :
OC::
lO
.. n p :
n R: : 1 q :
1 S
.. n p
p R: : 1 q :
qS
and
n p:
p R : : m p :
pC)
qCf
1 q:
q S : : k q
.*. mp :
pC : : k q •
qC
or
Fatp:
Fatq: : p C
SECTION
qC.
f III.
O S)
Q. e. d.
150. Prop. XI. This proposition we shall simplify by arranging the pro
portions one under another as follows :
LxQR( = Px): LxPv
But
LxPv
GvxPv
Qv^
Qx''
:GvxPv.:
: Q V '
.Qx*^ :
:QT^ :
PE
A C :
L
PC^
1
PE'^
CA^
PC
PC
G V
CD
1
PF^
PF^
C B^
I 3
134 A COMMENTARY ON [Sect. III.
..Lx QR: QT«: : ACxLxPC^xCD^ : PCxGvxCD^xCB^
and
QR _ ACx PC _ AC xPC _ AC
QT« ~ G V X C B * ~ 2 P C X C B^ ~ 2 C B ^
V QR / AC \ ^ 1
•' QT^ X SP«V 2 CB^x SPV SP^*
Q. e. d.
Hence, by expression (c) Art. 137, we have
^ 8A2 AC
F = — ,^„ X
gT'' 2CB2 X S P2
(a>
8 g^a 'b'^ a
^T2 ^ 2b2 X f2
4cr2a« 1
where the elements a and T are determinable by observation.
OTHERWISE.
A general expression for the force (g. 139) is
^ 4 A^ ,/d2 u , \
But the equation to the Ellipse gives
_ 1 _ 1 + e COS. ^
" " F ~ a(le2r
where a is the semiaxis major and a e the eccentricity,
d u e sin. 6
•*' dT ~ ~'a(l — e^)
and
d * u e cos. 6
dd'
""
a(l— e')
d^u .
=
1
a(l — e^)
[iU
. F
=
4A»
gT^"" a(le^)
But
A»
rr
ff^a^b^ = »^a^(a' —
a'e')
.. F
=r
4^*a^^ ,
ic same as before.
Book I.]
NEWTON'S PRINCIPIA.
13^
OTHERWISE.
Another expression is (k. 140)
4 A* dp
F =
X
gT'' p
Another equation to the Ellipse is also
1 _ 2 a —
2
P^ b^
** p'd f b2 gS
2a
"^ ~gT^^ b2g2
4cr2a2b2
gT'
4^^ a^ 1
151. Prop. XII. The same order of the proportions, which are also let
tered in the same manner, as in the case of the ellipse is preserved here.
Moreover the equations to the Hyperbola are
_ a(e^— 1)
S =
1 + e COS. 6
and
P = 1; —
which will give the same values of F as before excepting that it becomes
negative and thereby indicates the force to be repulsive.
152. Prop. XIII. By Conies
4SP.Pv = Qv2 = Qx2 ultimately.
But
Pv = Px = QR.
.. 4SP.Q R : Qx^
: : 1
1
and
Qx^ : QT^
:: SP«
. SN
: SP
SA
.. 4SP:QR : QT=^
:SP
S A
QR 1
1
~ L
" QT^ ~ 4SA
L being the latus rectum.
.•.F« QI^._
1
 a
or
QT^ X SP^ SP
F = — X ^PW'P <"• '^'^
g
II
13G A COMMENTARY ON [Sect. III.
Sa^ 1 2P2V2 1
X TTTTP or
gL'^SP* gL '"SP2
a being the area described by the radiusvector in a second, or P the per
pendicular upon the tangent and V the corresponding velocity.
OTHERWISE.
In the parabola we have
1 2 2.2
S
and
u =  = y (1 4 COS. 6) = y \ J COS.
p«L^7
wliich give
d^u
d^^ +"= L
and
dp _ 2 2
p^df L^ s'
and these giye, when substituted in
or
 P ' ^^ ' dp
~ g *P'dg
the same result, viz.
^ 2P^V^ 1 ...
^^n^""? ■ ■ • • •••••(")
Newton observes that the two latter propositions may easily be deduced
from Prop. XL
In that we have found (Art. 150)
F
=
4 A*
gT«
a
__
P«V
' a
X 1 o , o
Now when the section becomes an Hyperbola the force must be repul
sive the trajectory being convex towards the force, and the expression re
mains the same.
Book L] NEWTON'S PRINCIPIA. 137
Again by the property of the ellipse
which gives
a^ _ ^ ]_
.b^ ~ L ~"4 a
and if c be the eccentricity
b^ = a^ — c2 = (a + c) X (a — c)
. a _ 2 J_
* ' (a + c) X (a — c) ~ L 4a'
Now when the ellipse becomes a parabola a and c are infinite, a — c is
Jmite^ and a + c is of the same order of infinites as a. Consequently rj
\sjinite, and equating like quantities, we have
± 1
b«~ L'
which being substituted above gives
F = J— X —
the same as before.
Again, let the Ellipse merge into a circle ; then b =r a and
P^ V^ a
V« 1
X — „
g g
(c)
g X a
153. Prop. XIII. Cob. 1. For the focus, point of contact, and position of
the tangent being given, a conic section can be described having at that point
a given curvature.'}
For a geometrical construction see Jesuits' note, No. 268.
The elements of the Conic Section may also be thus found.
The expression for R in Art. 75 may easily be transformed to
« 6
R =
for
P =
d s ~ . / „ d p2
^U'^'m
138 A COMMENTARY ON [Sect. III.
Now the general equation to conic sections being
b* 1
f = r X
a 1 + e cos. 6
the denominator of the value of R is easily found to be
which gives
Hence
R =  X
b* g3
— *^ X R
is known.
Again, by
the equation to conic sections
P * = 2—
we
have
which, by aid of the above, gives
„ ±e'
"  2.e2 — p R'
And
2^'pR'
Whence the construction is easy.
154. The Curvature is given Jrom the Centripetal Force and Velocity being
given."]
If the circle of curvature be described passing through P, Q, V, and O
(P V being the chord of curvature passing through the center of force,
and P O the diameter of curvature) ; then from the similar triangles
P Q R, P V Q, we get
P Q2
Q R = TV"'
Also from the triangles P Q T and P S Y (S Y being the perpendicu
lar upon the tangent) we have
SPxQT
^^ SY
and from P S Y, P V O,
2Rx SY
PV =
SP
Book Ll NEWTON'S PRINCIPIA. 139
whence by substitution, &c.
Q R SP
QT^xSP^ ~2R X SY«
_ 2P^V^ QR _ V^xSP
** ~ g QT^xSP^RxSY
which gives
R _ SP V2
Hence, S P, S Y and g being given quantities, R is also given if V and
F are.
155. Two orbits which touch one another and have the same centripetal
Jbice and velocity cannot he describedJ]
This is clear from the " Principle of sufficient Reason." For it is a
truth axiomatic that any number of causes acting simultaneously under
given circiunstances, viz. the absolute force, law of force, velocity, direc
tion, and distance, can produce but one effect. In the present case that
one effect is the motion of the body in some one of the Conic Sections.
OTHERWISE.
Let the given law of force be denoted generally by f g, where f g means
any function; then (139)
„ P^V^ dp
F = X ^
and since P and V are given
„, P«V* dp'
g p'^dg'
But if A be the value of F at the given distance (r) from the center to
the point of contact ; then
F : A::fg :fr
and
and
F: A::fg':fr
...F = ^xfg
I r
F=^xf?'
f r ''
140 A COMMENTARY ON [Sect. III.
Hence
P^ V« dp _ A^ ^.
g • p ^ d f ~ f r ^
and
P'^ V^ dp^ _ A ^ ,
g * p' M / ~" f r ^
and integrating, we have
P' Vfr
2gA
and
p2V2fr
^ (f^^O =/d?ff
'•x(p2— .72)=/^?'f§'
2 g A • '^ VP 2 p'
Nowyd g f ^ and yd g' f g' are evidently the same functions of g and g',
which therefore assume
* pgandpg';
and adding the constant by referring to the point of contact of the two
orbits, and putting
pfV^fr
2gA
= M,
we get
^^ (f2~]^8) = ^^ — <p'^'
. J_ _ ^ , J fj'\
••p2 M* P« M^ (^,)
p/2  M "^ P2 M''
in which equations the constants being the same, and those with which
f and ^ are also involved, the curves which are thence descriptible are
identical. Q, e. d.
These explanations are sufficient to clear up the converse proposition
contained in this corollary.
156. It may be demonstrated generally and at once as follows :
By the question
. 1 .
Book I.] NEWTON'S PRINCIPIA. HI
then
and
f 1
. =/^^ =
and substituting in (d) we have
p2  r M ^ P2 ^ Mg'
But the general equation to Conic Sections is
J_ _ 2a 1
p2  b2g + b2*
Whence the orbit is a Conic Section whose axes are determinable from
2a _ J_ _ 2g A r^
b^ M ~ P2 V2
and
_ J_ 1_ J_
■^ b2 ~ r M "^ P^
 J_ 2 g A r ^
— P 2 p 2 Y? '
and the section is an Ellipse, Parabola or Hyperbola according as
V ^ is >, or = or < 2 g A r.
Before this subject is quitted it may not be amiss by these forms also to
demonstrate the converse of Prop. X, or Cor. 1, Prop. X.
Here
^i = i
f r = r
<Pi
= fi^^ =
_ r
■ 2
<pv
~ 2'
Whence
1
 '" 4
~ 2M ^
1 e
P^ 2 M*
But in the Conic Sections referred to the center, we liave
1
= A + i
a" — b^
which shows the orbit to be an Ellipse or Hyperbola and its axes may be
found as before.
142
A COMMENTARY ON
[Sect. III.
In tlie case of the Ellipse take the following geometrical solution and
construction
C, the center of force and distance C P are given. The body is projected
at P with the given velocity V. Hence P V is given, (for V ^ = g^ F . P V.)
Also the position of the tangent is given, .*. position of D C is given, and
2 C D ^
P V = — . Hence C D is given in magnitude. Draw P F per
pendicular to C D. Produce and take P f = CD. Join C f and bisect
in g. Join P g, and take g C, g f, g p, g q, all equal. Draw C p, C q.
These are the positions of the major and minor axis. Also ^ major axis
= P q, ^ minor axis = P p.
For from g describe a circle through C, f, p, q, and since C F f is a
right z_, it will pass through F.
.. Pp.Pq=PF.Pf=PF.CD
Also
PC +FP = Pg2 +gC2 + Pg2 +gf2, (since base of A bisected in g)
or
PC8+CD2 = Pg2+gq2 + Pg2 + gp2
= Pq2 — 2Pg.gq + Pp2 + 2Pg.gp
= Pq3 + Pp2
.'. Pp.Pq=PF.CD \ But a and b are determined by the same
pp2 + Pq2 = PC* + CD^J equations. .*. Pq = a, P p = b.
Also since p and F are right angles, the circle on x y will pass through p
and F, and Z.Ppx = Cpq= CFq = xFp, because ^xFC = pFq.
.. Z. Pp x = zin alternate segment. .*. P p is tangent.
Pp« = PF.Px .. PF.Px = b«.
But if in the Ellipse C x be the major axis, P F . P x =bK
Book I.] KEWTON'S PRINCIPIA. 143
.*. C X is the major axis, and .*. C q is the minor axis.
••. the Ellipse is constructed.
Prop. XIII, Cor. 2. See Jesuits' note. The case of the body's
descent in a straight line to the center is here omitted by Newton, be
cause it is possible in most laws of force, and is moreover reserved for a
full discussion in Section VII.
The value of the force is however easily obtained from 140.
157. Prop. XIV. L = ^^ a ^'
Q R F
a QT« X S P=by hypoth.
OTHERWISE.
By Art. 150,
gT^^b^^^'LgT^^g^
for the circle, ellipse, and hyperbola, and by 1 52.
Lg g*
for the parabola.
Now if ^ be the value of F at distance 1, we have ^
Whence in the former case
8 A^ 2P2 X V^
p^ = /., or = — ^j_ (a)
and in the latter
2P^ X V'
f = /^ (b)
gL ^ '
But
S P^ X Q T^ : 1 2 : A ^ : T ^
4
. ^ _ SP== X QT _ P^X V^
•*• X 2  4 ~ ^
..SP^x QT2 = ^L '. (c)
158. Prop. XIV. Cor. I. By the form (a) we have
A(= *ab) = J^ X V L X 1.
« T V L.
144. A COMMENTARY ON [Sect. III.
159. Prop. XV. From the preceding Art.
8 ff a b
But in the ellipse
"^ ~ 'V/tiff ^ VL •
L = H^
...T=^Xa^ (e)
160. Prop. XVI. For explanations of the text see Jesuits' notes.
By Art. 157 we get
OTHERWISE.
VL
for the circle, ellipse, hyperbola, and parabola.
But in the circle, L = 2 P.
,'.V = V~^X~ = VYf^X ^^ •••(g)
r being its radius.
In the ellipse and hyperbola
a
IT , b 1 ,, .
•*• V = V g ^ X ^ X p: (h)
161. Prop. XVI, Cor. I. By 157,
L = — X P^X V''.
162. Cor. 2. V = ^^ X :^,
D being the max. or min, distance.
163. Cor. 3. By Art. 160, and the preceding one,
V:V'::^i^xX^:V7^
VD
: : V L : V 2 D.
164. Cor. 4. By Art. 160,
Book L] NEWTON'S PRINCIPIA. 145
But
2 b^
L = , P = b, and r = a
^ b 111
.. V : \r : : r — 7—  —7— ; : 1 : 1.
b V a V a
165. Cor. 6. By the equations to the parabola, ellipse, and hyper
bola, viz.
P^ = 4 "^ ^' P = 2^^,' ^^ P = 2T+I
the Cor. is manifest
166. Cor. 7. By Art. 160 we have
2/2 1 L 1
2 P^ r
which by aid of the above equations to the curves proves the Cor.
OTHERWISE.
By Art. 140 generally for all curves
P V
^ 2
But generally
p V = ^P^ g
dp
and in the circle
P V = 2 g (rad. = g)
2/2 P d p
? d s
An analogy which will give the comparison between v and v' for any
curve whose equation is given.
167. CoR. 9. By Cor. 8,
. L
and
.'. ex equo
Vol. I.
'P
v : v" : : ^/ f : ^/ 
v:v:: Ai' : p.
V 2 ^
146 A COMMENTARY ON [Sect. III.
1G8. Prop. XVII. The " absolute quantity of the ford* must be
known, viz. the vatue of /it, or else the actual value of V in the assumed
orbit will not be determinable ; i. e.
L: L': : P'^ V^: P'^ V'«
will not give L'.
It must be observed that it has already been shown (Cor. 1, Prop.
XIII) that the orbit is a conic section.
See Jesuits' notes, and also Art. 153 of this Commentary.
169. Prop. XVII, Cor. 3. The two motions being compounded, the
position of the tangent to the new orbit will thence be given and therefore
tlie perpendicular upon it from the center. Also the new velocity.
Whence, as in Prop. XVII, the new orbit may be constructed.
OTHERWISE^
Let the velocity be augmented by tlie impulse m times.
Now, if jtt be the force at the distance 1, and P and V the perpendicu
lar and velocity at distance (R) of projection, by 156 the general equation
to the new orbit is such that its semiaxes are
R R
a = ^ „, or =
and
2_m^' "^  m2_2
m 2 P ni 2 P
b^ rr s, or
2 — m2' "^ m2 — 2
according as the orbit is an ellipse or hyperbola. Moreover it also
thence appears that when m ^ = 2, the orbit is a parabola, and that the
equations corresponding to these cases are
2 — m^ '
or
or
m^P X S
m'' — 2
= PX
Book L]
NEWTON'S PRINCIPIA.
147
DEDUCTIONS AND ADDITIONS
TO
SECTIONS II AND III.
170. In the parabola the force acting in lines parallel to the axis, required F,
4SP.QR:QT^::Qv»:QT2::YE2:YA*::SE:SA::SP:SA
Q R 1
• • QT* "■ 4 S A
, and S' P is constant, .•. F is constant.
Let u be the velocity lesolved parallel to P M then since the force acts
perpendicular to P M, u at any point must be same as at A. .*. if P Q be
S' P . O T
the velocity in the curve, Q T = u = constant quantity, and a = ^
S'P.u
.. F =
«^ = ««^ = ^'(seel57)
gS'P'.Qi g
which avoids the consideration of S P being infinite ; and
u^=2gF.t
.*. body must fall through — to acquire the velocity at vertex, which agrees
with Mechanics. (At any point V = u / qt*)
171. In the cycloid required the force when acting parallel to the axis.
148
A COMMENTARY ON
[Sect. III.
R P^ : QT« :: Z P» : ZT« :: V F* : E F* : : V B : BE
and since the chord of curvature (C. c) = 4 P M, R P* = 4 P M. R Q,
.. 4 P M. R Q : Q T* : : V B : (B E =) P M
QR _ VB
•*'QT* ~ 4PM2'
.*. F a pHTfi (since S P constant)
, 8a^ Q R u^.VB .^ , . „ i * r,
~ g.S P'.Q T' ~ 2 g. P M' ' " " = velocity parallel to A B.
(At any point V = u.^/^.)
172. In the cycloid the force is parallel to the base
RP»: QT*:: ZP': ZT«:: V E«: VM*:: VB: VM
and since C . c = 4 E M
R P« = 4E M.RQ,
.. 4 E M . R Q : Q T« : : V B : V M,
QR _ VB 1
•*• QT* ~ 4 E M. V M "^ E M. V M •
If V M = y, F =
VB>
gy V
2ry — v*V 2 J
II = velocity parallel to V B.
Book I.] NEWTON'S PRINCIPIA.
/'f  8a'Q R^ __ 2 uj. Q R _ u'. VB v
V^g.SP*QT^ g.QT^ 2g.EM.VMV
(At any point V = ^ • ^' ^')
173. Find F in a parabola tending to the vertex.
149
TAN
TP : FN : : TA : AE
or
V 4 X '^ + y * : y : ; X :
y^ _
= P, (A E),
V 4x'^+ y
1 _4x* + ax_ 4 X + a
'"p*"" ax^ ~ ax^
2 d p __ 4dx.ax' — 2axdx(4x + a)
p» ~" a^ X*
4x^ + 2 ax
ax
.dx =
2 2 X + a
. d X,
, dp _ 2x + a
. .  o — :— i — , u A .
a X'
Also
= /x^ + y%
a d X
1 >_xdx + ydy_ x d x + 2
Vx°fy* V'x^ + ax
dp _ 2 x 4 a 2Vx^+a"x _ 2Vx^4ax
*'p'dp~ ax' * 2x + a "~ ax^
. p AP
K3
150 A COMMENTARY ON • [Sect. III.
174. Geometrically. Let P Q O be the circle of curvature,
but
but
P V (C. c through the vertex of the parabola) = — ^^ —
PQ^ _ PO . A J
AP
QR
PQ^
QT2
QT« 
.. F =
AJP*
Az^
A P^
PO.Az^
8a^Q R
8a«. A P
g.A P^QT* " g.PO. A z'
c ps o Y« A T'
PO.Az' = 2 AS.~.3. gp3 = 2 A S.AN"
F =
4a'. A P
g.A S.A N^*
175. If the centripetal be changed into a repelling force, and the body
revolve in the opposite hyperbola, F « Tjpg .
Book I.]
NEWTON'S PllINCIPIA.
151
The body is projected in direction P R ; R Q is the deflection from the
Tangent due to repelling force H P, find the force.
L.Px : L.Pv :
Px:Pv::PE:PC::AC:PC
L . P V : P V . V G :
: L : 2 P C
P.v.vG : Q v'' :
PC^: CD»
Q v^ : Q x« : •
1 : 1
Qx*^ : Q T» : •
PE^ : PF»: : AC* : PF*^: ; CD": BC»
.. L.Px: QT'' :
AC.L.PC'.CD*: 2PC«.CD«.BC*
. . L. ^^ . . 1 . 1
"^  QR
.. F
8a^QR Sa^ 1
~ g.HP^QT^ ~ g.L.H P^"^ HP«'
R^^P
L. SP^
176. In any Conic Section the chord of curvature = — ^^
for
p V  Q P_'  QT^SP
^QRQR.SY^
L.SP^
177. Radius of curvature =
for
PW =
L.SP
~ S Y'
2r S"Y^'
PV.SP L.SP^
SY ~ SY'
8 a^
178. Hence in any curve F = — o  yt pt r
__ 8 a*' _ 4a^SP
~ g.SY' .2R.SY ~ gTHY^R
SP'
. see Art. 74.
152 A COMMENTARY ON [Sect. Ill
179. Hence in Conic Sections
_ 8a' _ 8a' _ 8a' 1_
^ ~ g.SY^PV~g.SY'. L.SP' ~ g.L.S P«°^ SP''
S Y^
L . S P' .
180. If the chord of curvature be proved =  o v «~ independently of
Q T
he proof that „ „ = L, this general proof of the variation of force in
tonic sections might supersede Newton's ; otherwise not.
181. ^ body attached to a strings whose length = b, is whirled round so as
to describe a circle whose center is the Jixed extremity of the string 'parallel
to the horizon in 'Y" \ required the ratio of the tension to the weight.
Gravity = 1 , .*. v of the revolving body = V' g F b, if b be the length of
the string ;
.'. F (= centripetal force = tension) = — r (131)
and
^ _ circumference __ 2 t b V b
V V g F b VgF
. F  ^^^
• " gT'
. 4 AT* b _
.♦. F : Gravity : : — rer^ : 1, or Tension : weight : : 4 w ' b : g T '.
If Tension = 3 weight ; required T.
4cr2b:gT': : 3 : 1,
•■'  3g •
If T be given, and the tension = 3 weight, required the length of the string.
^, _ £»'b
3g '
4 cr*
182. If a body suspended by a string from
any point describe a circle^ the string describes
a cone ; required the time of one r evolution or
of one oscillation.
Let A C = 1, B C = b,
The body is kept at rest by 3 forces, gra
vity in the direction of A B, tension in the
direction C A, and the centripetal force in q
the direction C B.
Book I.]
As before, centripetal force =
NEWTON'S PRINCIPIA.
4?r2 b
15S
gT
Tf
and centripetal force : gravity : b : v^ 1 * — b ^ (from a)
4 ff* V 1* — b^
4?r'=b
*. T* :=
g
.'. T = 2 ff ^ = a constant quantity if V 1 ^ — b *
be given.
.'. the time of oscillation is the same for all conical pendulums having a
common altitude.
183. V 171 the Ellipse at the perihelion : v in the circle e. d. : : n : 1, ^nd
the major axis, excentricity, and compare its T iioith that in the circle, and
Jind the limits qfn.
Let S A = c,
V in the Ellipse : that in the circle e. d. : : V H P : v^ A C
V H A : V A C in this case
n : 1 by supposition,
.2AC— AS = n«AC,
c
.. A C =
Excentricity = A C — A S = 5 j
2 — n*'
c
c =
c n'
2 — n'
s
c^
T: Tin the circle: : A C^: A S^:: ,
(2 — n«) 2
Also n must be < v' 2,
for if n = V 2, the orbit is a parabola
if n >• V 2, the orbit is an hyperbola.
184. Suppose ^ of the quantity of
matter qf^to be taken aiioay. How
much ivotdd T of D be increased, and
what the excentricity of her neiso orbit ?
the D '5 present orbit being considered
circular.
At any point A her direction is
perpendicular to S A,
.'. if the force be altered at any
point A, her v in the new orbit will
3
c^
1 : (2 — n«) 2
154 A COMMENTARY ON [Sect. III.
2 a
= her V in the circle, since v = ^ y > ^"^ S Y = S A, and a is the
same at A.
LetAS = c,PVatA = L,andF = ^^ a JL
in this case,
2 b'^
.*. 3 : 4 : : 2 c ( = L in the circle) : (= L in the ellipse)
2,
_ 3b^ _ 3 (a'— a — c ) __ 3(2ac— c') _ _ IsJ
'*""a'~ a ~" a ~ a
3c«
a =2^'
3c
c 2 /3 C\
And T in the circle : T' in the ellipse : : —^ : \~^ \
V_3 /£x 2 1 3
*'V4*\2.) ■'V2*2
: : V 2 : 3.
. , , . . 3 c
And the excentncity = a — c= c — c= .
185. What quantity must be destroyed that J 's T may be doubled^ and
what the excentricity of her new orbit ?
Let F of © :y(new force) : : n : 1
g
.. F«
.*. V = ^ ^ F . P V, and v is given,
1
P V
2b* „ a* — a — c 2ac — c'
.*. n : 1 : : — :2c:: : c : : — : c : : 2 a — c : a'
a a a
.*. n a = 2 a — c,
c
.*. a = ~ .
2 — n
Also T in the circle : T in the ellipse : : 1 : 2
\
5
C 2 C
"'^n'(2_n)i
: : (2 — n) ^ : n ^
.'. 1 : 4 : : (2 — n) ^ : n .*. n = 4 (2 — n) ^ whence n.
Book L] NEWTON'S PRINCIPIA.
And the excentricity
155
c =
_ c — (2c — nc) __c(n— 1)
2 — n ^ " 2 — n ~ 2 — n
186. What quantity must he destroyed that Ys orbit may become a
farabola P
L = 4 c,
.. F : / : : 4 c : 2 c : : 2 : 1,
.*. ^ the force must be destroyed.
187. Fa =rt' ^ ^^2/ ^^ projected at given D, v = v in the circle,
L. isoith S B = ^5° f find axis major, excentricity, and T.
Since v = v in the circle, .*. the body is projected from B,
and z. S B Y=r 45° ;
.. z. S B C, or B S C = 45°,
S B
S C = S B. COS. 45° =
• 2
But
S B = D = ^^^^ major
.•• axis major and excentricity are found.
And T may be found from Art. 159.
Y
P
188. Prove that the angular v round H : that round S : : S P : H P.
This is called Seth Ward's Hypothesis.
In the ellipse. Let P m, p n, be perpendicular to S p, H P,
.'. p m = Increment of S P = Decrement of H P = P n
.♦. triangles P m p, P n p, are equal,
.*. P m = p n, and angular v « j^—
^ ' ° distance
189. Similarly in the hyperbola.
Angular v of S P : angular vofSY::PV:2SP:: ?^^':2SP
•• ^^'' AC
: : HP : A C.
166 A COMMENTARY ON [Sect. III.
190. Compare the times of falling to the center of the logarithmic spiral
fiom different points.
The times are as the areas.
P
d . area =  , (^ = iL C S P), for d . area =  — 5 .
Also 7^ = ^^ = tan. z. Y P T = tan. «, (« being constant) = a
1 F d f
i
f'^dtf a.f.dg
••• y = 2 »
a . p '
.*. area = — j oc g S (for when ^ = 0, area = 0, .*. Cor. = 0)
.*. if P, p, be points given,
T from P to center : t from p to center : : S P * : S p *.
191. Compare v in a logarithmic spiral with that in a cit*cle, e. d.
9 V*
.. if F be given, Va V FY,
.*. V in spiral : v in the circle : : V P V in spiral : V 2 S P : : 1 : 1 .
192. Compare T in a logarithmic spiral with that in a circle^ e. d.
whole area a f ^ __ a g *
T in spiral =
area in 1'' 4 . v . S Y 2 v . f . sin. a
JL m circle ~ . — r^r — o •«»• — ^~ ■
area m \" v . b Y v . ^ v
2
rr»fr</ &£* Serf a „ .
.*. T : T : : :?: ^ : : : rr. : 2 * : : a : 4 * . sm. a.
2 V . f . sm. a V 2 sm. a
: : tan. « : 4 t . sin. a : : 1 : 4 ^ cos. a.
Book I.]
NEWTON'S PRINCIPIA.
157
192. In the Ellipse compare the time from the mean distance to the Aphe
lioUf •with the time from the mean distance to the Perihelion. Also given the
Excentricity, to find the diffei'ence of the times, a?2d conversely.
D
A D V is — ^— described on A V.
T of passing through Aphelion : t through Perihelion
: : SB V: SB A
:: SDV: SD A
:: CD V +
Let Q = quadrant C D V,
D C. SC
:CD V —
DC. S C
2
a. a e
2
.. (T+ t =) P: T — t: :2Q:a. ae
rp _ P a. a 6
• • ^ ^  ~2Qr
whence T — t, or, if T — t be given, a e may be found.
193. If the perihelion distance of a comet iri a parabola = 64, ©'5 mean
distance = 100, compare its velocity at the extremity of L "with ®'5 velocity
at mean dista?ice.
Since © moves in an ellipse, v at the mean distance = that in the circle
e . d . and v in the parabola at the exti'emity of L
: V in the circle rad. 2 S A : : V 2 : 1
v in the circle rad. 2 S A
: V in the circle rad. A C
'. V in the parabola at L
: V in the ellipse at B
v' A C : V S A
V2.AC: V'SA.2
10 V 2 : 8 V 2
5 : 4
194. TVhat is the difference between L of a parabola and ellipse, having
the same <" distance = 1, and axis major of the ellipse = 300? Compare
the V at the extremity of\, and <" distances.
In the parabola L = 4 A S = 4.
158
A COMMENTARY ON
[Sect. III.
In the ellipse L' = ^f^' = Jp. (A C^ — A C— S A')
AC
300
L' = 4
(4
1
150 J
V in the parabola at A : v in the circle rad. S A
V in the circle rad. S A : v in the ellipse e. d.
"" 150*
V 2 : 1
VAC:
V 150 :
VHP
V299
: V 300 : V 299.
: : VAC: V 2AC— SA
.*. V in the parabola at A : v in the ellipse e. d. ;
Similarly compare v*. at the extremity of Lat. R.
195. Suppose a body to oscillate in a
•whole cycloidal arCf compare the tension
of the string at the lowest point with
the weight of the body.
The tension of the string arises
from two causes, the weight of the
body, and the centrifugal force. At
V we may consider the body revolving
in the circular arc rad. D V, .•. the
centrifugal = centripetal force. Now
the velocity at V = that down C V by the force of grav.
= that with which the body revolves in the circle rad
2 C V.
.*. grav. : centrifugal force : : 1 : 1,
.*. tension : grav. : : 2 : 1
196. Suppose the body to oscillate
through the quadrant A B, compare the
tension at B with the weight.
AtBthestring will be in the direction of
gravity; .'. the whole weight will stretch
the string; .*. the tension will = centrifugal
force + weight. Now the centrifugal
force = centripetal force with which the
body would revolve in the circle e. d.
R
2
C
A
\
And v in the circle = V 2 g . F .
Book I.]
NEWTON'S PRINCIPIA.
159
.. F =
R
cCB
in this case,
also v' at B from grav. = V 2 g . C B, giav. = 1.
grav. = 1 =
2g C B
F : grav. : :
2gCB' g C B
2: 1,
since V = V .
.*. tension : grav. : : 3 : 1.
197. A body vibrates in a circular arc
from the center C ; through isohat arc must
it vibrate so that at the lowest point the
tension of the string = 2 X weight?
V from grav. = v d . N V, (if P
be the point required) v' of revo
lution in the circle = v d . ^r— .
H
N
centrifugal force : grav. : : y : V : : / — — : V N V
CV
.'. centrifugal force + grav. (= tension) : grav. : : J —^ + V N V : V N V
: : 2 : 1 by supposition.
^
C V
2
I CY
•'•% 2
.. N V =
+ V NV = 2 A/ N V
= V N V,
C V
198. There is a hollow vessel in form
of an invei'ted paraboloid down which
a body descends, the pi'esswe at lowest
point = n . weight, find from what point
it must descend.
At any point P, the body is in the
same situation as if suspended from G,
P G being normal, and revolving in the
circle whose rad. G P. Now P G =
V 4 A S . S P", .. at A, P G =
160
A COMMENTARY ON
[Sect. III.
V4AS* = 2AS. Also v ^ at A with which the body revolves =
Sg.F.LAS.
.'. centrifugal force =
2g A S
V
and grav. = ~ r , if h '= height fallen from.
But the whole pressure arises from grav. + centrifugal force, and=:n . grav.
.*. centrifugal force + grav. : grav. : : n : 1
or
1 _L 1 1 1
AS
1 ^
...^g:j^::n_l:l,
... h = n — 1 . A S.
199. Compare the time (T') in isohich a body de
scribes 90° of anomaly in a parabola with T in the
circle rad. = S A.
Time through A L : 1 : : area A S L : a in 1''
. ^ _ I A S. SL _ 4 A S'
a 3 a
T in the circle rad. S A : 1 : : whole circle : a' in \"
. ^ ^ ^A S^
S <t
' T' • T • •
3 a * a'
and
a: a':: \/L: '•2AS:: V4AS: V2AS
4
.. T' : T : :
3 V 2
ff : : 2 V 2 : 3 w.
V 2: 1
Compare the time of describing 90° in the parabola A L with that in the
parabola A 1, (fig. same.)
t : T in the circle rad. S A : : 4 : 3 V 2 . t
T in the circle S A : T' in the circle rad. <TA::SA^:ffA^
(smceT'oc R')
T' : X! through 1 A : : 3 V 2 . s : 4
.. t through S A : t' through <r A : : S A ^ : <r A ^.
See Sect. VI. Prop. XXX.
Book I.]
NEWTON'S PRINCIPIA.
161
200. Draw the diameter P p such that the time through P V p : time
through p A P : : n : 1, force <x .
Describe the circle on A V.
t =
Let t = time through P V p, and T the periodic time
n _ PVpS _ QVqS __ circle + a Q g S
n + ] ~ ellipse ~~ circle ~~ circle
circle . S R . 2 C Q
, (u =: excentric anomaly)
~ 2 ' 2
circle
*a^ ,
= —= H a e . sm. u .
a
cra^
= — + e. sm. u
.'. n ?r = n + 1 . (~ + e sin. u j
= n— + i + n + l.esin.u
sm. u =r
n + 1 • 2e
which determines u, &c.
201. The Moon revolves round the Earth in 30 dai/s, the mean distance
from the Earth = 240,000 miles. Jupiter^s Moon revolves in \ day, the
mean distance from Jupiter = 240,000 miles. Compare the absolute forces
of Jupiter and the Earth.
Vol. I. L .
162 A COMMENTARY ON [Sect. III.
A^
T « —  , A being the major axis of the ellipse,
.'. If A be given, fi (x. —;
^ Mass of Jup iter __ T' of the Earth's Moon _ 30j _ 14,400
*'* Mass of the Earth *" T' * of Jupiter's Moon ~ _j_ ~ T~'
42
202. A Comet jat perihelion is 400 times as near to the Sun as the Earth
at its mean distance. Compare their velocities at those points.
Velocity' of the Comet __ F.4 A S _ J_ ^ _ F I
Velocity* of the Earth ~ F^ 2 B S ~ F' * 2 . 400 ~ F' ' 200
_ 400 * J_ _
 "I^ • 200  ^^^
V V 2 . 20 30 ,
.*. — = , = 7 nearly.
V I 1 •'
203. Compare the Masses of the Sun and Earth, having the mean distance
of the Earth from the Sun = 400, the distance of the Moon from the Earth,
and Earth's V^. = 13. the Moon's V^.
T«a — ,
a
Mass of the Sun 400 » P 64,000,000 ,«^««« ,
••• M^i^fih^E^ = np • T3* = —169— = ^^^'^^^ ^^"^ly
1 a
204. If the force « , 5, where x is the distance from the center
1 a
of force, it mil be centripetal 'whilst — 5 > — 3 > or x > a ; there ivill be
1 a
a point of contrary flexure in the orbit 'when j = ^ , or x = a, and
afterwards 'when x < a, the force 'will be repulsive, and the curve change
its direction.
Book I.]
NEWTON'S PRINCIPIA.
163
205. JTie body revolving in an ellipse, at
B the force becomes n times as great. Find
the new orbit, and under ivhat values ofn it
•will be a parabola, ellipse, or hyperbola.
S being one focus since the force
the other focus must lie
0(
distance *'
in B H produced both ways, since
S B, H B, make equal angles with
the tangent. V* =  F.PV = jF.2ACinthe original ellipse, or
= ^ n F . P V in the new orbit
.. 2 AC = n. P V = n.
2 SB.h B
SB + h B
.. (S B + h B) A C = 2 n . S B . h B,
.. AC= +h B. AC = 2nAC.h B,
.•.hB = ,AC .
2 n — 1
If 2 n — 1 = 0, or n = ^, the orbit is a parabola ; if n > ^, the orbit
is an ellipse; if n <C i> the orbit is an hyperbola.
Let S C in the original ellipse be given = B C,
.. S R H = right angle, and S B or A C = B h . cot. B S h
whence the direction of a a', the new major axis ; also
/ QT^.Ri, ^c Sh VBh^ — SB^
a a'^ = S B + B h, and S c = —^ = _ .
If the orbit in the parabola a a' be parallel to B h, and L . R = 2 S B,
since S B h = right angle.
206. Suppose a Comet in its or
bit to impel the Earth from a cir
cular orbit in a direction making
an acute angle ivith the Earth's
distance from the Sun, the velo
city after impact being to the velo
city before : : V~S : V~2. Find
the alteration in the length of the
yea?:
Since V 3 : V 2 < ratio than V 2 : 1, .•. the new orbit will be an
ellipse,
L2
164 A COMMENTARY ON [Sect. III.
XI  ?. ^ _P y _ 2SP. HP _ H/P
v2  2 ~2SP~AC.2SP ~AC
= 2 A C — S P
AC
.. 3AC = 4AC — 2SP
.. 2 S P = A C
. T in ellipse _ 2^ S P^ _ £
* * T' in circle ~ S P ^ "" ^
207. A body revolves in an ellipse, at any given point the force becomes
diminished by ^^ part. Find the new orbit.
B'
v«a F. P V
in this case P V « — ,
F
But
P V in ellipse
pv in new orbit
_ 1
ition
— n
1 ~
~ 2SP
n ■
n
1
n
V * in conic sec
n
— IPV
V ^ in circle e.
d.
2S P
n HP
at P
n— r A C
if . . H P = A C, the new orbit is a Circle
n — 1 j
= 2 A C, Parabola j>
< 2 A C, Ellipse I
> 2 A C, HyperbolaJ
If 1 = 2, or n = 2, then when the orbit is a circle or an ellipse, P
must be between a, B ; when the orbit is a parabola, P must be at B ;
when the .orbit is an hyperbola, P must be between B, A.
Book I.} NEWTON'S PRINCIPIA. 165
208. If the curvature and inclination of the tangent to the radius be the
same at two points in the curve, the forces at those points are inversely as the
radii ^'
p_ 8a^ _ 8a^ _ 8 a' 1
~g.SY^PV"'g.SY.SP.Rg.sin.^SP2.R°^SP'^
This applies to the extremities of major axis in an ellipse (or circle) in
the center of force in the axis.
209. Required the angular velocity of ^.
By 46, & being the tracedangle,
dd
d t
But by Prop. I. or Art. 124,
dt:T::dA:A
^'^^A(ll3)
2
d ^ 2 A 1
'' = dl = T" ^ g^
or
_ PX V ,.
210. Required the Centrifugal Force (p) in any orbit.
When the revolving body is at any distance f from the center of force,
the Centrifugal Force, which arises from its inertia or tendency to persevere
in the direction of the tangent (most authors erroneously attribute this force
to the angular motion, see Vince's Flux. p. 283) is clearly the same as it
would be were the body with the same Centripetal Force revolving in a
circle whose radius is f. Moreover, since in a circle the body is always
at the same distance from the center, the Centrifugal Force must always
be equal to the Centripetal Force.
But in the circle
QT^ = Q R X 2SP
and .*. by 137 we have
1
F_8A^ \
4 A'^
gT'^
gT^ 2SP^
or
_ P^V 1
— ~ X a
g e
P and V belonging to the orbit.
L 3
166 A COMMENTARY ON [Sect. lU.
Hence then
9 = —z X 3 (a)
Hence also and by 209,
And 139,
^ = ^^^^^7^ (c)
211. Required the angular velocity of the perpendicular upon the
tangent.
If two consecutive points in the curve be taken ; tangents, perpendiculars
and the circle of curvature be described as in ArL 74, it will readily ap
pear that the incremental angle (d ']>) described by p = that described
by the radius of curvature. It will also be seen that
But from similar triangles
P V : 2 R : : p : g.
.. d ^ : d ^^ : : P V : 2 g
P V being the chord of curvature.
Hence
= « X 1^ (d)
or
2P X V , ,
= r3rpv (^)
or
_ P_>^V ^ dp
Pg
Ex. 1. In the circle P V = 2 g ; whence
PxV
H (^
,2
:= u.
Ex, 2. In the other Conic Sections, we have
P Sa + g
Book I.] NEWTON'S PRINCIPIA. 167
which gives by taking the logarithms
2lp = lb^ + 1^ — l(2a + f)
and (17 a.)
2dp^df, dj _ 2adf
P " T ~ 2~r:Ff "■ r(2"a + s)
whence
_ aP X V
""gMSa + g)'
212. Required the Paracentric Velocity in an orbit.
It readily appears from the fig. that
ds:d^::g:V* — p*.
.*. If u denote the velocity towards the center, we have
„ r_ d f\ _ d_s ^g' — p
"^Vdt^dt^ e
X ^^^P (125)
or
_ P X V ^ Vg' — p'
"" P g
= PVx^(pil) . . . . (g)
2 A //I In ,,,
Also since
2  g*d<?' + dg*
p* *" g*d^«
^ = P^^f^. (^)
213. Tojind 'where in an orbit the Paracentric Velocity is a maximum.
From the equation to the curve substitute in the expression (212. g)
for p *, then put d u = 0, and the resulting value of g will give the posi
tion required.
Thus in the ellipse
and
P =2ag
u^=P«V«x(?^^^^)=max.
2a 1 1
L4
168 A COMMENTARY ON [Sect. III.
and
_ b * _ Latus Rectum
or the point required is the extremity of the Latus Rectum.
OTHERWISE.
Generally, It neither increases nor decreases when F = p. Hence
when u = max. (see 210)
d p _ d g
p'  g^
which is also got from putting
d (u'') =
in the expression 212. h.
214. Tojind isohere the angular velocity increases fastest.
Bv Art 209 and 125,
d« „„ de PV 2P^V2 dg
5^ = 2PVxx 4:^ = 4—^ ^ rnS'
d t g3 g* d ^ g* g a fl
g^d^
But from similar triangles
p: V (g^ — p^)::QT:PT::gd<):dg
...^" = il^'x.,._p')=.ax.
•••S^'=F^^ =  <")
either of which equations, by aid of that to the curve, will give the point
required.
Ex. In the ellipse
b^g
p' =
2a — g
2 a — g 1
... ^—1 _ 3 = max. = m
, d m  .
and J— = gives
d e
7 4
Book I.] NEWTON'S PRINCIPIA. 169
which gives
f =  a+^ V (49 a' — 48b«)
for the maxima or minima positions.
If the equation
b* 1
e = — X
a 1 + e COS.
and the first form be used, we have
d e a e 5 . .
and
sin. i
= max. = m.
Whence and from d
m = 0, we get finally
cos. d
8 e  V V64 e «
+
I)
215. Tojind "where the lAnear Velocity increases fastest.
Here
dv
max.
But (125)
PX V
and
dt=^^Al 1 Pd
PxV"" P X V '" V j« — p«
7g'p^) ,
V(g'P^)
dv __ pgy V(g« — p^) dp
dt ~ I P'df
= gF X
P'df
or ^ = max. = m.
and
d m =
will give the point required.
170 A COMMENTARY ON [Sect. IV.
Thus in the ellipse
Fal
p* 1 b^
^^ — = max.
dm _ _4 10 a b'g^ — 6 b°g^ _
*** dg ~ g* "*" (2a — g)^f^<'
which gives
, , . J 8a* + 3b' ,5 , „ ^
g' + 4 a g' 31 g + — a b^ = 0,
whence the maxiiifa and minima positions.
In the case of the parabola, a is indefinitely great and the equation
becomes
4a2p — I ab^ =
'' 2
5 > b^ 5
.*. f=s X — szTX L.atusKectum.
* 8 a 16
Many other problems respecting velocities, &c. might be here added.
But instead of dwelling longer upon such matters, which are rather
curious than useful, and at best only calculated to exercise the student,
I shall refer him to my Solutions of the Cambridge Problems, where he
will find a great number of them as well as of problems of great and
essential importance.
SECTION IV.
216. Prop. XVIII. If the two points P, p, be given, then circles whose
centers are P, p, and radii AB+SP, AB^lSp, might be described
intersecting in H.
If the positions of two tangents T R, t r be given, then perpendiculars
S T, S t must be let fall and doubled, and from V and v with radii each
= A B, circles must be described intersecting in H.
Having thus in either of the three cases determined the other focus H,
the ellipse may be described mechanically^ by taking a thread = A B in
length, fixing its ends in S and H, and running the pen all round so as to
stretch the string.
Book I.] NEWTON'S PRINCIPIA. « ITl
This proposition may thus be demonstrated analytically.
1st. Let the focus S, the tangent T R, and the point P be given in
position ; and the axismajor be given in length, viz. 2 a. Then die per
pendicular S T ( = p), and the radiusvector S P ( = g) are known.
But the equation to Conic Sections is
,_ b^g
whence b is found.
Also the distance (2 c) between the foci is got by making p = g, thence
finding § and therefore c = a If g.
This gives the other focus ; and the two foci being known, and the axis
major, the curve is easily constructed.
217. 2d. Let two tangents T R, t r, and the focus S be given in position.
Then making S the origin of coordinates, the equations to the trajectory
are
h's , b* 1 ,.
P = 7i — =^» and p = — . z r: r • • . (a)
*^ 2a4.g' ^ a 1 + e cos. (tf — a) ^'
a being the inclination of the axismajor to that of the abscissae.
Now calling the angles which the tangents make with the axis of the ab
scissae T and T', by 31 we have
tan.T = iy.
d x
But
X =r I cos. 6f y =: § sin. 6
whence
rp __ d g sin. 6 + §d 6 cos. 6
"" d g cos. 6 — g d ^ sin. 6
^ tan. ^ + 1
= i^ (b)
Aitan.^
g d ^
Also from equations (a) we easily get
^4^ =  gsin.(._«) (1)
COS. (6a)= ^IjHAI (2)
^ aeg
sm. (tf — a) = — X V (2ag — g* — b«) . . (3)
^ 'aeg \ » »
and
 ^^P' . (4)
s =
p» + b
172 A COMMEMTARY ON [Sect. IV.
and putting
R z= V (2af — §2 — b^) . . . . (5)
we have
R * /. \ tan. — tan. a
= tan. (^ — a) = , . ,_ , ,_ , . v6)
b* — af~ 1 + tan. a . tan. 6
which gives tan. 6 in terms of a, b, f, and tan. a.
Hence by successive substitutions by means of these several expres
sions tan. T may be found in terms of a, b, p, tan. a, all of which are given
except b and tan. a. Let, therefore,
tan. T = f (a, p, b, tan. a).
In like manner we also get
tan. T' = f (a, p', b, tan. a)
p' belonging to the tangent whose inclination to the axis is T.
From these two equations b and tan. a may be found, which give
0= Va* — b* and a, or the distance between the foci and the position
of the axismajor; which being known the Trajectory is easily con
structed.
218. 3d. Let the focus and two points in the curve be ^ven in posi
tion, &c.
Then the corresponding radii f, /, and traced angles ^, 6', in the
equations
 ±a(I — e^)
^ "~ 1  e COS. {6 — a)
^ 1 + e COS. (^ — a)
are given ; and by the formula
COS. {6 — a) = COS. 6 . cos. a + sin. 6 sin. a
2 a e and a or the distance between the foci and the position of the axis
major may hence be found.
This is much less concise than Newton's geometrical method. But it
may still be useful to students to know both of them.
219. Prop. XIX. To make this clearer we will state the three cases
separately.
Case 1. Let a point P and tangent T R be given.
Then the figure in the text being taken, we double the perpendicular
S T, describe the circle F G, and draw F I touching the circle in F and
passing through V. But this last step is thus effected. Join V P, sup
pose it to cut the circle in M (not shown in the fig.), and take
V F^ = VM X (V P + PM).
The rest is easy.
Book I.]
NEWTON'S PRINCIPIA.
173
Case. 2. Let two tangents be given. Then V and v being determined
the locus of them is the directrix. Whence the rest is plain.
Case 3. Let two points (P, p) be given. Describe from P and p the
circles F G, f g intersecting in the focus S. Then draw F f a common
tangent to them, &c.
But this is done by describing from P with a radius = S P — S p, a
circle F' G', by drawing from p the tangent p F' as in the other case (or
by describing a semicircle upon P p, so as to intersect F' G' in F') by
producing P F' to F, and drawing F f parallel to F' p.
See my Solutions of the Cambridge Problems, vol. I. Geometry, where
tangencies are fully treated.
174 A COMMENTARY ON [Sect. IV.
These three cases may easily be deduced analytically from the general
solution above ; or in the same way may more simply be done at once,
from the equations
, _ L __ L 1
P  4^'^ 2 ^1+ COS. {6 — a)'
220. Prop. XX. Case 1. Given in species'] means the same as " simi
lar" in the 5th Lemma.
Since the Trajectory is given in specieSy &c.] From p. 36 it seems that
the ratio of the axes 2 a, 2 b is given in similar ellipses, and thence the
same is easily shown of hyperbolas. Hence, since
^ c^ = a^ + b^
2 c bemg the distance between the foci, if — = m, a given quantity, we
have
a a
which is also given.
With the centers B, C, &c.]
The common tangent L K is drawn as in 219.
Cases 2. 3. See Jesuits' Notes.
OTHERWISE.
221. Case 1. Let the two points B, C an4 the focus S be given.
Then
_ +a(l— e^) ..
^ ~ 1 + e cos. {d — a)r .,K
, _ +a(le') C * '
^ 1 + e cos. {&' — a))
a being the inclination of the axis of abscissae to the axis major.
But since the trajectory is given in species
e = — is known,
a
and in equations (1), g, ^ ; sf, ^, are given.
Hence, therefore, by the form
COS. {d — a) = COS. 6 . cos. a + sin. 6 sin. a,
a and a, or the semi axismajor and its position are found;
also c = a e is known ;
which gives the construction.
Book I.] NEWTON'S PRINCIPIA. 175
Case 2. By proceeding as in 220, in which expressions (e) will be
known, both a, a e, and a may be found.
Case 3. In this case
p« = XL = a^ X (le^)g
will give a. Hence c = a e is known and
__ +a(l~e^)
^ ~ 1 + e cos. {6 — a)
gives a.
Case 4. Since the trajectory to be described must be similar to a given
one whose a' and c' are given,
• = i —
~ a ~ a'
is known (217).
Also g and 6 belonging to the given point are known.
Hence we have
_ +a.(l— e^)
^ 1 + e COS. {6 — a)
And by means of the condition of touching the given line, another
equation involving a, a may be found (see 217) which with the former
will give a and a.
222. Scholium to Prop. XXI.
Given three points in the Trajectory and the focus to construct it.
ANOTHER solution.
Let the coordinates to the three points be f, ^ ; g', ^ ; f, tf', and a the
angle between the major axis and that of the abscissae. Then
_ +a.(l — e'')
1 + e cos. {& — a)
._ +a(l— e')
^ ~ 1 + e cos. {^ — a)
.,^ ±a(le^)
^ 1 + e COS. (^' — a)
i^ (A)
and eliminating + a ( 1 — e *) we get
I — I = e . COS. {S — a) — e cos. (^ — a) 1 ,t»»
g — g' =e . cos. {^' — a) — e cos. (^ — «) /
176 A COMMENTARY ON [Sect. V.
from which eliminating e, there results
^' . COS. {(/ — a) — ^ COS. {6 — a) "~ f COS. {^' — a) — ^ COS. {& — a)
Hence by the formula
COS. (P — Q) = cos. P . COS. Q + sin. P . sin. Q
_ (g— gOr COS. 6"—{^ — f) ^' COS. ^^ + g (g^  g^Qcos.^
''^  (g _g') f sin. r— (g— g'O g'sin.^' + g(/g")sin.^
which gives a.
Hence by means of equations (B) e will be known ; and then by substi
tution in eq. (A), a is known.
SECTION V.
The preliminary Lemmas of this section are rendered sufficiently intel
ligible by the Commentary of the Jesuits P.P. Le Seur, &c.
Moreover we shall be brief in our comments upon it (as we have been
upon the former section) for the reason that at Cambridge, the focus of
mathematical learning, the students scarcely even touch upon these sub
jects, but pass at once from the third to the sixth section.
223. Prop. XXII.
This proposition may be analytically resolved as follows :
The general equation to a conic section is that of two dimensions (see
Wood's Alg. Part IV.) viz.
y 2 + A X y + B X 2 H C y + D X + E =
in which if A, B, C, D, E were given the curve could be constructed.
Now since five points are given by the question, let their coordinates be
a, /3; a, /3; a, /3; a, ^; a, /3.
11 2 2 3 3 4 4
These being substituted for x, y, in the above equation will give us five
simple equations, involving the five unknown quantities A, B, C, D, E,
which may therefore be easily determined ; and then the' trajectory is
easily constructed by the ordinary rules (see Wood's Alg. Lacroix's DifF.
Cal. &c.)
224. Prop. XXIII. The analytical determination of the trajectory
from these conditions is also easy.
Let
a, /3; a, /3; a, ,3; a, /3
II 2 2 3 3
Book I.] NEWTON'S PRINCIPIA. 177
be the coordinates of the given point. Also let the tangent given in posi
tion be determinable from the equation
y' = m X' 4 n (a)
in which m, n are given.
Then first substituting the above given values of the coordinates in
y2 + Axy + Bx2+ Cy+Dx+E = . ..(b)
we get four simple equations involving the five unknown quantities
A, B, Cj D, E ; and secondly since the inclination of the curve to the axis
of abscissae is the same at the point of contact as that of the tangent,
d y __ ? y'
5x dx'
y = y'
X = x'
, Ay+ 2Bx + D _ _
**' 2y+Ax+C ~ ^
and substituting in this and the general equation for y its value
y' = m X + n
we have
A(mx + n)+2Bx + D
2(mx + n) + Ax+C
and
(mx + n)2 + Ax(mx + n) + Bx2+C(mx + n) + Dx+E = 0,
from the former of which
— n A + mC+ D
^~ 2(m« + mA + B)
and fiom the latter
^ =  2(m^+LA+B) ^ (nA + mC+D + 2mn
+ v'J(nA + mC + D + 2mn)*— (n2 + nC + E)(m=^ + mA + B)J
and equating these and reducing the result we get
4m*n* = (nA + mC + D+2mn)« — (n^nC + E) (m'^+m A+ B)
and this again reduces to
n2A» + m2C2 + D2 + mnAC + 2nAD
+ 2mCD — nBC— mAE — BE+ Smn^A
+ 3nm2C + 4mnD — n^B — m^E — n^m2 =
which is a fifth equation involving A, B, C, D, E.
From these five equations let the five unknown quantities be determined,
and then construct eq. (b) by the customary methods.
M
178 A COMMENTARY ON [Sect. V.
225. Prop. XXIV.
OTHERWISE.
Let
be the coordmates of the three given points, and
y' = m x' + n
y''=m'x" + n'
the equations to the two tangents. Then substituting in the general
equation for Conic Sections these pairs of values of x, y, we get three
si?)ij)le equations involving the unknown coefficients A, B, C, D, E ; and
from the conditions of contact, viz.
dy ^ d/ ^ ^^ dy ^ d/' _
d X d x'
) dx = 37' = ■"'(
( y = y" i
7 Y — v" y
y = y
X = x' ^ X = X
We also have two other equations (see 224) involving the same five un
knowns, whence by the usual methods they may be found, and then the
trajectory constructed.
226. Prop. XXV.
Proceeding as in the last two articles, we shall get two simple equations
and three quadratics involving A, B, C, I>, E, from whence to find them
and construct the trajectory.
227. Prop. XXVI.
In this case we shall have one simple equation and four quadratics to
find A, B, C, D, E, with, and wherewith to describe the orbit.
228. Prop. XXVII.
In the last case of the five tangents we shall have five quadratics,
wherewith to determine the coefficients of the general equation, and to
construct.
Book L] NEWTON'S PRINCIPIA. 179
SECTION VI.
229. Prop. XXX.
OTHERWISE.
j^ter a body has moved t" from the vertex of the parabola^ let it be re
quired iojind its position.
If A be the area described in that time by the radius vector, and P, V
the perpendicular or the tangent and velocity at any point, by 124 and
125 we have
c PV
^=^ x_t = _xt
and by 157,
pV=: .^1^
L being the latusrectum.
2 \/2
But
ASP = AOPSOP=AOxPO — ^SOxPO
= fxy — i.(x — r)y
where r = A S, &c. (see 21) and
y * = 4rx
.•.y3+ 12r^y = 12rt \^g/xr
by the resolution of which y may be found and therefore the position of P.
OTHERWISE.
230. By 46 and 125,
,ds_pds
~ V ~ c
Also
ds =
fd
c V (e^ — p*)
M2
180 A COMMENTARY ON [Sect. VI.
which is an expression of general use in determining the time in terms of
the radius vector, &c
In the parabola
whence
P* = rf,
dt=:i X
v' r f df
c V(^— r)
and integrating hy parts
2 V r 2 V r
t = f V(^~r)f4Vdf V(gr)
c
2V
J^^/(^_r)_(^_r)^}
c
3^' V(^— r)x(g+2r)
But
which gives
c= PV= V2g^r (229)
.•.t=^X(g+2r)(gr)^ . . . (b)
g' +3r^^ = 4r3+g/*t,
whence we have § and the point required.
By the last Article the value of M in Newton's Assumption is easily
obtained, and is
4r 4 ^ V 2r
231. Cor. 1. This readily appears upon drawing S Q the semilatus
rectum and by drawing through its point of bisection a perpendicular to
GH.
232. Cor. 2. This proportion can easily be obtained as in the note of
the Jesuits, by taking the ratio of the increments of G H and of the curve
at the vertex ; or the absolute value of the velocity of H is directly got
thus.
 dGH _ 3^ M __ £ Igji
^" dt'~dt~4N2r' ^
Also the velocity in the curve is given by (see 140)
v'* = 2g F X r = —2^
4) e
Book I.] NEWTON'S PRINCIPIA. 181
and at the vertex ^ = r,
••.v = V^
.. V ; v' : : 3 : 8.
233. Cor. 3. Either A P, or S P being bisected, &c. will determine
the point H and therefore
4 / 2r ^ TT
t= ^J X GH.
3N g^
234. Lemma XXVIII. That an oval cannot be squared is differently
demonstrated by several authors. See Vince's Fluxions, p. 356; also
Waring.
235. Prop. XXXI. This is rendered somewhat easier by the follow
ing arrangement of the proportions :
If G is taken so that
O G : O A : : O A : O S
or
and
or
GK: 2crOG::t:T
rv _ 2^x O A^^ t
Then, &c. &c. For
But
ASP = ASQX
a
= — X (OQA — OQS)
= ^(OQx AQ — OQx SR)
= ^(AQSR).
S R : sin. A Q : : S O : O A
:: OA : OG:: AQ: FG
A Q sin. A Q
SR =
FG
and
AQ — SR = ^^X(FG— sin.AQ)
M3
182 A COMMENTARY ON Sect. VI.
OS
OA
X(FG — sin.AQ)
.•.ASP = ^4^X (FGsin.AQ)
= ^^^^ XGK (b)
A Si,
(see the Jesuits' note q.) which is identical with (a), since
_t^_ AS^
T " JEUipse
_ ASP
"ffa b '
OTHERWISE.
236. By 230 we have
dt = — Eil^ 
But in the ellipse
p = = —
^ 2a — ^
• dt  ^^^^
• c V(2ae — b«— ^2)
and putting
g — a = u
it becomes
, b . (a + u) d u
cV(a2e2 — u*)
2 a e being the excentricity.
Hence
__ b a f du b / » udu
* ^y '•(a^e^— u^) "•" c^ V(a2e2 — u^)
= kf sin.'." —  Vra^e^ — u'^) + C.
c a e c ^ '
Let t = 0, when u = a e ; then
and we get
P _ba ft
^T ^2
t = ^x'"
c
r^ + sin.'— W. ^(a«e^ — u«)
V 2^ a e/ c ^
Book L] NEWTON'S PRINCIPIA. 183
which is the known form of the equation to the Trochoid, t being the ab
scissa, &c.
Hence by approximation or by construction u and therefore g may be
found, which will give the place of the body in the trajectory.
It need hardly be observed that (157)
OTHERWISE.
237. dtzz^";
but in the ellipse
b« 1
1 + e cos.
b* d^
.. d t =   X
a '^ c ( 1 + e cos. 6) ^
and (see Hirsch's Tables, or art. 110)
a^ri—e^) r 1 : e + cos. ^ esin.^ "I
t — —J L V J COS ~' — —  — — V
c ^\V(1 — e«) 'l+ecos.^ 1 + ecos. O
which also indicates the Trochoid.
To simplify this expression let
then
and
Hence
and
, e + COS. d
cos. 'T— . = u
1 + e cos. 6
e + cos. d
= cos. u
1 + e cos.
e — cos. u
cos. =
sm. d =
e cos. u — 1
V(l— e^)
1 — e COS. u
e sin. 6 e sin. u
1 + ecos.^ V(l— e'')
.*. t = ^ X Ju — e Sin. uf
But (157)
c = PV =b.^^= V(l — e*) V"^"^
M4
184 A COMMENTARY ON [Sect. VI.
X (u — e sin. u)
•t ^'
Let
Then
a^ 1
ut=.u — esin. u (1)
Again, 6 may be better expressed in terms of u, thus
^„„ 2 ^ 1 — COS. 6 1 + e 1 — COS. u 1 + e , u
tan. — ZZ — ~~ V — ' tan *
2 1 +cos.^~ 1— e^ 1 +cos.u~ 1 — e 2
^ / 1 + e u
''''^'2 = ^T::reX^'2 (2)
Moreover g is expressible in terms of u, for
a (1 — e*) ,, , ,„.
^= l + eco3> "('"°^"' (^>
In these three equations, n t is called the Mea?i Anomaly ,• u the
Excentric Anomaly, (because it = the angle at the center of the ellipse
subtended by the ordinate of the circle described upon the axismajor
corresponding to that of the ellipse) ; and 6 the True Anomaly.
238. SCHOLIUM.
Newton says that " the approximation is founded on the Theorem that
The area APSaAQ — SF, SF being the perpendicular let fall from
S upon O Q."]
First we have
APR=AQRx—
a
SPR=SQRx—
a
But
.•.ASP = ASQx —
a
ASQ = AOQ— SOQ
= MQxAO — iSFxOQ
= i AO X (AQ — SF).
.. A S P =  X (A Q — S F)
= — X (a u' — a e sin. u') (1)
u' being the /i. A O Q.
Book I.] NEWTON'S PRINCIPIA. 185
(Hence is suggested this easy determination of eq. 1. 237.
For 3 b / • ^
^ ASp 2^a^' 2(^"^^'^"")
t =T X ^pir^ = , ^ i
Ellipse \^ /tt g « a b
X (u — e sin. u). )
V g^
Again, supposing u' an approximate value of u, let
u = u' + ^
a
Then, by the Theorem, we have
iA^ = A q — S O X sin. A q
= AQ + Qq+ — SOx sin. (A Q + Q q)
to radius 1.
But A Q being an approximate value of A q, Q q is small compared
with A O, and we have
sin. ( A Q + Q q) = sin. A Q cos. Q q + cos. A Q sin. Q q
= sin. A Q + Q q cos. A Q nearly.
J_
.. Qq = (^AP_AQ+SOsin.AQ) X j— ^^ nearly
^+cos. AQ
which points out the use of these assumptions
N' = r —  = rT=s X area of the Ellipse
B' = s o =2^*
and
D' = S O . sin. A Q = B' sin. A Q
^  SO
Then
Qq=: (N^_AQ + ly) X T/ ^' A r>
^ ^ L' + cos. A Q
in which it is easily seen B', N', D', \J
are identical with B, N, D, L.
Hence
E = Qq = (N_AQ + D)^=^^.
186
A COMMENTARY ON
[Sect. VII.
Having augmented or diminished the assumed arc A Q by E, then re
peat the process, and thus find successively
G, I, &c.
For a developement of the other mode of approximation in this
Scholium, see the Jesuits* note 386. Also see Woodhouse's Plane Astro
nomy for other methods.
SECTION VII.
239. Prop. XXXII. F oc
. Determine the spaces which a
distance
body descending from A in a straight line towards the center of
force describes in a given time.
If the body did not fall in a straight line to the center, it would
describe some conic section round the center of force, as focus
C ellipse '\
(which would be < parabola >• if the velocity at any point were to
(.hyperbola J
the velocity in the circle, the same distance and force, in R'
■{=}
V 2 : 1.)
(I) Let the Conic Section be an Ellipse A R P B.
Describe a circle on Major Axis A B, draw
C P D through the place of the body perpendi
cular to A B.
The time of describing A P a area A S P a
area A S D, whatever may be the excentricity
of the ellipse.
Let the Axis Minor of the ellipse be diminish
ed sine llmite and the ellipse becomes a straight
line ultimately, A B being constant, and since
A S . S B = (Minor Axis) ^ = 0, and A S finite
.•.SB = 0, or B ultimately comes to S, and
time d . A C a area A D B. .*. if A D B be taken proportional to time,
C is found by the ordinate D C.
(T . A C a area ADBaADO + ODBaarcAD + CD
.'. take 6 + sin. d proportional to time, and D and C are determined.)
Book I.]
NEWTON'S PRINCIPIA.
187
(Hence
the time down A O
T.OB
+ 1
flr
'§ +
<!r
11
_2
+ 1
7
+
1
v
— 1
11
1
2
7
18 9 , ,
=  =  nearly)
1
N. B. The time in this case is the time
from the beginning of the fall, or the time
from A.
(II) Let the conic section be the hyperbola
B F P. Describe a rectangular hyperbola on
Major Axis A B.
T a area S B F P a area S B E D.
Let the Minor Axis be diminished sine
limite, and the hyperbola becomes a straight
line, and T a area B D E.
N. The time in this case is the time from
the end of motion or time to S.
Let the conic section be the parabola B F P.
Describe any fixed parabola BED.
T a area S B F P a area S B E D.
Let L . R. of B F P be diminished sine
limite the parabola becomes a straight line,
and T a area B D E.
N. The time in this case is the time from
the end of motion, or time to S.
Objection to Newton's method. If a
straight line be considered as an evanescent
conic section, when the body comes to peri
helion i. e. to the center it ought to return to aphelion i. e. to the original
point, whereas it will go through the center to the distance below the
center = the original point.
240. We shall find by Prop. XXXIX, that the distance from a center from
which the body must fall, acted on by a '''^ force, to acquire the velocity such
as to make it describe an ellipse = A B (finite distance), for the hyperbola
= — A B, for the parabola = a .
241. Case 1. vdv = — g«,dx, f= force distance 1,
x^
^)
/a
a X
if a be the original point
v
V a
V2g(ji.
dx.
V ax — x'
188
A COMMENTARY ON
[Sect. VII.
.'. t = ^/ . ^ Va X — X* — I
+ C, when t == 0, X = a, 2 J
/a rVax — x*+ /circumference
.*. t = ^ / ,i— . 1 ( vers.  ' X, I
V 2g
•1
vers,
rad.
if the circle be described on B A = a,
_ l ~~r~ 4 / CD.OB AD.
Case 2. V * = 2 g /u . ^ ^ > if — a be an original point,
•)=
.BAD.
a X
2»
_ / a /• X d X
~V2g^Vvax + x
for t in this case is the time to the center, not the time from the original
poin^
. A t — ^ d t — —
~" V ' "" V *
Now if with the Major Axis A B = a, we describe the rectangulai
hyperbola,
B
we
have
Book L]
NEWTON'S PRINCIPIA.
189
d.BED = d.BEDC — d.ABDC=ydx —
d.xy _ydx" — xdy
2
X d X
Vax + x^, /a, \j axdx j^
= ^ .dx — x.{+ X) dx = ^===^ = d t .
Va
gfi
2 V 2
2Vax+x«
.'. t from B = /
^ 2gfi
.BED, for they begin or end together at B.
Case 3. v ^ = 2 g /* — , if a be ex ,
.'. t =
,. dx Vxdx,,. . ,„
.•• d t = = — ■■ • , t beuiff time to B,
V V2gfi, ^
1 2 ^
 . X ^ + C, when t = 0, x = 0, .. C ^ 0.
V 2g/i3
Describe a parabola on the line of fall, vertex B, L . R. = any fixed
distance a,
" '" .BED.
1
2 V2
V2g/i
7. V x.x = .j. V ax.x =
2 V2
V agfA
Vag /(i
2 V 2
. curvili
Hence in general, in Newton Prop. XXXII, t =
V a g /A
near area, a being L . R. of the figure described.
T 1 . . T T. 2 (Ax. Min.)* .^ .
In the evanescent conic sections, L . K. = — K — ^t^ » .*. 11 Ax.
Min. be indefinitely small, L. R. will be indefinitely small with respect to the
Ax. Min. The chord of curvature at the finite distance from A to B is ulti
mately finite, for P V = .1. ; but at A or B, P V = L, = in
finitesimal of the second order. Hence S B is also ultimately of the second
A B
order, for at B, S B = L. —  — .
2 AS
1
Prop. XXXIII. Force a
VatC
V in the circle distance S C ~ V'~SA
(distance) * *
VAC
in the ellipse and hyperbola.
190 A COMMENTARY ON [Sect. VU.
/V V H P v' A C
(— = . ,. = — =: when the conic sectionbecomes a straight line^
^ V v Maj. Ax. /■ SA ^ ;
2 V 2
Newton's method.
but
L
V* SY« L SP
v^ ~2SP ~ 2 • SY*
SP»
AC.CB AO^ 2AO
2 AO
C P* ~ /Min. Ax.>, « ~ 2/Min. Ax.x »
 L
AO
L AO.CP»
"2 ~ AC.C B
V» AO.CP^SP
••'v* "" AC.CB.SV
CO BO
BO ■" TO'
CO C B comp. in the ellipse
• • B O "" B T ' div. in the hyperbola,
, A C C T div. in the ellipse __
CP
" B O ~ B T ' comp. in the hyperbola "~
BQ'
AC2 CP»
•• AO' BQ'"
BQ^AC AO.CP'
AO  AC '
\' BQ^AC.SP
V*  AO.B C.S Y*^'
but ultimately
BQ = SY, SP = BC,
1 • t ^ V ^" ^ straight line _ A C
^ V * in the circle " A O '
AC
AO*
AC C T
Cor, 1. It appeared in the proof that jrr = ofp*
•'• X ~ ^
Book I.] NEWTON'S PRINCIPTA. 191
 . , A C C T
.'. ultimately ^^ = g?j, .
(This will be used to prove next Prop.)
Cor. 2. Let C come to O, then A C = A O and V = v,
.•. the velocity in the circle = the velocity acquired by falling externally
through distance = rad. towards the center of the force a jr — — , .
° distance *
242. Find actual Velocity at C.
V ^ at C _ AC
v^ in the circle distance B C ~ B A '
~2~"
. y. 2 AC 2 AC g/.
'BABA* BC^' '
if At = the force at distance 1,
. V2 — 9 ff «, A C
V a. — X
•. V = V 2 g /* . , if B A = a, B C = X.
V a X
. . . ,T ^ space described
If a IS given, V a — ^
V space to be described
In descents from different points,
, V space described
V a — ; .. =
V space to be described x initial height *
In descents from different points to different centers,
_^ V space described X absolute force
V a
V space to be described X initial height
243. Otherwise. vdv = — ^dx,
X*
.*. v^ = 2 g /i . , when a is positive, as in the elhpse
a X
= 2 g /ct . , when a is negative as in the hyperbola
a X
1
= 2 g At . — 5 when a is a , as in parabola
(when X = 0, V is infinite)
V ^ in the circle radius x (in the ellipse and hyperbola)
= S_^.x = ^
.*. y^ = in the ellipse, =
(f)
192
A COMMENTARY ON
[Sect. VII.
11 = ^^ + ^ in the hyperbola,
_ a + X
V * in the circle radius = — (in the parabola) =
X'
Sff/tt
.. =^ = — in the parabola.
244. In the hyperbola not evanescent
Velocity at the infinite distance __ S A
velocity at A " S Y ^
finite R°., but when the hyperbola van
ishes, S Y ultimately = Min. Ax. for
■jrr = r— p , and ultimately S C =
A C, and b C = A C, .. ultimately S Y = A b = C B, .. ultimately
S Y _ infinitesimal of the first order
ST ~ =
of the 2d order
__ velocity at A
~ velocity at a distance
245. Prop. XXXIV.
the parabola.
Velocity at C
velocity in the circle, distance S C
~2~
= y , for
S P
For the velocity in the parabola at P = velocity in the circle — — what
ever be L . R . of the parabola.
246. Prop. XXXV. Force a
(distance) ^ '
The same things being assumed, the area swept out by the indefinite
T T?
radius S D in fig. D E S = area of a circular sector (rad. = — '^—
of fig.) uniformly described about the center S in the same time.
Whilst the falling body describes C c indefinitely small, let K k be the
arc described by the body uniformly revolving in the circle.
L.R_ S A
2 ~ 2 '
Cc _ CT
Dd ~ DT
CD _ DT
S Y ~ TS
Case ] . If D E S be an ellipse or rectangular hyperbola.
m
Book I.] NEWTON'S PRINCIPIA. 193
Cc.CD CT AC,. ,
DdTSY = TS = AO "^^^"^^tely.
(Cor. Prop. XXXIII.)
But
velocity at C _ V A C
V in the circle rad. S C ~ V A O
V in the circle rad. S C _ ,SK_ /A O
V in the circle rad. SK "" V S~C ~" 'V S~C
/ velocity at C \ __ C c _ /A C _ A C
. •'• Vv in the circle rad. S K/ "" K k ~ V SC ~ CD
.. Cc. CD = Kk. AC
Kk. A C _ AC
•'• D d . S Y " A O '
.. AO. Kk = Dd. S Y,
.*. the area S K k = the area S D d,
.*. the nascent areas traced out by S D and S K are equal
.*. the sums of these areas are equal.
Case 2. If D E S be a parabola S K = ^^IL^.
As above
Cc.CD CT 2
Dd.
SY ■
~ T S ~ 1
als
o
Cc
Kk
_ velocity at C
velocity in the circle
velocity in the circle L
SC
2
~~ velocity in the circle L .
R"
,. R
2
2
V SK SK
V SC CD
2 2
.. Cc.CD = 2.Kk.SK
.. Kk. S K = Dd. S Y.
247. Prop. XXXVI. Force a
(distance) ^ *
71? determine the times of descent of a body Jailing from the given {and
,'. finite) altitude A S
On A S describe a circle and an equal circle round the center S.
From any point of descent C erect the ordinate C D, join S D. Make
the sector O S K = the area A D S (O K = A D + D C) the body
will fall from A to C in the time of describing O K about the center S
V©L. I. N
^
194
A COMMENTARY ON
[Sect. VII.
uniformly, the force oe _— , Also S K being given, the period
in the circle may be found, (P = / — . «• . S K ^), and the time through
o
OK = P..
O K
^ . .*. the time through O K is known. .*. the time
circumterence °
through A C is known.
248. Find the time in lohich a Planet would Jail from any point in its
orbit to the Sun.
f*1 VOX ^ ^ ^^
Time of fall = time of describing — ^— O K H> S O = —^ ,
period in the circle O K H _ period in the circle rad. S O _ S O ^
period in the ellipse "" period in the circle rad. AC ~~ a r; i
3
.'. the time of fall = ^ . P. (rp) » P= period of the planet. If the orbit
be considered a circle
I
VACv' ^2/ V8
and the time of fall
P r> V 2 „ 4 .
= p. ^ = p. o nearly.
4 V 2
= — nearly.
Book L]
NEWTON'S PRINCIPIA.
195
249. The time down A C a (arc
= A D + C D), a C L, if the cy
cloid be described on A S. Hence,
having given the place of a body at a
given time, we can determine the
place at another given time.
CutoffSm = CL.$?^^4^.
time d. A C
Draw the ordinate m 1 ; 1 c will deter
mine c the place of the body.
250. Prop. XXXVII. To determine the times of ascent and descent of a
body projected upwards or downisoards from a given pointy F a . ^.
Let the body move off from the point G with a given velocity. Let
—r — ii — • 1 J = Tj (V and v known, .*. m known).
V * m the circJe e. d. 1 ^ '
To determine the point A, take
G A mj
1
S A
2
GA
G A + G S
GA
m "^
 ™'
•• G S ~ 2 — m*'
.*. if m* = 2, G A is + and 00 , .'. the parabola
ifm*<2, GAis+ and fin. .•. the circle
^ must be des
^ , . scribed on the
if m*> 2, G A is — and fin. .*. therectangular hyperbola J axis S A.
With the center S and rad. =   of the conic section, describe the
circle k K H, and erecting the ordinates G I, C D, c d, from any places
of the body, the body wUl describe G C, G c, in times of describing the
areas S K k, S K k', which are respectively = S I D, S I d. 
25 L Prop. XXXVIII. Force « distance.
Let a body fall from A to any point C,
by a force tending to S, and ««. as the
distance. Time a arc A D, and V acquir
ed a C D. Conceive a body to fall in an
evanescent ellipse about S as the center.
.*. the time down A P or A C
A_S
2
a A D for the same descent, i. e. when
A is given.
a ASP a ASD a AD.
196
A COMMENTARY ON
[Sect. VII.
The velocity at any point P
a V F. P V
/' ^ 2 A C . C a , . ^ ,
a / S P . ^p ultimately.
a CD.
Cor. I. T. from A to S = ^^ period in an evanescent ellipse.
= ^ period in the circle A D E.
= T. through A E.
Cor. 2. T. from different altitudes to
S a time of describing different quadrants
about S as the center a 1.
N. In the common cycloid A C S it is
proved in Mechanics that ifSca=SCA
and the circle be described on 2 . Sea,
and if a c = AC, the space fallen through,
then the time through A C a arc a d,
and V acquired a c d, which is analogous
to Newton's Prop.
Newton's Prop, might be proved in the
same way that the properties of the cycloid
are proved.
OTHERWISE.
252.
vdv = — g/ix.dx,
.♦. V* = 2 g /«. (a* — X*), if a = the height fallen from
.. V = V2gA(.. V a«— x« = V2g(i. C D.
d X _ d X 1
V ~"
dt =
V2i
:f^
V a
.. t = +
arc
a V2gfM' ^rad
I
^COS. = XV ^^ ^ ^ ^^
. = a/
.AD.
a V2g/A
.*. velocity a sine of the arc whose versed sine = space, and the arc
a time, (rad. = original distance.)
253. The velocity is velocity from ajinite altitude.
If the velocity had been that from infinity, it would have been infinite
Book I.] NEWTON'S PRINCIPIA. 197
d X X
and constant. .*. d t = , and t = , ■ + C, when t = 0,
" ' g /^
V a. V
= Vg(««.a, a= a.
1
x = a, .'. c is finite, .•.t= C =
V g ^
Similarly if the velocity had been > velocity from infinity, it would
have been infinite.
254. Prop. XXXIX. Force a (distance)'^, or any Junction of distance.
Assuming any ex"*, of the centripetal force, and also that quadratures of
all curves can be determined (i. e. that all fluents can be taken) ; Re
quired the velocity of a body, when ascending or descending perpendicu
larly, at different points, and the time in which a body will arrive at any
point.
(The proof of the Prop, is inverse. Newton assumes the area A B F D
to ot V * and A D to « space described, whence he shows that the force
a D F the ordinate. Conversely, he concludes, ifFaDF, ABFD
V* a/vd va/F. d s.
Let D E be a small given increment of space, and I a corresponding
increment of velocity. By hypothesis
ABFD _ Vj _ V
ABGE v'* V«+2V.HP
ABFD V« V« ,. 
••• DFGE = 2V.I + P = 2Vri ^t^^^^t^^y
But
ABFDcxV^.. DFGEa2V.I
.. D E . D F ultimately, a 2 . V . I
But in motions where the forces are constant if I be the velocity gene
rated inT, Fa?p, (F (X. j—\ and if S be the space described with uni
form velocity V in T, ^ = —, j (d t = — ) . Also when the force is
I. V
a^'*^, the same holds for nascent spaces.' .*. F « ■ ' , and D E re
presents S. .*. D F represents F.
N 3
]98
A COMMENTARY ON
[Sect. VII.
Let D L at
1
V ABFD V
^ « ^ , .. D L M E ultimately = D L . D E
DE
a time through D E ultimately.
.*. Increment of the area A T V M E « increment of the time down A D.
.. A T V M E a T.
(Since ABFD vanishes at A, .•• A T is an asymptote to the time
curve. And since E M becomes indefinitely small when A B F D is in
finite, .*. A E is also an asymptote.)
255. CoR. 1. Let a body fall from P, and be acted on by a constant
force given. If the velocity at D = the velocity of a body falling by the
action of a''*^ fowe, then A, the point of fall, will be found by making
ABFD = PQRD.
For
ABFD
DFGE
DFGE
gj by Prop.
DP _ I
DR ~ i
= ^ ultimately.
D R SE
if i be the increment of the velocity generated through D E by a constant
force.
DRS E __ V'(V + i)' _ 2_i
PQRD~ V
ABFD _ 1
•'• PQRD  1 •
256. Cor. 2. If a body be projected up or down in a straight line
from the center of force with a given velocity, and the law offeree given;
Find the velocity at any other point E'. Take E' g' for the force at E'.
Book I.] NEWTON'S PRINCIPIA. 199
velocity at t/ = velocity at D . ^^ — — "  + if pro
jected down, — if projected up.
(Yor ^ PQJ ^P±DFg^E^ __ V A B g^ E \
257. Cor. 3. Find the time through D W.
Take E' m inversely proportional to VPQRD + DFg'E' (or
to the velocity at E').
T.PD _V"PD_ V TB _ V"FD
T.PE" vTE~ V(PD + DE)~ ^p^ . _^E_^ '
PD . .
PD + ^^
2
T.PD 2PD 2PD.DL
"T.DE ~ DE ~DLME
also
T.D Eby PC bie force _ D LM E
T.DE'by do. ~ D L m E"
but T . D E by a constant force = T . D E by a'''* force since the velo
cities at D are equal (d t = — )
T. PD _ 2PD.DL
•'•T.DE' ~ DLmE' •
d V
258. It is taken for granted in Prop. XXXIX, that F a g^ (46),
d s . . d V
and that v = j— , whence it follows that if c . F = t— , d v = c . F. d t,
and vdv = cF.ds.
.. v» = 2c/Fds
Newton representsy F d s by the area A B F D, whose ordinate D T
always = F.
•••»=/
V V2c./Fds
ds
V 2 c/Fd s
N4
200 A COMMENTARY ON [Sect. VII.
— r by the area A B T U M E, whose or
V/Fds ^
dinate D L always = r^*
\ V2e.ABFD>'
V 2g. A B F
In Cor. 1. If F' be a constant force V * = 2 g F' . P D, by Mechanics
but
V'^=:2c./Fds
And F^ P D or P Q R D is proved =/F d s or A B F D,
.. c = g
and
V* = 2g./Fds.
y p velo city at E^ _ V y F d s when s = A E'
• velocity at D ~ V/Fd s when s = A D
_ V A B g^ E^
V A B F D
In Cor. 3. t=time through D E' = /*— •= f ^ = D L m E',
^ J y J V2g/Fds
T, ,. ,, , T3T. 2PD 2PD
1 rrtime through P JJ = • ,, ,t> = _
"S VatD V 2g. PQR D
= 2 PD. DL
T^_ 2 P D . D L
**• t ~ D L m E' *
259. The force a x «.
.*. vdv == — g(«,x"dx5/ct = the force distance 1.
n + 1
if a be the original height.
Let n be positive.
V from a finite distance to the center is finite 1
V from CO to a finite distance is infinite. i
Let n be negative but less than 1.
V from a finite distance to the center is finite 1
V from 00 to a finite distance is infinite. J
Let n = — 1 the above Integral fails, because x disappears, which
cannot be.
Book 1.1 NEWTON'S PR INCIPI A. 201
dx
V d V = — g fi
X
a
X
.'. V from a finite distance to the center is infinite 1
V from » to a finite distance is infinite. /
.Ix
1 X
But the log. of an infinite quantity is x '^ Jess than the quantity itself — when
X is infinite = — . DifF. and it becomes — = — = — .
XX
"dl^
Let n be negative and greater than 1.
V from a finite distance to the center is infinite \
» V from CD to a finite distance is finite. J
260. If the force be constant, the velocitycurve is a straight line parallel
to the line of fall, as Q R in Prop. XXXIX.
DEDUCTIONS.
261. To find under what laws of force the velocity from oo to a finite
distance will be infinite or finite, and from a finite distance to. the center
will be finite or infinite.
If (1) F a X % V a V'a'~~x''
(2) X V a^ — x'
(3) 1 V a — X
w—i — V>T
(5) —2 ^ ax
^^^ x^ V a^x^
m — A — J
x° 'N a^'x**'
In the former cases, or in all cases where F cc some direct power of
distance, the velocity acquired in falling from oo to a finite distance or to
the center will be infinite, and from a finite distance to the center will be
finite.
202
A COMMENTARY ON
[Sect. VII.
In the 4th case, the velocity from oo to a finite, and from a finite dis
tance to the center will be infinite.
In the following cases, when the force a as some inverse power of
distance, the velocity from oo to a finite distance will be finite, for
a°^ — X"' _ / 1
V a^^x"^^ ~ Vx"°i
when a is infinite. And the velocity from a finite distance to the center
will be infinite, for
/ a°^ — x""^ _ FT_
S a'^^x'^i ~N~0
wh^n x = 0.
262. On the Velocity and TimeCurves.
(1) Let F a D, the area which represents V* becomes a a.
For D F a D C.
(2) Let Fa V D, .♦. D F* a D C and Vcurve is a parabola.
(3) Let F a DS .. D F a D C^ and Vcui*ve is a parabola the
axis parallel to A B.
(4) Let F a Yj J ••• D F a ^r^ , .*. Vcurve is an hyperbola referred
to the asymptotes A C, C H.
(6) If F a D, and be repulsive, V« a DC.DF a DC,
.•.V a D C, .*. the ordinate of the tune curve a ^ a ^^ ,
.*. Tcurve is an hyperbola between asymptotes.
(6) If a body fall from oo distance, and F a jjj , V a ^ ,
.♦. the ordinate of the timecurve D, .*. Tcurve is a straight line.
(7) If a body fall from oo , and F a ^ , V a — ,
.'. the ordinate of Tcurve x \^ D C, .*. Tcurve is a parabola.
(8) If a body fall from cc, and F a ^3, V a ^,
.. the ordinate of Tcurve a D C*, .*. Tcurve is a parabola as in case 3.
Book L]
NEWTON'S PRINCIPIA.
203
EXTERNAL AND INTERNAL FALLS.
263. Find the external fall in the ellipse, the force in the focus.
Let X P be the space required to acquire the velocity in the curve at P.
V ' down P X _ Pjc
S3
2
Aa
V ^ in* the circle distance S P
V ' in the circle distance S P
V ' in the ellipse at P ~ 2. H P
V * down P X _ A a . P X
**• V * in the ellipse at P ~ Sx. H P
•• Sx ~ Aa
P^x _ HP
•'• S P ~ S P
.. P X = H P
.. Sx= SP + Px = Aa, and the locus of x is the circle on 2 A a,
the center S.
264. Find the internal fall in the ellipse, the force in the focus.
V * down P X __
V * in the circle S x ~
V * in the circle S x _
V 2 in the circle S P ~
P^
2
SP . 1
ci — , lorce a tt =
S X distance*
204
A COMMENTARY ON
[Sect. VII.
V * in the circle S P __ A a
V ^ in the ellipse at P ~
V * down P X _
* * V * in the ellipse at P ~~
• — =
' ' S X """
P X
•'•SP~Aa+HP
Describe a circle from H with the radius A a. Produce P H to the
circumference in F. Join F S. Draw H x parallel to F S.
265. Generally/.
2HP
Px. A a
Sx.HP
HP
A a
HP
For external falls.
V * down P 2c 2 g . ^rea AB F D Newton's fig.
V * in the circle distance S P ~ g F . S P F = force at distance S P
V ^ in the circle S P 2 S P
V ^ in the curve at P
V  down P X
PV
4. A B FD
*• VMn the curve F. P V
.. 4 . A B F D = F . P V
Find the area in general ! , . ~ >
° (abscissa = space J
In the general expression make the distance from the center = S P,
and a the origuial height, S x will be found.
266. For internal falls.
V^down Px _ 2 g . A B F D Newton's fig.
2 g F . S P F = force at P
2 SP
V * in the circle S P
V 2 in the circle S P
in the curve at P
V 2 down P X
PV
4 A BFD
F. P V
* * V^in the curve at P
.'. if the velocities ar6 equal, 4ABFD = F.PV.
Book L] NEWTON'S PRINCIPIA.
267. Ex. For internal and external falls.
205
In the ellipse the force tending to the center.
In this case, D F a D S. Take A B for the force at A. Join B S.
.♦. D F is the force at D, and the area A B F D =
AD
(A B + D F)
= AS — SD.AB+ DF. Let im equal the absolute force at the dis
tance 1. Let SA = a,SD = x, .'. AB = a/4
D F = x/!«
■ 2 v2
AT» x^ TV ^ """ X • a ^ X a '
B FD =/t. — s — ■ — = iJ^.—
and
or
or
4ABFD = F.PV,
C D^
a ^ — X * = C P . pp in the ellipse,
a « — X ^ = C D ».
For the external fall, make x = C P, then a = Cx, and C x * — C P' = CD*,
or Cx* = CP^ + CD^
= AC* + BC*
= AB*
.. C X = A B.
For the internal fall, make a = C P, then x = C x', and
CP« — Cx'« = CD*,
or
Cx'* = CP* — CD*,
.. C x' = V CP* — CD*.
268. Similarly, in all cases where the velocity in the curve is quadrable,
without the Integral Calculus we may find internal and external falls.
But generally the process must be by that method.
206
A COMMENTARY ON
[Sect. VII.
Thus in the above Ex.
vdv = — g/ix.dx
.. v^ = g^ (a^ — x^)
.. A B F D = „— = /*
2g
, as above, &c.
269. And in general.
Also
v^ = ^frCa""^'— x" + '), if the force a xN
n4 1
y' = F.PV = £^f» ^^^
n + I
2
xn+i) = g/Ag
• dp
dp
..^(a« + » — x" + = ga.p.^^
n + 1 ^ ^ *^ d p
And to find the external fall, make x = ^, and from the equation find a,
the distance required.
And to find the internal fall make a = r, and from the equation find x,
the distance required.
270. Find the external fall in the hyperbola^ the force oc .^from the focus.
V* down OP: VMn the circle rad. S P : : O P : ^
V« in the circle S P : V in the hyperbola at P : : A C : H P
Book L] NEWTON'S P^INCIPIA. 207
.. V ^ down OP: V "^ in the hyperbola : : A C . O P : ^^'^^
..2 A COP = SO. HP
.. 2AC.SO — 2AC.SP = SO.HP
...SO = — HP— 2AC = — 2AC
To find what this denotes, find the actual velocity in the hyperbola.
Let the force = j8, at a distance = r, .*. the force at the distance
~ x' •
Also
V ^ in the circle S P __ jS. r' x ._ 8 x^
2 g ~ ~x^ ' 2" ~ ^x"
V* in the hyperbola _ (2 a + x) /S r "
** 2g ~ a . 2 X
 til ?ll
~ X "*" 2~a
V 2 B r^ V ^
But ^— when the body has been projected from cc = h p — of
projection from oo , .♦. ^ — of projection from go = ^ — = — down 2 a,
/3 r ^
F being constant and = 7 — 5 , or= V * from x to O', when S 0' = 2 A C.
4 a
.*. V in the hyperbola is such as would be acquired by the body ascend
ing from the distance x to 00 by the action of force considered as repul
sive, and then being projected from cd back to O', S O being = 2 A C.
In the opposite hyperbola the velocity is found in the same way, the
force repulsive, p externally = „ ' „ „ .
271. Internal fall.
V^ down P O : V* in the circle rad. S O : : P O : ^
V in the circle S O : V ^ in the circle S P : : S P : S O
V2 in the circle S P : V^ in the hyperbola at P : : A C : H P
.. V* down P O : V^ in the hyperbola : : A C . P O
.. 2 AC. PO = SO. HP
or
2AC(SP— SO) = SO.HP
<i O  2 A C. SP
•'•^^"~2AC + HP*
SO. HP
208
A COMMENTARY ON
[Sect. VII.
and
PO=SP— SO=
SP.HP
2 A C + HP*
Hence make HE = 2 A C, join S E, and draw H O parallel to S E.
Hence the external and internal falls are found, by making V acquired
down a certain space p with a ^^^ force equal that down : . P V by a
constant force, P V being known from the curve.
272. Find how Jar the body must fall externally to the cir
cumference to acquire V in the circle, F « distance to*wards the q ^
center of the circle.
Let OC = p, OB = x, OA=a, C being the point re
t^uired from which a body falls.
B4
p..
vdv= — g.F.dx, (for the velocity increases as x decreases)
Let the force at A = 1, .•. the force at B = —
= — e — . d X
^ a
.. V " =
a
and when v = 0, x = p,
... C = i
+ c
o
.1 (p2__x^)
and when x = a,
at A = s (p « — a *=).
But
at A = 2
the force at A being constant, and
a _ P V
"2  4 '
= ga
2 _ o 2
= 2 a
p * — a ^ = a % .'. p ' = JJ a % .'. p = V 2. a.
273. Find how Jar the body must Jail internally from the circutnference to
acquire V in the circle, F a distance towards the center of the circle.
Let P be the point to which the body must fall, O A = a, O P = p,
O Q = X, F at A = ], .. the force at Q = — .
Book I.] NEWTON'S PRINCIPIA. 209
.'. V d V = — Of . — . d X
^ a
.. v2 = ^ .x^ + C,
a
and when v = 0, x = a.
.. C = ^.a^
a
.♦. v^ = ^ (a^ — x^)
a ^ '
and when x = p,
V* = — (a* — p") from a'''^ force
and
V ^ = g . a, from the constant force 1 at A.
.'. a ' — P '^ = a % .*. p = 0, .*. the body falls from the circumference
to the center.
274. Similarly, when F a p .
•^ distance
O C, or p externally = a V e, (e = base of hyp. log.)
and
OP, or p internally = ^— . '
275. When F a
distance* "
p externally = 2 a
2 a
p internally = — .
276. When F a ^
277. When F a
distance ' *
p externally = x .
p internally = —jq •
1
distance " + ' '
II
p externally = a ^ ^ZITn
n
p internally = a /j
'V 2 + n
If the force be repulsive, the velocity increases as the distance increases,
.*. vdv = gF.dx
Vol r. O
210 A COMMENTARY ON [Sect. VII.
278. Find haw far a body must fall externally to any point P in the
parabola, to acquire v in the curve. F a ^v, , to'wards the focus.
P V = 4 S P = c, S Q = p, S B = X, S P = a, force at P = 1,
.. FatB = ,,
.. V d V = — g '^2 . d X
.2
• • 2  "IT + ^'
when V = 0, X = p
.. c = ti!.
v' = 2ga«(— i)=2ga'(iL)atP,
but
v* = 2g.
4=3ga,
1 _ 1 _ 1
a p ~ a '
4=.,.
p = ».
279. Similarly, internally, p = — .
280. In the ellipse, F a vyj towards a focus
p externally = P H + P S. (.*. describe a circle with the center S, rad. = 2 A C)
. , ,, PH.PS
pmternally= g^^_^p^ .
(Hence V at P = V in the circle e. d.)
281. In the hyperbola, F a yrz towards focus
pexteni^lly= — 2 A C (Hence V at P = V in the circle e. d.)
P H . P S
p internally = ' « a n !i P R * (^^^"^^ V at P = V in the circle e.d., p. 190)
282. In the ellipse F cc D from the center
pexternally= V A C* + B C^ (= A B)] (Hence construction)
or (= V CD* + CP^)
(Hence also V at P = Y in the circle radius C P, when C D = C P)
p internally = V C P^ — C D^
Book I.]
NEWTON'S PRINCIPIA.
211
(Hence if C P = C D, p = 0, and V at P = V in the circle e. d., as
was deduced before)
(If C P < C D, p impossible, .•. the body cannot fall from any distance
to C and thus acquire the V in the curve)
283. In the ellipse, F a D from the center.
External fall.
The velocitycurve is a straight line, (since D F a C D, also
sijHce F = 0, when C P = 0, this straight line comes to C, as
Cdh,V a VCOb a CO, O being the point fallen from, to acquire
V at P.
.. V from O to C : V from P to C : : O C : P C
Also since vdv=: — gF.dx, and if the force at the distance 1 = 1,
the force at x = x. .*. v d v = — g x d x, and integrating and correct
ing, V ' = g (p * — x ^), where p = the distance fallen from.
.•; V a V p ^ — x^, and if a circle be described, with center C, rad. C O
a P N (the right sine of the arc whose versed P O is tlie space fallen
through).
.. V from O to P : V from O to C : : P N : (C M =) O C
and
V from P to C : V in the circle rad. C P : : 1 : 1
(for if P v = ^ P C, v d = C d P) and
V m the circle C P : V in the ellipse : : C P : C D.
Compounding the 4 ratios,
V down O P : V in the ellipse : : P N : C D
.. Take P N = C D, and
V down O P s= V in the ellipse,
.. C O = C N = V C P^ + C D'.
02
212
Internal fall.
A COMMENTARY ON
[Sect. II.
V in the eUipse : V in the circle rad. C P : : C D : C P
V in the circle : V down C P : : 1 : I
V down C P : V down P O : : (C M =) C P : O N
.. V in the eUipse : V down P O : : C D : O N
.. Take O N = C D, and V in the curve = V down P O, and C O
= V C P 2 — C D '^.
284. Find the point in the ellipse, the f wee in the cente?, wheir V = the
velocity in the circle, e. d.
In this case C P = C D, whencethe construction,
circle
Join A B, describe
on it, bisect the circumference in D', join
'' "^^ 2
B jy, A D'. From C with A D' cut the ellipse in P.
2AD'^(=2PC^) = AB^=AC^ + BC'(=CP« + CD^)
.♦. 2 CP^= C P*+ CD*
.. C P = C D". (C P will pass through E.)
A simpler construction is to bisect A B in E, B M in F, then C P is
the diameter to the ordinate A B, and from the triangles C E B, C F B,
C F is parallel to A B, .♦. C D' is a conjugate to C P and = C P.
p externally (to which body must
285. In the hyperbola,
force repulsive, a D, from the center
rise from the center) = VC PC D*
(Hence if the hyperbola be rectangular p internally = 0, or the body must
rise through C P.)
rise from P,)= V C D ^ + C P *
p internally (to which body must
Book I.] NEWTON'S PRINCIPIA. 213
286. Li any curve, F a tYIT+i i^^d p externally.
a
p= a
n c I
where a = S P, c = P V.
287. K the curve be a logarithmic spiral, c = 2 a,
= a
also
so Fa i,,) ...p=:a(jL)H = co .
.. n = 2 J
288. In any curve, F a ^^ ^^  j ,^«</ p internamy.
289. If the curve be a logarithmic spiral, c = 2 a, n = 2,
/ a \ i. a
•••P = nHra)'^ = V2
290. If tlie curve be a circle, F in the circumference, c = a, and n = 4,
. .'. p externally = a ( \'^ = »
and p internally = a { \ * = i — .
^ ^ Va + a; ^2
291. In the ellipse, F a yr^ from focus. External fall.
V * down O P : V 2 in the circle radius S P : : O P : ^ , Sect. VII.
V * in the circle S P : V Mn the ellipse at P : : A C : H P,
03
214 A COMMENTARY ON
.. V« down O P : V^ in the ellipse : : A C . O P
[Sect. VII,
SO. HP
.. S O =
.. 2 AC.OP = SO.HP
2 A COP 2AC.SO — 2AC.SP
HP
TTF
... S O = J.^J^^J^ = 2 A C.
Internal Jail.
2 A C — H P
V ^ down P O : V 8 in the circle radius S O : : P O : ^? ,
V« in the circle S O : V* in the circle S P : : S P : S O
V* in the circle S P : V « in the ellipse at P : : A C : H P
.. V^ down P O : V* in the ellipse : : P O . A C :
.. 2 P O . A C = S O . H P
.. 2SP.AC — 2SO.AC = SO.HP
2 AC.SP
SO.HP
.. S O =
2 AC + H P
Hence, make H E = 2 A C, join S E, and draw H O parallel to E S.
292. External Jail in the parabola^ T O
F a lYi ft'oni focus.
V* d . O P : V Mn the circle radius S P
::OP: ^, Sect. VII.
V « in the circle S P : V Mn the parabola
atP:: 1 :2,
Book I.]
NEWTON'S PRINCIPIA.
215
Internal fall.
.. V^ down O P : V = in the parabola : : O P : S O
.. O P = S O, .. S O = a
SP
V 2 down O P : V Mn the circle S O : : O P : "_
V = V down
V^ in the circle S O : VMn the circle S P
V ' in the circle S P : V^ in the parabola at P
.', V ^ down OP: V " in the parabola
.. O P = s o,
SP
2 *
P V
S P: SO
1 : 2
O P : S O,
.. S O =
V down S P = V . down E P = V of a body describ
ing the parabola by a constant vertical force = force at P.
293. Find the external fall so that the velocity^ ac
quired = n' . velocity in the curve. Fax".
V dv = — g/«..x". dx, (a4 = force distance I),
.'. V 2 = ~~T .(a°+'— x" + ')a=: original heiglit,
V* in the curve = g /a . S — ~ = i .<* • c, if c = —2 — i
*= d p 2 ' dp'
.. w' .% fi. c =?44(a"+'— x°+0, orn'.c = ^. (a" + '— x'^ + O
Make x = S P = ^, and from the equation we get a, which = S x.
For the internal fall, make a = S P = ^, and from the equation we get
x, which = S x'.
294. Fitid the external fall in a lemniscata.
(x^ + y^)= = a^(x^ — y^)
is a rectangular equation whence we must get a polar one
Let z. N S P = ^,
•*• y = ?• sin. ^, X = . cos. 6, ^^ — (x" + y^)
.. ^* = a^. (g2(cos.Msin.*^)) = a"g^cos. 2 tf,
.. ^ '^ =r a " . COS. 2 6
.. 2 « = ^ (cos. = 1^),
a^ V a.* — g*
V a*'
04
216 A COMMENTARY ON [Sect. VII.
a*^
" d6'
__a*g*
but
in general
g. d 6
_ df.p
V^^_p«'
•'i
M ^2_^2jJ^2p8
= dg
^PV
.•.p«
f*d^«
r
 g*d^^ + dg^
r
S*
§^ +
a* —
•r

.•.p«
1
a*
.*. force to S a — t;
S
V d V = — ^Af . d X,
.. v« = 1^^ ''*
Vx6 a^y^
Also
PV_ 2pdg a^ _2.g' a' _ 2g
^^ dp "'P'Sg'^" a« *3g*~ 3'
Make x in the formula above = g,
•••g6 a^~ g«'
.«. — ^ = 0, .*. a is infinite.
a ^
NEWTON'S PRINCIPIA.
Book I.]
295. Ft
CY^=CP«YP2=CP« — CA^
217
295. Find the force and external Jail in an EPICYCLOID
YB' B
CB^
Let ^'
C Y = p, C P = f, C B = c, C A = b,
.♦. c* p* = c^ g * — b^ c^ + b* p
••P  c*^ — b^
2dp _ c^ — b' / — 2 d g . g \
P'
.*. force «
« =t:
{s' — hV P*
(as in the Involute of the circle which is an Epicycloid, when the radius
of the rota becomes infinite.)
Having got a° of force, we can easily get the external (or internal) folL
296. Fijid in *what cases we can integrate for the Velocity and Time.
Case 1. Let force a x °,
.*. V d V = g (« . X " d X,
.. v2 = AiJt (an + I _ x'' + I)
n + 1 ^ '
, /» — dx __ / n+ 1 ^ — dx
~J V ~V 2g/AVV(a"+i — x"*'
Now in general we can integrate x'^dx.(a + bx''^)— , when
4
is whole or — — 1 ^ whole.
n n q
.•. in this case, we can integrate, when
Let
1
—  — = , or —  — r^ — « > is whole,
n + 1' n + 1 2 *
— j—r: = p any whole number
.•.n+l = I,
.». n = * , (p being positive), (a)
218 A COMMENTARY ON [Sect. VIl. ?
. Let
1
— o = P'
n + 1 2
• • 11 + 1 ^ ^ 2 2 '
1 2 p r,s
.*. these formulae admit only and 1 for integer positive values of n, and
no positive fractional values, .'.we can integrate when F cc x, or Fa 1.
297. Case 2. Let force oc —. ,
, d X
.. V d V = — g /i — ,
X'
n — 1
2 _ ^ g /^ / a"~^ — x"s
— r — ^^— / " — I.a"~^ p — d x . X j—
~~ .y V ~ ^ 2 g /" '•' Va""^ x"~'
2 ^ 2 ^ 2
in which case we can integrate, when ■ '  . — , or — , whole,
i. e. if  H or r , be whole.
2 ^ n — 1, n — 1 '
Let ^ = p, any whole positive No.,
..."nI,= l,...„=P^\(»')
^^^2 +ir=ri=P' ■
* * n
1
1
=
2p
2
1
>
.. n
—
1
=
2
2Y
1'
.•.n = p:^;.(.',
.'. these formulae admit any values of n, in which the numerator ex
ceeds the denominator by 1, or in which the numerator and denominator
are any two successive odd numbers, the numerator being the greater.
, T^ 11 1 Ion
.•. we can integrate, when h ^ — j j —35 —45 —59 &c.
X X §• X J x^ I
or J>
1 JL i 1 «.
P'xf'xf'^'^'^J
Book I.] NEWTON'S PRINCIPIA. 219
298. Case 3. The formulae (a') (/3'), in which p is positive, cannot be
come negative. But the formulae (a) and (/S) may. From which we can
integrate, when F oc _,_,_,_, &c.
Xj X^ ■ii^ X.J
• or when F oc — ^ &c.
x^ x^ xf x
299. When the force oc yi"^, Jind a", of times from different altitudes
to the center of force. Find the same, force a « — .
Fax", .*. vdv = — g/ix°dx,
••. d t = a — ^ which is of — dimensions,
V V a° + ^ x" + ' 2
.*. t will be of — dimensions.
and when x = 0, t will oc — j^^^ .
F « — S » ••• t a _n,i a a a
x" a ^—
. _ — d X 1 ^ — dx
t a /^ ■ .1 a /^ — — —
J Va» + '— x»+^ a24^y /, /xx" + '
«a4^{'(Tr'}"'
when t = 0, X = a,
. n f.l a, 1.3 a . o \
•••a4>({'+i„i2W^+M^:^.+^)
.. when X = 0, t a ^ a — ^
220 A COMMENTARY ON [Sect. VII.
, . . 1 ^ n+J
when n is negative t a — r a a 2 .
^ a"'
2
Cor. If n be positive and greater than 1, the greater the altitude, the
less the time to the center.
300. A body is projected up P A isoith the velocity V Jrom the given
po'nt A, force in S « yi\yjind the height to >which the body "jcill rise.
vdv = — g/u,x"dx,
for the velocity decreases as x increases, A
V. v2 = l^.x"+> + C
n 4 1
when v = V, X = a,
.•.C=: V« + ^^.a" + ^
... lUt., (x» + i— a"+0 = V2_v2
n ■+■ 1 ^
Let v = 0,
n + 1 ^
n^ V2.II+1
2g/i
. xn + i = V^n+ 1+ 2g^.a"+^
2g^
. _ . V2.n + 1 + 2g^.a " + yi.,.
g
Cor. Let n = — 2, and V = the velocity down — , force at A con
slant, = velocity in the circle distance S A.
. X = /_ V2 + ^g^ \~' = ^g^
( ^] 2g^ ^.^
2 g/to / a
2g/t 2
2g^ g^ a — — i
a a'' * a a
= 2 a.
Book I.] NEWTON'S PRINCIPIA. 221
SECTION VIII. ,
301. Prop. XLI. Resolving the centripetal force I N, or D E (F)
into the tangential one I T (F') and the perpendicular one T N, we
have (46)
I N : I T : : F : F : : i^ : ^'
d t d r
.. d V : d v' : ; d t X I N : d t' X I T.
But since (46)
, ^ d s , , d s'
d t = — , d t' = —r
V v'
and by hypothesis
V = v'
.. d t : d t'
.*. d V : d v'
: d s ; d s' : : I N : I K
: IN^ : IK X IT
: 1 : 1
or
d V = d v',
&c. &c.
OTHERWISE.
302. By 46, we have generally
vdv = gFds
s being the direction of the force F, Hence if s' be the straight line and
s the trajectory, &c. we have
vdv =: gFds
v' d v' = g F' d s'
... v^ — V = 2g/Fd s
v'* — V'» = 2g/Fds'
V and y being the given values of v and v' at given distances by which
the integrals are corrected.
Now since the central body is the same at the same distance the central
force must be the same in both curve and line. Therefore, resolving F
222 A COMMENTARY ON [Sect. VIU.
when at the distance s into the tangential and perpendicular forces, we
have
^~^*^IN~^^IK
_, d s
Z= ¥ X r—,
a s
.. F d s' = F d s
and
v"^ — V'^ _ 2g/Fds = v^ — V
which shows that if the velocities be the same at any two equal distances^
they are equal at all equal distances — i. e. if
V = V
then
V = v'.
303. CoR. 2. By Prop. XXXIX,
v^a A B GE.
But in the curve
y a F oc A"*
.♦. ydxoc A"^dA
Therefore (112)
ABGE=/ydxa— i^+C
a
n
P" — A
n
Hence
v2 a P" — A".
OTHERWISE.
304. Generally (46)
vdv = — gFds
and if
F = ya S"i
then
v^ = ?^(C^s")
n ^ ^
But when v = 0, let s = P ; then
= ig_^(C — P'')
n ^
and
C = P".
Book L] NEWTON'S PRINCIPIA. 223
n ^
in which s is any quantity whatever and may therefore be the radius vector
of the Trajectory A ; thai is
v2 = i»i'(pn_ A")or = — ^^(Dn — J")
in more convenient notation.
N. B. From this formula may be found the spaces through which a
body must fall externally to acquire the velocity in the curve (286, &c.)
305. Prop. XLI. Given the centripetal farce to constnict the Trajec
torry, and tojind the time of describing any portion of it.
By Prop. XXXIX,
V = VFi. V A B F D = ^^ (46) = 1^
But
1 X ^ Tr Time ^ /« rr wt Timc
d t = I C K X . = I C X K N X 0. —
Area 2 Arga
= p ryj (P being the perpendicular upon the
tangent when the velocity is V. See 125, &c.)
Moreover, if V be the velocity at V, by Prop. XXXIX,
V = V~2y. V A B L V.
Whence
/ToEtf: PVABLV IK
VABFD= ^ X j^
/. putting
^^PVABLV/ ^ Q^ PxVx
A V A V 2 g A>' . ^ '
we have
ABFD : Z* : : IK^ : KN'
.. ABFD — Z2> Z^ : : IK^— K N^ : KN^
and
V A B F D — Z ^ : Z = ^ : : I N : K N
A
■•■axkn= ^^^Qbfdz') • • • • (2)
Also since similar triangles are to one another in the duplicate ratio oi
their homologous sides
YXxXC = AxKNx ^^
:24 A COMMENTARY ON [Sect. Vlll.
_ Q X CX^ X IN .
~ A« V (ABFD — Z^) • • • ^"^^
and putting
y~^'^2V(ABFD — Z«)
and
/_ n _ Q X CX^
y  ^ <^  2 A' ^^ (A B F D — Z')'
Then
Area VCI=/ydx = VDba"» .^.
AreaVCX=/y'dx = VDcaJ • • * • »J
Now (124)
2VCI 2VDba
* ~ P X V ~ P X V
or
2 VDba
V2g.Px VABLV
the time of describing V I.
Also, if iL V C I = ^, we have
XVxCV «xCV*
(5)
VDca=: VCX =
. 2VDca
2
which gives the Trajectory.
306. To express equations (5) ajid (6) m /e7"ws </g and &, {§ = A).
First
ABFD = ^
(6)
and
V2
ABLV = 
V2g
•• Z  ,.  2g^«
2 P2 X V*
..ABFD — Z» = ^
2g 2gr
Book I.] NEWTON'S PRINCIPIAs 226
Hence
and
PxVg
y ~2V(f«v^— P«V*)
P^x V
y =
and
2g V(f2v«— P^V^)
2 ♦>' V(f«v« — P«V*)
...VDba = ?^/ ^^^
vr» _P'V /  dg
... t = / ^ii
But by Prop. XL.
v«=2/gFdg
the integral being taken from v = 0, or from g =D, D being the same as
P in 304.
// p a, p /• yfdp *iy»
^(SgVgFdg— P^V^)'"'' =y V(g2v«— P«V^) • • ^ ^
. _ r Px Vdg _ / » PVdg .
*ygV(__2gygFdg— P*V2)'°'/gV(g«v« — P*V=) • ^ ^
307. Tojind t aw^ ^ m ^^rws of g and p.
Since (125)
v^=^p:=2/gFdg
. t  f Jvi^i
„. I)
and
d?
/,
But previous to using these forms we must find the equation to the tra
jectory, thus ( 139)
P^V^ dp „ „,,
X r^ = F = f (g)
f denoting the law of force.
Vol. I. P
226 A COMMENTARY ON [Sect. VIIL
or
pays
P = V^_2g/d^fg (^^)
308. To these different methods the following are examples :
1st. Let F a g = fi I. Then (see 304)
.. v^ = g^(D^r)
and if P and V belong to an apse or when P = ^ ;
V^ = g/.(D" — P^)
_ 1 /• g d g
D*
Let g ^ — = u. Then we easily get
u ^
= sin.^ D^ + C
p2 td
2
and making t = at an apse or when g = P, we find
pa ^
2
C =; — sin.  K ^^ = — sin. " * 1
ps ^
2
IT
' 2 '
1 ). ' T rr)
.*. t = = X J sm. ' ~ — _ V . .
2Vgf^ I P3_D^ sr
(1)
Also
'V=>?'=2vV/(u+^y(pL^V_u^}
and assuming
2 ^  • > V* 2
P^ — ^ — u = v2x (P2 — :^ + u)
we get
ttTt = = X < sin.' r^, h C?
PV 2Vg/*P.VD2P2 I g2rp2_2J\ 3
Book I.] NEWTON'S PRINCIPIA. 227
and making ^ = 0, when ^ = P we find
C=— sin.'l= ^.
Also
V= VYa^. V (D2 — P2)
•■• —. ; nv = sin. (2 « + g)
= COS. 2 6=2 COS. 2^—1
which gives
^ P2_(2 P^ — D=^) cos.^tf  ^ ''
Now the equation to the ellipse, g and 6 being referred to its center, is
^ 1 — e' COS.* d
Therefore the trajectory is an ellipse the center of force being in its
center, and we have its semiaxes from
b2 = D^— P*
c« a« — b^ 2P' — D*
i
e' =
a
'}
viz.
b = V(D^ — P«)
and J (3)
a = P
which latter value was already assumed.
Tojind the Periodic time.
From (3) it appears that when
t=^,or^= ,g = b= V(D«P2)
and substituting in (1) we have
1 ) . . 2 ff f
= — 7= X <sm. ^ T^r^^'irr
2Vg^ I P2_5I 2
= _i=x {siii.'(l)}
P2
2S8
A (
But
sin. 
'(
1)
=
3or
2 
•••
T
4
=
«•
and
2 V git.
T
__
2«r
A COMMENTARY ON TSect. VIll
(4)
which has already been found otherwise (see 147).
To apply (9) and (10) of 307 to this example we must first integrate
(11) where f ^ :=: fig; that is since
we have
1p2
2
p2 =
But
V2=g^(D2_P2)
PMJD^Pf) ..
••P  D^=V~ (5)
which also indicates an ellipse referred to its center, the equation being
generally
2_ a^b^
P  a^+h^ — S^'
Hence
p2 ^ P2(D2_P2)
... t = i/
?dg
v'i;;^.' vjg2 (D^ — f 2)  p«(D2 — p2)]
the same as before.
With regard to 6, the axes of the ellipse being known from (5) we have
the polar equation, viz.
b2
S' =
1 — e 2 cos.
309. Ex. 2. Let F = 4 • Then (304)
^ = ^~x(D'r')
Book I.] NEWTON'S PRINCIPIA. ^
V22ff /(i X
and
D — P
DP
P and V belonging to an apse.
J k/ 9 IT „.
D^
whicfe, adding and subtracting —— , transforms to
^T^ ^ ^(Dfe^DP + P^)
4
V D
t=:
^^g'''/^{(p£)"lo^)»}
and making f g" = "
2
D
t =
VD /_ (^ + 2)^"
V2
=VsT.x Jc.'{(p2:).«^}+sin..^^
(see 86).
Let t — 0, when g = P. Then
C = ^sin.'l = ^X
D
^2
Also
But assuming
P— ^ — U=V2X (P— ^ +u)
the above becomes rationalized, and we readily find
P3
230 A COMMENTARY ON [Sect. Vlll.
/(" + t)v''{(pt)^"1^
. 5 (''f)'+ °" )
and making ^ = 0, when g == P, or when u = P — „ , we get
C = tan..i.=.
Hence, since moreover
+ ^=tan.
or
= sin.
_ pg— PD
_ 2P.(D — P) 1
~ D , . /, 2P;
= sin. \6 + _ j = COS. 6
1+ (l— ^)cos.<J
(2)
But the equation to the ellipse referred to its focus is
1)2 1
a 1 + e cos. ^
b«_ 2P(D — P)
•*• a ~ D
and
a'*  * a«"" V D^
Book L] NEWTON'S PRINCIPIA. 231
• . 2 = TJ — D^  C"2 X (^ — ^)
b2 2
= — X rJ
a D
^ }
b = >/ P X rD — P))
and r ^'^
X (D— P).
To find the Periodic Time ; let ^ = ie. Then g = 2a— P=D— P,
and equation (1) gives
T ; D D / • 1 1 «f \
1)
^ 2a*
••^ ^^gA6'
see
159.
OTHERWISE.
p2
First find the Trajectory by formula (11. 307) ; then substitute for =g
in 9 and 10, &c.
LO. JReqi
By 304
310. Required the Time and Trajectcnywhen F= j
— g/.X (D2— g2)
— D3 ^ g2
.'. if V and P belong to an apse we have
,.o g** D2— P2
V 2 = g2 X p 2
I,:(P
e)

^ J
V P^
P4,
232 A COMMENTARY ON [Sect. VI 11.
^ X (C± Vpa — ^^)
and taking t = at an apse or when ^ = P, 0=0,
D
t==^ X VP2_^2 . (1)
V g/tt
also
6 _ r d_t _ D  d g
PVy ^«  V~^ ^/f V(P«g^)
But
r—^L—  J X li ^(P^g') + p ", cl
and
V = ^^ X V {D^P').
TVs r>2\ — *• ^ T ^
•• V (D« — P«) ~ • s
and making tf z= at the apse or where f = Pj
p
C = 1. =
"  V D«— P* ' g
. P^ _ V (pg — g8)q:p
•*e V(D2— P2) f
which gives
2 P e V'^*^'''
2Pe
(2)
311. Required the Trajectory and circumstances of motion iiohen
F = a
or for any inverse law of the distance.
The readiest method is this ; By (11) 307, if r, and P be the values of
f and J) for the given velocity V (P is no longer an apsidal distance)
the equation to the Trajectory.
Also since
vdv = — gFdf
Book L]
NEWTON'S PRINCIPIA.
233
Hence
and if we put
•'• "' ' = (n— f)^"i (^'■°™ "^ ^"^ ^^
V2 =
2 m g/A
(n— 1) r"!
in which ra may be > = or < 1 we easily get
2
/ m Pea ^ 
70" + ^)
■^ " — 1
P 2
P=^zri X f
/
P =
>^ m— 1
n — 1
X P^
VVl— m ^""^
Tb ^«d ^ ore this hypothesis.
We have (307)
in= 1
m< 1
which gives by substitution
d^=±r /— f^Px
N' m — 1
? 2 dg
(2)
//■ m n — 3 n — Z.
3y..m>l
d^ =
n_5
rg 2 dg
V(^^"1
m= 1
d^ = +
/
VI — m
X Px
n3
g 3 d g
«V(t^t^p^«"^«"')
the positive or negative sign being used according as the body ascends or
descends.
Ex. If n = 2, we get
/ 111 T1
V0'+5^0
. . . . m>l
234 A COMMENTARY ON [Sect. VIII.
P i
P = Tr m = 1
r i
the equations to the ellipse, parabola and hyperbola respectively.
Also we have correspondingly
cVG^+sr^^sr^^O
dtf = +r P.
di
fV(rgP2)
dtf=+r /= . — ^
which are easily integrated.
Ex. 2. Let n = 3. Then we get
P = J — — T X P X ^ s . . m > 1
V = ~ s m=l
P = J^ X P X — i . . m < 1
d6 = + /— ^.Prx ^^2 2 . m>l
— Vm — 1 //o mP2 — t\
d^ = ± V(r^lp ^)7 "='
d^=±>^'T^XrPx 'Jp. . m<l
312. In the first of these values of ^, m P ^ may be > = or < r^.
(1). Let m P 2 > r ^. Then (see 86)
and at an apse or when r = P
Book 1.] NEWTON'S PRINCIPIA. 235
for
/ m — 1 __ j_ _ 1^
N m P* — r^ ~ P ®^ ~ r •
(2) Let m P^ = r*. Then we have
r
V (m — 1) ^ /, V
^ 0' + sr^)
— 4_Jll___ /"^^C
"~  V (m— 1) ^y 72'
= ± , X ( )
X ^— (c)
~ • m — 1 i
which indicates the Reciprocal or Hyperbolic Spiral,
(3) LetmP2be<r«. Then
v^Cn^ + sO
/ e^U'+ ml )
J. L/
+rP /— ^ .., r ^(ml.f +r^mP)V(rmP^)
Wr'— mP^'g Vm .(i^— P) — V (r*— m P«) '*^^
at an apse r = P ; and then
, = +P /J^xl. ^''^''^ . . . (f)
— ^ 1 — ra i
Thus the first of the values of 6 has been split into three, and integrat
ing the other two we also get
"" = + V(r'^P') ^''g~'''>
 a: V(r'— P') ^ r
«. = +rP /tJ5 / , . ''^i
— >/l — m/ //r^ — mP* a
236
A COMMENTARY ON
m
= +rP /
//r*mP_,\ ,r^ — mP
[Sect? VIII.
mP
and if * is measured from an apse or r = P it reduces to
< = + P /J!^l/+^g'— ^'.
— >r 1 — m g
313. Hence recapitulating we have these pairs of equations, viz.
(1)P=
or
" = ±^ Viirp?=:?><0^^~'^V
^ = + p /_?5_. X sec.^4.
— ^ m — 1 P
To construct the Trajectory^
put tf = 0, then
g = P= SA;
let g = CD, then
and
m — 1
m P— r^'
sec."
m
V m P2.
—r^)
m
 2 Vj
m
m— 1'
and taking A S B, A S B' for these values of ^,
and S B, S B' for those of p and drawing B Z,
B' 7/ at right angles we have two asymptotes ; S C being found by put
ting d zz V. Thus and by the rules in (35, 36, 37, 38.) the curve may
be traced in all its ramifications.
2. p =
V (m — 1)
?
VG' + sr^)
and
I — a = +
S —
V (m — 1) g
Book I.]
NEWTON'S PRINCIPIA.
837
This equation becomes more simple when
we make Q originate from 1=00; for then
it is
• _ r' }_
V (m . 1) ^ g
and following the above hinted method the
curve, viz. the Reciprocal Sj^iral, may easily be
described as in the annexed diagram.
m .. e
'■p=pVt^
m
Jii~^^1
and
,_» = +rP /^IL
— ^ r — m
mP =
' i' Vm(r2_p2)_V (r^ — mP^)
and when 6 is measured from an apse or when P' = r
, = + P ./J!L_.i^C' + fr
— ^ 1 — m g
Whence may easily be traced this figure.*
A
P
V(r='— P'^) r*
From which may be described the Logarithmic Spiral.f
m— 1
X
V(t^^')
_ / m , r \/(r^— mP— 1 — rn.g^)— A/ (r^ — mP)
?«J:rP^^^pXl.. v(m.r^ g^)  V (r ^ — m P^)
238
or
A COMMENTARY ON
[Sect. VIIT
'=±'Vt
m
1.
r — V ii' — r')
m
when P = r.
Whence this spiral.
These several spirals are called Cotes' SpiralSi
because he was the first to construct them as
Trajectories.
314. If n = 4. Then the Trajectory, &c.
are had by the following equations, viz.
5
d^
= r P ^/ ^ X
S m — 1
*V(^
m— 1 ^^Iir:=i;
315. If n = 6. Then
p = P V m
V (m— l.?«+r*)
d^
V m — 1 // 4
which as well as the former expression is not integrable by the usual
methods.
When
m— 1
is a perfect square, or when
,^^^J^^^^,^+ '
m— 1
m^P*
dg
m — I ~ 4 (m — 1) *
^ then we have
^ • 2 (m — 1)
Therefore (87)
/ m P
„ / m ^ /2(m— 1)^, N 2 (m — 1) '
Vv ""2(m— 1)/
Book I.]
6 — a = rV2 Xl.
NEWTON'S PRINCIPIA.
F V m — § V 2 (m — f)
V(2.m — l.g2— m P^)
, ./oi f V2 (m— 1) + P Vm
a — ^or=r v 21. = ^^ ^ ' ■
V(mP2_2.m — l.f^)
and these being constructed will be as subjoined.
316. Cor. 1. otherwise.
To find the apses of an orbit 'where F = ^,
Let
P = f
Then
i
m
m— 1
n — 1
r n — 3 ,
+ = = m > 1
m — 1 ^
m = 1
Pn3
and
+
m
pspn;
. . . m < 1
239
1— m " 1— m
which being resolved all the possible values of f will be discovered in each
case, and thence by substituting in ^, we get the position as well as the
number of apses.
Ex. 1. Let n = 2. Then
, , r mV'
m
^ = T =
L
=
g*
1 — m
m
4
mP«
g + T =
240 A COMMENTARY ON [Sect. VIII.
which give
r + 4 m P . (m — 1)
2(m— 1)— V 4(m — i;
L,
4.
and
4mP^{l — m)
^  2 (1 — m)  V 4.(1— m) ^
Whence in the ellipse and hyperbola there are two apses (force in the
focus) ; in the former lying on different sides of the focus ; in the latter
both lying on the same side of the focus, as is seen by substituting the
values of ^ in those of ^. Also there is but one in the parabola.
Ex, 2. Let n = 3. Then eq. (A) become
m P^ 4 r*
(1) s' = ,
^ ' ^ m — 1
which indicate two apses in the same straight line, and on different sides
of the center, whose position will be given by hence finding 6 ;
(2)
r
S = 2  QO
po
because r is > P,
ienc(
(8)
I there is no apse.
, r^ — mP^
which gives two apses, r * being > m P ^ because m is < 1 and P < r ;
and their position is found from 6.
317. Cor. 2. This is done also by the equation
P
p =r g. sm. <Pf or sin. f = ^
f being the z. required.
Ex. When n = 3, and m = 1, we have (4. 313)
P
P=^
P
.*. sm. (p = 7^
.*. (p is constant, a known property of the logarithmic spiral.
318. To find isohen the body reaches the center of force.
Put in the equations to the Trajectory involving p, ^ ; or g, ^
J = 0.
Ex. 1. When n = 3, in all the five cases it is found that
p =
Book I.] NEWTON'S PRINCIPIA. 341
and
6 = — X.
Ex. 2. When n = 5 in the particular case of 315, the former value of
d becomes impossible, being the logarithm of a negative quantity, and the
latter is = — oo .
319. Tojind 'when the Trajectory has an asymptotic circle.
If at an apse for ^ = cc the velocity be the same as that in a circle at
the same distance (166), or if when
^ = CD
and
P = f
we also have
p  1p
then it is clear there is an asymptotic circle.
Examples are in hypothesis of 315.
320. Tojind the number of revohitions from an apse to § = co .
Let 6' be the value of ^ — a when ^ = p or at an apse, and 0" when
^ = 00. Then
V = — = the number of revolutions required.
Ex. By 313, we have
^ rv ^' = P J^— ^ sec. » %
>f m — 1 P
_ / m ir
 Vm— 1 • T
1 / ^
321. CoR. 3. First let V R S be an hyperbola whose equation, x being
measured from C, is
Then
But
b*
V C R = y^^ /y d X
/ydx = ^/dx ^/(x2a2)
£1
= ° X V x«a^ g/
a a, ^
h r x« d X
Vou I. Q
242 A COMMENTARY ON [Sect. VIII.
=xv'(x2— a»)— /dx V(x2— a^) —  f^A^
a ^ ^ a^ ^ ^ a«/ V(x*— a^)
•• «7 y a X = — xV(x* — a*) — abl. — ■ ^ '
a a
and
VCR=!^l."+^ '"'"'> .... (1)
2 a ^ '
Again
g = CP = CT = x — subtangent
= xLdi^(29)
d y ^ '
_ x' — a^ _ a"
~ x ~ X
aiid substituting for x in (1) we have
.•■< = VCP«VCR=iJNl. ''+ ^(»'s') . . . . (2)
2 a f ^ '
N being a constant quantity.
322. Hence conversely
and differentiating (17) we get
d u^ _ 4 / 2 1 \ .
dJa'  a^b^N^ ^ ^" "■ ^; * ' ' " V*;
and again differentiating (d d being constant)
d^u 4
d<>2  a^b^N"
Hence (139)
F =
X U
P'V / 4 x2_ i_
g • Va^b^Ns + ^J g^'^ g»
322. By the text it would appear that the body must proceed from V
in a direction perpendicular to C V — i. e. that V is an apse.
From (1) 322, we easily get
d g ^ _ 4 / 2 2 4\
dtf2~" a^b^N*^" S —S )
Book L] NEWTON'S PRINCIPIA. 243
and since generally
d^2 p8 \S P ^
4
 a^b^'N' Xp*X(a'g^) =g==p
.•.P^ = ^— ^^ . ... (1)
which is another equation to the trajectory involving the perpendicular
upon the tangent.
Now at an apse
P = S
and substituting in equation (1) we get easily
S = a .
which shows V to be an apse.
OTHERWISE.
Put d ^ = 0, for f is then = max. or min.
324. With a proper velocity.']
The velocity with which the body must be projected from V is found
from
vdv = — gFdf.
325. Descend to the center']. When
g = 0, p = (1. 323) and ^ = CO (2. 321). ^
326. Secondly, let V R S be an ellipse, whose equation referred to the
center C is
b*
y« = „. (a«— x«)'
•^ a^ ^ '
Then
VCR=y^+/_yd
and as above, integrating by parts,
rA /,! t\ X V (a' — x') , a' /» dx
/d X V (a*x^) = ^2 ^ + ^/ v(a'_^.)
Q2
W* A COMMENTARY ON [Sect. VIII.
X V (a' — x^) . ay. ,x ^.
.•.VCR = ^r^.sin.iiV
2 \2 a/
Also
dy
a« — x^ a*
= X H = —
X X
and
a b N / <!r
= N. VCR =
a T 2 6
.(^ sin.. !)...(„
«•. sm.  * — = — —
2 ab N
a . /«• 2 ^ \ 2 <)
• • "~ — sm. I tr r^ ) = COS.  , .. y
f \2 abN/ abN
and
f=asec.^ (2)
Conversely by the expression for F in 139, we have
F ex ^
327. To Jind 'when the hody is at an apse^ eithei' proceed as in 323,
or put
d g = 0.
„ . , d x . sin. X
13y (27) d. sec. x = ^
sin. 6
=
COS. ^ 6
or
'6=0
that is the point V is an apse.
328. The proper velocity of projection is easily found as indicated
in 324^
329. Will ascend perpetually and go off' to infinity. 1
From (2) 327, we learn that when
2 6 _ It
HTN ~ 2
f is oo;
also that f can never = 0.
Book L] NEWTON'S PRINCIPIA. 245
330. When the force is changed from centripetal to centrifugal, the
sign of its expression (139) must be changed.
331. Prop. XLII. The preceding comments together with the Jesuits'
notes will render this proposition .easily intelligible.
The expression (139)
g P'df
or rather (307)
p2 y2
P°' = V^2g/d7Tf
in which P and V are given, will lead to a more direct and convenient
resolution of the problem.
It must, however, be remarked, that the difference between the first
part of Prop. XLI. and this, is that the force itself is given in the former
and only the law of force in the latter. That is, if for instance F = /* f " ~ ^,
in the former fi is given, in the latter not. But since V is given in the
latter, we have /x from 304.
SECTION IX.
332. Prop. XLIII. To make a body move in an orbit revolving about
the center of force^ in the same 'way as iti the same orbit quiescent^
that is. To adjust the angular velocity of the orbit, and centripetal force
so that the body may be at any time at the same point in the revolving
orbit as in the orbit at rest, and tend to the same center.
That it may tend to the same center (see Prop. II), the area of the new
orbit in a fixed plane (V C p) must a time a area in the given orbit
( V C P) ; and since these areas begin together their increments must also
be proportional, that is (see fig. next Prop.)
CPxKRaCpxkr
But
KR = CK X Z.KCP
k r= Ck X zkCp
and CP= Cp, andCK = Ck
..ziKCPakCp
and the angles V C P, V C p begin together
.. /lVCP a ^VCp.
Q3
246 A COMMENTARY ON [Sect. IX.
Hence in order that the centripetal force in the new orbit may tend to
C, it is necessary that
^VCpa^VCP.
Again, taking always
CP = Cp
and
VCp:VCP::G:F
G : F being an invaa'iable ratio, the equation to the locus of p or the orbit
in fixed space can be determined; and thence (by 137, 139, or by Cor.
1, 2, 3 of Prop. VI) may be found the centripetal force in that locus.
333. Tojind the orbit infixed space or the locus qf\i.
Let the equation to the given orbit V C P be
where f = C P, and ^ = V C P, and f means any function ; then that of
the locus is
f = f(0 (')
which will give the orbit required.
OTHERWISE.
Let p' be the perpendicular upon the tangent in the given orbit, and p
that in the locus ; then it is easily got by drawing the incremental figures
and similar triangles (see fig. Prop. XLIV) that
K R : k r : : F : G
k r : p r : : p : V (g ^ — p ^)
pr :PR:: 1 : 1
PR:KR:: V(f^ — p'^) : p'
whence
and
1:1 :: F.p V(f2 — p'2): Gp' V(f* — p«)
••P  F2g2 + (G2_F2)p'2 v;
334. Ex. 1. Let the given Trajectory be the ellipse with the force in
its focus ; then
P 2T^' ''"'*^" 1 + ecos.d'
and therefore
b^G^(2ae)g^
P b2(G2— F'^) + F''^(2ag — g«)
Book I.] NEWTON'S PRINCIPIA. 247
and
a.(l— e^)
s = 7T"T'
1 + e COS. ( p dj
Hence since the force is (139)
"'a"+")
g
and here we have
F
a(l — e*)u=l + e cos. ^ 6
2 F* F«
= Q + aG(le) "G'^"'
and again differentiating, &c. we have
d^u _ F^ G^ — F"
dT^ + " ~ G»a(l— e*) ■*" G*^ ^ "'
But if 2 R = latusrectum we have
.♦. the force in the new orbit is
P»V' jF'^ , RG'— RF« ^
gRG«^t^^+ e j
335. Ex. 2. Generally let the equations to the given trajectory be
f = f(^)
and
Then since
G*
... d^^ = ^dd'«
d«u _ F'd'u ,
F^ /d^u . \ . F«
and if the centripetal forces in the given trajectory and locus be named
X, X', by 139 we have
gX^ _ FJ gX G' — F' 2.
Q4
}
248 A COMMENTARY ON [Sect. IX.
p« v« / F^X G'F ' 1 X
Also from (2. 333) we liave
J^ _ Fj J_ G'— F g J^
p« ~ G^^ p"* "^ G* ^ g«
••p'dg~ G^ P"d§ G^ g^
.. by 139
gX' F^gX , G — F« I
p2y2— p/ay/aT^ Qs ^ ^j
the same as before.
This second general example includes the first, as well as Prop. XLIV,
&c. of the text.
336. Anothej' determination of the force tending to C and txihich shall
make the body describe the loctis of^.
First, as before, we must show that in order to make the force X tend
to C, the ratio /iVCP: iiVCp must be constant or = F : G.
Next, since they begin together the corresponding angular velocities
w, w' of C P, C p are in th^t same ratio ; i. e.
« : <w' : : F ; G.
Now in order to exactly counteract the centrifugal force which arises
from the angular motion of the orbit, we must add the same quantity to
the centripetal force. Hence if f, f ' denote the centrifugal forces in the
given orbit and the locus, we have
X' = X + 9' — p
X being the force in the given orbit.
But (210)
p 2 V * ]
f = X —
g f
and
a «*
when I is given.
«'» G*P*V«G« 1
•*. P' = ? X T = f X ,^, = — —  X V.Y X
(a
F« g ^F'^g
p£V2 G^— F* 1
.'. f' — f = X rr^ X 7
p^V G^— F^ _ 1
X' = X + i^ x^^^pr^ xL . .... (1)
Book I.] NEWTON'S PRINCIPIA. 8t9
or
^i^h^^^) (3)
or
__ P' V^ /_d^p ^ G' — F'
"■ g ^ ^P'
337. Prop. XLIV. Take u p, u k similar and equal to V P and V K ;
also
mr:kr::^VCp:VCP.
Then since always C P = p c, we have
p r = P R.
Resolve the motions P K, p k into P R, R K and p r, r k. Then
RK(=rk):rm::z.VCP:^VCp
and therefore when the centripetal forces PR, p r are equal, the body
would be at m. But if
pCn:pCk::VCp:VCP
and
Cn = Ck
the body will really be in n.
Kence the difference of the forces is
mkxms (mr — kr).(mr+kr)
m n = = i — ^ ■ ^ .
m t m t
But since the triangles p C k, p C n are given,
K r a m r a j^ —
Cp
1 1
.*. m n 05 7s — i X —  .
C p* m t
Again since
p Ck: p Cn
: PCK:pCn:: VCP: V Cp
: k r : m r by construction
: p C k : p C m ultimately
.*. p C n = p C m
and m n ultimately passes through the center. Consequently
m t = 2 C p ultimately
and
1
Cp
260 A COMMENTARY ON [Sect IX.
OTHERWISE.
338. By 336,
~ g ^ F^ ^P
1
a — .
i'
339. To trace the variatiofis of sign of mn.
If the orbit move in coiisequentia, that is in the same direction as C P,
the new centrifugal force would throw the body farther from the center,
that is
Cmis>CnorCk .
or m n is positive.
Again, when the orbit is projected in antecedentia with a velocity <
than twice that of C P, the velocity of C p is less than that of C P.
Therefore
C m is < C n
or m n is negative.
Again, when the orbit is projected in antecedentia with a velocity =
twice that of C P, the angular velocity of the orbit just counteracts the
velocity of C P, and
m n = 0.
And finally, when the orbit is projected in antecedentia with a velocity
> 2 vel. of.C P, the velocity of C p is > vel. of C P or C m is > C n, or
m n is positive.
OTHERWISE.
By 338,
But
m n oc <p' — p
w' = « + W
W being the angular velocity of the orbit.
.•. m n cx42 wW+ W^
a + 2 w + W
4" or — being used according as W is in consequentia or antecedentia.
Book L] NEWTON'S PRINCIPIA. 251
Hence m n is positive or negative according as W is positive, and nega
tive and > 2 w ; or negative and <C 2 w. That is, &c. &c.
Also when W is negative and =: 2 w, m = 0. Therefore, &c.
340. CoR. 1. Let D be the difference of the forces in the orbit and in
the locus, and f the force in the circle K R, we have
D: f : : m n : z r
^^mkXms.rk'
m t * 2kc
(m r 4" r k) (m r — r k) , r k '
* * 2 k c • 2'kc
:: mr* — rk^ : rk'^
:: G«— F* : F«.
341. CoR. 2. In the ellipse *with the force in the focus, we have
F ' R G2 R F^
A«^ A^
For (C V being put = T)
V* v'*
Force at V in Ellipse : Do. in circle : : ; tttxt : t^, ,t /
'^ chord P V _r V
_ 1 1
• ' 2 R* 2T
::T: R
Also F in Circle :mnatV::F*:G«— F*
m n at V : m n at p : : Tpj : rj
. T? .T7 11 . TF^ RG« — RF«
.'. 1^ at V m ellipse : m n at p : : — j^
3
Hence
F'
we have
F*
F in ellipse at V = ™j
and
RG«— RF
m n =
and
X' = X + m n
F« , RG«— RF*
T^+ A^
see 834.
252 A COMMENTARY ON [Sect. IX.
OTHERWISE.
342. By 336,
But
X = ^.
and
P* V* L
= ~ /* = R f* (157)
g 2
345. Cor. 3. In the ellipse with the force in the center.
V, F«A , RG* — RF«
For here X a A and the force generally a p^ (140)
Force in ellipse at V : Force in circle at V : : T : R
I
Fin circle : m n at V ::F«:G* — F*
m n at V : m n at p : : ipg : ^3
r 11 .xr . F« ^ RG' — RF
.*. F m ellipse at V : m n at p : : «r3 • 1 ' T3 ■
F* A
.*. assuming F in ellipse at P = ^^  3 > we have
F in ellipse at V = ^3 X T
and
RG* — R F«
.. m n = ^^3
.•.X' a X 4 m n a, &c.
OTHERWISE.
„ ,P«V« 4 (Area of Ellipse)
344. X = /* p, and — = — ^ ,^^ ■ ./,
'' g g (Period)*
= Ata«b« (147)
g( Period)*
Book I.] NEWTON'S PRINCIPIA.
Therefore by 336
X' = /*g + ^a«b* X
S58
^ X I
 F^ ^ 1
F'S
G^ —
F2 1
b^ X
X gs
(G« — F
RG«
— RF*
}
F
345. Con. 4. Gena^ally let X he the force at P, V ttt^ at V, R the
radius of cui'vature m V, C V = T, &c. then
V RG^— VR F2
X'oc Xf
A^
For
{
F in orbit at V : F' in circle at V
F' : m n at V
m n at V : m n
.'. F in orbit at V : m n
.'. since by the assumption
T : R
F": G*^ — F«
A': T'
V F'' G^ F*
44 : VR.
A =
F in orbit at V =
VF'
m n
_ VR(G^ — F»)
and
A^
OTHERWISE.
This may better be done after 336, where it must be observed V is not
the same as the indeterminate quantity V in this corollary.
346. CoR. 5. The equation to the new orbit is (333)
P  F2g« + (G* — F=^)p'*
p' belonging to the given orbit.
Ex. 1. Let the given orbit be a common parabola.
Then
p' ^ = r f
.•.p« =
G^re'
F^^ + (G^— F*)f
and the new force is obtained from 336.
864 A COMMENTARY ON [Sect. IX.
Elx. 2. Let the given orbit be any one of Cotei SpiralS) ivhose general
equation is
b* P*
P a^ + s^'
Then the equation of 333 becomes
P' = rL2
— b* p*
gb^+a'^ — b« + j«
F
which being of the same form as the former shows the locus to be similar
in each case to the given spiral.
This is also evident from the law of force being in each case the same
(see 336) viz.
fi , P2V« _ G« — F* ^^ J
X' — 3 + g X Qi X ,3
1
Ex. 3. If the given orbit be a circle, the new one is also.
Ex. 4. Let the given trajectory be a straight line.
Here p' is constant. Therefore
V — Y^ G* F^
? H JM P
the equation to the elliptic spiral, &c. &c.
Ex. 5. Let the given orbit be a circle 'with the force in its circumference.
Here
P ~ 4r2
and we have from 333
G'g*
P*
4r«F*+ (G«— F*)g**
Ex. 6. Let the given orbit be an ellipse laith force in the focus.
Here
t
and this gives
*^ 2 a — g
F'g(2a — g)+ b*(G* F*)'
Book I.] NEWTON'S PRINCIPIA. $^
347. To find the points of contrary jlexurCi in the locus put
dp = 0;
and this gives in the case of the ellipse
_ b' F' — G»
^ ~ T' F^
OTHERWISE.
In passing from convex to concave towards the center, the force in the
locus must have changed signs. That is, at the point of contrary flexure,
the force equals nothing or in this same case
F' A + RG* — R F* =
'•• A =^, X(F«G»)
 k! F^ — G'
~" a • F* •
And generally by (336) we have in the case of a contrary flexure
pa V2 G'^ F* 1
which will give aU the points of that nature in the locus.
348. To find the "points 'where the locus and given Trajectory intersect
one another.
It is clear that at such points
g = g', and (J' = 2 W T + /J
W being any integer whatever. But
F
. . . __ 2 W*
m+ 1
This is independent of either the Trajectory or Locus.
349. To find the number of such intersections during an entire revolution
of C P.
Since 6 cannot be > 2 «•
W is < m + 1 and also < m — 1
.. 2 W is < 2 m.
2 G
Or the number required is the greatest integer in 2 m or p .
This is also independent of either Trajectory or Locus.
S56
A COMMENTARY ON
[Sect. IX.
350. To Jind the number and position of the double points of the Loctts,
i. e. of those points where it cuts or touches itself.
Having obtained the equation to the Locus find its singular points
whether double, triple, &c. by the usual methods; or more simply,
consider the double points which are owing to apses and pairs of equal
values of C P, one on one side of C V and the other on the other, thus :
The given Trajectory V W being
symmetrical on either side of V W, let
W be the point in the locus correspond
ing to W. Join C W and produce
it indefinitely both ways. Then it is
clear that W is an apse; also that the
angle subtended by V v' x' W is
5= = Xffrrwff+^VCy', w being
the greatest whole number in ^ (this
supposes the motion to be in consequentia). Hence it appears that where
ever the Locus cuts the line C W there is a double point or an apse, and
also that there are w + 1 such points.
pi
Ex. L Let =r = 2 ; i. e. let the orbit move in conse
quentia 'with a velocity = the velocity of C F. Then z,
V C y' = 0, y' coincides with V, and the double points
are y' V, x' and W.
The course of the Locus is indicated by the order of
the figures 1, 2, 3, 4.
Ex. 2. Let % =S.
F
Then the Locus resembles this figure, i, 2, 3,
4, 5, 6. showing the course of the curve in which
V, x', A, W are double points and also apses.
Ex. 3. Let ^ = 4. '
Tlien this figure sufficiently traces the Locus.
Its five double points, viz.
also apses.
V, x'. A, B, W are
G
Higher integer values of p will give the Locus
Book I.l
NEWTON'S PRINCIPIA.
257
still more complicated. If p be not integer, the
figure will be as in the first of this article, the
double points lying out of the line C V. More
over if ^r be less than 1, or if the orbit move in
F '
antecedentia this method must be somewhat
varied, but not greatly. These and other curio
sities hence deducible, we leave to the student.
351. To investigate the motion of (p) 'when the
ellipse, the force being in the focus, moves in ante
cedentia with a velocity = velocity of C V in
consequentia.
Since in this case
G =
., (333) also
P =
or the Locus is the straight line C V.
Also (342)
X' = ^ r^
R F'
f
= /i X
?~R
Hence
vdvcx X'dga
dg ■ Rd
2 I .3
2g.
R , 2g— 1
1 OC 2
e^ — e'
(where o^~^ = 1 and the body stops when
2g— 1 + e'' — A^ = 0,
or when
g = 1 + e.
Hence then the body moves in a straight line C V, the force increasing
3
to — of the latusrectum from the center, when it = max. Then it
4
decreases until the distance = — or R. Here the centrifugal force pre
vails, but the velocity being then = max. the body goes forward till tVie
Vol. I. R
268 A COMMENTARY ON [Sect. IX.
distance = the least distance when v = 0, and afterwards it is repelled
and so on in infinitum.
352. Tojind isolien the velocity in the Locus = max. or min.
Since in either case
d.v2 = 2vdv =
and
V d V = X' d f
.. X' =
.. (336)
Ex. In the ellipse with the force in the focus, we have (342)
.. ? = R X
pa
 b^ F' — G'
~ V ^ F^ •
b 2 L
If G = 0, V = max. when g = — o" > °^ when P is at the extie
a /i
mity of the latusrectum.
If F = 2 G, V = max. when ? = R . ^^^~;— = ?^ R = f 
' 1 (jr  4 8
lat. rectum.
353. To find nahen the force X' in the Locus = max. or min.
Put d X' = 0, which gives (see 336)
, ^ 3P^V^ ^ G' — F' 1
— g F* g*
Ex. In the ellipse
and (157)
x = A
= /i R
g
— 2F'dg 3RG*df— 3RF«dg _
which gives
r
3R F* — G

S = 2 X
Book I.] NEWTON'S PRINCIPIA. 259
Hence when
G =
X, 3 R
= max. when § = —^ .
When f = R, and G = 0. Then
R^ ~BJ' ~
When F = 2 G, or the eUipse moves m consequentia with ^ the velo
city of C p ; then ^
X = max. when
 i^ 4G^ — G^ _ _^ T?
^ ~ 2 • 4 G^  8
354. CoR. 6. Since the given trajectory is a straight line and the center
of force C not in it, this force cannot act at all upon the body, or (336)
X = 0.
Hence in this case
^, _ P^V^ ^ G^F^ 1
^  — g— X pi ^ p
where P = C V and V the given uniform velocity along V P.
In this case the Locus is found as in 346.
355. If the given Trajectory is a circle, it is clear that the Locus of p
is likewise a circle, the radiusvector being in both cases invariable.
356. Prop. XLV. The orbits (round the same center of force) acquire
the same form, if the centripetal forces by which they are desciibed at equal
altitudes be rendered proportional.']
Let f and f be two forces, then if at all equal altitudes
f a f
the orbits are of the same form.
For (46)
d«e 1 1
f a Tr? a T— ■„ a
and
QT^
a
1
d 6^'
1
a
1
d6'^
d^
a
d^.
dt^ dt'' S P'^ X QT^
1 _ 1
SP'^X d^^
R 2
860 A COMMENTARY ON [Sect. IX.
But they begin together and therefore
6 a: 6'
and
f = (.
Hence it is clear the orbits have the same form, and hence is also sug
gested the necessity for making the angles 6, ^ proportional.
Hence then X', and X being given, we can find ^ such as shall ren
der the Trajectory traced by p, very nearly a circle. This is done ap
proximately by considering the given fixed orbit nearly a circle, and
equating as in 336.
357. Ex. 1. Tojind the angle hetisoeen the apsides lahen X' is constant.
In this case (342)
X' a 1 a ^ a — ^__! .
Now making g = T — x, where x is indefinitely diminishable, and
equating, we have
(T — x)^ = F2T — F2x + RG^—RF^
= T3 — 3T2x43Tx2 — x^
and equating homologous terms (6)
T3=F2T+RG2_RF2=F2 X (T — R) + RG*
and
F*= ST"
G_2 _ T^ T — R
•*• F 2  R F 2 R
_ T^ T — R
~ 3 RT=^ R
_ _T T — R _ 3 R — 2T
3 R li 3 li
=r —nearly
3 •'
since R is = T nearly.
Hence when F = 180° = cr
r = G = ;3 . . . (I)
the angle between the apsides of the Locus in which the force is constant.
358. Ex. 2. Let X' a g''^. Then as before
(T — x)n = F^(T — x) + RG^ — RF*
and expanding and equating homologous terms
T° = F«T + RG' — RF*
Book I.] NEWTON'S PRINCIPIA. 261
and
But since T nearly = R
'J' n — 1 _. Q. 2
•*• F 2 ~ n
and when F = t
Thus when n — 3 = 1, we have
^=;^i = f = ««"
When n — 3 = — 1, n = 2, and 7 = ^ = 1270. 16'. 45'^
When n — 3 = — ^,n = 4 ,and7 = 2cr= 360°.
4 4
359. Let X' a ^^"^^^° . Then
b.(T — x)'« + c(T — xj'^zr F^(T— x)+ R.(G2 — F^)
and expanding and equating homologous terms we get
bT'" + cT« = F2(T — R) + RG^
and
bmT'»i + cnT"^ = F^.
But R being nearly = T, we have
b T'^ii cT"i = G^
G^ _ bT'^i + c T°' _ bT'^ + cT"
•*'F2~ bmT"»i + cnT°i "■mbT">±ncT''
which is more simply expressed by putting T = 1. Then we have
9l  b+c
F^ "" mb + nc
and when F = w •
r / b + c
' N m b + n c
360. CoR. 1. Given the l. between the apsides to Jind the force.
Let n : m : : 360° : 2 y
: : 180° =r ^ : 7
m
.". 7 = — «■
But if X' a gP
7 =
Vp
R3
262 A COMMENTARY ON [Sect. IX.
n
n2
Ex. 1. If n : m : : 1 : 1,
as in the ellipse about the focus.
2. If n : ra : : 363 : 360
X' a mS"
X'«l
■ 3
X'a gCl2i; ^
3. Ifn : m : : 1 : 2
1
X'«
And so on.
S
Again if X^ « — j
and the body having reached one apse can never reach another.
1
IfX'oc _ . ^
7 =
V — q
.*. the body never reaches another apse, and since the centrifugal force
^'f — 5 , if the body depart from an apse and centrifugal force be > centri
petal force, then centrifugal is always > centripetal force and the body
will continue to ascend in infinitum.
Again if at an apse the centrifugal be <^ the centripetal force, the centri
fugal is aWays < centripetal force and the body will descend to the center.
The same is true if X' a ^ and in all these cases, if
centrifugal = centripetal
the body describes a circle.
361. CoR. 2. First let us compare the force j^ — c A, belonging to
the moon's orbit, with
A'^ ■*■ A^"
Since the moon's apse proceeds, (n m) is positive.
Book I.] NEWTON'S PRINCIPIA. 263
.*. — c A does not correspond to n m and .•. ^ does not correspond
A.
Now
1 . A — cA* bA" — cA
__ c A « — A»  °^ A3
1— 4c „ F« „
.. X'a Al^^"^a Ag13
1 — 4c _ Fj
•*• 1 — 2 ~ G^
£! KG' — RF'' _ 1 —4c 3cR
•"'A'''*" A' ~ A' "^ A^
F« 1 —4c , 1
•*• A« "^ — AT^^ ^" A^
3c R
m n = ■ . „ .
and
Hence also
y =» /— — —  . &c. &c. &c.
' 'VI — 4 c.
362. To determine the angle between the apsides generally.
Let
x«£^ ' . . .
f (A) meaning any function whatever of A. Then for Trajectories which
are nearly circular, put
f(A) _ F'^ A + R.(G''— F')
A' ~ A'
.. f. A = F' A + R(G*— F^)
or
f.(T — x) =5 F2(T — X) + R(G' — F*)
But expanding f (T — x) by Maclaurin's Theorem (32)
u = f (T — x) =U — U'x + U""^— &c.
U, U' &c. being the values of u, t— , , — . &c.
° d x d x^
when X = 0, and therefore independent of x. Hence compaiing
homologous terms (6) we have
U = F'^T + R(G' — F')
U' = F''
R4
264 A COMMENTARY ON [Sect. IX.
Also since R = T nearly
U = TG^
G« U
F '^ ~ T . U'
Hence when F = ff, the angle between the apsides is
y = G = 'rJ,^jjA
or
making T = 1.
Ex. 1. Let f (A) = b A •" + c A " = u
Then
T— = mbA'»'+ncA'»"'.
d X
Hence since A = T when x =
U = f T = b T " + c T "
U'= mbT"' + ncT"i
G« bT'» + cT«
or
and
F^ "" mbT^+ncT"
GJ _ b+ c
F 2 ~ m b + n c
/ b+c
V m b + n
(1)
(2)
as in 359.
Ex. 2. Let f . (A) = b A •» + c A » + e A ' + &c.
.. ~ = mb A'^' + ncA"' + re A'' + &c.
d X
.. U = bT"* +cT"+eT^ + &c.
and «
T X U' = m b T "» + n c T " + r e T' + &c.
. G^ _ b T" + c T» + e T^ + &c.
•*• F^~mbT™+ncT"+reT^+&c.
or
when T = L
Also
— b + c+e + f+&c.
■~mb + nc+re + sf+ &c.
J
b + c + e . . .
ni b + n c + J' e +
Book I,] NEWTON'S PRINCIPIA. 865
A
Here (17)
^ = A«aA X (3+ Ala)
Hence
U = T=^aT X (3 + Tla)
T X U' = T3aT(3 + Tla)
G^ _ 1
F 2 ~ T X (3 + T 1 a)
and when T = I
21 i
F2~ 3 + la
Hence if a = e the hyperbolic base, since I e = 1, we have
Ex. 4. Let f (A) = e A = u.
Then
^  eA
dx  ^
.. U = e T
and
T.U' = TeT
••• p 2  T
.*. y = T.
f (A)
Ex. 5. Let ■ ). J = sin. A.
A^
and
u = f (A) = A^sin. A
.. U = T^sm. T
T^ = 3 A 2 sin. A + A 'cos. A
d X
T U' = 3 T^sin. T + T*cos. T
. 21 _ sm. T
'• F * ~ 3 sin. T + T cos. T
_ / sin. T
•*'^ ''V3sin.T + Tcos.T*
266 A COMMENTARY ON [Sect. IX.
IfT = J. Then
4
= 'V7^
^ + 7
363. To j^cfoe that
bA"+cA» 1 inb + nc_3
in = I — i — 'A i* + <=
A^ b + c
bA'° + cA" = b.(l — x)'" + c.(l — x)"
= b + c — (rab + nc)x+ &c.
mb+ nc
= bTT(>
b + c
mb + nc
+ &C.)
b + c
X(l — x) b + .
1 m b +_n c
A b + c^
b + c
364. To Jind the apsides when the excentricity is infinitely great.
Make
2 q : v^ (n + 1) : : velocity in the curve : velocity in tlie circle of the
same distance a.
Then (306) it easily appears that when F « ^n
n + 3
, q a 2 d ^
~ ^ V (a » + 1 — g » + 1 yp^ZT^slT" +^"(a«^^'*)
and
dp
d^
gives the equation to the apsides, viz.
(a» + i — ^n + i)^2_q2an + i (^2 — ^2) _
whose roots are
a (and — a when n is odd) and a positive and negative quantity (and when
n is odd another negative quantity).
Now when q =
(an + i — f " + ^)f'^ =
two of whose roots are 0, 0, and the roots abovementioned consequently
arise from q, which will be very small when q is.
Again since
1 5 ** 4 , „ 2
a^ + i ^^ i q  "
when q and ^ are both very small
t
Book I.] NEWTON'S PRINCIPIA. 267
and
^ = + a q.
.'. the lower apsidal distance is a q.
A nearer approximation is
S=±
aq
Hence
n + 3
rl^ qa ^ dg
^V(g2_a2q^ + /3) X Q
where jS contains q * &c. &c., and this must be integrated from g = b to
g = a (b = a q).
But since in the variation of g from b to c, Q may be considered con
stant, we get
p c
= sec.  '. ^ 4 C = sec. ~K — .
aq a q
and
7 = ^, g , — , &c. ultimately
the apsidal distances required.
Next let
1 fa" •
F«!and= — .
Then again, make
V : V in a circle of the same distance : : q V 2 : V {n — 1)
and we get (306)
ov/a"ig3_n — ^i — q2)g2 — a*q*
and for the apsidal distances
^^="1 + ' ^ n^^l .3_n — ^
an — 1 ' a" — "^ p3 — n
which gives (n > 1 and < 3)
2
f = a q 3 — u'
Hence
=/;
aqdg
= ^ . f ^^^^
VQv^y (p3n q2a^""
268 A COMMENTARY ON Sect. IX.
and
3 — n
y = 3=71 ^"^ ■ lE^ = 3=11 = 3=11' ^^
qa 2
Hence, the orbit being indefinitely excentric, when
F « g . ... we have . . . . y "=■ ^
for
JToe \ y=5^
any number < 1 '2
Fa1 y='
g ^ 2
1 — 1 ^ '"' ^
£ number between 1 and 2 * " * ' ' 2
F^pWs 7>*.
But by the principles of this 9th Section when the excentricity is inde
finitely small, and Fag"
^~ V (n + 3)
(see 358), and when
F a — .
y  V (3 — n)
Wherefore when n is > I
7 increases as the excentricity from
V (3 + n) ^^ "2 •
When F OC g
y = — is the same for all excentricities.
When F a g  «i
7 decreases as the excentricity increases from
It It
'/(3 — n) '"^ 2"
which is also true for Fa—.
i
Book i.j NEWTON'S PRINCIPIA. 269
WhenFot_J_
y decreases as the excentricity increases from
to
V (3 — n) 3
When F « Jr
i
1
When F a
g >2<3
7 increases with the excentricity from
to
V (3 — n) 3 — n *
If the above concise view of the method of finding the apsides in this
particular case, the opposite of the one in the text, should prove obscure ;
the student is referred to the original paper from which it is drawn, viz.
a very able one in the Cambridge Philosophical Transactions, Vol. I,
Part I, p. 179, by Mr. Whewell.
365. We shall terminate our remarks upon this Section by a brief dis
cussion of the general apsidal equations, or rather a recapitulation of re
sults — the details being developed in Leybourne's Mathematical Repository,
— by Mr. Dawson of Sedburgh.
It will have been seen that the equation of the apsides is of the form
x" — Ax*" — B = (1)
the equation of Limits to which is (see Wood's Algeb.)
nx^J — mAx™^ = (2)
and gives
(^AV
1
— m
If n and m are even and A positive, i; has two values, and the number
of real roots cannot exceed 4 in that case.
Multiply (1) by n and (2) by x and then we have
(m — n)Ax™ — nB =
which gives
/ B \ m"
(^)"(i)
and this will give two other limits if A, B be positive and m even ; and if
(1) have two real roots they must each =: x.
270 A COMMENTARY ON [Sect. X.
If m, n be even and B, A positive, there wiD be two pairs of equal roots.
Make them so and we get
(m — n)"". /n\ „
^ —^ A«— ( — ) «B"™ =
which will give the number of real roots.
(1). If n be even and B positive there are two real roots.
(2). If n be even, m odd, and B negative and (M), the coefficient to
A ", negative, there are two ; otherwise none. .
(3). If n, m, be even. A, B, negative, there are no real roots.
(4). If m, n be even, B negative, and A positive, and (M) positive there
are four real roots ; otherwise none.
(5). If m, n be odd, and (M) positive there will be three or one real.
(6). If m be even, n odd, and A, B have the same sign, there will be
but one.
(7). If m be even, n odd, and A, B have different signs, and M's sign
differs from B's, there will be three or only one.
(8). If
x° + An™ — B =
then
(^")
n — m
A°.
is positive, and it must be > B, and the whole must be positive.
If
x^ — Ax^^. B =
tlie result is negative.
SECTION X.
366. Prop. XL VI. The shortest line that can be drawn to a plane
from a given point is the perpendicular let fall upon it. For since
Q C S = right ^L, any line Q S which subtends it must be > than either
of the others in the same triangle, or S C is < than any other S C.
A familiar application of this proposition is this :
367. Let SQ be a sling with a body Q at the end of it^ and by the hand
S let it be whirled so as to describe a right cone whose altitude is S C, a7id
base the circle xsohose radius is Q C ; required the time of a revolution.
Let S C = h, S Q = 1, Q C = r = '^V — \\\
h
P = 2J^ (I)
Book I.] NEWTON'S PRINCIPIA. 271
Then if F denote the resolved part of the tension S Q in the direction
Q C, or that part which would cause the body to describe the circle P Q,
and gravity be denoted by 1, we have
F : 1 : : r : li
...F = ^.
But by 134, or Prop. IV,
g
the time required.
If the time of revolution (P) be obsetved, then h may be hence obtained.
If a body were to revolve round a circle in a paraboloidal surface, whose
axis is vertical, then the reaction of the surface in the direction of the
normal will correspond to the tension of the string, and the subnormal,
which is constant, will represent h. Consequently the times of all such
revolutions is constant for every such circle.
368. Prop. XLVII. When the excentricity of the ellipse is indefi
nitely diminished it becomes a straight line in the limit, &c. &c. &c.
369. Scholium. In these cases it is sufficient to consider the motion
in the generating curves.]
Since the surface is supposed perfectly smooth, whilst the body moves
through the generating curve, the surface, always in contact with the
body, may revolve about the axis of the curve with any velocity whatever,
without deranging in the least the motion of the body ; and thus by ad
justing the angular velocity of the surface, the body may be made to trace
any proposed path on the surface.
If the surface were not perfectly smooth the friction would give the
body a tangential velocity, and thence a centrifugal force, which would
cause a departure from both the curve and surface, unless opposed by
their material ; and even then in consequence of the resolved pressure a
rise or fall in the surface.
Hence it is clear that the time of describing any portion of a path in a
surface of revolution, is equal to the time of describing the corresponding
portion of the generating curve.
Thus when the force is in the center of a sphere, and whilst this force
causes the body to describe a fixed greatcircle, the sphere itsej^ revolves
with a uniform angular velocity, the path described by t^©: body on the
surface of the sphere will be the Spiral of Pappus. A" \
872 A COMMENTARY ON [Sect. X.
370. Prop. XL VIII and XLIX. Li the Epicycloid and Hypocycloid,
s: 2 vers. "I:: a(R + r) : R
•wJiere s is any arc of the curve, s,' the corresponding one of the wheel, and R
the radius of the globe and r that of the wheel, the + sign being used for
the former and — in the Hypocycloid. (See Jesuits' notes.)
OTHERWISE.
If p be the perpendicular let fall from C upon the tangent V P, we
have from similar triangles in the Epicycloid and Hypocycloid
PY:CB::VY:VC
or
^!_p8:R2 :: (R + 2r)' — p^: (R + 2r)2
which gives
Now from the incremental figure of a curve we have generally
d s g
But
er^
V(g« — p»)
R'
(R + Sr)*^ — R
(1)
(2)
, 2Vr2 + Rr^
...ds = — — X
X {(R±2r)^— g^^
V (R±2r)2 — g2
and integrating from
s = 0, when ^ = R
we get
s = g^r^±R^ X W(R±2r)^— R^— V(R±2r)^g'J
which is easily transformed to the proportion enunciated.
The subsequent propositions of this section shall now be headed by a
succinct view of the analytical method of treating the same subject.
371. Generally, A body being constrained to move along a given curve by
knffjon forces, required its velocity.
Let the body P move along the curve
P A, referred to the coordinates x, y
originating in A ; and let the forces be
resolved into others which shall act
parallel to x, y and call the respective
aggregates X, Y. Besides these we
have to consider the reaction (R) of the
Book L] NEWTON'S PRINCIPIA. 273
curve along the normal P K, which being resolved into the same .direc
tions gives (d s, being the element of the curve)
R r— , and R ^^ .
d s as
Hence the whole forces along x and y are (see 46)
d
d_«
d
Again, eliminating R, we get
j~ =^ ^ = 2Xdx + 2Ydy
and
dx*+dy'' ^..,xr I . ^r 1 \
ji— ^ =2y(Xdx + Ydy)
But
ds^_ dx' + dy
^ dt« dt« ^^^'
.•.v=' = 2/(Xdx + Ydy) (1)
Hence it appears that The velocity is independent of the reaction of the
curve.
372. If the force be constant and in parallel lintis, such as gravity, and
X be vertical ; then
X = g
and
Y =
and we have
v2 = 2/— gdx
= 2g(c— x)
= 2g(hx)
h being the value of x, when v = ; and the height from which it begins to
fall.
373. To determine the motiofi in a common cycloid, ixhen the force is gravity.
The equation to the curve A P is
'2r— X
dy = dx^:
X
r being the radius of the generating circle.
.•.ds=:dx^^
Vol. I. S
aT4 A. COMMENTARY ON [Sect. X.
and
dt =
ds / r d X
= ^ — X,
V2g.'v/(h — x) ^g V(hx — x^)
.•.t = C,^I'vers.^^ji^(86)
t being = 0, when x = h.
Hence the whole time of descent to the lowest point is
T ;r"
which also gives the time of an oscillation.
374. Required the time of an oscillation in a small circular' arc.
Here
y=v'(2rx— x^)
r being the radius of the circle, and
1 r d X
us
V (2rx— x^)
.. dt =
ds
V 2 g V (h 
x)
 V2 g "^
dx
V{\i
— x) (2 r X —
x^)}
r
X
V2g
dx
V{{h
^ X — X ^) (2 r
X)}
to integrate which,
put
' =
sin. ~ ^ ^ / (
^ h
>
.. d ^ =
dx
2 V rhx — x*)
and since
Jl =
sin. 6
X =
h sin. ^ ^, 2 r 
— X =
2 r — h sin. '
d
=
2r(l — a^sin. 2^),
5 '^ being put=
h
"2r
.. dt =
 /i^x
d^
V g V ( 1—3 'sin. 2^)*
Now since the circular arc is small, h is small ; and therefore 3 is so.
And by expanding the denominator we get
Book I.] NEWTON'S PRINCIPIA. 275
and integrating by parts or by the foi'mula
yd ^. sin. n»^ = COS. ^ sin. "»' ^ + ^^i^n_ fddsin.^^6
m m ^
and taking it from
^ = to ^ = J
2
we get
/ d d sin. "i 6 = ^~^ X ^ ^ sin, "^ 6
the accented^ denoting the Definite Integration from ^ = 0, totf= * .
In like manner
^ m — 2^
/; d ^ sin. «  2 ^ = l!^ r /; d tf sin. » * ^
and so on to
/d^sin.^5 = ^/d^^ 2
Hence
y;d^sin.^=(V^^("^^) 1x4
*^' m (ra — 2) 2 2
and
yd ^ ^ d =.
wTi — ^"~^~^' ^''''^
(1 — S^sin.^^
is the same as
:!}
V (1 —32 sin. '^ 6) from
whence then
and taking the first term only as an approximate value
. ' = Wi (')
. . r
which equals the time down a cycloidal arc whose radius is j.
If we take two terms we have
: Wi('+x)
= Wio + s^) ••••■••• w
S2
276 A COMMENTARY ON [Sect. X.
375. To determine the velocity and time in a Hypocycloid, the force
tending to the center of the globe and « ^.
By (370)
the equation to the Hypocycloid is
 R«_D«
by hypothesis.
Now calling the force tending to the center F, we have
X= — F X ,Y = — F x^
.■./(Xdx+Ydy)=/F "^" + y<^y
^/Fdf
.•.v» = C — 2/Fdf (1)
But by the supposition
F = t^s
.'.v' = /.{h' — s') (2)
Hence
V
—
V R^ — D*
RV/M ^
To integrate it, put
S^ — B' = u'
^"^^ du
D'
^ — u«
and
VU§'D«)(h^g')]
VR^ — D^ du
at = —
RVfx, /(h«— D*— u«)
Hence making g = D, we have
Oscill. cr /R2 — D
 2V RV ^^^
2
376. Since h does not enter the above expression the descents are
Isochronous.
We also have it in another form, viz.
T
2 ""V VR/tt R'fJ
Book L] • NEWTON'S PRINCIPIA. 277
IfRfjk = g or force of gravity and R be large compared with b,
T /r
2=Wg
the same^as in the common cycloid.
377. Required to ^nd the value of the reaction R, 'ivhen a body is con
strained to move along a given curve.
As before (46)
^^ = X + R^
dt' ^ dx
i!y = YR^.
dt
ds
Hence
dyd*x — dxd^y ^j ttj .tjj
—^ V2 ^ = Xdy — Ydx+Rds
.R_ Xdy — Ydx , dyd^x — dxd»y
ds
But if r be the radius of curvature, we have (74)
ds^
dyd*x — dxd^y *
dt^ds
r =
Hence
R _ Ydx— Xdy ,ds^
^ dl "^rdt*
Another expression is
_ Ydx— Xdy . v'
^ = dl ■*■ 7
or
_ Ydx— Xdy
 ds
f being the centrifugal force.
If the body be acted on by gravity only *
_gdy ds^
 ds "^rdt*
+ 9
(1)
(2)
R
or
or
 ds "^ r
_gdy
■" ds
+ P
(3)
If the body be moved by a constant force in the origin of x, y, we hav6
xri xri T^xdy — ydx
Ydx— Xdy= F ^ '
= Fed
e
S3
278
A COMMENTARY ON
[Sect. X.
for
xdy — ydx = f*d
or
or
• " ^ d s "^ r d t *
_ Fgd <? vj
 ds "•" r
Fgd ^
ds
+ f
(4^)
378. To Jind the tension of the string in the oscillation of a common
cycloid.
Here
but
^d s^ rdt'
and
d y _ 2 a — x
d s ~ V 23"
r = 2V2a's/(2a — x)
d s^
Jti = 2g(hx)
.••R = gV
2 a — x _^ g (h^— x)
2 a ' V2 a V (2 a — x)
_ 2 a + h — 2 X
When X = h
R =
• V(4a^ — 2ax)*
2a — h
V(2a — h)
When x =
;^* V (4.a* — 2ah) ~ ^ V (2 a) '
J, 2a + h/',h\
When moreover h = 2 a, the pressure at A the lowest point is = 2 g.
379. To Jind the tension ivhen the body oscillates in a circular arc by
gravity. •
Book L] NEWTON'S PRINCIPIA. 279
Here
dv  (^  ^) ^ ^
^ ~ V(2cx— X*)
J c d X
d s =
V (2 ex — x*^)
dj' _ c — X
d X c
r = c
d s'^
i— = 2 ff (h — x)
n c — x . 2 g (h — x)
° C C
When X =
= g
R = g
c + 2 h — 3 X
c + 2 h
= 3 g or h = c.
If it fall through the whole semicircle from the highest point
h = 2 c,
and
R = 5g
or the tension at the lowest point is five times the weight.
When this tension = 0,
c + 2 h — 3 X = 0, or X = ^ \^^ .
A body moving along a curve whose plane is vertical will quit it when
R =
that is when
c + 2 h
and then proceed to describe a parabola.
/ 380. To Jind the motion of a body upon a surface of revolution^ when
acted on by forces in a plane passing through the axis.
Referring the surface to three rectangular axes x, y, z, one of whicli (z)
is the axis of revolution, another is also situated in the plane of forces, and
the third perpendicular to the other two.
Let the forces which act in the plane be resolved into two, one parallel
to the axis of revolution Z, and the other F, into the direction of the
radiusvector, projected upon the plane perpendicular to this axis. Then,
S4
280
A COMMENTARY ON
[Sect. X.
calling this projected radius f, and resolving the reaction R (which also
takes place in the sanie plane as the forces) into the same directions, these
components are
a s
d s
supposing ds=:V(dz^ + df^) and the whole force in the direction
of f' is
d s
and resolving this again parallel to x and y, we have
d t^ V a sJ P
and
d^z
= — Z + R
d 1 2 "" " • *•' d s
Hence we get
X d' y — y d' x _ ^v __ i xd y — y d x
dt =
dt
and
dxd'x+dyd^y+dzd*z__ ^ xdx+ y dy
dl^  •
S
— Zdz
^ dz f xdx + y dy dgdz\
^•dll ~1 ds j
df
(1)
(2)
(3^
Which, since
X dx + y d y
i
,/dxHdy^+dx% ^_, ^_,
\ \i'[2 ) =— 2Fdf — 2Zdz.
Again
d z^ _ d z^ dg'
dt2dg**dt«'
and from the nature of the section of the surface made by a plane passing
through the axis and body, t— is known in terms of g. Let therefore
dz
.d7=P
Book I.] NEWTON'S PRINCIPI A. "281
and we have
d_z^ _ ^ dg'
d t* ~ P dt^ •
Also let the angle corresponding to f be ^, then
xdy — ydx = ^*d^ *^
and
dx^ + dy^ = di^ + g^ddS
and substituting the equations (2) and (3) become
d. 4^11 =
Integrating the first we have
g 2 d ^ = h d t
h being the arbitrary constant,
or
at = i^ • (4)
The second can be integrated when
— 2Fd^ — 2Zdz
is integrable. Now if for F, Z, z we substitute their values in terms of ^,
the expression will become a function of § and its integral will be also a
function of g. Let therefore
/(F d f + Z d z) = Q
and we get
dg' g'd^' dg' _ /
dT^ + "TT^ + PaF^2Q (5)
which gives, putting for d t its value
• >/Uc2Q)g^h^} ^^^
Hence also
d t  V(l +P^)»gdg
^^  V f(c — 2Q)g2 — h^j • • • • ; • • ^'^
If the force be always parallel to the axis, we have
F =
and if also Z be a constant force, or if
Z = g
we then have
Q = /Zdz It: gz . (8)
282 A COMMENTARY ON [Sect. X.
Z being to be expressed in terms of g.
381. Tojind under tohat circumstances a body will describe a circle on a
surface of revolution.
For this purpose it must always move in a plane perpendicular to the
axis of revolution ; p, z will be constant; also (Prop. IV)
I COS. ^ = X
_ d * X _ I COS. ^ d ^ 
•"• dT^ ~ d~P
Also
ed 6
V = ^—, —
d t
d * X _ V ^ COS. d
•'* dlF ~ ~i •
Hence as in the last art.
f ds f
ds
^ ...I'^F+Z^^ ........ (,)
If the force be gravity acting vertically along z, we have
Z = g
V* __ d z
Hence may be found the time of revolution of a Conical Pendulum,
(See also 367.)
382. To determine the motion of a body moving so as not to describe a
circle, when acted on by gravity.
Here
Q = gz
and
C — 2Q = 2g.(k — z)
k being an arbitrary quantity.
Also
g« = 2 r z — z*
z being measmed from the surface.
.. D(\g = (r — z) d z
and
r
1 + p2 = 1 +
(r _ z) '^  (r
s •
BookL] NEWTON'S PRINCIPI a. 283
Hence (390)
dt= ^{^ + P')X^S
Vj2g(k — z). (2 rz — z*) — h^ '
In order that
d t
the denominator of the above must be put = ; i. e.
2g(k — z)(2rz — z2)_h2 =
or
z^" — (k + 2r)z2 + 2krz — 2l = o
2g
which has two possible roots ; because as the body moves, it will reach
one highest and one lowest point, and therefore two places when
dz
di = »
Hence the equation has also a third root. Suppose these roots to be
S ^> 7
where a is the greatest value of z, and/3 the least, which occur during the
body's motion.
Hence
1 . rd^
 V(2g) V J(«z).{z^)(7z)
To integrate which let
^=:sm.^^3.
Then
dz
d^ =
2V(z/3).(a^)J^{l_^^}
 2V J(«  z) (2  /3)}
Also
sm. * d = 3
.. z = j8 + (a — i8) sin. * 6
and
■
y — z = 7 — J^ + (a — /3) sin. ' 6]
= (7 — ^) n — ^'sin.M^,
if
3 /"^
^Vy_(3
J284 A COMMENTARY ON [Sect. X.
. , 2r«d 6
"" V2 g . (7 — /3) . Vfl — a^ sin. 2 ^}
which is to be integrated from z = /8, to z = a ; that is from
tf = to tf = .
this expanded in the same way as in 374 gives
2r
t =
=Sx{'+(l)'(^)Hacc.};
V2g(7
which is the time of a whole oscillation from the least to the greatest
distance.
Also
, , h d t h dt
g^ 2 r z — z*
and 6 is hence known in teims of z.
383. A body acted on by gravity moroes on a surface of revolution xohose
axis is vertical : 'when its path is nearly circular^ it is requiied to find the
angle between the apsides of the path projected in the plane of'SL, y.
In this case
/ZdzrrgzrrQ
and if at an apse
g = a, z = k
we have
(C — 2gk)a* — h«=
Hence (380)
h'
•. C = ^, + 2 g k.
d,^^ V(J^+p«)hd^
^{2g(k_z).h^(V— ^)}
Let L = ± + «
S a
__ V (1 + pgjhdfe
^{2g(kz)h»(li)}
Book 1.] NEWTON'S PRINCIPIA. 285
^ _ 8g(kz)h'(l(^)
" d&^ ~ h2(l + pi)
It is requisite to express the righthand side of this equation in terms
of w
Now since at an apse we have
« = 0, z = k, and g = a
we have generally
,dz d^zw^
d u d w2 1.2
the values of the differential coefficients being taken for
w = (see 32)
And
dz = pdg = — pf'^dw
d^z = — 2pfdgdw — g*d«Mdp
or, making
dp = qdf
d^zzr — (2p + qg)^d^d«=(2p + qf)g»d«*.
And if P/ and q, be the values which p and q assume when « = 0,
f = a, we have for that case,
3^2= (2p,+ q,a)a'
Z = k— p,a«« + (2p + q,a) a^ ^ — &c.
e* \a / a' a ^
Hence
2g(kz)h'(il)
becomes
' 2 g (p, a* «  (2 p,+ q, a) a^^ + &c.)h«(^ + a,*).
But when a body moves in a circle of radius = a, we have
h2 = ggSp _ ga'p,
in this case. And when the body moves nearly in a circle, h * will have
nearly this value. If we put
h« = (1 + a)ga^p,
we shall finally have to put
5 =
280 A COMMENTARY ON [Sect. X.
•in order to get the ultimate angle when the orbit becomes mdefinitely near
a circle. Hence we may put
h^ = ga>,
and
2g{kz)_h(ll)
becomes
 {3ga3p, + ga*q,]a>2 + &c.
in which the higher powers of a may be neglected in comparison of w * ;
. d^ _ _ ga^(3p, + q,a)u>' _ — (3 p, + g, a) a; ^
"dd'2 h^l+P') ~ P. (l+P')
_ (3p,+ q.a)a,^
P/(l+P/)
again omitting powers above « ^ : for p = p; + A w + &C'
Differentiate and divide by 2 d w, and we have
^L" =  ^P'ig^. . = _ N «
d <j»  p, (1 + p/)
suppose; of which the integral is taken so that
^ = 0, when w =
is
w = C sin. 6 V N.
And w passes from to its greatest value, and consequently § passes
from the value a. to another maximum or minimum, while the arc ^ V N
passes from to sr. Hence, for the angle A between the apsides we have
AVN=i«orA=,T,T
V N
where
_ 3 p, + q, a
~R(1+P/)
384. Let the surface he a sphere and let the path described be iiearly a
circle ; to Jind the horizontal angle between the apsides.
Supposing the origin to be at the lowest point of the surface, we have
z = r — • V (r 2 — g ^)
_ d z _ f
P ~ d7 ~ vTT^ — fO
_ d p _ r'
__ a
•*• P' ~ V (r* — a~«)
Book I.l NEWTON'S PRINCIPIA. 287
9/ = 3.
(r^ — a^)^
. ^ 4 r 2 _ 3 ca 2
.'. N = .
Hence the angle between the apsides is
A   "^
/(4r2_3a2)'
The motion of a point on a spherical surface is manifestly the same as
the motion of a simple pendulum or heavy body, suspended by an inex
tensible string from a fixed point ; the body being considered as a point
and the string without weight. If the pendulum begin to move in a ver
tical plane, it will go on oscillating in the same plane in the manner al
ready considered. But if the pendulum have any lateral motion it will
go on revolving about the lowest point, and generally alternately ap
proaching to it, and receding from it. By a proper adjustment of the velocity
and direction it may describe a circle (134) ; and if the velocity when it
is moving parallel to the horizon be nearly equal to the velocity in a cir
cle, it will describe a curve little differing from a circle. In this case we
can find the angle between the greatest and least distances, by the for
mula just deduced.
Since
w r
A =
V (4 r 2 — 3 a 2)
if a = 0, A = ^ 5 the apsides are 90° from each other, which also ap
pears from observing that when the amplitude of the pendulum's revolu
tion is very small, the force is nearly as the distance ; and the body de
scribes ellipses nearly ; of which the lowest point is the center.
If a = r,
A = ^ = 180°
this is when the pendulum string is horizontal ; and requires an infinite
velocity.
If a =  ; so that the string is inclined 30° to the vertical ;
2
A = ~— = 99° 50'
V 13
288 A COMMENTARY ON [Sect. X
If a ' = — ; so that the string is inclined 45° to the vertical ; •
A = ,^ =n3°.S6'. I
3 r '
If a ' = J— ; so that the string is inclined 60° to the vertical ;
2 V
A = —^ = 136° nearly.
385. Let the surface be an inverted cone, with its axis vertical : to find
the horizontal angle between the apsides when the orbit is nearly a circle.
Let r be the radius of the circle and y the angle which the slant side
makes with the horizon. Then
z = g tan. y
p = tan. 7
q =
T., 3 tan. 7
N = : ^5 = 3 cos. ^ y
tan. y. sec. ^ y '
and
A =
cos. y V 3 *
If 7 = 60°
A = ^^ = 120°.
386. Let the surface be an inverted paraboloid whose parameter is c.
f* = cz
d z 2 e
*^ dg c
2
^= c
6 a 2a
VT c c 4 c
.•. N =
2a/ . 4a \ c*+4a
^a + V)
2 *
c
If a = Q > or the body revolve at the extremity of the focal ordinate,
to
N = 2
and
A  —
^ V2
Book I] NEWTON'S PRINCIPIA. 281)
387. When a body moves on a conical surface^ acted on by a force tend
ing to the vertex ; its motion in the surface 'will be the same, as if the sur
face "were un'wrapped, and made plane, the force remaining at the vertex.
Measuring the radiusvector (g) from the vertex, let the force be F,
and the angle which the slant side makes with the base = y : then
z = g tan. y
p = tan. y
1 + p '^ = sec. ^ y
also
Q=/(Fd^ + Zdz) =/Fdg'.
Hence (380)
or putting
and
we have
, __ sec, y h d g
" i V \{C2f¥'d^')em
h' cos. y for h
d ^ sec. 7 for d ^
^ COS. 7 for g
g' VJ(C— 2/F'dg')g'2 — h'^r
Now d ^ is the differential of the angle described along the conical sur
face, and it appears that the relation between ^ and g' will be the same as
in a plane, where a body is acted upon by a central force F. For in that
case we have
and integrating
h'^dg'^ h
/2
J 4
d^'
+ ^ =C2/Fdg'
which agrees with the equation just found.
388. JVJien a body moves on a surface of revolution, to find the reac
tion R.
Take the three original equations (380) and multiply them by x d z,
y d z, g d g ; and the two first become
xd^xdz F'xMz p dz' x'
d t* g ds ' g
i*ydz_ F'y^dz r^Iz" y
d t^ — e "dT'T
Vol. 1.
290 A COMMENTARY ON [Sect. X.
add these, observing that
and we have
(xd'x.fyd'y)dz^__^,^^^_^ dz^^
d t* ' ^ ds
Also the third is
d t* 5 5 ^ 5 d s
Subtract this, observing that dz* + dg* = ds*, and we liave
(xd'x + yd^y) dz — gdgd'z _
dT^ "■
f (Z d g — F' d z) — R g d s.
But
x^ + y« = S'
xdx+ydy = gdg
xd*x + yd*y + dx2+dy2 = gd''g + df*.
Hence
(dg' — dx" — dy') dz g d z d" g — g dgd^z _
dt^ "*" dt^ ^
g (Z d g — F' d z) — R g d s
and
dg2 = d s'' — d z*.
Hence
P __ Z d g — F d z dgd'^z— dzd^g
ds "*■ dt^ds
(dx^ + dy^ + dz^— ds') dz
■*■ gdt'ds
Now if r be the radius of curvature, we have (74)
_ d s^
~ dgd'^z — dzd'^g
and
d x« + dy«4 dz* = d tf«
a being the arc described.
Hence
P _ Z d g  F d z ds«
^ ~ dl + rdt«
d g' — d s' d z . .
+ ■ ^n^ •d's ^*''
Here it is manifest that
ds«
are
Book I.] NEWTON'S PRINCIPIA. 291
is the square of the velocity resolved into the generating curve, and that
da — d s 
dT^
is the square of the velocity resolved perpendicular to §. Tlie two last
terms which involve these quantities, form that part of the resistance
which is due to the centrifugal force ; the first term is that which arises
from the resolved part of the forces.
From this expression we know the value of R ; for we have, as before
^^; = C2/(Fd^ + Zdz).
Also
d_(j= — d s = _ gMr  _ hj
dt* ' ~ dt^ " §•'
Hence
^'=C2/(Fdg+Zdz)
889. To find the tension of a pendulum moving in a spherical surface.
C2/(Fdg + Zdz) = 2g(kz)
1= V (2rz — z^)
d _ r — z
d~z ~ V (2rz — z^)
d s _ r
dg ~ r — z
d s _ r r
cTz ~ V(2rz— z^) ~ Y '
Hence
, . 2g(k — z) z I J
R = glLl^) . __L\i_ !i_ . _L
r r ?'* 
_ g(r+2k — .3z)
r
and hence it is the same as that of the pendulum oscillating in a vertical
plane with the same velocity at the same distances.
390. To find the Velocity^ Reaction^ and Motion of a body upon any
surface xvhatevei:
Let R be the reaction of the surface, which is in the direction of a nor
mal lo it at each point. Also let «, «', t" be the angles which this normal
T2
292
A COMMENTARY ON
[Sect. X.
makes with the axes of x, y, z respectively ; we shall then have, consider
ing the resolved parts of R among the forces which act on the point
d*x
T— 2 = X + R cos. i
d^v
^ = Y+R.cos..'
^,= Z+R.cos..
Now the nature of the surface is expressed by an equation between
X, y, z : and if we suppose that we have deduced from this equation
dz =pdx + qdy
. dz , d z
where p = ^p and q = ^ — 
^ dx ^ dy
p and q being taken on the supposition of y and x being constants respec
tively ; we have for the equations to the normal of the points whose co
ordinates are
X, y, z
x' — X 4 p (z' — z) = 01
y' — y + q(z' — z) = 0i
x', y', z' being coordinates to any point in the n
No. 143.)
Hence it appears that if P K be the normal,
P G, P H its projections on planes parallel to
orm
z
1
al (see
1
Lacr<
^ix,
p
X z, y z respectively.
yl
^
^
The equation of P G is
x' — X + p . (z' — z) = 0,
'
i/
si
N
and hence
ly
X
GN+pPN=0 ^
and
GN = — p.PN.
Similarly the equation of P H is
/ — y + q(z'— z) =
wheDce
HN+q.PN =
HN = — q.PN.
And hence,
cos. s = cos. K P h = ^^
GN
C
I
1
V(PN«+NG=+ HN«)
Book I.] NEWTON'S PRINCIPIA. 2Sf8
_ P
COS. i' = COS. K P g = p .°
HN
V(PN* + NG'^.f HN«)
q
V(l+p« + q^)
Whence, since
COS. ^ « + COS. ^ g' + COS. ' s" = 1
COS. '^ ?" = V (1 COS. ^ f COS. ® ?')
1
V(l+p2 + q2)
Substituting these values ; multiplying by d x, d y, d z respectively, iu
the three equations ; and observing that
dz — pdx — qdy =
we have
!l^iii + ili;y±ii^^ = X d X + Y d y + Z d z
and integrating
dx'+dy* + dz* ^ ^ ,^ ■, tt, r, i ^
^l^^ = 2/(Xdx+ Ydy+ Zdz)
and if this can be integrated, we have the velocity.
If we take the three original equations, and multiply them respectively
by — p, — q, and 1, and then add, we obtain
d'^x d*y d'^z __
~" P dF* — ^ • dT^ ■*■ dT« ~
— pX — qY+ Z + R V(l + p^+q^).
But
dz = pdx + qdy.
Hence
d^z _ d^x d'^y dpdx + dq d y
dF ~ P d~r + ^dl^ ■*" dF •
Substituting this on the first side of the above equation, and taking
the value of R, we find
_ pX + qY — Z dpdx + dqd y
^ V(l+p^4 q=)f dtW(Hp'^+q*)
If m the three original equations we eliminate R, we find two second
differential equations, involving the known forces
X,Y, Z
T3
iil>4 A COMMENTARY ON [Sect. X.
and p, q, which are also known when the surface is known, combining
with these the equation to the surface, by which z is known in terms of
X, y, we have equations from which we can find the relation between tlie
time and the three coordinates.
391. To find the path "johich a body raill describe upon a given surface^
'jchen acted upon by no force.
In this case we must make
X, Y, Z each = 0.
Then, if we multiply the three equations of the last art. respectively by
— (qdz + dy), pdz + dx, qdx — pdy
and add them, we find,
— (qdz + dy)d«x + (pdz+dx)d«y+ (qdx — pdy)d*z
/— (q dz + dy) cos. e ^
= Rdt2+(pdz+dx) COS. i' \
(.4. (qdx — pdy) cos. ^'J
or putting for cos. e, cos. «', cos. il' their values
■p J i 2
Hence, for the curve described in this case, we have
(pdz + dx)d2y = (pdy — qdx)d2z+(qdz+dy)d'^x.
This equation expresses a relation between x, y, z, without any regard
to the time. Hence, we may suppose x the independent variable, and
d * X r= ; whence we have
(pdz4dx)d*y = (pdy — qdx)d*z.
This equation, combined with
dzrrpdx + qdy,
gives the curve described, where the body is left to itself, and moves along
the surface.
The curve thus described is the shortest line which can be drawTi from
one of its points to another, upon the surface.
The velocity is constant as appears from the equation
v« = 2/(X d X + Y d y + Z d z).
By methods somewhat similar we might determine the motion of a point
upon a given curve of double curvature, or such as lies not in one plane
when acted upon by given forces.
392. To find the curve qfi equal pressure, or that on 'which a body descend'
infr by the farce of gravity, pesses equally at all points.
Book I.]
NEWTON'S PRINCIPJA.
(1)
But if H M be the height due to the velocity at P,
A H = h, we have
ds°
dt'
= 2g(h — x).
Also, if we suppose d s constant, we have (74)
d s d X
Let A M be the vertical abscissa = x, M P the hori
zontal ordinate = y ; the arc of the curve s, the time t, g
and the radius of curvature at P = r, r being positive /^T
when the curve is concave to the axis ; then R being the ^"""'""/
reaction at P, we have by what has preceded. V
R = Sdj:+ ds^ . ....
d s r dt^
295
H
M
and if the constant value of R be k, equation ( 1 ) becomes
k ^gdy 2g(hx)d^y
d s d s d x
h_ dx __^/i^ \ ^^y dy ^^
g •2'V7h — x) ^ ^^~''^^'l^~Ts'2V (h^l^
The righthand side is obviously the differential of
V(hK),
hence, integrating
. V{hx) = V(hx).^ + C,
dy ^ k C
d s g V {h — x)
If C = 0, the curve becomes a straight line inclined to the horizoii,
. . . . k
which obviously answers the condition. The sine of inclination is — .
•" a
o
In other cases the curve is found by equation (2), putting
V(dx2+dy2) for ds
and integrating.
If we differentiate equation (2), d s being constant, we have
(3)
ds
y_
Cdx
2 (h — x) 2
, dsdx _ 2(h — x)^
(3)
d^y " C
And if C be positive, r is positive, and the curve is concave to the axis.
T4
296
A COMMENTARY ON
[Sect. X.
We have the curve parallel to the axis, as at C, when j^ = 0, that is,
us '
C
when —
g
V (h
X)
X =:h
; when
When X increases beyond this, the curve approaches the axis, and j^
is negative ; it can never become < — 1 ; hence B the limit of x is
found by making
g V(h_x)
or
If k be < g, as the curve descends towards Z, it approximates perpe
k
tuallv to the inclination, the sine of which is — .
g
If k be > g there will be a point at which the curve becomes horizontal.
C is known from (2), (3), if we knew the pressure or the radius of cur
vature at a given point.
If C be negative, the curve is convex to the axis. In this case the part
of the pressure arising from centrifugal force diminishes the part arising
from gravity, and k must be less than g.
393. Tojind the curve *which cuts a given assemblage of ctirves, so as to
make them Synchronous, or descriptible by the force of gravity in the same
time.
Let A P, A P', A P", &c. be curves of the
same kind, referred to a common base A D,
and differing only in their parameters, (or the
constants in their equations, such as the radius
of a circle, the axes of an ellipse, &c.)
Let the vertical A M = x, M P (horizontal)
r= y ; y and x being connected by an equation
involving a. The time down A P is
/dx
V(2gx)
the integral being taken between
X = and X = A M ;
and this must be the same for all curves, whatever (a) may be.
Book L] NEWTON'S PRINCIPIA.
Hence, we may put
/V{2gx)=^ <^^
k being a constant quantity, and in differentiating, we must suppose (a)
variable as well as x and s.
Let
d s = pd X
p being a function of x, and a which will be of dimensions, because d x,
and d s are quantities of the same dimensions. Hence
/  Pdx _,
'V(2gx)^
and differentiating
Pf "" , +qda = (2)
V(2 gx) ^ ^ ^ '
Now, since p is of dimensions in x, and a, it is easily seen that
/pdx
is a function whose dimensions in x and a are ^, because the dimensions
of an expression are increased by 1 in integrating. Hence by a known
property of homogeneous functions, we have
px
+ qa = ^ k;
V(2gx)
k X) V X
.. q =
2a aV(2g)
substituting this in equation (2) it becomes
pdx kdapdaVx _ ._.
V (2 g x) "^ 2 a a V (2 g) ~ '' " ' ^^
in which, if we put for (a) its value in x and y, we have an equation to the
curve P F F'.
If the given time (k) be that of falling down a vertical height (h), we
have
'2h
g
and hence, equation (3) becomes
p (a d x — X d a) + d a V (h x) = . . . . (4}
Ex. Let the curves A P, A P', A P'' be all cycloids of iiohich the bases
coincide 'with A D.
Let C D be the axis of any one of these cycloids and = 2 a, a being
the radius of the generating circle. If C N = x', we shall have as before
— ds zr dx' / —J
'S x'
=v=
298 A COMMENTARY ON [Sect. X.
and since
x' = 2 a — X
ds = dx ^ ./
2a
Hence
N 2a — X*
 V 2a — x'
P
and equation (4) becomes
^^^^^E^ + ^^i^) = .... (5)
Let — = u
a
so that
adx — xda = a^du
X = au ;
and substituting
a«du V 2
V(2 — u)
+ dav'(hau) =
du V2 da Vh
T 3 — = "
•• V(2u — u'')
•. V2x vers.»u— 2^ ^ =: C (6)
When a is infinite, the portion A P of the cycloid becomes a vertical
line, and
X = h, .. u = 0, .. C = 0.
Hence
 rrvers. /— . . (7)
a N a
From this equation (a) should be eliminated by the equation to tlie
cycloid, which is
y = avers.' — — V(2ax — x'^) .... (8)
Si
and we should have the equation to the curve required.
Substituting in (8) from (7), we have
y= V (2ah) — V (2ax — x2)
J _dav^h xda + adx — xdx
^ ~ V~(2a) V {2ax — x«)
and eliminating d a by (5)
d y _ 2a — x __ 2a — x
dx"~ v'(2ax — x*)~ ^ x
Book I.]
NEWTON'S PRINCIPIA.
299
But differentiating (8) supposing (a) constant, we have in the cycloid
dy= / .
y /S 2 a — X
And hence (31) the curve P P' F" cuts the cycloids all at right angles,
the subnormal of the former coinciding with the subtangent of the latter,
each being
2 a — X
y
/:
AOC
The curve P P' F" will meet A D in the point B, such that the given
time is that of describing the whole cycloid A B. It will meet the vertical
line in E, so that the body falls through A E in the given time.
394. If instead of supposing all the cycloids
to meet in the point A, we suppose them all to
pass through any point C, their bases still being
in the same line A D ; a curve P P' drawn so
that the times down PC, P C, &c. are all
equal, will cut ail the cycloids at right angles.
This may easily be demonstrated.
39.5. Tojind Tautochronoiis curves or those down which to a given Jixed
point a body descending all distances shall move in the same time.
(1) let the force be constant and act in parallel lines.
Let A the lowest point be the fixed point, D that
from which the body falls, A B vertical, B D, M P
horizontal. A M = x, A P = s, A B = h, and the
constant force = g.
Then the velocity at P is
V = V (2 g . h — x)
and
, As _ — ds
V~ V2g^/(h — x)
and the whole time of descent will be found by integrating this from
X rr h, to X = 0.
Now, since the time is to be the same, from whatever point D the body
falls, that is whatever be h, the integral just mentioned, taken between the
limits, must be independent of h. That is, if we take the integral so as
to vanish when
X =
and then put h for x, h will disappear altogether from the result. This
must manifestly arise from its being possible to put the result in a form
300 ■ A COMMENTARY ON [Sect. X.
involving only p , as fj , &c. ; that is from its being of dimensions in
X and h.
Let
ds = p dx
where p depends only on the curve, and does not involve h. Then, we
have
_ p pdx
^V{2g(h— x)}
1 ^rpdx 1 pxdx 1.3 pxnix ■)
~ ^{2gK I h^ 2 • 1^1 ^2.4 i^f
nnd from what has been said, it is evident, that each of the quantities
/ »p d x rpxdx /»px°dx
h^ h^ h2
must be of the form
that is
hence
or if
2n + l
C X 2
in + l '*
hi~
8n + l
yp x"d X must = cx 2 ;
p x " d X = ^ — ex ii d X ;
2n + 1 c
P = 2 '"l'
2n + 1 i
c =r a
2
P
=V^
and
which is a property of the cycloid.
Without expanding, the thing may thus be proved. If p be a function
of m dimensions in x, ■ ; ,. r is of m — \ dimensions ; and as the
' V (h — x) ^
dimensions of an expression are increased by 1 in integrating
y V (h — x)
Book I.]
NEWTON'S PKINCIPIA.
301
is of m + 1 dimensions in x, and when h is put for x, of m + ^ dimen
sions in h. But it ought to be independent of h or of dimensions
Hence
m+i =
.. p = a
2 V fi
as before. . "•
396. (2) Let the force tend to a center and vary as ant/ Junction of the
distance. Required the Tautochronous Curve.
Let S be the center of force, A the point to
which the body must descend ; D the point from
which it descends. Let also
SA = e, SD = f, SP=^, AP = s
P being any point whatever.
Now we have
v^= C — 2/Fdf
or if
2Pd^=f (g)
v' = f(f) — ?>(g)
the velocity being when g — f.
Hence the time of describing D A is
t=/
V(?>f— pg)
taken from g = f, to ^ = e. And since the time must be the same what
ever is D, the integral so taken must be independent of f.
Let
9 S — 9^ = z
^f — pe = h
d s = p d z
p depending on the nature of the curve, and not involving f. •
, from z = h to z =
Then
/p d z
V fh — z)
V (h
/p dz
V (h — z)
from z = to z = h.
And this must be independent of f, and therefore of <p f, and of h«
Hence, after taking the integral the result must be when z = 0, and
independent of h, when h is put for z. Therefore it must be of dimen
sions in z and h. But if p be of n dimensions in z, or if
p = cz°
P
V(h — z)
will be of n — ^ dimensions,
302 A COMMENTARY ON [Sect. X.
and
/,^, r of n 4 I dimensions.
V {h — z) . ^ ^
Hence, n + ^ = 0, n = — ^5 and
Therefore
ds = dz / — =p'e^eu 7;
whence the curve is known.
If 6 be the angle A S O, we have
ds^zz dg2 + ?'d^*
and
whence may be found a polar equation to the curve.
397. Ex. 1. Let the force vary as the distance, and be attractive.
Then
F = /(A g, p g = /i g '^ ;
z = fs — (pe = /A(g2_e');
dz = 2^0^ d^
d s
when P = e, J— is infinite or the curve is perpendicular to S A at A.
If S Y, perpendicular upon the tangent P Y, be called p, we have
p* _ d s' — dg'
P ~ ds"2
 ds^
= 1
4c/ig*
e^ — (1 — 4c/ti)g2
p* = ^^j .
If e = 0, or the body descend to the center, this gives the logarithmic
spiral.
In other cases let
e*
1 — 4c^ = ^,
Book I.]
NEWTON'S PRINCIPIA.
308
.•. 4lC fL ZZ
and
the equation to the Hypocycloid (370)
If 4 c ti = 1, the cuiTe becomes a straight line, to which S A is per
pendicular at A.
If 4 c i«. be > 1 the curve will be concave to the center and go off to
infinity.
398. Ex. 2. Let the force vary inversely as the square of the distance.
Then
F =
and as before we shall find
r!»
V' = i'~
? Mf  e)
2 /i c e
399. A body being acted upon by a fmce in parallel lines^ in its descent
from one point to another, to find the Brachystochron, or the curve of' quick
est descent between them.
Let A, B be the given points, and A O P Q B
the required curve. Since the time down
A O P Q B is less than down any other curve, if
we take another as A O p Q B, which coincides
with the former, except for the arc O P Q, we
shall have
Time down A O : T. O P Q + T. Q B, less than
Time down A 0+ T. O p Q + T. Q B
and if the times down Q B be the same on the two suppositions, we shall
have
T. O P Q less than the time down any other arc O p Q.
The times down Q B will be the same in the two cases if the velocity
at Q be the same. But we know that the velocity acquired at Q is the
same, whether the body descend down
A O P Q, or A O p Q.
Hence it appears that if the time down A O P Q B 6^ a minimum, the
time down any portion O V Q,is also a minimum.
304 A COMMENTARY ON [Sect. X.
Let a vertical line of abscissas be taken in the direction of the force;
and perpendicular ordinates, O L, P M, Q N be drawn, it being sup
posed that
L M = M N.
Then, if L M, M N be taken indefinitely small, we may consider them
as representing the differential of x : On this supposition, O P, P Q, will
represent the differentials of the curve, and the velocity may be supposed
constant in O P, and in P Q. Let
AL = x, LO = y, OA = s,
and let d X, d y, d s be the differentials of the abscissa, ordinate, and
curve at Q, and v the velocity there ; and d x', d y', d s', v' be the cor
responding quantities at P. Hence the time of describing O P Q will
be (46)
d s d s'
which is a minimum ; and consequently its differential = 0. This dif
ferential is that which arises from supposing P to assume any position as
p out of the curve O P Q; and as the differentials indicated by d arise
from supposing P to vary its position along the curve O P Q, we shall
use 8 to indicate the differentiation, on hypothesis of passing from one
curve to another, or the variations of the quantities to which it is
prefixed.
We shall also suppose p to be in the line M P, so that d x is not sup
posed to vary. These considerations being introduced, we may pro
ceed thus,
Hv + ^'} = ° (')
And V, v' are the same whether we take O P Q, or O p Q ; for the
velocity at p = velocity at P. Hence
d V = 0, 3 v' =
and
ads 8 d s^ __
V + v' ~
Now
Similarly
d s=^ = d x* + dy«
.•. ds3ds = dy3dy,
(for 3 d X = 0).
d s' 3 d s' = d y' a d y .
Book L] NEWTON'S PRINCIPIA. 305
Substituting the value of 3 d s, a d s' which these equations give,
we have
dy3dy dyady ' _
vds ■*■ v'ds' ~
And since the points O, Q, remain fixed during the variation of P's
position, we have
d y + d y' = const,
a d y' = — 3 d y.
Substituting, and omitting 3 d y,
dy _ d/ ^0,
vds v' d s'
Or, since the two terms belong to the successive points O, P, their
difference will be the differential indicated by d ; hence,
vds
.. ^ = const. . (2)
vds
Which is the property of the curve ; and v being known in terms of x,
we may determine its nature.
Let the force be gravity ; then
v= V(2gx);
dy
ds \^ (2gx)
Ay 
const.
ds \/ X V a
a being a constant.
. ly _ II
•* ds ~ V a
which is a property of the cycloid, of which the axis is parallel to x,
and of which the base passes through the point from which the body
falls.
If the body fall from a given point to another given point, setting off
with the velocity acquired down a given height; the curve of quickest
descent is a cycloid, of which the base coincides with the horizontal line,
fi'om which the body acquires its velocity.
400. If a body he acted on by gravity, the curve of its quickest descent
from a given point to a given curves cuts the latter at right angles.
Let A be the given point, and B M the given curve; A B the curve of
quickest descent cuts B M at right angles.
Voi,. I. u
306 A COMMENTARY ON [Sect. X.
It is manifest the curve A B must be a cycloid, for
otherwise a cycloid might be drawn from A to B, in ^
which the descent would be shorter. If possible, let
A Q be the cycloid of quickest descent, the angle
A Q B being acute. Draw another cycloid A P, and
let P P' be the curve which cuts A P, A Q so as to
make the arcs A P, A P' synchronous. Then (394) P P'
is perpendicular to A Q, and therefore manifestly P' is
between A and Q, and the time down A P is less than the time down
A Q ; therefore, this latter is not the curve of quickest descent. Hence,
if A Q be not perpendicular to B M, it is not the curve of quickest
descent.
The cycloid which is perpendicular to B M may be the cycloid of
longest descent from A to B M.
401. If a body be acted on by gravity ^ and if A "Q be the
curve of quickest descent from the curve A L to the point B ;
A T, the tangent of A 1^ at A, is parallel /o B V, a peipen
dicidar to the curve A B a^ B.
If B V be not parallel to A T, draw B X parallel to
A T, and falling between B V and A. In the curve A L
take a point a near to A. Let a B be the cycloid of quick
est descent from the point a to the point B ; and B b being
taken equal and parallel to a A, let A b be a cycloid equal
and similar to a B. Since A B V is a right angle, the
curve B P, which cuts off A P synchronous to A B, has B V for a tan
gent. Also, ultimately A a coincides with A T, and therefore B b with
B X. Hence B is between A and P. Hence, the time down A b is less
than the time down A P, and therefore, than that down A B. And
hence the time down a B (which is the same as that down A b) is less
than that down A B. Hence, if B V be not parallel to A T, A B is not
the line of quickest descent from A L to B.
402. Supposing a body to be acted on by any forces whatever, to determine
the Brachystochron.
Making the same notations and suppositions as before, A L, L O, (see
a preceding figure) being any rectangular coordinates ; since, as before,
the time down O P Q is a minimum, we have
^{t + v'}=» <•'
Book I.] NEWTON'S PRINCIPIA. 307
ads ads^ ds_3v_dVa_v^_ Q
y y' y 2 y' 2
Now as before we also have
ad s = —^ — ^
ds
supposing a d X = 0, and 
.^./ d/.ad/ _ d/.ady
"^"^  d7~ ~ ds' •
dv =
for V is the velocity at O and does not vary by altering the curve.
v' = V + d V
dv' = av + adv=radv.
Hence
dyady dy'ady d s' a d v _
v d s v' d s' v' * "~ *
Also
1 _ 1 __ 1 _dv
v'~ v+dv"~v V*'
for d V ^, &c. must be omitted. Substituting this in the second term of
the above equation, we have
dy.ady d y' a d y d y' d v a d y ds^dv _
yds vds' v*ds' v'* ~
or
/dy' dy\ 1 dy'.dv ds' adv_
Vd~?~dlJ* 7 +ds'.v2~7^ • ad^  "
Now as before
d s' d s * d s *
And in the other terms we may, since O, P, are indefinitely near, put
d s, d y, v for d s', d y', v' :
if we do this, and multiply by — v, we have
d.dy_ll^ + isadv^^ (2)
ds ds.vvady ^ '
which will give the nature of the curve.
If the forces which act on the body at O, be equivalent to X in the
direction of x, and Y in the direction of y, we have (371)
vdv=Xdx+Ydy
, Xdx+ Ydy
.*. d v = ^
V
>A Yady
.*. a d V = ^
V
U2
308
A COMMENTARY ON
[Sect. X.
because 3v = 0, 5dx = 0; also X and Y are functions of A L, and L O,
and therefore not affected by 8.
Substituting these values in the equation to the curve, we have
, dy dy Xdx+Ydy , ds Y ^
dsds V* vv
or
, dy dx Xdy — Ydx ^
^•A —A' ^—z = ^
ds d s v'^
which will give the nature of the curve.
If r be the radius of curvature, and d s constant, we have (from "74)
d s d X
r being positive when the curve is convex to A M ;
r =
A d y _ d X
' d s "~ r
and hence
Xdy — Ydx
ds
V
The quantity — is the centrifugal force (210), and therefore that part
of the pressure which arises from it. And ^ is the
pressure
which arises from resolving the forces perpendicular to the axis. Hence,
it appears then in the Brachystochron for any given forces, the parts of
the pressure which arise from the given forces and from the centrifugal
force must be equal. ^
403. If we suppose the force to tend to a center S,
which may be assumed to be in the line A M, and F
to be the whole force ; also if
SA= a,SP = g,SY = p;
then we have
^^ = force in P S resolved parallel to
and
YS = F X ^
g
v2= C — 2/gFd»
. C2g/Fdg ^Fp
•• r s
also
r = — ,— ^
dp
Book I.] NEWTON'S PRINCIPIA. 309
.•.C_2g/Fdf = Si
2dp_ — 2Fdg
•'• p C2g/Fdg
and integrating
P^ = CqC2g/Fdgl
whence the relation of p and j is known.
If the body begin to descend from A
C — 2g/Fdg =
when f = a.
404. Ex. 1. Let the force vary directly as the distance.
Here
F = iu.g
C2g/Fdg=v^ = /*g(a« — g^)
p2= CXa«— g^)
which agrees with the equation to the Hypocycloid (370).
405. Ex. 2. Let the force vary i7iversely as the square of the distance ,•
then
1? _ '^
by supposition.
2^' a— _f_.^, a— I
f' + c^e — c*a
.'. £ * — P ' = ^ — ^
d^ =
pdg
?^(?^P^)
_ c V (a — g).dg
> V(g3 + c«g — c'a)
cdg
When g = a, d t) = ; when
g3 + c^g — c^a =
d ^ is infinite, and the curve is perpendicular to the radius as at B. This
equation has only one real root.
If we have c = — , S B = 5
B being an apse.
U3
310 A COMMENTARY ON [Sect. X.
i^^ = 3o'S^ = m
Ifc= ,^ ,SB =
n3+ n' n''+ r
406. Wf%«* a body moves on a given surface, to determine the Brachy
stochron.
Let X, y, z be rectangular coordinates, x being vertical ; and as before
let d s, d s' be two successive elements of the curve ; and let
d X, d y, d z,
d x', d y', d z'
be the corresponding elements of x, y, z; then since the minimum pro
perty will be true of the indefinitely small portion of the curve, we have
as before, supposing v, v' the velocities,
ds . ds'
— H — = mm.
■■■^■{4^'}']='> •••(')
The variations indicated by 3 are those which arise, supposing d x, d x'
to be equal and constant, and d y, d z, d y', d z' to vary
Now
ds^ = dx^ + dy^ + dz^
.*. ds3ds = dySdy + dz3dz.
Similarly
ds'ads'= dy'3dy'+dz'adz.
Also, the extremities of the arc
d s + d s'
being fixed, we have
d y + d y' = const.
.. 3 d y + 3 d y' =
d z + d z' = const.
.. 3 d z + 3 d z' = 0.
Hence
And the surface is defined by an equation between x, y, z, which we
may call
L = 0.
Book I.] NEWTON'S PRINCIPIA. 311
Let this differentiated give
dz=pdx + qdy (3)
Hence, since d x, p, q are not affected by 8
adz = q3dy (4)
For the sake of simplicity, we will suppose the body to be acted on
only by a force in the direction of x, so that v, v' will depend on x alone,
and will not be affected by the variation of d y, d z. Hence, we have by (1)
3 d s , 8 d s' ^
J— =
V V
which, by substituting from (2) becomes
\v'ds' vdsj •'(^v'ds' vdsj
Therefore we shall have, as before
d.J^3dy + d. ^,adz=0;
v d s "^ v d s
and by equation (4), this becomes
J d y , d z  ,^v
d.— f + qd.— ^ = (5)
vds^vds ^ '
whence the equation to the curve is known.
If we suppose the body not to be acted on by any force, v will be con
stant, and the path described will manifestly be the shortest line which
can be drawn on the given surface, and will be determined by
'••^^<i0as = » <«)
If we suppose d s to be constant, we have
d'y + qd^zrz
which agrees with the equation there deduced for the path, when the
body is acted on by no forces.
Hence, it appears that when a body moves along a surface undisturbed,
it will describe the shortest Ime which can be drawn on that surface, be
tween any points of its path.
407. Let P and Q be two bodies, of which the Jirst hangs
from afxedpoint and the second from the first by means of
inextensiUe strings A P, P Q; it is required to determine the
small oscillations.
Let
A M = X, M P = y,
A N = x', N Q = /
A P = a, P Q = a'
, mass of P = /i, of Q = ^'
tension of A P =p,ofPQ= p'.
U4
312 A COMMENTARY ON [Sect. X.
Tlien resolving the forces p, p', we have
d t« ~ At' • a' ^
By combining these with the equations in x, x' and with the two
x' + y« = a«,
(x' — x)«+(y' — y)=^ = a'«;
we should, by eliminating p, p' find the motion. But when the oscilla
tions are small, we may approximate in a mor« simple manner.
Let jS, jS' be the initial values of y, y'. Then manifestly, p, p' will de
pend on the initial position of the bodies, and on their position at the time
t : and hence we may suppose
p = M + P/3 + Q/3' + R y + S y' + &c.
and similarly for p'.
Now, in the equations of motion above, p, p' are multiplied by y, y' — y
which, since the oscillations are very small are also very small quantities,
(viz. of the order /3). Hence their products with /3 will be of the order
6\ and may be neglected, and we may suppose p reduced to its first
term M.
M is the tension of A P, when /3, /3' &c. are all = 0. Hence it is the
tension when P, Q, hang at rest from A, and consequently
M = /tt + ^'.
Similarly, the first tenn of p', which may be put for it is m'. Substi
tuting these values and dividing by g, equations (1) become
d t^ V^a' ^ fia J^ ^ /!,&' y
d'y' _ Z.__ y^
gdt* a' af
Multiply the second of these equations by X and add it to the first, and
we have
d
}■
gdt^ ~ V^a'**" fia a' ) ^ \a'' (j.a')^
and manifestly this can be solved if the second member can be put in
the form
— k.(yHXyO
that is, if
fia' lia a' *
Book I.] NEWTON'S PRINCIPIA. 818
or
k X = ,
a
—
fjb a'
+
a'_^/.'a'
a fi a
_:=(a'
k
l)x
1
(3)
Eliminating X we have
(a'kl)a'k— ^' = (a'kl)(^'+ + ^')
Hence
(^'•=)=(i+^)(i+:)»'k=3^^....(4)
From this equation we obtain two values of k. Let these be de
noted by
and let the corresponding values of x, be
'X,2X.
Then, we have these equations.
and it is easily seen that the integrals of these equations are
y + ^Xy' = »Ccos. t V (^kg) + 'D sin. t V ('kg)
y + ^Xy' = ^Ccos. t V fkg) + ''D sin. t V (%g)
'C, 'D, ^C, *D being arbitrary constants. But we may suppose
'C = 'E cos. 'e
'D = 'E sin. 'e
^C = ^E cos. «e
D« = ^E sin. 2e
By introducing these values we find
y + 'x x' = 'E cos. Jt v' ('k g) + 'e} i ,^.
y + 8X y' = 2E COS. {t V ^k g) + ^e] }
From these we easily find
(6)
The arbitrary quantities 'E, *e, &c. depend on the initial position and
i.^^cos. [t V {'kg)+'e] + ^^^cos. [t V (^kg)+^ej
I
314 A COMMENTARY ON [Sect. X.
velocity of the points. If the velocities of P, Q = 0, when t = 0, we
shall have
'E, ^e, each z=
as appears by taking the Differentials of y, y'.
If either of the two ^E, E be = 0, we shall have (supposing the latter
case and omitting ^e)
'^ 'E , , „ \
y = ;^_,^ COS. t V ( Ik g)
*E
y' = j^^— ^cos.t V^kg).
Hence it appears that the oscillations in this case are si/mmetrical : that
is, tlie bodies P, Q come to the vertical line at the same time, have similar
and equal motions on the two sides of it, and reach their greatest dis
tances from it at the same time. It is easy to see that in this case, the
motion has the same law of time and velocity as in a cycloidal pendulum;
and the time of an oscillation, in this case, extends from when t = to
when t V ('k g) = v. Also if /3, /S' be the greatest horizontal deviation
of P, Q, we shall have
y = /3 . cos. t V ( *k g)
f = jS'.cos. t V (%g).
In order to find the original relation of /3, /3', (the oscillations will be
symmetrical if the forces which urge P, Q to the vertical be as P M, Q N,
as is easily seen. Hence the conditions for symmetrical oscillation might
be determined by finding the position of P, Q that this might originally
be the relation of the forces) that the oscillations may be of this kind, the
original velocities being 0, we must have by equation (5) since *E = 0.
/3 + ^X i8' = 0.
Similarly, if we had
i8 + 'X /3' =
we should have 'E = 0, and the oscillations would be symmetrical, and
would employ a time
V(^kg)
"When neither of these relations obtains, the oscillations may be consi
dered as compounded of two in the following manner : Suppose that we
put
y = H cos. t V ( ^k g) + k cos. t V ( *k g) . . . (7)
omitting 'e, *e, and altering the constants in equation (6) ; and suppose
that we take
M p = H . cos. t V ( »k g) ;
Book I.] NEWTON'S PRINCIPIA. 315
Then p will oscillate about M according to the law of a cycloidal pen
dulum (neglecting the vertical motion). Also
p P will = K . COS. t V ( % g).
Hence, P oscillates about p according to a similar law, while p oscil
lates about M. And in the same way, we may have a point q so moved,
that Q shall oscillate about q in a time
V(^kg)
while q oscillates about N in a time
V('kg)
And hence, the motion of the pendulum A P Q is compounded of the
motion A p q oscillating symmetrically about a vertical line, and of A P Q
oscillating symmetrically about A p q, as if that were a fixed vertical line»
When a pendulum oscillates in this manner it will never return exactly
to its original position if V % V °k are incommensurable.
If */ 'k, V % are commensurable so that we have
m V 'k = n V == k
m and n being whole numbers, the pendulum will at certain intervals, re
turn to its original position. For let
t V ( ^k g) = 2 n T
then
t >/ ( «k g) = 2 ra ^
and by (7)
y = H cos. 2 n ff } K . cos. 2 m ?r
= H + K,
which is the same as when
t = 0.
And similarly, after an interval such that
t • ( *k g) = 4 n ff, 6 n cr, &c.
the pendulum will return to its original position, having described in the
intermediate times, similar cycles of oscillations.
408. Ex. Let (j! = (j^
a' = a
to determine the oscillations.
Here equation (4) becomes
a^k* — 4 ak = — 2
and
a k = 2 + V 2.
316 A COMMENTARY ON
Also, by equation (3)
[Sect. X
ak = 3 — X
.'.'X = 1 + V 2, 'X = I — V 2.
Hence, in order that the oscillations may be symmetrical, we must
either have
/3 + (I + V 2) i8' = 0, whence /3' = — ( V 2 —1) /3
or
/3— ( V 2 — 1) iS' = 0, whence ^' = ( V 2 + 1) iS.
The two arrangements indicated by these equations are thus repre
sented.
Q' N Q QNQ'
The first corresponds to
/3' = (V2 + l)/3
or
QN = (V 2 + 1)PM.
In this case, the pendulum will oscillate into the position A P' Q', simi
larly situated on the other side of the line ; and the time of this complete
oscillation will be
In the other case, corresponding to
/3' = — (V 2 1)^
Q is on the other side of the vertical line, and
QN = ('/2 — 1)PM.
The pendulum oscillates into the position A P' Q', the point O remain 
ing always in the vertical line ; and the time of an oscillation is
<r /a
V(2 + V2)^J'
The lengths of simple pendulums which would oscillate respectively in
these times would be
2— V 2
and
2+^2
Book 1]
NEWTON'S PRINCIPIA.
317
or
1 .707 a and .293 a.
If neither of these arrangements exist originally, let ^, ^' be the origi.
nal values of y, f when t is 0. Then making t = in equation (5), we
have
»E = /3+ (V 2 + l)/3'
and
*E = /3— (\^ 2— l)j8'.
And these being known, we have the motion by equation (6).
409. Any number of material points Px, P2J P3. . . Q,
hang by means of a string without weight, from a 'point
A ; it is required to determine their small oscillations in
a vertical plane.
Let A N be a vertical abscissa, and Pi Mi, P2 M2,
&c. horizontal ordinates ; so that
A Ml = Xi, A M2 = X2, &c.
Pi Ml = yi, P2 M2 = yg, &c.
A Pi = a„ Pi P2 = a2, &c.
tension of A Pi = pi, of PiPgr: p^, &c.
mass of Pi = ^1, of Pg = Aaj &c.
Hence, we have three equations, by resolving the forces parallel to the
horizon.
d
yi _
dt
d
Pig yi . P2g
72 — yi
y2 _ P2g
dt^ . ^ ■
ya— yi
dt^ ~
Psg y3— y2
/*3
P3g 73 — ya
Pig y^— y3
f^3
a*
(1)
^lln = Eaj yn — yni
d 1 2 A^n ' an
And as in the last, it will appear that pi, p2, &c. may, for these small
oscillations, be considered constant, and the same as in the state of rest.
Hence if
fJ^l+ f^+ . . . . A*n = M,
then
Pi = M, P2 = M — ^1, p3 = M ~ /(ij  ;t2j &c.
Also, dividing by g, and arranging, the above equations may be put in
this form :
318
A COMMENTARY ON
g d t^ Vj ai "*■ fLi a.J ^' ■*"
TSect. X.
f^i %
gdt^ ~ A^ Eg
g d t * /«3 ag ^/^ 83 "^ /i3 a*/ ^^ "^ ^ a^
K
(1)
d'yn _. Pnyn1 _ Pn Jn
gdt* /A„a„ ./!ina„
The first and last of these equations become symmetrical with the rest
if we observe that
yo =
and
Pn + l = 0.
Now if we multiply these equations respectively by
1, X, X', X'', &c.
and add them, we have
d'^yi + Xd' yg + X^d' y3 + &c. _
gdt*
P2
r Pi P2_ . iPlXy
I /*i aa ^^^2 ag f^ ag/ /^s^jj
It^as ^^^3 a3 f^ 34/ fii a^ )
ys
I ^n1 a„ /(in a„ i^"
and this will be integrable, if the righthand side of the equation be redu
cible to this form
— k (yi + X y2 + X' y3 + &c.).
That is, if
_ Pi
^Pz
/ii ai /»! aj (Wa ag
kX = —
kX' =
P2
+ x(
Pa .. P3
"■2 aa i<A2 as
?^'P3
> _ri_P3
/ A3 a
/*! 3.2 ' ^"■a aa i<a2 ag/ /is
_ >P3 , ^ / /' Pa . P4 \ , ^'' P4
~ /*2 33
X'(nS) p
k X^">2) = ^ +
/*n  1 an
+ ?/ (_P3_ + ^) + ^B
An an
(3)
Book I.] NEWTON'S PRINCIPIA. 319
If we now eliminate
X, X', X", &c.
from these n equations, it is easily seen that we shall have an equation of
n dimensions in k.
Let
% % % "k
be the n values of k ; then for each of these there is a value of
X', X'', X'"
easily deducible from equations (3), which we may represent by
^X, IX', 'X", &C.
«X', «X", V, &C.
Hence we have these equations by taking corresponding values X and k,
d*yi + *Xd2y2+2X"d2y3 + &c. ,, , ,2^ ,2^, , e \
— ii?= ^^2 ^^^^ — =— k (ya + '>^y2+'?^' y3+ &c.)
and so on, making n equations.
Integrating each of these equations we get, as in the last problem
Ji + '^ y2 + '^' ya + &c. = IE cosjt V ('k g) + »e^  .^.
yi + ''^y2 + '^'y3 + &c. = 2Ecos4t Vfkg) + 2ei/ * \' ^ '
*E, *E, &c. 'e, % &c. being arbitrary constants.
From these n simple equations, we can, without difficulty, obtain the n
quantities yj, yg, &c. And it is manifest that the results will be of this
form
yi='HiCos.{t V (»kg) + »e}+'^HiCOs.{tVfkg) + «e} + &c.j
y2=iH2Cos.{t V Ckg) + 'e] +^H2Cos.Jt V^kg) + «e] + &c. V . . . (6)
&c. = &c. J
where ^Hi,'H2, &c. must be deduced from /Sj, jSg, &c. the original values
of yi, y2j &c.
If the points have no initial velocities (i. e. when t = 0) we shall have
»E = 0, ^E = 0, &c.
We may have symmetrical oscillations in the following manner. If,
of the quantities 'E, ^E, ^E, &c. all vanish except one, for instance "E ; we
have
yi + 'Xy2 + 'X'y3 + &c. = ^.
yi + '^y2 + '^'y3 + &c. = o
yi + '^y2 + '^'y3 + &c. = > . . . . (7)
yi+"?^y2+°?^'y3+&c.=°Ecos.tv^("kg).
omitting "E.
320 A COMMENTARY OlSf [Sect. X.
From the n — 1 of these equations, it appears that yg, ya, &c. are in a
given ratio to yi ; and hence
yi + ">^y2 + ''>^'y3 + &c.
is a given multiple of yj and = m yi suppose. Hence, we have
m yi = "E cos. v^ ("k g) ;
or, omitting the index n, which is now unnecessary,
m yi = E cos. t V (k g).
Also if y2 = 62 yi,
m ya = E e2 cos. t V (k g)
and similarly for y^ &c.
Hence, it appears that in this case the oscillations are symmetrical. All
the points come into the vertical line at the same time, and move similar
ly, and contemporaneously on the two sides of it. The relation among
the original ordinates (Sj, /S^, /Sg, &c. which must subsist in order that the
oscillations may be of this kind, is given by the n — 1 equations (7),
5i + '?^/32 + iX'i83+&c. =
i8l+'X^2 + V/33 + &C. =
^l+'?^^2 + '?^'/33 + &C. =
&c. = &c.
These give the proportion of /3i /Sg, &c ; the arbitrary constant "E, in
the remaining equation, gives the actual quantity of the original displace
ment
Also, we may take any one of the quarttities *E, *E, ^E, &c. for that
which does not vanish ; and hence obtain, in a different way, such a sys
tem of n — 1 equations as has just been described. Hence, there are n
different relations among /Sj /S^, &c. or n different modes of arrangement,
in which the points may be placed, so as to oscillate symmetrically.
( We might here also find these positions, which give symmetrical oscil
lations, by requiring the force in each of the ordinates Pi Mi, P2 M2 to
be as the distance; in which case the points Pi, P2, &c. would all come
to the vertical at the same time.
If the quantities V 'k, V ^k have one common measure, there will be
a time after which the pendulum will come into its original position. And
it will describe similar successive cycles of vibrations. If these quantities
be not commensurable, no portion of its motion will be similar to any
preceding portion.)
The time of oscillation in each of these arrangements is easily known ;
the equation
m yi = "E cos. t V (°k g)
Book L] NEWTON'S PRINCIPIA. 321
shows that an oscillation employs a time
And hence, if all the roots 'k, *k, ^k, &c. be different, the time is dif
ferent for each different arrangement.
If the initial arrangement of the points be different from all those thus
obtained, the oscillations of the pendulum may always be considered as
compounded of n symmetrical oscillations. That is, if an imaginary pen
dulum oscillate symmetrically about the vertical line in ^ time
vckg)'
and a second imaginary pendulum oscillate about the place of the first,
considered as a fixed line, in the time
vekg)'
and a third about the second, in the same manner, in the tiiac
Vfkg)'
and so on; the n'^'' pendulum may always be made to coincide per
petually with the real pendulum, by properly adjusting the amplitudes of
the imaginary oscillations. This appears by considering the equations
(6), viz.
yi = 'Hi cos. t V Qk g) + ^Hi cos. t V {'k g) + &c.
&c. = &c.
This principle of the coexistence of vibrations is applicable in all cases
where the vibrations are indefinitely small. In all such cases each set of
symmetrical vibrations takes place, and affects the system as if that were
the only motion which it experienced.
A familiar instance of this principle is seen in the manner in which the
circular vibrations, produced by dropping stones into still water, spread
from their respective centers, and cross without disfiguring each other.
If the oscillations be not all made in one vertical plane, we may take a
horizontal ordinate 2 perpendicular to y. The oscillations in the direc
tion of y will be the same as before, and there will be similar results ob
tained with respect to the oscillations in the direction of z.
We have supposed that the motion in the direction of x, the vertical
axis, may be neglected, which is true when the oscillations are very
small.
410. Ex. Let there be three bodies all equal (each = /t), and also their
distances aj, aj, s.^ all equal (each =: a).
Vol. I. X
322 A COMMENTARY ON [Sect. X.
Here
p = 3 /(A, p2 = 2 ^, P3 = a
and equations (3) become
a k = 5 — 2 X
akX=r — 2 + 3X — X'
a k X' = — X + \'.
Eliminating k, we have
5X — 2X* = — 2 + 3X — X',
5 X' — 2 X X' = — X + X',
or
X' = 2 X^ — 2X — 2,
4. X' — 2 X X' = — X
.'. X' =
or
2 X — 4
.. (2X2_2X — 2)(2X— 4) = X
X3— 3X2+ 3 ^ + 2=0,
4
which may be solved by Trigonometrical Tables. We shall find three
values of X.
Hence, we have a value of X' corresponding to each value of X ; and
then by equations (7)
P +^X/3, + Vi83=0J ^ '
whence we find /Sgj ^3 in terms of /3,.
We shall thus find
^2 = 2. 295 /3i
or
^2 = 1.348/3,
or
j82=— .643/3,
according as we take the different values of X.
And the times of oscillation in each case will be found by taking tlie
value of
a k = 5 — 2 X;
tliat value of \ being taken which is not used in equation (7'). For the
time of oscillation will be given by making
t V (k g) = cr.
If the values of /Sj, /S^, ^3 have not this initial relation, the oscillations
Book I.l
NEWTON'S PRINCIPIA.
323
will be compounded in a manner similar to that described in the example
for two bodies only.
411. A Jlexihle cham, of uniform thickness, hangs f^om a fxed point :
to find its initial form, that its small oscillations may be symmetrical.
Let A Mj the vertical abscissa = x ; M P the hori
zontal ordinate = y; A P = s, and the whole length
A C = a;
.. A P = a — s.
And as before, the tension at P, when the oscillations
are small, will be the weight of P C, and may be represent
ed by a — s. This tension will act in the direction of a
tangent at P, and hence the part of it in the direction
P M will be
. • dy
tension X 7^
d s
or
(as)iy.
d s
Now, if we take any portion P Q = h, we shall find the horizontal
force at Q in the same manner. For the point Q, supposing d s constant
r^ becomes ^^ + ,
d s d s d s
d^y h . d^y h^ „
1 d S3
S3 1.2
(see 32).
Also, the tension will be a — s + J^' Hence the horizontal force in
the direction N Q, is
. Subtracting from this the force in P M, we have the force on P Q
horizontally.
^^ 'Hds^ 1 +ds^1.2 + ^''V
^ Vds + ds^' 1 +^7
and the mass of P Q being represented by h, the accelerating force
( =z " — 5^ is found. But since the different points of P Q move
V. mass / ^
with different velocities, this expression is only applicable when h is inde
finitely small. Hence, supposing Q to approach to and coincide with P,
we have, when h vanishes
d * V d v
accelerating force on P = (a — s) ^—^ — t= .
X 2
324 A COMMENTARY ON [Sect. X.
But since the oscillations are indefinitely small, x coincides with s and
we have
d * V d V
accelerating force on P = (a — x) 3—^ — 3^.
Now, in order that the oscillations may be symmetrical, this force must
be in the direction P M, and proportional to P M, in which case all the
points of A C, will come to the vertical A B at once. Hence, we must
have
('''')a^»^x="y (')
k being some constant quantity to be determined.
This equation cannot be integrated in finite terms. To obtain a
series let
y = A+ B.(a — x)+C(a — x)2 + &c.
..^ =_B — 2C(a — x) — 3D(a — x)'
d x ^ ' ^
.. —?, = 1. 2. C + 2. 3 D (a — x) + &c
dx* ^
Hence
= (ax)^.^ + ky
d X* dx
gives.
= 1.2. C (a — x) + 2.3D(a — x)« + &c.
+ B + 2 C (a — x) + 3 D (a — x) * 4 &c.
+ kA + kB(a — x) + kC(a — x)* + &c.
Equating coefficients ; we have
B = — kA,
22 C= — k B
S^D = — k C
&c. = &c.
.. B = k A
r _k^A
^  gT
2^32
&c. = &c.
and
v = Alk(ax)+](ax)^^^.(ax) + &c.} ..(2)
Book I.] NEWTON'S PRINCIPIA. S26
Here
A is B C, the value of y when x = a. When x = 0, y = ;
.•.lka + ^^ __ + &c. = ..... (3)
From this equation (k) may be found. The equation has an infinite
number of dimensions, and hence k will have an infinite number of values,
which we may call
% % .. .°k... 1,
and these give an infinite number of initial forms, for which the chain
may perform symmetrical oscillations.
The time of oscillation for each of these forms will be found thus. At
the distance y, the force is k g y : hence by what has preceded, the time
to the vertical is
2v'(kg)
and the time of oscillation is
v^ (k g) •
(The greatest value of k a is about 1.44 (Euler Com. Acad. Petrop.
tom. viii. p. 43). And the time of oscillation for this value is the same as
2
that of a simple pendulum, whose length is —a nearly.)
The points where the curve cuts the axis will be found by putting y = 0.
Hence taking the value °k of k, we have
= l_»k(a.) + °Jil(^^' + "Jiy2^+ &c.
which will manifestly be verified, if
° k (a — x) = 'k a
or
« k (a — x) = % a
or
° k (a — x) = % a
&c. = &c.
because ^k a, *k a, &c. are roots of equation (3).
That is if
X = a (l — 5^) or = a (l — ^) or = &c.
Suppose 'k, %, ^k, &c. to be the roots in the order of their magnitude
k being the least.
Then if for "k, we take 'k, all these values of x will be negative, and
the curve will never cut the vertical axis below A.
X3
326 A COMMENTARY ON [Sect. X.
If for "k, we take *k, all the values of x will be negative except the
first ; therefore, the curve will cut A B In one point. If we take ^k, all
the values will be negative except the two first, and the curve cuts A B
in two points ; and so on.
Hence, the forms for which the oscillations will be
symmetrical, are of the kind thus represented.
And there are an infinite number of them, each
cutting the axis in a different number of points.
If we represent equation (2) in this manner
y = A p (k, x)
it is evident that
y = 'A p ('k, X)
y = 'A<p {% X)
 &c. = &c.
will each satisfy equation (1). Hence as before, if we put
y = ^A p ('k, x) + ^A p ('k, x) + &c.
and if 'A, *A, &c. can be so assumed that this shall represent a given
initial form of the chain, its oscillations shall be compounded of as many
coexisting symmetrical ones, as there are terms 'A, '^A, &c.
We shall now terminate this long digression upon constrained mo
tion. The reader who wishes for more complete information may con
sult Whewell's Dynamics, one of the most useful and elegant treatises
ever written, the various speculations of Euler in the work above quoted,
or rather the comprehensive methods of Lagrange in his Mecanique
Analytique.
We now proceed to simplify the text of this Xth Section.
412. Prop. L. First, S R Q is formed by an entire revolution of the
generating circle or wheel, whose diameter ie O R, upon the globe
SOQ.
413. Secondly, by taking
C A : C O : : C O : C R
we have
CA:CO::CA — CO.CO — CR
: : A O : O R
and therefore if C S be joined and produced to meet the exterior globe
in D, we have also
AD : SO(:: C A: CO):: AO: OR.
But
S O = the semicircumference of the wheel O R = — —^ — .
Book I.] NEWTON'S PRINCIPIA. 327
.♦.AD = ^^ — =h the circumference of the wheel whose diameter is
2 ^
A O. That is S is the vertex of the Hypoc}xloid A S, and A S is per
pendicular in S to C S. But O S is also perpendicular to C S. There
fore A S touches O S in S, &c.
414. The similai Jigures A S, S R.]
By 39 it readily appears that Hypocycloids are similar when
R : r : : R' : r'
R and r being the radii of the globe and wheel ; that is when
C A: AO :;CO : O R
or when
C A : C O : : C O : C R
.*. A S, S R are similar
415. V B, V W are equal to O A, O R.] .
If B be not in the circumference AD let C V meet it in B'. Then
V P being a tangent at P, and since the element of the curve A P is the
same as would be generated by the revolution of B' P around B' as a
center, and .*. B' P is perpendicular both to the curve and its tangent
P V, therefore P B, P B' and .. B, B' coincide. That is
V B := O A.
Also if the wheel O R describes O V whilst A O describes A B, the
angular velocity B P in each must be the same, although at first, viz. at
O and A, they are at right angles to each other. Hence when they shall
have arrived at V and B their distances from C B must be complements
of each other. But
z.TVW = BVP=^— PBV
.*. T V is a chord in the wheel O R, and
.. V W = O R.
See also the Jesuits' note.
• OTHERWISE.
416. Construct the curve S P, to which the radius of curvature to every
point of S R Q is a tangent ; or which is the same, find S A the Locus of
the Centers of Curvature to S R Q.
Hence is suggested the following generalization of the Problem, viz.
417. To make a body oscillate in any given curve.
Let S R Q (Newton's fig.) the given curve be symmetrical on both sides
X4
328 A COMMENTARY ON [Sect. X.
of R. Theu if X, y be the rectangular coordinates referred to the vertex
R, and ot, j8 those of the centers of curvature (P) we have
r 2 _ p X 2 = (y — ^) s 4. (X — a) 2.
Hence, the contact being of the second order (74)
xa+ iy^)^ = . (1)
and
d V* d^v
i + f^. + (y^)^/. = o (2)
These two equations by means of that of the given curve, will give us
3 in terms of a, or the equation to the Locus of the centers of curvature.
Let S A be the Locus corresponding to S R, and A Q the other half.
Then suspending a body from A attached to a string whose length is R A,
when this string shall be stretched into any position APT, it is evident
that P being the point where the string quits the locus is a tangent to it,
and that T is a point in S R Q.
Ex. 1. Let S R Q &<? the common parabola.
Here
y^ =
: 2 a X
"dx ~
a
d*y
dx« ~
a
dy
dx
= ■
a"
.*. substituting we get
X — a
+ (y
^).
a
v
=
and
^^k
(y/3)
• 2
a
xy
=
.'. X
« + tO
n^)
2
xy
a
=
2xy
zz 3 x — a 4 a
or
and
But
a = 3 X + av
..13* =
= 2 ax
4 x^y^ _ 8 X 3
a
Book IJ NEWTON'S PRINCIPIA. 329
^ 8_^(«ar^ 8 ^
a 27 27 a ^ ' • \^)
Now when /3 = 0, a = a ; which shows that A R the length of the
string must equal a. Also making A the origin of abscissas, that is, aug
menting a by a, we have
^' = i^ X »"
the equation to the semicubical parabola A S, A Q, which may be traced
by the ordinaiy rules (35, &c.); and thereby the body be made to oscillate
in the common parabola S Q R.
Ex. 2. JLet S R Q 5e an ellipse.
Then, referring to its center, instead of the vertex,
b^
or
and
a^y^ + b^x* = a^b'
.\ a^y ^ + b^x =
•^ d X
•^ d X dx*
These give
and
Hence
and
d y _ b* X
dx "" a* y
d*y _ b
dx*^ ~ a^y'*
_ (a^ — b')
(a'b'')y
Hence substituting the values of y and x in
a^y'' + b*x2 = a*b'
we get
/3 b xf
c&r + (iT^.r = ' w
the equation to tlie Locus of the centers of curvature.
330 A COMMENTARY ON [Sect. X.
In the annexed figure let
SC = b, CR = a
C M = X, T M = y.
Then
P N = ^, C N = a.
And to construct A S' by points, first put
^ =
whence by equation (a)
. a«— b«
a = +
— a
the value of A C. Let
a =
then
a* — b«
/3 = +
b
the value of S' C or C Q'.
Hence to make a body oscillate in the semiellipse S R Q we must
take a pendulum of the length A R, (part = A P S' flexible, and part
= S S' rigid ; because S S' is horizontal, and no string however stretched
can be horizontal — see Whewell's Mechanics,) and suspend it at A.
Then A P being in contact with the Locus AS', P T will also touch
A S in P, &c. &c.
Ex. 3. Lei S K Q be the common cycloid ,•
The equation to the cycloid is
^^ = >v/^^ = V(t')
• ill — — L
*• dx* "^ y^
whence it is found that
Hence
and
a = x42V(2ry— y2)
/3 = y i
da _ 2 r — y
dx ~ y
d5__dv__ /2r — y
dx ~ dx~ V
y
"da" >\2r — y~ 'S2rf)8
which is also the equation of a cycloid, of which the generating circle is
Book I.] NEWTON'S PRINCIPIA. 331
precisely the same as the former, the only difference consisting in a change
of sign of the ordinate, and of the origin of the abscissae.
The rest of this section is rendered sufficiently intelligible by the
Notes of P. P. Le Seur and Jacquier ; and by the ample supplementary
matter we have inserted.
SECTION XI.
417. Prop. LVII. Two bodies attracting one another, describe round
each other and round the center of gravity similar figures.
Q
Since the mutual actions will not affect the center of gravity, the bodies
will always lie in a straight line passing through C, and their distances
from C will always be in the same proportion.
.. S G : T C : : P C : Q C
and
z.SCT = QCP.
.% the figures described round each other are similar.
Also if T t be taken = S P, the figure which P seems to describe
round S will be t Q, and
Tt: TQ:: SP: TQ
::CP:CQ
and
z. t T Q = P C Q.
.•. the figures t Q, P Q, are similar ; and the figure which S seems to
describe round P is similar, and equal to the figure which P seems to
describe round S.
418. Prop. LVIII. If S remained at rest, a figure might be de
scribed by P round S, similar and equal to the figures which P and S
seem to describe round each other, and by an equal force.
332
A COMMENTARY ON
[Sect. XI.
Curves are supposed similar and Q R, q r indefinitely small. Let P and
p be projected in directions P R, p r (making equal angles C P R, s p r)
with such velocities that
V
V S
VCP
• sp
= 1
/PQ
V S + P
Vpq
Then
f since d t
ds\
~ V J
T
* t
_PQ Vpq
pq VPQ
_ VPQ
_ /QR
V qr
But in the beginning of the motion f =
pt
I
2
. F _ Q R jL*" _ ±
•*• f ~ qr • QR ~ 1 •
The same thing takes place if the center of gravity and the whole system
move uniformly forward in a straight line in fixed space.
419. CoE. 1. If F Qc D, the bodies will describe round the common
center of gi'avity, and round each other, concentric ellipses, for such would
be described by P round S at rest with the same force.
Conversely, if the figures be ellipses concentric, F « D.
420. CoR. 2. If F a ,t— the figures will be conic sections, the foci in
the centers of force, and the converse.
421. CoR. 3. Equal areas are described round the center of gravity,
and round each other, in equal times.
V
422. CoR. 3. Otherwise. Since the curves are similar, the areas, bounded
by similar parts of the curves, are similar or proportional.
.. spq : C P Q : : sp*^ : C P^ : : (S+P)'' : s^ in a given ratio;
Book I.t NEWTON'S PRINCIPIA. 333
and T. through s p q : T. through CPQ::VS+P:VS, ina given ratio
and .. : : T. through spv: T. through CPV
.. T. through C P Q : T. through CPV:: T. through spq : T. through spv
: : s p q : s p V (by Sect. II.)
::CPQ:CPV
.*. the areas described round C are proportional to the times, and the
areas described round each other in the same times, which are similar to
the areas round C, are also proportional to the times.
423. Prop. LIX. The period in the figure described in last Prop.
: the period round C : : v' S + P : V~S ; for the tunes through shnilar
arcs p q, P Q, are in that proportion.
424. Prop. LX. The major axis of an ellipse which P seems to de
scribe round S in motion ^^ Force a Yil) • niajor axis of an ellipse which
would be described by P in the same time roimd S at rest : : S + P • first
of two mean proportionals between S + P and S.
Let A =r major axis of an ellipse described (or seemed to be described)
roimd S in motion, and which is similar and equal to the ellipse de
scribed in Prop. LVIII.
Let X = major axis of an ellipse which would be described round S at
rest in the same time.
period in ellipse round S in motion V S ,t» t tv\
.'.  — : — T 11 m = — ■ (Prop. LIX)
period m same ellipse round bat rest V S + P
and by Sect. Ill,
period in ellipse round S at rest _ ^
period in required ellipse round S at rest ~ yrl
5
period in ellipse round S in motion _ A ^ V S
period in required ellipse round S at rest ~~ « I ^ o . p
but these periods are to be equal,
.. A^ s = x^s"TP
A:x::VSfP: v'S::S+P: first of two mean proportionals
(for if a, a r, a r % a r % be proportionals, V a.: V a r ' : : a : a r.)
425. At what mean distance from the earth would the moon revolve
round the earth at rest, in the same time as she now revolves round the
earth in motion ? This is easily resolved.
426. Prop. LXI. The bodies will move as if acted upon by bodies at
the center of gravity with the same force, and the law of force with re
334 A COMMENTARY ON [Sect. XI.
spect to the distances from the center of gravity will be the same as with
respect to the distances from each other.
For the force is always in the line of the center of gravity, and .*. the
bodies will be acted upon as if it came from the center of gravity.
And the distance from the center of gravity is in a given ratio to the
distance from each other, .•. the forces which are the same functions of
these distances will be proportional 
427. Prop. LXII. Problem of two bodies with no initial Velocities.
F oc — — . Two bodies are let fall towards each other. Determine the
motions.
The center of gravity will remain at rest, and the bodies wiU move as
if acted on by bodies placed at the center of gravity, (and exerting the
same force at any given distance that the real bodies exert),
.. the motions may be determined by the 7th Sect.
428. Prop. LXIII. Problem of two bodies with given initial Velo
cities.
F a Y\l ' Two bodies are projected in given directions, with given
velocities. Determine the motions.
The motion of the center of gravity is known from the velocities and
directions of projection. Subtract the velocity of the center of gravity
from each of the given velocities, and the remainders will be the velocities
with which the bodies will move in respect of each other, and of the cen
ter of gravity, as if the center of gravity were at rest. Hence since they
are acted upon as if by bodies at the center of gravity, (whose magnitudes
are determined by the equality of the forces), the motions may be deter
mined by Prop. XVII, Sect. Ill, (velocities being supposed to be acquired
down the finite distance), if the directions of projection do not tend to the
center, or by Prop. XXXVII, Sect. VII, if they tend to or directly from
the center. Thus the motions of the bodies with respect to the center of
gravity will be determined, and these motions compounded with the uni
form motion of the center of gravity will determine the motions of the
bodies in absolute space.
429. Prop. LXIV. F a D, determine the motions of any number of
bodies attracting each other.
Book I.] NEWTON'S PRINCIPIA. 335
T and L will describe concentric
ellipses round D.
Now add a third body S.
Attraction of S on T riiay be re
presented by the distance T S, and
on L by L S, (attraction at distance
being 1) resolve T S, L S, into
T D, D S ; L D, D S, whereof the
parts T D, L D, being in given
ratios to the whole, T L, L T, v/ill
only increase the forces with which
L and T act on each other, and
the bodies L and T will continue to describe ellipses (as far as respects
these new forces) but with accelerated velocities, (for in similar parts of
similar figures V^ « F.R Prop. IV. Cor. 1 and 8.) The remaining
forces D S, and D S, being equal and parallel, will not alter the relative
motions of the bodies L and T, .*. they will continue to describe ellipses
round D, which will move towards the line I K, but will be impeded in
its approach by making the bodies S and D (D being T + L) describe
concentric ellipses round the center of gravity C, being projected with
proper velocities, in opposite and parallel directions. Now add a fourth
body V, and all the previous motions will continue the same, only accel
erated, and C and V will describe ellipses round B, being projected with
proper velocities.
And so on, for any number of bodies.
Also the periods in all the ellipses will be the same, for the accelerating
forceonT = L.TL S . TD = (T+L). TD + S. TD = (T+L +S).
T D, i. e. when a third body S is added, T is acted on as if by the sum
of the three bodies at the distance T D, and the accelerating force on D
towards C = S.SD = S.C S+ S.D C = (T + L).DC+ S. D C
= (T + L + S). D C.
.. accelerating force on T towards D : do. on D towards C : : T D : D C
.'. the absolute accelerating forces on T and D are equal, or T and D
move as if they revolved round a common center, the absolute force the
same, and varying as the distance from the center, i. e. they describe el
lipses, in the same periods.
Similarly when a fourth body V is added, T, L, D, S, C, and V, move
as if the four bodies were placed at D, C, B, L e. as if the absolute forces
were the same, and with forces proportional to their respective distances
from the centers of gravity, and .*. in equal periods.
336
A COMMENTARY ON
CSeci'. XI.
And so on, for any number of bodies.
430. Prop. LXVI. S and P revolve round T, S in the exterior orbit,
P in the interior,
F oc — ^ , find when P will describe round T an orbit nearest to the
ellipse, and areas most nearly proportional to the times.
(1st) Let S, P, revolve round the greatest body T in the same plane.
Take K S for the force of S on P at the mean distance S K,
and
LS = SK.
SK
= force at P,
SP^
resolve L S into L M, M S,
L M is parallel to P T, and .. tends to the center T, .*. P will con
tinue to describe areas round T proportional to the times, as when acted
on only by P T, but since L M does not a pPjr^ j the sum of L M and
P T will not a ^^ , .*. the form of the elhptic orbit P A B will be
disturbed by this force, L M, M S neither tends from P to the center
T, nor « prrTj , .*. from the force M S both the proportionality of areas
to times, and the elliptic form of the orbit, will be disturbed, and the
elliptic form on two accounts, because M S does not tend to C, and be
cause it does not « "pfpi •
.'. the areas will be most proportional to the times, when the force
M S is least, and the elliptic form will be most complete, when the forces
M S, L M, but particularly L M, are least
Now^ let the force of S on T = N S, then this first part of the force
M S being common to P and T wiU not affect their mutual motions, .•. the
Book I.]
NEWTON'S PRINCIPIA.
SSt
disturbing forces will be least when L M, M N, are least, or L M remain
ing, when M N is least, i. e. when the forces of S on P and T are nearly
equal, or S N nearly = S K.
(2dly) Let S and P revolve round T in different planes.
Then L M will act as before.
But M N acting parallel to T S, when S is not in the line of the Nodes,
(and M N does not pass through T), will cause a disturbance not only
in the longitude as before, but also in the latitude, by deflecting P from
the plane of its orbit. And this disturbance will be least, when M N is
least, or S N nearly = S K.
431. Cor. 1. If more bodies revolve round the greatest body T, the
motion of the inmost body P will be least disturbed when T is attracted
by the others equally, according to the distances, as they are attracted by
each other.
432. Cor. 2. In the system of T, if the attractions of any two on the
third be as yp j P will describe areas round T with greater velocity near
conjunction and opposition, than near the quadratures.
433. To prove this, the following investigation is necessary.
.
A
,
1 ,
^/^
V.
si —  —
,_^7
^
"YX
\
n
m
. B\ 1
T j
D
Take 1 S to represent the attraction of S on P,
nS T,
Then the disturbing forces are 1 m (parallel to P T) and m n.
Now
S 1 = g^2 (force a ~^^
S
.•.Sm = Sl.S^=
R*— 2Rrcos. A + r*'
S.R
(R = ST,r = PT)A = ^STP
SP~ (R* — SRrcos. A + r=')
S.R
V R'^— 2Rrcos. A + r«
= R»(l —
2 V cos. A
R
R
W
2 r COS. A T*_
R
R
Vol. I.
SS8 A COMMENTARY ON [Sect XT.
 S /, , 3/2r . r*v ^ 3.5 /2rcos.A r«N« x
 S / 3r . /3 3.5 . ^\r' \
= E^C^ + R*^^' ^ b  1:^ COS.* A)g,&c.)
 R«v*+ R >
where R is indefinitely great with respect to r.
Also
S /, . 3 r COS. A\ S S.Srcos.
r„„_e^ c!„ ^ ^1 , ^^ COS. A\ ft>
ultimately
andlm = SI. ~ = —^ (R« — 2Rrcos. A + r«)
= ^.(R« — 2Rrcos. A + r«)»
_ S^ . S.2r* _
~ R^* "*" R* »^*^ . .
= n^ ultimately. •
434. Call 1 m the addititious force
and m n the ablatitious force
and m n = 1 ra 3 cos. A.
Resolve m n into m q, q n.
The part of the ablatitious force which acts in the direction m q
= m n . cos. A
* — = central ablatitious force.
3 S r
The tangential part = m n . sin. A = "WT" • ^^^' ^ • cos. A
=r  . r^g . sin. 2 A = tangential ablatitious force
'*. the whole force in the direction PT = lm — mq =
R3 ■ R
R
S r
= ^ (1 — 3 COS.* A) and the
3 S . r
whole force in the direction of the Tangent = q n = o" • ~^^ • s^"* ^ A.
435. Hence Cor. 2. is manifest, for of the four forces acting on P, the
Book I.]
NEWTON'S PRINCIPIA.
389
three first, namely, attraction of T, addititious force, and central ablatiti
ous force, do not disturb the equable description of areas, but the fourth
or tangential ablatitious force does, and this is + from A to B, — from B
to C, + from C to D, — from D to A. .*. the velocity is accelerated from A
to B, and retarded from B to C, .*. it is greatest at B. Similarly it is a
maximum at D. And it is a minimum at A and C. This is Cor. 3.
436. To otheriaise calculate the central and tangefitial ablititious forces.
On account of the great distance of S, S M, P L may be considered
parallel, and
.. P T = L M, and S P = S K = ST.
.*. the ablatitious force = 3 P T. sin. <) = 3 P K.
Take P m = 3 P K, and resolve it into P n, n m.
P n = P m . sin. ^ = 3 P T. sin. * ^ = central ablatitious force
= 3 P T. ^ — ^Qs ^ ^
n m = P m . cosv 5 = 3 P T. sin. 8 cos. ^ = ^ . P T. sin. 2 ^ = tangential
ablatitious force.
The same conclusions may be got in terms of 1 m from the fig. in Art
433, which would be better.
437. Find the disturbing force on P in the direction P T.
This = (addititious + central ablatitious) force = 1 m + 3 1 m . sin. ' 6
1 o 1 f^ — cos. 2 ^ \
= lm3lm( )
438. To Jind the mean disturbing force of S during a lohole revolution
in the direction P T.
Let P T at the mean distance = m, then — 1 m T
Y2
1 — 3 cos. 2 ey
340 A COMMENTARY ON [Sect. XI.
2
1 m in . ^ . . , II 1 ,
=r — = — ~ since COS. 2 ^ is destroyed during a whole revo
lution.
439. The disturbing forces on P are
S r
(1) addititious = ^3 = A.
(2) ablatitious = 3 . A . sin. 6
3
8
3 . A
which is (1) tangential ablatitious force — '^ . cos. 2
and (2) central ablatitious force = 3 A . 5— ^
3 A 3 A
.*. whole disturbing force in the direction P T = A — f — . cos. 2 6
= 2 +2. COS. 2^.
But in a whole revolution cos. 2 6 will destroy itself, .*. the whole dis
turbing force in the direction P T in a complete revolution is ablatitious
and = ^ addititious force.
S r
The whole force in the direction P T = ~ (1 — 3 sm.^6) (Art. 433)
= It('t(i™^2»))
S r / 3 3
multiply this by d.^, and the integral = .^ \^$ — ^ ^ + — . sin. 2 ^)
S r v
= sum of the disturbing forces; and this when^=ff becomes :^j . — .
This must be divided by t, and it gives the mean disturbing force act
S r
ing on P in the direction of radius vector = — ^ ^y ,
440. The 2d Cor. will appear from Art. 433 and 434.
3
For the tangential ablatitious force = — . sin 2 ^ . x addititious force,
.*. this force will accelerate the description of the areas from the quadra'
tures to the syzygies and retard it from the syzygies to the quadratures,
since in the former case sin. 2 is +, and in the latter — .
441. Cor. 3 is contained in Cor. 2. (Hence the Variation in as
tronomy.)
Book I.]
NEWTON'S PRINCIPIA.
3i]
442. P V is equivalent to P T, T V, and accelerates the motion ;
p V is equivalent to p T, T V, and retards the motion.
443. CoR. 4. Cast, par., the curve is of greater curvature in the quadra
tures than in the syzygies.
For since the velocity is greatest in the syzygies, (and the central abla
titious force being the greatest, the remaining force of P to T is the least)
the body will be less deflected from a right line, and the orbit will be less
curved. The contrary takes place in the quadratures.
444. The whole force from Sin the direction P T=^^ (1 — 3 sin. 2^)
T
(see 433) and the force from T in the direction P T = ~ .
.*. the whole force in the direction P T =
T S r
and at A this becomes —5 + ^r
r^ R^
„2 ~
R^ ^
3 sin. = 6)
at B
at C
atD
2.S.r
T_
J. 2
R'
S.r
R3
2 S.r
R^
(for though sin. 270 is — , yet its syzygy is +).
Thus it appears that on two accounts the orbit is more curved in the
quadratures than in the syzygies, and assumes the form of an ellipse at
the major axis A C.
Y3
342 A COMMENTARY ON [Sect. XI.
.'. the body is at a greater distance from the center in the quadratures
than in the syzygies, which is Cor. 5.
445. Cor. 5. Hence the body P, caet. par., will recede farther fiom
T in the quadratures than in the syzygies ; for since the orbit is less
curved in the syzygies than in the quadratures, it is evident that the body
must be farther from the center in the quadratures than in the syzygies.
446. CoR. 6. The addititious central force is greater than the ablati
tious from Q' to P, and from P' to Q, but less from P to P', and from
Q to Q', .'. on the whole, the central attraction is diminished. But it
may be said, that the areas are accelerated towards B and D, and .*. the
time through P P' may not exceed the time through P' Q, or the time
through Q Q' exceed that through Q' P. But in all the corollories, since
the errors are very small, when we are seeking the quantity of an error,
and have ascertained it without taking into account some other error,
there will be an error in our error, but this error in the error will be an
error of the second order, and may .*. be neglected.
The attraction of P to T being diminished in the course of a revolution,
the absolute force towards T is diminished, (being diminished by the
S r . . r ^
mean disturbing force — ^ y^ , 439,) .•. the period which « — — , is
increased, supposing r constant.
But as T approaches S (which it will do from its higher apse to the
lower) R is diminished, the disturbing force Twhich involves ^j will be
increased, and the gravity of P to T still more diminished, and .*. r will
be increased; .•. on both accounts (the diminution of f and increase of r)
the period will be increased.
(Thus the period of the moon round the earth is shorter in summer
than in winter. Hence the Annual equation in astronomy.)
When T recedes from S, R is increased, and the disturbing force di
minished and r diminished. .*. the period will be diminished (not in com
parison with the period round T if there were no body S, but in compari
son with what the period was before, from the actual disturbance.)
rp Q
447. Cor. 6. The whole force of P to T in the quadratures =—5+^
, . T 2Sr
the syzygies =^, ^
. . on the whole the attraction of P to T is diminished in a revolution.
For the ablatitious force in the syzygies equals twice the addititious force
in the quadratures.
Book I.] NEWTON'S PRINCIPIA. 343
At a certain point the ablatitious force = the addititious ; when
1 = 3 sin. * S
or
^"••^=V3
and
A = 55°, &c.
P
(the whole force being then = 77 •)
Up to this point from the quadratures the addititious force is greater
than the ablatitious force, and from this point to one equally distant from
the syzygies on the other side, the ablatitious is greater than the addititious ;
.•. in a whole revolution P's gravity to T is diminished.
Again since T alternately approaches to and recedes from S, the radius
r^
P T is increased when T approaches S, and the period a — ^^:r=.
V absolute force
and since f is diminished, and .*. r increased, .\ the periodic time is in
creased on both accounts, (for f is diminished by the increase of the dis
r \
turbing forces which involve ^A If the distance of S be diminished, the
absolute force of S on P will be increased, .'.the disturbing forces which gctyj
from S are increased, and P's gravity to T diminished, and .*. the periodic
time is increased in a greater ratio than r * (because of the diminution of
r^x
fin the expression —7?) and when the distance of S is increased, the dis
turbing force will be diminished, (but still the attraction of P to T will be
diminished by the disturbance of S) and r will be decreased, .•. the
5
period will be diminished in a less ratio than r ^.
448. CoR. 7. To find the effect of the disturbing force on the motion
of the apsides of P's orbit during one whole revolution.
T S . r
Whole force in the direction P T = ^ + V^" ^^ — ^ ^°^* ^ "^^
= , + T.c.r, (if T.c =^3(1— 3 cos." A) = :^:3 ,
1 4 c
.*. the Z between the apsides =180 " by the IXth Sect, which
is less than 180 when c is positive, i. e. from Q' to P and from P' to P^
Y4.
344
A COMMENTARY ON
[Sect. XL
(fig. (446,)) and greater than 180 when c is negative, i. e. from P to F
and from Q to Q',
.*. upon the whole the apsides are progressive, (regressive in the quadra
tures and progressive in the syzygies) ;
T 3 S r
force = — ^ ^3 = force in conjunction
Now
r'^ R
R'T~3Sr3
r^R'
3 . Sr'
y — = force in opposition
and
R'T — 3Sr^^
r'^^R'
differ most from —5 and .^
when r is least with respect to r',
which is the case when the Apsides are in the syzygies.
But
R^T+ Sr^ R^T+ Sr^»
r«R' r'^R^
differ least from — ^ and jj when r is most nearly equal to r',
449. CoR. 7. Ex. Find the angle from the quadratures, when the apses
are stationary.
Draw P m parallel to T S, and = 3 P K, m n perpendicular to T P,
resolve P m into P n, n m, whereof n m neither increases nor diminishes
the accelerating force of P to T, but P n lessens that force, .'. when P n
= P T, the accelerating force of P is neither increased nor diminished,
and the apses are quiescent,
by the triangles PT: PK::PM = 3PK: Pn = PT
.*. in the required position 3 P K^ = P T'
or
P T
PK=^^=PT.sin.tf,
Book I.] NEWTON'S PRINCIPIA. 346
or
6 = 35° 26'.
The addititious force P T — P n is a maximum in quadratures.
ForPT:PK::3PK:Pn = ^p^'
3 P K*
.'.FT — Pn = PT p „ , which is a maximum when P K = 0,
or the body is in syzygy.
450. Cor. 8. Since the progression or regression of the Apsides de
pends on the decrement of the force in a greater or less ratio than D % from
the lower apse to the upper, and on a similar increment from the upper
to the lower, (by the IXth Sect.), and is .*. greatest when the proportion
of the force in the upper apse to that in the lower, recedes the most from the
inverse square of D, it is manifest that the Apsides progress the fastest from
the ablatitious force, when they are in the syzygies, (because the whole forces
in conjunction and opposition, i. e. at the upper and lower apses being
—^ 53 , when the apsides are in the syzygies and when r is greatest
T
at the upper apse, — being least, and the negative part of the expression
2 S r
^ 3 being greatest, the whole expression is .*. least, and when r is least,
T
at the lower apse, —5 being greatest, and the negative part least, .*. the
whole expression is greatest, and .•. the disproportion between the forces at
the upper and lower apse is greatest), and that they regress the slowest
T S r
in that case from the addititious force, (for ~„ } ^j^ , which is the whole
force in the quadratures, both before and after conjunction, r being the
semi minor axis in each case, differs least from the inverse square) ; there
fore, on the whole the progression in the course of a revolution is greatest
when the apsides are in the syzygies.
Similarly the regression is greatest when the apsides are in the quadra
tures, but still it is not equal to the progression in the course of the re
volution.
451. CoR. 8. Let the apsides be in the syzygies, and let the force
at the upper apse : that at the lower, : : D E : A B, DA'
346
A COMMENTARY ON
being the curve whose ordinate is inversely
as the distance * from C, .*. these forces being
diminished, the force D E at the upper apse
2 r S
by the greatest quantity  ^ 3 , and tlie force
A B at the lower apse by the least quantity
p 3 ; the curve a d which is the new force
curve has its ordinates decreasing in a
greater ratio than w^ .
Let the apsides be in the quadratures, then the force E D will be increased
S r
by the greatest quantity rrji and the force A B by the least quantity
S r'
pj , .*. the curve a' d' which is the new force curve will have its
ordinates decreasing in a less ratio than ^p— .
451. Cor. 9. Suppose the line of apsides to be in quadratures, then while the
body moves from a higher to a lower apse, it is acted on by a force which
1 R^T+Sr^
does not increase so fast as r=r^ (for the force = ^tt^ » •*• the
jjz \ r ^ R^
numerator decreases as the denominator increases), .*. the orbit will be
exterior to the elliptic orbit and the excentricity will be decreased. Also as
Sr
the descent is caused by the force
R
(1 — 3^ cos. ^ A), the less this
force is with respect to —^ , the less will the excentricity be diminished.
Now while the line of the apsides moves from the line of quadratures, the
S r
force t> 3 ( 1 — 3 cos. ^ A) is diminished, and when it is inclined at z. 35"
16' the disturbing force = 0, and .*. at those four points the excentricity
is unaltered. After this, it may be shown in the same manner that the
excentricity will be continually increased until the line of apsides coin
cides with the line of syzygies. Here it is a maximum, since the disturb
ing force is negative. Afterwards it will decrease as before it increased
until the line of apsides again coincides with the quadrature, and then the
excentricity = maximum.
(Hence Evection in Astron.)
Book I.] NEWTON'S PRINCIPIA. 347
452. Lemma. To calculate that part of the ablatitious force which is
employed in drawing P from the plane of its orbit.
Let A = angular distance from syzygy.
Q = angular distance of nodes from syzygy.
I = inclination of orbit to orbit of S and T.
3 S r
Then the force required = „ 3 • . cos. A . sin. Q . sin. L (not quite
accurately.)
When P is in quadratures, this force vanishes, since 00s. A = 0.
When nodes are in syzygy, since sin. Q = 0,
quadratures, this force (cast, par.) = maxi
mum, since sin. Q = sin. 90 = rad.
453. CoR. 12. The effects produced by the disturbing forces are all
greater when P is in conjunction than when in opposition.
For they involve „^, .•. when R is least, they are greatest.
454. CoR. 13. Let S be supposed so great that the system Pand T re
volve round S fixed. Then the disturbing forces will be of the same kind
as before, when we supposed S to revolve round Tat rest.
The only difference will be in the magnitude of these forces, which will
be increased in the same ratio as S is increased.
455. Cor. 14. If we suppose the different systems in which S and S T
a, but P T and P and T remain the same, and the period (p) of P round
T remains the same, all the errors « ^^ ex ^ , if a = density of S,
and d its diameter,
a 3 3, if A given, and B = apparent diam.
also
1 S
■pi ^ R^ if P = period of T round S,
.*. the errors <^ p^ . *
These are the linear errors, and angular errors oc in the same ratio,
since P T is given.
456. Cor. 15. If S and T be varied in the same ratio,
S T
Accelerating force of S : that of T : : ^r^ : — , the same ratio as before.
° K* r ^
.*. the disturbances remain the same as before.
(The same will hold if R and r be also varied proportionally.)
.*. the linear errors described in P's orbit oc P T, (since they involve r),
if P T a,the rest remaining constant.
31.8
A COMMENTARY ON
[Sect. XL
also the angular errors of P as seen from T oc — oc __ « i,
and are .*. the same in the two systems.
The sunilar linear errors oc f . T % .*. P T oc f . T ^ and f «
P T P T
Tpv , but f a accelerating force of T on P oc — ~ , (p = period of P
round T,)
.'.Tap and .*. « P
(forP^a^aflJ ap2)
Cor. 14. In the systems
S, T, P, Radii R, r Periods P, p
S', T, P R',r PVp.
Linear errors dato t. in 1st. : do. in second
.*. angular errors in the period of P 
Cor. 15. In the systems
S, T, P, R, r
S', Ty P R', r'
1
p2
1
p/ i
1
p/2*
 S' T' , R'
so that ^v = rp and ^ =
• • p/  p/ •
Linear errors in a revolution of P in 1 st. :
angular errors
CoR. 16. In the systems
S, T, P, R, r
S, T', P, R, I''
F,P'
r
do. in second
: r : r
: 1 : 1.
— P,P
— P,P'.
Linear errors in a revolution of P in 1st. : do in second
angular errors in a revolution of P :
To compare the systems
(1) S, T, P R, r P, p
(2) S', T', P' R', r' P', p'.
Assume the system
(3) S', T, P R', r P', p
r p
P'
r' p'
•p
2
p/«
.*. by (14) angular errors in P S revolution in (1) : in (3) : ; ^ : p;
by (16) angular errors in (3) : in (2) : : p^ : p"^
P^ P'*
therefore errors in (1) : in (2) : ". ^j * pTS*
Book I.] NEWTON'S PRINCIPIA. . 349
Or assume the system (3) 2, T, P — ^ , r — II, p
so that g, = ^, R> = 7>
1 1 S . 1 . . S^ . R '
R3 • e3 •• 2 • p3
/. the errors in (1) ; errors in (3) : : p^ : —
(3) : (2) : : 1 : 1
.S^S^. R3 R;^^.^ T' R' r^3
••S' 2 • R''* s' '*S' • T • R'3* r^
" R 3 T ■ R' 3 * T * * P 2 • P' 2 •
457. CoR. 16. In the different systems the mean angular errors of
P a — whether we consider the motion of apses or of nodes (or errors
in latitude and longitude.)
For first, suppose every thing in the two different systems to be the same
except P T, .*. p will vary. Divide the whole times p, p', into the same
number of indefinitely small portions proportional to the wholes. Then if
the position of P be given, the disturbing forces all a each other a P T ;
and the space a f . T ^ .*. the Linear errors generated in any two corre
sponding portions of time oc P T . p ^.
.*. the angular errors generated in these portions, as seen from T, « p *.
.•• Comp°. the periodic angular errors as seen from T x p ^
Now by Cor. 14, if in two different systems P T and .*. p be the same,
every thing else varying, the angular errors generated in a given time, as in
1
.*. neutris datis, in different systems the angular errors generated in the
tune p oc SI •
Now
■n/f . Iff . .
pa p
_/, . p, . . e: . i
.*. the angular errors generated in V (or the mean angular errors) or p^.
Hence the mean motion of the nodes as seen from T oc mean motion
of the apses, for each oc ^ •
458. CoR. 17.
Mean addititious force : mean force of P on T : : p * : P*.
For
mean addititious force : force of S on T : : P T : S T,
350
A COMMENTARY ON
[Sect. XI.
Sr
\" R
force of S on T : mean force of T on P: :
ST
S
PT
R
)
(force a ^)
.*. mean addititious force : mean force of T on P: : p '^ : P '^
.*. ablatitious force : mean force of T on P: :3 cos. tf * p ^ : P.
Similarly, the tangential and central ablatitious and all the forces may
be found in terms of the mean force of T on P.
459. Prop. LXVII. Things behig as in Prop. LXVI, S describes
the areas more nearly proportional to the times, and the orbit more ellipti
cal round the center of gravity of P and T than round T.
P , T
For the forces on S are
PS'
and
TS
.*. the direction of the compound force lies between S P, ST; and T
attracts S more than P.
.*. it lies nearer T than P, and .*. nearer C the center of gravity of T
and P.
.*. the areas round C are more proportional to the times, than when
round T.
Also as S P increases or decreases, S C increases or decreases, but S T
remains the same ; .*. the compoimd force is more nearly proportional to
the inverse square of S C than of S T ; .*. also the orbit round C is more
nearly elliptic (having C in the focus) than the orbit round T.
/
A
SECOND COMMENTARY
ON
SECTION XI.
460. To find the axis major of an ellipse, whose periodic time round
S at rest would equal the periodic time of P round S in motion.
Let A equal the axis major of an ellipse described round P at rest
equal the axis major of P Q v.
Let X equal the axis major required,
P. T. of P round S in motion : p S at rest : : V S : \^ S + P
P. T. of p in the elliptic axis A : P. T. in the elliptic axis x : : A « : x *
.. p. T. of P round S in motion : P.T. in the elax. x : : VATS : Vx'(«+P).
By hyp. the 1st term equals the 2d,
.. A»S = x'. S + P
.. A:x::(S+P)*: si
461. Prop. LXIII. Having given the velocity, places, and directions
of two bodies attracted to their common center of gravity, the forces vary
ing inversely as the distance % to determine the actual motions of bodies in
fixed space.
Since the initial motions of the bodies are given, the motions of the center
of gravity are given. And the bodies describe the same moveable curve
round the center of gravity as if the center were at rest, while the center
moves uniformly in a right line.
♦ Take therefore the motion of the center proportional to the time,
i. e. proportional to the area described in moveable orbits.
* Since a body describes some cunre in fixed space, it describes areas in proportion to the times
in this curve, and since the center moves umformly forward, the spaca described by it is is pro
portion to the time, therefore, &c.
352
A COMMENTARY ON
[Sect. XI.
462. Ex. 1. Let the body P describe a circle round C, while the center C
moves uniformly forward. Take C G : C P : : v of C : v of P, and with the
center C and rad. C G describe a circle G C N, and suppose it to move
round along G H, then P will describe the trochoid P L T, and when P
has described the semicircle P A B, P will be at the summit of the trochoid
.*. every point of the semicircumference G F N will have touched G H,
.•. G H equals the semicircumference G F N,
.. V of P : V of C : : P A B semicircumference : C ll = G F N semicircle
* : : C P : C G Q. e. d,
463. Ex. 2. Let the moveable curve ^^^P
be a parabola, and let the center of gravity
move in the direction of its primitive
axis. When the body is at the vertex
A', let S' be the position of the center
of gravity, and while S' has described
uniformly S' S, let A have described the
arc of the parabola A P.
Let A' N = X, N P = y, be the abA' S'
scissa and ordinate of the curve A P in fixed space.
Let 4 p equal the parameter of the parabola A P.
.. A N = ^, A' S = S'S = X _y = iEil3!
4p 4p 4p
SN = AN — AS= AN
xL^w y*"~^p*
4 p ^ .4 p
AreaASP=ANP— SNP=ANx N P— i N S X NP
^ "» — 4p'y __ y^+ 12p«y
4p
.9 y:
"3
ly
4 p 24 p
By Prop. S' S cd A S P ; therefore they are in some given ratio.
y^ + i2p'y 4 px — y«
24 p * 4 p
Let A S P : S' S : : a : b
• If C P = C G the curve in fixed space becomes the common cycloid.
If C P >. C G the ollongated trochoid.
Book I.]
NEWTON'S PRINGIPIA.
y'f 12p*y = 4pax — ay'
353
.. y'+ ay2+ 12 p^ y — 4 pa x = C.
Equation to the curve in fixed space.
464. Ex. 3. * Let B B' be the orbit of the earth round the sun, M A
that of the moon round the earth, then the moon will, during a revolution,
trace out a contracted or protracted epicycloid according as A L has a
greater or less circumference than A M, and the orbit of the moon round
tlie sun will consist of twelve epicycloids, and it will be always concave to
the sun. For
F of the earth to the sun : F of the mdbn to the earth : : rr^
400
1
"•(365) 2* (27)^
in a greater ratio than 2 : 1. But the force of the earth to the sun is
nearly equal to the force of the moon to the sun, .*. the force of the moon
to the earth, .. the deflection to the sun will always be within the tan
gential or the curve is always concave towards the sun.
465. Prop. LXVI. If three bodies attract each other with forces
varying inversely as the square of the distance, but the two least revolve
• To determine the nature of the curve described by the moon with respect to the sun.
Tot. I. Z
354 A COMMENTARY ON [Sect. XI.
about the greatest, the innermost of the two will more nearly describe the
areas proportional to the time, and a figure more nearly similar to an el
lipse, if the greatest body be attracted by the others, than if it were at rest,
or than if it were attracted much more or much less than the other bodies.
(L M : P T : : S L : S P,
PT
.'. L M Qc
SP
3 »
T M  PT X SL _ SK^xPTv
.. SK' : SP» :: SL : SP).
Let P and S revolve in the same plane about the greatest body T, and
P describe the orbit P A B, and S, E S E. Take S K the mean distance
of P from S, and let S K represent the attraction of P to S at that dis
tance. Take SL : SK :: SR* : SPS and SL will represent the
attraction of S on P at the distance S P. Resolve it into two S M, and
L M parallel to P T, and P will be acted upon by three forces P T, L M,
S M. The first force P T tends to T', and varies inversely as the dis
tance % .*. P ought by this force to describe an ellipse, whose focus is T.
The second, L M, being parallel to P T may be made to coincide with it
in this direction, and .*. the body P will still, being acted upon by a centri
petal force to T, describe areas proportional to the time. But since L M
does not vary inversely as P T, it will make P describe a curve different
from an ellipse, and .*. the longer L M is compared with P T, the more
will the curves differ from an ellipse. The third force S M, being neither
in the direction P T, nor varying in the inverse square of the distance, will
make the body no longer describe areas in proportion to the times, and the
curve differ more from the form of an ellipse. The body P will .*. describe
areas most nearly proportional to the times, when this third force is a
minimum, and P A B will approach nearest to the form of an ellipse, when
both second and third forces are minima. Now let S N represent the
attraction of S on T towards S, and if S N and S M were equal, P and
T being equally attracted in parallel directions would have relatively the
same situation, and if S N be greater or less then S M, their difference
M 'N is the disturbing force, and the body P will approach most nearly
the equable description of areas, and P A B to the form of an ellipse,
when M N is either nothing or a minimum.
Case 2. If the bodies P and S revolve about T in different planes, L M
being parallel to P S will have the same effect as before, and will not
Book I.]
NEWTON'S PRINCIPIA.
355
tend to move P from its plane. But N M acting in a different plane,
will tend to draw P out of its plane, besides disturbing the equable des
cription of areas, &c. and as before this disturbing force is a minimum,
when M N is a minimum, or when S N = nearly S K.
466. To estimate the magnitude of the disturbing forces on P, when P
moves in a circular orbit, and in the same plane with S and T.
Let the angle from the quadratures P C T = ^,
P C
S T = d, P T = r, F at the distance (a) = M,
t;, t3 Ma*
.*. From P in the direction S P : P T : : S P
.*. F in the direction P T = ^^' v £5^
ButSP* = d^ + r*— 2drsin. ^,
.. F m the direction P T =
M a*r
PT,
(d^ + r' — Sdrsin. Of
Ma^rf, r« — 2drsin. rf
{ii^
}
"■ d^ I « d
_ Ma' r . , .
— 33 = A nearly, smce d bemg indefinitely great compared with r
in the expansion, all the terms may be neglected except two. First i
d
vanishes when compared with ^3, .. the addititious force in the direction
F T = A. By proportion as before, force in the direction S T
__M^ST___ Ma ^d f
SP'SP d^ (1 + rr*_2dr sin. ^,
)
Ma^ / 1 _ 1 r« — 2drs in. 6y
d^ \ 2 d^ J
Ma
d«
3 M a * r 2 3 Ma'r sin. <?
2d
22
356
A COMMENTARY ON
[Sect. XL
f ■ *u r .■ en. Ma«3Ma«r. ^ , .
.. torce in the direction b 1 = — y^ — j jj — sin. t nearly, since
Tj vanishes when compared with r , and the force of S on T = — ~ ,
,, . . ^ Ma' , 3Ma r . ^ Ma*
.*. ablatitious F = — rj 1 rj — • ""• ^ TT"
= 3 A . sin. 6.
If P T equal the addititious force, then the ablatitious force equals 3 P K,
for PK: PT::sin. ^: (1 = r),
.. 3 P K = 3 P T . sin. ^ = 3 A . sin. 6.
To resolve the ablatitious force. Take
P m : P n : : P T : T K : : 1 : cos. 6,
3 A
.. P n = P m X cos. ^ = 3 A X sin. 6 cos. 9 = — . sin. 2 &
mn = PmX PK = 3A. sin. « = 3 A . ^ — ^Q^. 2 6 ^
.*. the disturbing forces of S on P are
M a ' r
1. The addititious force = — p — = A.
2. The ablatitious force which is resolved into the tangential part
= —^ . sin. 2 6f and that in the direction T P = 3 A . ^ —  — ,
.*. whole disturbing force in the direction P T = A — 3 A . —  —
= A Q — I — 5— . cos. 2 6 = — ■] — —  . COS. 2 ^, and in the whole
revolution the positive cosine destroys the negative, therefore the whole
disturbing force in a complete revolution is ablatitious, and equal to one
half of the mean addititious force.
467. To compare N M and L M.
L M : P T : : (S L = ^') : S P,
.. L M = g p, X P T
Book I.]
NEWTON'S PRINCIPIA.
357
MN = ^3XSTST = ^^g~3^^' xST
__SK^— (SK — KP)^
SP
X ST
_ SK^— SK' + 3SR'xK P^^^^ ,
= q p3 X o 1 nearly
3SK«xPK^^„  3SK3^„^
— X S r nearly = ^ ^, X P K
SP^
3SK3 „^
= g^3 X P T X sm. 6,
.. M N : L M : : 1 : 3 sin. 6.
SP =
468. Next let S and P revolve about T in different planes, and let
N P N' be P's orbit, N N' the line of the nodes. Take T K in T S =
3 A . sin. 6. Pass a plane through T K and turn it round till it is per
pendicular to P's orbit. Let T e be the intersection of it with P's orbit.
Produce T E and draw K F perpendicular to it, .*. K F is perpendicular
to the plane of P's orbit, and therefore perpendicular to every line meet
ing it in that orbit, T in the plane of S's orbit ; draw K H perpendicular
to N' N produced ; join H F, then F H K equals the inclination of the
planes of the two orbits. For KHT, KFT, KFH being all right angles,
KT« = KH* + HT»
K F*+ H« = K F« + FH« + HT^
*.% FT* = F H» + HT*,
.*. F H is peipendicular to H T.
Since PT=A, TK = Ax sin. i
• Let the angle KHT=T, HTKc=^ = angular distance of the line cf the ncdt«
from S y z.
Z3
358
A COMMENTARY ON
[Sect. XI.
P T : T K : : 1 : 3 sin. d
T K : K H : : 1 : sin. <p
K H : K F : : 1 : sin. T,
.*. P T : K F : : 1 : 3 sin. ^. sin. f . sin. T,
.% ablatitious force perpendicular to P's orbit = K F
= 3 P T X sin. 6. sin. (p x i^in. T = 3 A X sin. (J. sin. <p X sin. T.
2d. Hence it appears that there are four forces acting on P.
C
m
\
f
\
"~^
Nyp///
m'"
\
V^„/
^
^
D^
V
^.^
T
T
^
m'
v
yp"
m"
D
1. Attraction of P to T a
2. Addititious F in the direction P T =:
M
a' r
3. Ablatitious F in the direction P T =
3 Ma^r
sin. * L
4. Tangential part of the ablatitious force =
Ma'
sm.
Of these the three first acting in the direction of the radiusvector do
not disturb the equable description of areas, the fourth acting in the di
rection of a tangent at P does interrupt it.
Since the tangential part of F is formed by the revolution of P M = 3 A X
sin. ^ at C, tf = 0, therefore P m = 0, and consequently the tangential
F = ; from C to A, P n is in consequentia, and therefore accelerates
the body P at A, it again equals 0, and from A to D is in antecedentia,
and therefore retards P; from D to B it accelerates; from B to C it re
tards.
Therefore the velocity of P is greatest at A and B, because these are
the points at which the accelerations cease and retardations begin, and
the velocity is least at D and C. To find the velocity gained by the ac
tion of the tangential force.*
dZ= Fdx = fA. sin. 2 ^ d ^
* F in the direction P T is a maxunum at the quadratare, because the ablatitious F in the
quadrature Is 0, and at every other point it is something.
Book L] NEWTON'S PRINCIPIA. 959
sin. 2 ^ X 2 f)' = — (cos. 2 6)',
V
.*. Z =  — = Cor. — x A. COS. 2 ^.
2g ^
But when ^ = 0, the tangential F = 0, and no velocity is produced,
.*. COS. 2 <i = R = ],
V
3 A
.. ^— =— r (1 — COS. 2 6) = I A. 2 sin. ^ ^,
2 g 4 ^ ' *
.'. V* = 3 g A, sin.* 6,
.*. V = V 3 g A. sin. &,
.'. v' oc (sin. dy,
.*. whole f on the moon at the mean distance : f of S on T
1 A
and the force of S on T : add. f at the mean distance (m) : : ^ : ^^ ,
.*. whole f at the mean distance : m : : P * : p * and —^ x whole f &c. = m.
f ci r
Now f on the moon at any distance (r) = — ^ — ^^rj and at the mean
distance (1) = f — ^^3 = f — ^ ,
p2f mp2
. . Ill — p J .J p 2 '
.*. m ==
ps 2 P'
2p^f
2 P^ + p
2»
and therefore nearly = 2P~* '
.'. m r
f p2 2 p* 1
(which equals the addititious force) = f. r.  p^ W*'\ '
469. To compare the ablatitious and addititious forces upon the moon,
with the force of gravity upon the earth's surface. (Newton, Vol. III.
Prop. XXV.)
add. f : fofSonT : : P T : S T
f of S on T : f of the earth on the moon : : ^rr • ^r — — ir »
P* p" p
.'. add. f : f of the earth on the moon : : p'^ : P*
f of the earth on the moon : force of gravity : : 1 : 60 ^,
.. add. f : force of gravity : : p« : P^ 60» . . . (I)
Also ablat. f : addititious force : : 3 P K : P T,
.. ablat. f : addititious force : : 3 P K . p « : 60 ^ P T. P * . (2)
470. Cor. 2. In a system of three bodies S, P, T, force oc^ ^, the
Z4
360 A COMMENTARY ON [Sect. XI.
body P will describe greater areas in a given time at the syzygies than at
the quadrature.
The tangent ablatitious f = f . P T . sin. 2 6 ; therefore this force will
accelerate the description of areas from quadratures to syzygies and retard
it from syzygies to quadratures, since in the former case sin. 2 ^is positive,
and in the latter negative.
CoR. 3. is contained in Cor. 2.
The first quadrant d. sin. being positive the velocity increases,
in the second d. sin. negative the velocity decreases, &c. for the 1st Cor.
2d Cor. &c.
Also V is a maximum when sin. 6 is a maximum, i. e. at A and B.
471. Cor. 4. The curvature of P*s orbit is greater in quadratures than
in the syzygy. .
mi. 1 1 T1 T^ Ma^ , Ma^r 3Ma«r,, _ ,. ^
The whole F on P = ^ + —^ gjj (1 — cos. 2 0) X
/3 M a ^ r . sin. 2 ^\
V 2"d^ )'
In quadratures sin. 3^=0,
••• ^  r« + d^
And in syz. 2 9= 180,
.*. sin. 2^ = 0, cos. 2 ^= 1
SMa^r ,, ^,, 3Ma2r
*u u 1 T? T» • xu Ma* 2Ma*r
.'. the whole 1? on P in the syz. = — ^ ?
.'. F is greater in the quadratures than in the syzygies; and the velocity
is greater in the syzygies than in the quadratures.
1 F
But the curvature a p^ a ^ ^ , .*. is greatest in the quadratures and
least in the syzygies.
472. CoR. 5 Since the curvature of P's orbit is greatest in the quadra
ture and least in the syzygy, the circular orbit must assume the form of an
ellipse whose major axis is C D and minor A B
.*. P recedes farther from T in the quadrature than in the syzygy.
473. Cor. 6.
MflS Ma'r SMa'r
The whole F on P in thelinePT=:^+^^^ — • ^3 'Sin.»^
, M a*^ . Ma*r
= m quad. —5 + — js—
Book 1.]
NEWTON'S PRINCIPIA.
set
M a « 2 M a * r
and m syz. = ~^^ jj—
let the ablatitious force on P equal the addititious, and
Ma«r 3 M a^r
.*. sin. 6 =
1
V 3
. sin. * 6
sin. aS". 16.
Therefore up to this point from quadrature the ablatitious force is less
than the addititious, and from this to one equally distant from the other
point of quadrature, the ablatitious is greater than the addititious, therefore
in a whole revolution the gravity of P to T is diminutive from what it
Ri
would be if the orbit were circular or if S did not act, and P a , . ,— ^
V abl. F
and since the action of S is alternately increased or diminished, therefore
P a from what it would be were P T constant, both on account of the
variation, and of the absolute force.
474. CoR. 7. ♦ Let P revolve round T in an elliptic orbit, the force on
„., , Ma=Ma*r.b.
P in the quad. = ^j H jj— + jjtj + c r.
' b 4 c
•'• G + 180 / . and since the number is greater than the de
nomination G is less than 180. .♦. the apsides are regressive if the same
effect is produced as long as the addititious force is greater than the abla
titious, i. e. through 35°. 16'.
The force on P in the syz. = M^' ^ ^ f " = J^ 2 cr
• Since P a
rI
— and in winter the sun is nearer the earth than in summer.
y' ablatitious force
R is Increased in winter, and A is diminished, therefore the lunar months are shoiter in winter
than in summer.
362 A COMMENTARY ON [Sect. XI.
.. G = 180 . 1 ,1'' > 1800
.•. in the sjz. the apsides are progressive, and since ^ r will be
ah improper fraction as long as the ablatitious force is greater than the
addititious, and when the disturbing fdrces are equal, m c =r n c, therefore
G = 180°, i. e. the hue of apsides is at rest (or it lies in V C produced
9th.) .*. since they are regressive through 141°. 4' and progressive
218°. 56' they are on the whole progressive.
To find the effect produced by the tangential ablatitious force, on the
velocity of P in its orbit. Assume u = velocity of a body at the mean
distance 1, then —  = velocity at any other distance r nearly, the orbit
being nearly circular.
Let V be the true velocity of P at any distance (r), vdv = gFdx
(I = 16 jg . For the tangent ablatitious f = f . P T . 2 ^, and x' = r ^')
= 3 P T.mr.sin. 2 6.6',
.. v=' = — 3PTmr cos. 2 ^ + C,
and
C = 2
r
2»
v''=^ — &c.
r
.2
Hence it appears that the velocity is greatest in syzygy and least in
quadrature, since in the former case, cos. 2 6 is greatest and negative, and
in the latter, greatest and positive.
To find the increment of the moon's velocity by the tangential force
while she moves from quadrature to syzygy.
v2 = —3 PT.m.r. cos. 2 ^ + C,
but (v) the increment = 0, when ^ = 0,
.. C = 3 P T . m . r,
.'. v« = 3 P T . m . r (1— cos. 2 0) = 6 P T. m. r. sin.«^,
and when 6 = 90°, or the body is in syzygy v ' = 6 P T m . r.
475. Cob. 6. Since the gravity of P to T is twice as much diminished
in syzygy as it is increased in quadrature, by the action of the disturbing
force S, the gravity of P to T during a whole revolution is diminished.
Now the disturbing forces depend on the proportion between P T and
T S, and therefore they become less or greater as T S becomes greater
Book 10 NEWTON'S PRINCIPIA. 363
or less. If therefore T approach S, the gravity of P to T will be still
more diminished, and therefore P T will be the increment.
5
R^
Now P . T a ■ ; since, therefore, when S T is di
V absolute force
minished, R is increased and the absolute force diminished (for the ab
solute force to T is diminished by the increase of the disturbing force) the
P . T is increased. In the same way when S T is increased the P . T is
diminished, therefore P . T is increased or diminished according as S T
is diminished or increased. Hence per. t of the moon is shorter in winter
than in summer.
OTHERWISE.
476. CoH. 7. To find the effect of the disturbing force on the motion
of the apsides of P's orbit during a whole revolution.
Let f = gravity of P to T at the mean distance (1), then — = gravity
of P at any other distance r.
f f
Now in quadrature the whole force of P to T = — + add. f = — j + r
f r + r * . . . . /'f+i
■ 4
and with this force the distance of the apsides = 180° / w—
which is less than 180°, therefore the apsides are regressive when the
f
body is in quadrature. Now in syz. the whole force of P to T = — —
f J. 2 r"*
2 r =r 3 , therefore the distance between the apsides = 180°
^ Ti which is greater than 180°, therefore the apsides are progressive
when the body is in syzygy.
But as the force (2 r) which causes the progression in syzygy is double
the force (r) which causes the regression in quadrature, the progressive
motion in syzygy is greater than the regressive motion in the quadrature.
Hence, upon the whole, the motion of the apsides will be progressive
during a whole revolution.
At any other point, the motion of the apsides will be progressive or
P T 3 P T
retrograde, according as the whole central force 5 —  5 — . cos. 2 6
is negative or positive.
364
A COMMENTARY ON
[Sect. XI.
477. Cor. 8. To calculate the disturbing force when P*s orbit is ex
centric
P T 3 P T
The whole central disturbing force = \ cos. 2 ^ =
+ — rt— • COS. 2 ^ (ra IS the mean add. f). Now r = ^
2 ' 2
z= by div. 1 — e ^ + e . cos. u + e *. cos.
e* e
volving e^, &c. = 1 ~ + e. cos. u + — . cos. 2 u; therefore the
e cos. u
u, &c. neglecting terms in
m
whole central disturbing force = — ^ +
2
m e'
m
COS. u
me* COS. 2 u
m COS. 2 d ■
3 m e'
. COS. 26 •\ —m e. cos. u . cos. 2 6
4 • 2
+ f m e . COS. 2 u . cos. 2 6.
478. Cor. 8. It has been shown that the upsides are progressive in
syzygy in consequence of the ablatitious force, and that they are regres
sive in quadrature from the effect of the ablatitious force, and also, that
they are upon the whole progressive. It follows, therefore, that the
greater the excess of the ablatitious over the addititious force, the more will
the apsides be progressive in the course of a revolution. Now in any
position m M of the line of the apsides, the excess of the ablatitious in
conjunction =^ 2 A T in opposition = T B, therefore the whole excess
= 2 A B. Again, the excess of the addititious above the ablatitious force
in quadrature = C D. Therefore the apsides in a whole revolution will
be retrograde if 2 A B be less than C D, and progressive if 2 A B be
greater than C D. Also their progression will be greater, the greater the
excess of 2 A B above C D ; but the excess is the greatest when M m is
in syzygy, for then A B is greatest and C D the least. Also, when M m
is in syzygy the apsides being progressive are moving in the same direc
tion with S, and therefore will remain for some length of time in syzygy.
Again, when the apsides are in quadrature A B = P p, and C D = M ni,
Book I.] NEWTON'S PRINCIPIA. 366
but if the orbit be nearly circular, 2 A B is greater than C D ; therefore
the apsides are still in a whole revolution progressive, though not so
much as in the former case.
F
In orbits nearly circular it follows from G = —7= when F a A p  ^,
V r
that if the force vary in a greater ratio than the inverse square, the
apsides are progressive. If therefore in the inverse square they are sta
tionary, — if in a less ratio they are regressive. Now from quadrature to
35° a force which oc the distance is added to one varying inversely as
the square, therefore the compound varies in a less ratio than the inverse
square, therefore the apsides are regressive up to this point. At this point
F a r. ^ , therefore they are stationary. From this to 35*> from
Qistance
another D a quantity varying as the distance is subtracted from one
varying inversely as the square, therefore the resulting quantity varies
in a greater ratio than the inverse square, therefore the apsides axe
progressive through 218°.
OTHERWISE.
4T9. CoR. 8. It has been shown that the apsides are progressive m
syzygy in consequence of the ablatitious force, and that they are regressive
in the quadratures on account of the addititious force, and they are on the
whole progressive, because the ablatitious force is on the whole greater
than the addititious. .. the greater the excess of the ablatitious force
above the addititious the more will be the apsides progressive.
In any position of the line A B in conjunction the excess of the ablati
tious force above the addititious is 2 FT, in opposition 2 p t. .*. the whole
excess in the syzygies = 2 P p. In the quadratures at C the ablatitious
force vanishes. .*. the excess of the addititious = additions = C T.
.*. the whole addititious in the quadratures = C D.
Now the apsides will, in the whole revolution, be progressive or regres
sive, according as 2 P p is greater or less than C D, and then the progres
sion will be greatest in that position of the hne of the apses when 2 P p —
C D is the greatest, i. e. when A B is in the syzygy, for then 2 P p =
2 A B, the greatest line in the ellipse, and C D = R r = ordinate =
least through the focus. .*. 2 P p — CD is a maximum. Also when
A B is in the syzygy, the line of apsides being progressive, will move the
same way as S. .*. it will remain in the syzygy longer, and on this account
the apsides will be more progressive. But when the apsides are in the
quadratures S P = R r and C D = A B, and the orbit being nearly
circular, R r nearly equals A B. .'. 2 P p — C D is positive, and the
366 A COMMENTARY ON [Sect. XI.
apsides are progressive on the whole, though not so much as in the last
case ; and the apsides being regressive in tlje quadratures move in the op
posite direction to S, .*. are sooner out of the quadratures, .*. the regres
sion in the quadrature is less than the progression in the syzygy.
480. Cor. 9. Lemma. If from a quantity which gc tj any quantity
be subtracted which oc A the remainder will vary in a higher ratio than
the inverse square of A, but if to a quantity varying, as ^^ another be
added which oc A, the sum will vary in a lower ratio than ^ .
J ... 1 c A*
If , . be diminished C A = j; . If A increases 1 — c A '
A* A'' .
decreases, and rj increases. .*. the quantity decreases, I — c A increases
1
and Tr increases. .. increases from both these accounts. .*. the whole
^ .... 1
quantity varies in a higher ratio than ^ .
1 4 c A *
If C A be added ^ — , as A is increased the numerator increases,
and ^ decreases. .*. the quantity does not decrease so fast as ^^ , and
if A be diminished 1 + c A * is diminished, and ^ increased. .'. the
quantity is not increased as fast as ^^ . .•. &c. Q. e. d.
OTHERWISE.
481. Cor. 9. To find the effect of the disturbing force on the excen
tricity of P's orbit. If P were acted on by a force a p , the excentricity
of its orbit would not be altered. But since P is acted on by a force vary
ing partly as r^ and partly as the distance, the excentricity will continual
ly vary.
Suppose the line of the apsides to coincide with the quadrature, then
while the body moves from the higher to the lower apse, it is acted upon
by a force which does not increase so fast as r, , for the force at the quad
f
rature = — + m r, and .*. the body veill describe an orbit exterior to the
elliptic which would be described by the force a rj . Hence the body
Book L] NEWTON'S PRINCIPIA. 367
will be farther from the focus at the lower apse than it would have been
had it moved in an elliptic orbit, or the excentricity is diminished. Also
as the decrease in excentricity is caused by the force (m r), the less this
f
force is with respect to —^ , the less will be the diminution of excentricity.
Now while the line of apsides moves from the line of quadratures, the force
(m r) is diminished, and when it is inclined at an angle of 35° 16' the
disturbing force is nothing, and .*. at those four points the excentricity
remains unaltered. After this it may be shown in the same manner that
the excentricity will be continually increased, until the line of apsides
coincides with the syzygies. Hence it is a maximum, since the disturbing
force in these is negative. Afterwards it will decrease as before it in
creased, until the line of apsides again coincides with the line of quadra
ture, and the excentricity is a minimum.
CoR. 14. Let P T = r, S T = d, f = force of T on P at the distance
1, g = force of S on T at the distance, then the ablatitious force
3 ff r sin. d .^ . , . /. t^ i • . i . i , , .
= — ^^— p ; II .'. the position ot P be given, and d vanes, the ablati
tious force a Vg . But when the position of P is given, the ablatitious
: addititious : : in a given ratio, .*. addititious force a ^ , or the dis
turbing force a t^ . Hence if the absolute force of S should oc the dis
turbing force cc ^r~ . Let P = the periodical time of T about S,
1 1 1 P
•*• pT ^ — A3 • ^^^ ^ ~ density, d = diameter of the sun, then the
A ^ 3 1
absolute force a A ^ ', then the disturbing force a — j^cc p^ a A (ap
parent diameter)^ of the sun. Or since P T is constant, the linear as well
as the angular errors a in the same ratio.
483. Cor. 15. If the bodies S and T either remain unchanged, or their
absolute forces are changed in any given ratio, and the magnitude of the
orbits described by S and P be so changed that they remain similar to
what they were before, and their inclination be unaltered, since the accel
erating force of P to T : accelerating force of S : : p~T"2 '
c~*T''2 > ^^^ *^^ numerators and denominators of the last
terms are changed in the same given ratio, the accelerating forces remain
in the same ratio as before, and the linear or angular errors a as before.
368
A COMMENTARY ON
[Sect. XI.
i e. ns the diameter of the orbits, and the times of those errors oc P T's
of the bodies.
Cor. 1 6. Hence if the forms and inclinations of the orbits remain, and
the magnitude of the foixes and the distances of the bodies be changed ; to
find the variation of the errors and the times of the errors. In Cor. 14.
it was shown, how that when P T remained constant, tiie errors a ^Ti •
sr
Now let P T also a , then since the addititious force in a given position
of P (X P T, and in a given position of P the addititious : ablatitious in
a ffiven ratio.
CoK. If a body in an ellipse be acted u}x>n bv a force which varies
in a ratio greater than the inverse
square of the distance, it will in de
scending from the higher apse B to the
lower apse A, be drawn nearer to the
center. .*. as S is fixed, the excen
tricity is increased, and from A to B
the excentricity will be increased
also, because the force decreases the faster the distance* increases.
484. (CoR. 10.) Let the plane of P's orbit be inclined to the plane of T's
orbit remaining fixed. Then the addititious force being parallel to P T,
is in the same plane with it, and .'. does not alter the inclination of the
plane. But the ablatitious force acting from P to S may be resolved into
two, one parallel, and one perpendicular to the plane of P's orbit. The
force perpendicular to P's orbit = 3 A X sin. 6 X sin. Q x sin. T
when d = perpendicular distance of P from the quadratures, Q = angular
distance of the line of the nodes from the syzygy, T = first inclination of
the planes.
Hence when the line of the nodes is in the syzygy, ^ = 0,
.*. no force acts perpendicular to the plane,
tmd the inclination b not changed. When
the line of the nodes is in the quadratures,
= 90", .*. sin. is a maximum, .*. force per
pendicular produces the greatest change
in the inclination, and sin. being posi
tive from C to D, the force to change the
inclination continually acts from C to D
pulling the plane down from D to C. Sin. d
is negative, .*. force which before was posi
sin. =
Book L] NEWTON'S PRINCIPIA. 369
tive pulling down to the plane of S's orbit (or to the plane of the paper)
now is negative, and .*. pulls up to the plane of the paper. But P's orbit is
now below the plane of the paper, .•. force still acts to change the inclina
tion. "Now since the force from C to D 'continually draws P towards the
plane of S's orbit, P will arrive at that plane before it gets to D.
If the nodes be in the octants past the quadrature, that is between C
and A. Then from N to D, sin. 6 being positive, the inclination is di
minished, and from D to N' increased, .•. inclination is diminished through
270°, and increased through 90", .*. in this, as in the former case, it is
more diminished than increased. When the nodes are in the octants be
fore the quadratures, i. e. in G H, inclination is decreased from H to C,
diminished from C to N, (and at N the body having got to the highest
point) increased from N to D, diminished from D' to N', and increased
from 2 N' to H, .*. inclination is increased through 270°, and diminished
through 90°, .*. it is increased upon the whole. Now the inclination of
P's orbit is a maximum when the force perpendicular to it is a minimum,
i. e. when (by expression) the line of the nodes is in the syzygies. When
is the quadratures, and the body is in the syzygies, the least it is increased
when the apsides move from the syzygies to the quadratures ; it is dimin
ished and again increased as they return to the syzygies.
485. (Cor. 11.) While P moves from the quadrature in C, the nodes
being in the quadrature it is drawn towards S, and .*. comes to the plane
of S's orbit at a point nearer S than N or D, i. e. cuts the plane before it
arrives at the node. .•. in this case the line of the nodes is regressive. In
the syzygies the nodes rest, and in the points between the syzygies and
quadratures, they are sometimes progressive and sometimes regressive,
but on the whole regressive; .*. they are either retrograde or stationary.
486. (CoR. 12.) All the errors mentioned in the preceding corollaries are
greater in the syzygies than in any other points, because the disturbing
force is greater at the conjunction and opposition.
487. (CoR. 13.) And since in deducing the preceding corollaries, no re
gard was had to the magnitude of S, the principles are true if S be so
great that P and T revolve about it, and since S is increased, the disturbing
force is increased ; .*. irregularities will be greater than they were before.
488. (CoR. 14.) L M = ^^^ = N N M = ^ ^^^f "" sin. 6, .. in
a given position of P, if P T remain unaltered, the forces N M and L M
Voi„ I. A a
S70 A COMMENTARY ON [Sect. XI.
1 1 ^3
« j3 X absolute force oc ^^^^^^, of T for (sect. 3 . P* oc absolute f. )
whether the absolute force vary or be constant. Let D = diameter of S,
d =z density of S, and attractive force of S a magnitude or quantity of
matter oc D ^ 3,
.*. forces L M and N M a
d
But— r = apparent diameter of S,
.'. forces Qc (apparent diameter) ' d another expression.
489. (Cor. 15.) Let another body as P' revolve round T' in an orbit
similar to the orbit of P round T, while T' is carried round S' in an orbit
similar to that of T round S, and let the orbit of P' be equally inclined to
that of T' with the orbit P to that of T. Let A, a, be the absolute forces
of S, T, A', a', of S', T',
A a
accelerating force of P by S : that of P by T : : c^pi : pFpa ,
and the orbits being similar
A' a
accelerating force of P' by S' : that of P' by T' : : ^pm • prrpTi »
.•. if A' : a' : : A : a, and the orbits being similar,
SP : PT* :: S' F : FT^
accelerating force of P' by S' : that of P' by T'
: : force on P by S : force on P' by T',
and the errors due to the disturbing forces in the case of P are as
A . A'
■^rjTs X r, in the case of P' and S' are as ^71^73 X R,
.•. linear errors in the first case : that in the second : : r : R.
. , sin. errors
Angular errors oc ^5 ,
angular errors in the first case : that in the second : : 1 : L
XT /. o T V ^ s linear errors
Now Cor. 2. Lem. X. T* a 7^^
angular errors „
a^ — ^ X R,
.. T * QC angular errors,
.. angular errors : 360 : : T ^ : P *,
.'. T ' a P * X angular errors,
.'.Tec P for = angular errors. •
Book I.] NEWTON'S PRINCIPIA. 371
490. (Cor. 16.) Suppose the forces of S, P T, ST to vary in any man
ner, it is required to compare the angular errors that P describes in simi
lar, and similarly situated orbits. Suppose the force of S and T to be
constant, .\ addititious force oc P T, .*. if two bodies describe in similar
orbits = evanescent arcs. Linear errors oc p * X P T.
.. angular errors cc p ^ (p = per. time of P round T, P = that of T
round S). But by Cor. 14. if P T be given, the absolute force of A and
SToc.
Angular errors cc pv
.'. if P T, ST and the absolute force alternately vary,
angular errors a ^ ,
•P = per. time of P round T) ^ M a= r
/ r = per. time ot P round T "> ^
Vp = per. time of T round S J
1 linear errors
angular errors x
radius
M a^ r
.. lin. errors oc force T» * — ^l — X P* by last Cor.
I rP« P\
.. angular errors oc ^ ,^^cc ^ j .
Now the errors d t X p = whole angular errors x ~ .
.'. error d t x ^^ thence the mean motion of the apsides x mean motion
of the nodes, for each x p^ , for each error is formed by forces varying as
proof of the preceding corollaries, both the disturbing forces, and .•. the
errors produced by them in a given time will a P T. Let P describe an
indefinite small angle about T (in a given position of P), then the linear
errors generated in that time x force T P time % but the time of describ
ing = angles about T x whole periodic time (p), .*. linear errors x
P T p ^ and as the same is true for every small portion, similar; the
linear errors during a whole revolution x P T p ^ Angular errors
x j ' .'. oc p * .'. when S T, P T, and the absolute force vary, the
angular errors x ^j a — ^^ r.. 3 ■■ a q 'Ps (^'^^" ^^^ absolute force is
Aa8
872 A COMMENTARY ON [Sect. XL
given.) Now the error in any given timexp varies the whole errors during
a revolution a ^ • .*. the errors in any given time a ^^ . Hence the
mean motion of the apsides of P's orbit varies the mean motion of the
nodes, and each will a ^ the excentricities and inclination being small
and remaining the same.
491. (CoR. 17.) To compare the disturbing forces with the force of
PtoT.
F of S on T : F of P on T
absolute F
a
ST' • T P'
absolute F .. A. S T . aT P
axis major ' * * S S ' ' T P '
.. ST . TP .. A . JL
• p* • p, =: pg ' p.
mean add. F : F of S on T : : ~^ : ^^ : : r : d
.. mean add. F : F P on T : : p « : P «.
492. To compare the densities of different planets.
Let P and P' be the periodic times of A and B, r and r' their distances
from the body round which they revolve.
F of A to S : F of B to S : : ^, : ~
quantity of matter in A do. in B D
distance *
!r in A do. in B
D 3 of Ax density ^ D ^ of Bx density
* distance ^ '
distance' ' distance'
r r'
•
• p 2 • p/ a
D'Xd D''xd'
1 1
r' • r" '
• p « • p/ 2
.. d : d' :
r' r'*
• J)3 pS J)/3p/2
1 1
§3 p2 S'jp/t
where S and S' represent the apparent diameters of the two planets.
493. In what part of the moon's orbit is her gravity towards the earth
unaffected by the action of the sun.
„ Ma'' . Ma'r 3Ma*r U — cos.'^ . 3Ma'r . ,.
^=r + d' d^ — 2— +^^^'"^
M a'
and when it is acted upon only by the force of gravity = for die
other forces then have no effect.
Book I.]
NEWTON'S PRINCIPIA.
373
M a» r 3 M a^r 1 — cos. 2 6 3 M a' r .
=
1 — 3.
i
COS. 26,2.^^
g + — sm. 2^ =
3 3 3
1 —  + S COS. 26 + ~ sin. 2 () =
3.31 — sin.'C . 8 . „
8 + 2 2 + 2«''2«=0
Let X = sin. 6,
(.. 1
and
+
I sin. * ^ + I X 2 sin. 6 x cos. 6 = 0)
3x
+ 3xVl — x* = 0.
An equation from which x may be found.
494. Lemma. If a body moving towards a plane given in position, be
acted upon by a force perpendicular to its motion tending towards that plane,
the inclination of the orbit to the plane will be increased. Again, if the body
be moving from the plane, and the force acts from the plane, the inclina
tion is also increased. But if the body be moving towards the plane, and
the force tends from the plane, or if the body be moving from the plane,
and the force tends towards the plane, the inclination of the orbit to the
plane is diminished.
495. To calculate that part of the ablatitious tangential force which is
employed in drawing P from the plane of its orbit.
Let the dotted line upon the ecliptic N A P N' be that part of P's orbit
which lies above it. Let C D be the intersection of a plane drawn per
\)endicular to the ecliptic ; P K perpendicular to this plane, and there
Aa3
S74
A COMMENTARY ON
[Sect. XI.
fore parallel to the ecliptic. Take T F = 3 P K ; join P F and it will
represent the disturbing force of the sun. Draw P i a tangent to, and
F i perpendicular to the plane of the orbit. Complete the rectangle i m,
and P F may be resolved into P m, P i, of which P m is the effective force
to alter the inclination. Draw the plane F G i perpendicular to N N' ;
then F G is perpendicular to N N'. Also F i G is a right angle. As
sume P T tabular rad. Then
::R:3g^..
: : R : s > .
: : R : i J '*
PT: Pm :: R3 : 3g. s. i
_ PT.3g. s. i
R'
Pm =
PT : TF:: R:3g•
T F : F G
FG : Pm
g = sin. 6 = sin. a dist. from quad.
s = sin. p = sin. l. dist. of nodes from syz.
i = sin. F T i = sin. F G i = sin. inclination of orbit to ecliptic.
Hence the force to draw P from its orbit =
P. 3
R
when P is in
the quadratures. Since g vanishes this force vanishes. "When the nodes
are in the syzygies s vanishes, and when in the quadratures this force is a
maximum. Since s = rad. cotan. parte.
496. To calculate the quantity of the forces.
Let S T = d, P T = r, the mean distance from T = 1. The force
of T on P at the mean distance = f ; the force of S on P at the mean
distance = g.
Then the force S T = ^,, and the force S T : f P T : : d : r,
.. force P T = , hence the add. f = ^; ablat. f = ^ sin. 6, the
mean add. force at distance J = ^s> the central ablat. = jg sin. * d, the
tangential ablat. f = 5^^ . sin. 2 6.
Book I.l NEWTON'S PRINCIPIA. 375
The whole disturbing force of S on P = Kp + orfT • <^os. 2 6; the
s r , . ^ . • 1 V ni
— g r 3 g r
~2dJ' "^ 2 d '
mean disturbing f = ■ ^3  (since cos. 2 ^ vanishes) = — — by supposi
tion.
Hence we have the whole gravitation of P to T = — 5 — ^71 + q^t ^
COS. 2 ^, and the mean = —1 — #r; (since cos. 2 tf vanishes).
r' 2 d* ^
PROBLEM.
497. Required the whole effect, and also the mean effect of the sun to
diminish the lunar gravity; and show that if P and p be the periodic
times of the earth and moon, f the earth's attraction at the mean
distance of the moon, r the radiusvector of the moon's orbit ; the additi
tious force will be nearly represented by the formula j p^ — 2P*J ^^*
Pn=3PT. sin.«^,andPT — 3PT.sin.M = — ^ +.PT x
cos. 2 d = whole diminution of gravity of the moon, and the mean di
= ^l
Again,
T) rip
minution = * —^  — ^3 by supposition.
P* a d'
ab. f d ,,, J
•■•"d^ « pi • V)d. seq.
498. To find the central and ablatitious tangential forces.
Take Pm = 3PK = 3PT. sin. = ablatitious force.
Then P n = P m . sin. ^ = 3 P T . sin. * 6 = central force
m n = P m . cos. 6 =z 3 P T . sin. 6 . cos. 6
= I . P T sin. 2 6 = tangential ablatitious force.
To find what is the disturbing force of S on P.
S76 A COMMENTARY ON [Sect. XI.
'— l + 3cos.2 0\
The disturbing force = P T — 3 P T . sin. '6 = Q
P T ^
PT = — ig^ + I^P T. COS. 2 6.
To find the mean disturbing force of S during a whole revolution.
P T 3
Let P T at the mean distance = m, then — + — . P T cos. 2 &
= —  = —  — since cos. 2 6 is destroyed during a whole revolution.
499. To find the disturbing force in syzygy.
SAT — AT = 2AT = disturbing force in syzygy ;
the force in quadrature is wholly effective and equal P T,
.*. force in quadrature : f in syzygy : : P T : 2 P T : : 1 : 2.
To find that point in P's orbit when the force of P to T is neither
increased nor diminished by the force of S to T.
In this point Pn= PTor3PT sin. « tf = P T,
.•. sm. 6 =  —
V 3
and
6 = 35° 16'.
To find when the central ablatitious force is a maximum.
P n = 3 P T . sin. * 5 = maximum,
.*. d . (sin. * ^) or 2 sin. 6 . cos. & — d ^ = 0,
.*. sin. d . cos. ^ = 0,
or
sm. 6. V I — sin. * ^ = 0,
and
sin. 6 =z ly
or the body is in opposition.
Then (Prop. LVIII, LIX,)
T « : t « : : S P : C P : : S + P : S
and
and
T' : f^ :: A' : x'
A' : x' :: S+ P : S
A : X ::(S+P)^ : si
500. Prob. Hence to correct for the axis major of the moon's orbit.
Let S be the earth, P the moon, and let per. t of a body moving in a
secondary at the earth's surface be found, and also the periodic time of
Book L] NEWTON'S PRINCIPIA. 377
the moon. Then we may find the axis major of the moon's orbit round
the earth supposed at rest = x, by supposition. Then the corrected axis
or axis major round the earth in motion : x : : ( S + P) ^ : S ^
1
(S + P) ^
.•. axis major round the earth in motion = x . ^ = y.
S^
Hence to compare the quantity of matter in the earth and moon,
y : x : : V S + F : V S
.•.y ^ — X"
: : P : S.
501. To define the addititious and ablatitious forces. Let S T repre
sent the attractive force of T to S. Take
1 1
S L : S T
ST*: SP*
S P^ ■ S T^
and S L will represent the attractive force of P to S. Resolve this into
S M, and L M ; then L M, that part of the force in the direction P T
is called the addititious force, and S M — S T = N M is the ablatitious
force.
502. To compare these forces.
Since S L : S T : : S T« : S P^ .. S L = ^^ = attractive force of
P to S in the direction S P, and S P : S T
ST' ST
JS p2 g p3
= attractive
force of P to S in the direction TS=ST*(ST — PK)"' =ST
+ 3 P K = S M nearly,
.\3PK = TM = PL = ablatitious force = 3 P T . sin. 6.
O 'T>3 Q TS
Also SP:PT::~: 1^3.
P T = attractive force of P to S in the direction L M = P T nearly.
Hence the addititious force : ablatitious force : : P T : 3 P T . sin. 6 : 1
: 3 sin. $. Q. e. d.
BOOK III.
1. Prop. I. All secondaries are found to describe areas round the
primary proportional to the time, and these periodic times to be to each
other in the sesquiplicate ratio of their radii. Therefore the center of
force is in the primary, and the force a
i'
2. Prop. II. In the same way, it may be proved, that the sun is the
center of force to the primaries, and that the forces a r — » . Also the
dist. *
Aphelion points are nearly at rest, which would not be the case if the
force varied in a greater or less ratio than the inverse square of the dis
tance, by principles of the 9th Section, Book 1st.
3. Prop. III. The foregoing applies to the moon. The motion of the
moon's apogee is very slow — about 3° 3' in a revolution, whence the force
will X j^p2 2irs • It was proved in the 9th Section, that if the ablatitious
force of the sun were to the centripetal force of the earth : : I : 357.45,
that the motion of the moon's apogee would be ^ the real motion.
.*. the ablatitious force of the sun : centripetal force : : 2 : 357.45
: : 1 : 178 f^.
This being very small may be neglected, the remainder oc yyi •
4. CoR. The mean force of the earth on the moon : force of attraction
::177^: 178§.
The centripetal force at the distance of the moon : centripetal force at
the earth : : I : D *.
5. Prop. IV. By the best observations, the distance of the moon from
the earth equals about 60 semidiameters of the earth in syzygies. If the
moon or any heavy body at the same distance were deprived of motion in
the space of one minute, it would fall through a space = 16 ,V f^et. For the
380
A COMMENTARY ON
[Book III.
deflexion from the tangent in the same time = ^^ rs feet. Therefore the
space fallen through at the surface of the earth in I" =: 16 ^^ feet.
For 60" : t : : D : 1,
60'' _ , .
— *■ t
.: t =
60
thence the moon is retained in its orbit by the force of the earth's gravity
like heavy bodies on the earth's surface.
6. Piiop. XIX. By the figure of the earth, the force of gravity at
the pole : force of gravity at the equator : ; 289 : 288. Suppose A B Q q
a spheroid revolving, the lesser diameter P Q, and A C Q q c a a canal
filled with water. Then the weight of the arm Q q c C : ditto of
A a c C : : 288 : 289. The centrifugal force at the equator, therefore 1
suppose 2^^ of the weight.
Again, supposing the ratio of the diameters to be 100 : 101. By com
putation, the attraction to the earth at Q : attraction to a sphere whose
radius = Q C : : 126 : 125. And the attraction to a sphere whose ra
dius A C : attraction of a spheroid at A formed by the revolution of an
ellipse about its major axis : ; 126 : 125.
The attraction to the earth at A is a mean proportional between the at
tractions to the sphere whose radius = A C, and the oblong spheroid,
since the attraction varies as the quantity of matter, and the quantity of
matter in the oblate spheroid is a mean to the quantities of matter in the
oblong spheroid and the circumscribing sphere.
Hence the attraction to the sphere whose radius = A C : attraction to
the earth at A : : 126 : 125 .
.*. attraction to the earth at the pole : attraction to the earth at the equa
tor : : 501 : 500.
Now the weights in the canals a whole weights oc magnitudes X gra
Book III.] NEWTON'S PRINCIPIA. 381
vity, therefore the weight of the equatorial arm : weight of the polar
: : 500 X 101 : 501 X 100
: : 505 : 501.
4
Therefore the centrifugal force at the equator supports ^^r^ to make an
equilibrium.
But the centrifugal force of the earth supports —^ ,
= the excess of the equatorial over the polar
Hence the equatorial radius : polar : : 1 + ^^ : 1
: : 230 ; 229.
Again, since when the times of rotation and density are diflPerent the
V*
difference of the diameter a j , and that the time of the earth's rota
dens.
tion = 23h. 56'.
The time of Jupiter's rotation = 9h. 56'.
The ratio of the squares of the velocity are as 29 : 5, and the density
of the earth : density of Jupiter : : 400 : 94.5.
d the difference of Jupiter's diameter is as — X ^t= X 5^ ,
4 1
• ' 505 • 289 '
1
• 100
1
229
radius.
.*. d : Jupiter's least diameter : : — X ^^r X qHq
The polar diameter : equatorial diameter
29 X 80 : 94.5 X 229
2320 : 21640
232 : 2164
I : n
H : 10^
ON THE TIDES.
7. THE PHENOMENA OF THE TIDES.
1. The interval between two succeeding high waters is 12 hours 25
minutes. The diminution varies nearly as the squares of the times from
high water.
2. Twentyfour hours 50 minutes may be called the lunar day. The
interval between two complete tides, the tide day. The first may be call
S8«
A COMMENTARY ON
[Book III.
O
ed the snperior, the other inferior, and at the time of new moon, the
morning and evening.
3. Tlie high water is when the moon is in S. W. to us. The highest tide
at Brest is a day and a half after full or change. The third full sea after
the high water at the full moon is the highest ; the third after quadrature
is the lowest or neap tide.
4. Also the highest spring tide is when the moon is in perigee, the next
spring tide is the lowest, since the moon is nearly in the apogee.
5. In winter the spring tides are greater than in summer, and from the
same reasoning the neap tides are lower.
6. In north latitude, when the moon's declination is north, that tide in
which the moon is above the horizon is greater than the other of the same
day in which the moon is below the horizon. The contrary will take
place if either the observer be in south latitude or the moon's declination
south.
7. Prop. I. Suppose P to be any
particle attracted towards a center E,
and let the gravity of E to S be repre
sented by E S. Draw B A perpendi
cular to E S, which will therefore re
present the diameter of the plane of il B
lumination. Draw Q P N perpendicu
lar to B A, P M perpendicular to E C.
Then take E I = 3 P N, and join P I,
P I will represent the disturbing force
of P. PI may be resolved into the
two P E, P Q, of which P E is counter
balanced by an equal and opposite force,
P Q acts in the direction N P.
Hence if the whole body be supposed
to be fluid, the fluid in the canal N P
will lose its equilibrium, and therefore
cannot remain at rest. Now, the equi
librium may be restored by adding a
small portion P p to the canal, or by
supposing the water to subside round
the circle B A, and to be collected to
wards O and C, so that the earth may put on the form of a prolate sphe
roid, whose axis is in the line O C, and poles in O and C, which may be
( N
E
P\
/
iVi y^^
P
Q
c
I
s
Book III.] NEWTON'S PRINCIPIA. 383
the case since the forces which are superadded a N P, or the distance
from B A, so that this mass may acquire such a protuberancy at O and C,
that the force at O shall be to the force at B : : E A : E C ; and by the
above formula
x^ _ 5C _ EC — E A
r ~ 4g " E A
8. Prop. II. Let W equal the terrestrial gravitation of C; G equal its
gravitation to the sun; F the disturbing force of a particle acting at O and
C ; S and E the quantities of matter in the sun and earth.
3 S C
.. F : W
• CS* X CGCE^
1
Since the gravitation to the sun « ,. — ,
° dist. '
CS^rES^:: ES: CG
.. CG X CS^ = ES'.
3 S E
.. F : W :
ES^* CE^
and
E : S : : 1 : 338343
E C:ES: : 1 : 23668
3 S E
~: : 1 : 12773541 : : F : W.
"ES^" CE =
.. 4W: 5F ::CE:EC — EA.
4 d 3d
Attraction to the pole : attraction to the equator : : 1 k • ^ k~
Quantity of matter at the pole : do. at equator : : 1 : 1 — d.
Weight of the polar arm : weight of the equatorial arm : : 1 ^ ^ 1 k^
.'. Excess of the polar = attractive force : weight of the equator or
A
5
4 d
mean weight W : : —^ : 1
. . _ 5F
* 4W *
9. Prop. III. Let A E a Q be the spheroid, B E b Q the inscribed
384 A COMMENTARY ON [Book III.
sphere, A G a g the circumscribed sphere, and D F d f the sphere equal
(in capacity) to the spheroid.
Then since spheres and spheroids are equal to f of their circumscribing
cylinder, and that the spheroid = sphere D F d f.
CF^xCD = CE2xCA
CE'':CF«:: CD:CA,
and make
but
Also
CE:CF::CF:Cx
..CE'^: CF^: : C E : Cx
.♦. CD:CA::CE:Cx
.. C D : C E : : C A : C X
C D = C E nearly
... C A = C X.
E X = 2 E F nearly
.. A D = 2 E F.*
LetCE = a,CF = a + x,
.. Cx =
a*42axfx» __ a*42 x
a a
= a + 2 X neaily
. •. E X = 2 X nearly.
Book III.] NEWTON'S PRINCIPIA. 385
Prop. IV. By the triangles p I L, C I N,
A B: IL::r': (cos.)«z.TC A
.. I L = A B X (cos.) '^ ^. I C A = S X (cos.) ' x
(if S = A B and x = angular distance from the sun's place.)
Again,
G E : K I : : r « : (sin.) « ^ T C A
.. K I = S X (sin.) ^^K.
Cor. 1. The elevation of a spheroid above the level of the undisturbed
g
ocean = 11 — 1 m = S X (cos.) ^ x —  = S X (cos.) ^ x — ^.
The depression of the same = S X (sin.) * x — S = S X (sin.) '^ x — .
Cor. 2. The spheroid cuts the sphere equal in capacity to itself in a
S
point where S X (cos.) * x = — = 0, or (cos.) ^ x = ^.
.. cos. X = .57734, &c.
= COS. 54°. 44'.
10. Prop. V. The unequal gravitation of the earth to the moon is
(4000) ^ times greater than towards the sun.
Let M equal the elevation above the inscribed sphere at the pole of
the spheroid, 7 equal the angular distance from the pole.
.'. the elevation above the equally capacious sphere = Mx (cos.) ^7 — ^
the depression = M X (sin.) '^ 7 — .
Hence the effect of the joint action of the sun and moon is equal to the
sum or difference of their separate actions.
.. the elevation at any place = S X (cos.) ' x4 M X (cos.) ^7 — ^ S + M
the depression = S X (sin.) ^ x + M X (sin.) ^ 7 —  S+M.
1. Suppose the sun and moon in the same place in the heavens.
Then the elevation at the pole = S + M — i S + M =  S + M, and
the depression at the equator = S + M —  S + M = J S + M,
,'. the elevation above the inscribed sphere = S + M.
2. Suppose the moon to be in the quadratures.
The elevation at S = S — J S+M = I S —  M.
the depression at M = S — f S+M = i S — § M,
the elevation at S above the inscribed sphere = S — M,
the elevation at M (by the same reasoning) = M — S.
But (by observation) it is found that it is high water under the moon
when it is in the quadratures, also that the depression at S is below the
natural level of the ocean ; hence M is more than twice S, and although
Vol. I. B b
386
A COMMENTARY ON
[Book III.
the high water is never directly under the sun or moon, when the moon is
in the quadratures high water is always 6 hours after the high water at
full or change.
Suppose the moon to be m neither of the former positions.
Then the place of high water is where the elevation = maxim urn,
or when S X cos. ' x + M X cos. * y = maximum,
and since
cos. * X = ^ + ^ cos. 2 X,
and
COS. y = ^ + ^ cos. 2 y,
elevation = maximum, when S X cos. 2 x + M X cos. 2 y = max
imum.
Therefore, let A B S D be a great circle of the earth passing through
S and M, (those places on its surface which have the sun and moon in the
zenith). Join C M, cutting the circle described on C S in (m). Make
S d : d a : : force of the moon : force of the sun (which force is supposed
Book III.] NEWTON'S PRINCIPIA. 387
kuown). Join m a, m d, and let H be any point on the surface of the
ocean. Join C H cutting the circle C m S in (h) ; draw the diameter
h d h', and draw m t, a x perpendicular to h h', and a y parallel to it.
Then
M = Sd, 8= ad
and
and
AMCH = y, aSCH = x,
.. ^mdh = 2zMCH = 2y
^adx = /^SdH = 2x.
.. d t = M X COS. 2 y, d X = ^ X cos. 2 x,
.♦. elevation = maximum when t x = a y =: maximum,
or wlien a y = a m, i. e. when h h' is parallel to a m, hence
CONSTRUCTION.
Make
S d : d a : : M : S,
and join m a, draw h h' parallel to a m, and from C draw C h H cutting
the surface of the ocean in H, which is the point of high water.
Again, through h' draw L C h', meeting the circle in L, U; these are
the points of low water. For let
LCS = u, LCM = z.
COS. Z. a dx = COS. a S d h' = cos. 2 z:. S C h' = cos. 2 u =r d x
and
cos. 2 z = COS. 2 L C M = d t.
.♦. S X COS. 2 u + M X COS. 2 z = max.
Cor. If d f be drawn perpendicular to a m, a m represents the whole
difference between high and low water, a f equals the point effected by the
sun, m f that by the moon.
For
sin. ^ u = cos. * X,
sin. * y = cos. * x.
*. elevation + depression = S X : cos. ^ x — ^ + M X : cos. * y
+ S X COS. ^ X — §
+ M X : COS. * y — f = S X : 2 cos. * x — 1 + M X : 2 cos. ^ y
= S X cos. 2 X + M X cos. 2 y
and
d t = M X COS. 2 y
d X = S X COS. 2 X.
Bb2
388
A COMMENTARY ON
[Book III.
12. Conclusions deduced from the above (supposing that both the sun
and moon are in the equator.)
1. At new and full moon, high water will be at noon and midnight.
For in this case C M, a m, C S, d h, C H, all coincide.
2. When the moon is in the quadrature at B, the place of high water is
also at B under the moon, when the moon is on the meridian, for C M is
perpendicular to C S, (m) coincides with C, (a m) with (a C), d h with
dC.
3. While the moon passes from the syzygy to the quadrature the place
of high water follows the moon's place, and is to the westward of it, over
takes the moon at the quadratures, and is again overtaken at the next
syzygy. Hence in the first and third quadrants high water is after noon
or midnight, but before the moon's southing, and in second and fourth vice
versa.
4. iL M C H = max. when S C H = 45o. S d h' = 90°. and m' a
perpendicular to S C, and /I a m' d rr max., and a m' d — m' d h'rr 2 y'.
Book III.] NEWTON'S PRINCIPIA. 389
Hence in the octants, the motion of the high water = moon's easterly
motion; in syzygy it is slower; in quadratures faster. Therefore the tide
day in the octants = 24h. 50' = the lunar day ; in syzygy it is less = 24h.
35'; in quadratures = 25h. 25'.
For take any point (u) near (m), draw u a, u d, and d i parallel to a u
and with the center (a) and radius a u, describe an arc (u v) which may
be considered as a straight line' perpendicular to am; u m and i h are
respectively equal to the motions of M and H, and by triangles u m v,
d m f.
um:ih::ma:mf.
Therefore the synodic motion of the moon's place : synodic motion ot
high water : : m a : m f.
Cor. 1. At new or full moon, m a coincides with S a, and m f with S d ;
at the quadratures, m a coincides with C a, and m f with C d ; therefore
the retardation of the tides at new or full moon : retardation at quadra
tures ::Sa:Ca::M + S:M — S.
Cor. 2. In the octants, m a is perpendicular to S a, therefore m a, m f
coincide. Therefore the synodic motion of high water equals the synodic
motion of the moon.
CoR. 3. The variation of the tide during a lunation is represented by
(m a) ; at S, m a = S a, at C = C a.
Therefore the spring tide : neap tide : : M + S : M — S.
CoR. 4. The sun contributes to the elevation, till the high water is in
the octants, after which (a f ) is — v e, therefore the sun diminishes the
elevation.
CoR. 5. Let m u be a given arc of the moon's synodic motion, m v is
the difference between the tides m a, u a corresponding to it.
Therefore by the triangles m u v, m d f.
mu:mv::md:df.
.*. m v Qc d f ;
and since
m d : d f : : r : sin. d m f : : r : sin. m d h : : r : sin. 2 M C H
m v a sifi. 2 arc M H.
13. Prop. VI. In the triangle m d a, m d, d a and ^ m d a arc known
when the proportion M : S is known and the moon's elongation.
Let the angle m d a = a,
and make
M + S : M — S : : tan. a : tan. b
Bb3
890 A COMMENTARY ON [Book III.
then
__ a — b __ a + b
y  —2—' ^  ""F"*
For
M + S : M — S : : m d + d a : m d — d a
mad+amd mad — amd
: : tan. ^ : tan. —
2
2x + 2y 2x — 2y
: : tan. = i ^ : tan. ^
: : tan. x + y : tan. x — y
: : tan. a : tan. b,
x+y:x — y::a:b,
2x=ra + b, 2y = a — b,
a + b
and
a— b
y = 2
14. Prop. VII. To find the proportion between the accelerating forces
of the moon and sun. 1st. By comparing the tide day at new and full
moon with the tide day at quadratures.
35 : 85 : : M : S,
nr lix 35 + 85 85 — 85 ^ „ .
... M : M : : ^ : ^ : : 5 : 2^2.
Also, at the time of the greatest separation of high water from the moon
in the triangle m' d a, m d : d a : : r : sin. 2 y : : M : S,
.•.jj = sm.2y,
at the octants y is found =12° 30',
... 2 = sin. 25°,
M
.*. M : S : : 5 : 2^ nearly.
) Hence taking this as the mean proportion at the mean distances of the
moon and sun (if the earth =1) the moon = «77 •
Cor. 1. If the disturbing forces were equal there would be no high or
low water at quadratures, but there would be an elevation above the in
scribed spheroid all round the circle, passing through the sun and moon
=: f M + S.
Book 111.] NEWTON'S PRINCIPIA. 391
Cor. The gravitation of the sun produces an elevation of 24 inches,
the gravitation of the moon produces an elevation of 58 inches.
.'. the spring tide = 82 inches, and the neap tide = 33 inches.
15. CoR. 3. Though M : S : : 5 : 2, this ratio varies nearly from (6 : 2)
to 4 : 2, for supposing the sun and moon's distance each = 1000.
In January, the distance of the sun = 983, perigee distance of the
moon = 945.
In July, the distance of the sun = 1017, apogee distance of the moon
= 1055.
1
Disturbing force oc j^,; hence
,S M
apogee 1.901 4.258
mean 2 5
perigee 2.105 5.925.*
5 a' d'
The general expression isM= — 8x^73X7^.
To find the general expression above.
Disturbing force of different bodies (See Newton, Sect. 11th, p. 66,
Cor. 14.) a i,
.*. disturbing force S : disturbing force at mean distance : : D^ : A'
disturbing force M : disturbing force at mean distance : : d ^ : 3 ^
. M
5
d^
d'
.. g .
2
'"j).
' A
M 5 A3 d^
S ~ 2 ^ D^ ^ 63'
T%/r 5 ^ A^ d'
.. M = 2^ X S X ^3 X p
(or supposing that the absolute force of the sun and moon are the same).
16. Prop. VIII. Let N Q S E be the earth, N S its axis, E Q its equa
tor, O its center ; let the moon be in the direction O M having the de
clination B Q.
* The solar force may be neglected, but the variation of the moon's distance, and proportion
ally the variation of its action, produces as eifect on the times, and a much greater on the heighta
of the tides.
Bb4
Sd3
A COMMENTARY ON
[Book III.
Let D be any point on the surface of the earth, D C L its parallel of
latitude, N D S its meridian ; and let B' F b' f be the elliptical spheroid
of the ocean, having its poles in O M, and its equator F O f.
As the point D is carried along its parallel of latitude, it will pass
through all the states of the tide, having high water at C and L, and low
water when it comes to (d) the intersection of its parallel of latitude with
the equator of the watery spheroid.
Draw the meridian N d G cutting the terrestrial equator in G. Then
the arc Q G (converted into lunar hours) will give the duration of the
ebb of the superior tide, G E in the same way the flood of the inferior.
N. B., the whole tide G Q C, consisting of the ebb Q G, and the flood
G Q is more than four times G O greater than the inferior tide.
Cor. If the spheroid touch the sphere in F and f, C C is the height
at C, L L' the height at L, hence if L' q be a concentric circle C q will
be the difference of superior and inferior tides.
CONCLUSIONS DRAWN FROM PROP. VIII.
1. If the moon has no declination, the duration of the inferior and su
perior tides is equal for one day over all the earth.
2. If the moon has declination, the duration of the superior will be
longer or shorter than the duration of the inferior according as the
moon's declination and the latitude of the place are of the same or differ
ent denominations.
3. When the moon's declination equals the colatitude or exceeds it,
Book III.] NEWTON'S PRINCIPIA. 393
there will only be a superior or inferior tide in the same day, (the paral
lel of latitude passing through f or between N and f.)
4. The sin. of arc G O = tan. of latitude X tan. declination.
For
rad. : cot. d O G : : tan. d G : sin. G O,
.'. sin. G O =r cot. d O G X tan. G d
= tan. declination X tan. latitude.
17. Prop. IX. With the center C and radius C Q (representing the
P
whole elevation of the lunar tide) describe a circle which may represent
the terrestrial meridian of any place, whose poles are P, p, and equator
E Q. Bisect P C in O, and round O describe a circle P B C D ; let M
be the place on the earth's surface which has the moon in its zenith, Z
the place of the observer. Draw M C m, cutting the small circle in A,
and Z C N cutting the small circle in B ; draw the diameter BOD and
A I parallel to E Q, draw A F, G H, IK perpendicular to B D, and
join I D, A B, A D, and through I draw C M' cutting the meridian in
M'. Then after J a diurnal revolution the moon will come into the
situation M', and the angle M' C N ( = the nadir distance) = supplement
the angle ICB = zlIDB.
Also the ^ADB = BCA = zenith distance of the moon.
394 A COMMENTARY ON [Book III.
Hence D F, D K a cos. * of the zenith and nadir distances to rad. D B.
a elevation of the superior and inferior tides.
CONCLUSIONS FROM PROP. IX.
1. The greatest tides are when the moon is in the zenith or nadir of the
observer. For in this case (when M approaches to Z) A and I move to
wards D, B, and F coincides with B ; but in this case, the medium tide
which is represented by D H (an arithmetic mean to D K, D F) is di
minished.
If Z approach to M, D and I separate ; and hence, the superior and
iriferior and the medium tides all increase.
2. If the moon be in the equator, the inferior and superior tides are
equal, and equal M X (cos) * latitude. For since A and I coincide with
C, and F and K with (i) D i = D B X (cos.) « B D C = M X (cos.) *
latitude.
3. If the observer be in the equator, the superior and inferior tides are
equal every where, and =r M X (cos.) ^ of the declination of the moon.
For B coincides with C, and F and K with G ; P G = P C X cos. * of
the moon's declination = M x (cos.) * of the moon's declination.
4. The superior tides are greater or less than the inferior, according as
the moon and place of the observer are on the same or different sides of
the equator.
5. If the colatitude of the place equal the moon's declination or is less
than it, there will be no superior or inferior tide, according as the latitude
and the declination have the same or different denominations. For when
P Z=M Q, D coincides with I, and if it be less than M Q, D falls between
I and C, so that Z will not pass through the equator of the watery spheroid.
6. At the pole there are no diurnal tides, but a rise and subsidence
of the water twice in the month, owing to the moon's declining to both
sides of the equator.
18. Prop. X. To find the value of the mean tide.
A G = sin. 2 declination (to rad. = O C.)
and
O G = cos. 2 declination (to the same radius).
M
.'.OH = cos. 2 declination X cos. 2 lat. X g,
.•.DH= OD + OH
1 4 COS. 2 lat. X COS. 2 declination
= M X —^ 7i — ,
Book III.] NEWTON'S FRINCIPIA. 395
Now as the moon's declination never exceeds 30°, the cos. 2 declination
is always + v ^ and never greater than  ; if the latitude be less than 45°,
the cos. 2 lat. is + v e, after which it becomes — v e.
Hence
1. The mean tide is equally affected by north and south declination of
the moon.
2. If the latitude = 45°, the mean tide ^ M.
3. If the lat. be less than 45°, the mean tide decreases as the declina
tion increases.
4. If the latitude be greater than 45°, the mean tide decreases as the
declination diminishes.
^ Tr.i 1 .. J r^ .i_ .J TVfl 1 + cos. 2 declination
5. If the latitude = 0, the mean tide = M X — ' 5
BOOK I.
SECTION XII.
503. Prop. LXX. To find the attraction on a particle placed within
a spherical surface, force <x^. ^r~ .,
'^ distance *
Let P be a particle, and through P draw H P K,
I P L making a very small angle, and let them j
revolve and generate conical surfaces I P H, H
L P K. Now since the angles at P are equal
and the angles at H and L are also equal (for
both are on the same segment of the circle),
therefore the triangles H I P, P L K, are similar.
.. H I : K L : : H P : P L
Now since the surface of a cone a (slant side) \
.'. surface intercepted by revolution of I P H : that of L P K :
and attractions of each particle in I P H : that of L P K :
* Hpin?
but the whole attraction of P oc the number of particles X attraction of
each,
HI' K r *
• .*, the whole attraction on P from H I : from K L : : tt~t\ 'Kit
:: J :1;
and the same may be proved of any other part of the spherical surface ;
.*. P is at rest.
504. Prop. LXXL To find the attraction on a particle placed mthout
a spherical surface, force a^ p— ,.
^ distance '
PH*:
;PL«
HI«;
:KL»
I
1
HP^'
PL^
1
1
398
A COMMENTARY ON
[Sect. XII.
Let A B, a b, be two equal spherical surfaces, and let P, p be two
particles at any distances P S, p s from their centers; draw P H K,
P I L very near each other, and S F D, S E perpendicular upon them, and
from (p) draw p h k, p i 1, so that h k, i 1 may equal H K, I L respective
ly, and s f d, s e, i r perpendiculars upon them may equal S F D, S E,
I R respectively ; then ultimately PE = PF = pe = pf, and D ¥
= d f. Draw I Q, i q perpendicular upon P S, p s.
Now
PI:PF::IR:DF
and J..PI pf:pi.PF::IR:ir::IH:ih
I:PF::IR:DF"J
f:p i :: df :ir )"
: SF"J
:iq J
PI:PS::IQ: SF
and J.. Pl.psrpi. P S: :IQ:iq
p s : p i : : s f
.. P I*, p f . p s : (p i) ^ P F . P S : : I Q. I H : i q. i h
: : circumfer. of circle rad. I Q X I H : circumfer. of circle rad i q X i h
: : annulus described by revolution of 1 Q : that by revolution of i q.
Now
attraction on 1st annulus : attraction on 2d
And
1st annulus 2d annulus
distance ? * distance *
PP.pf.ps fpi)».PF.PS
Pl^ • (pi)*
pf.ps :PF.PS.
attraction on the annulus : attraction in the direction P S : : P I : P Q
: : P S : P F
P F
.'. attraction in direction PS = p f. p s. p^
PF ^„ „„ pf
ps
. . p S . r k5 . . pgj . ;;^
.. whole aU". of P to S : whole att°. of (p) to s : : p f . p s . p^ : P F . P S • ^
p s
Book I.] NEWTON'S PRINCIPIA. 399
and the same may be proved of all the annul! of which the surfaces are
composed, and therefore the attraction of P oc p^ cc rr ^ from
the center.
Cor. The attraction of the particles within the surface on P equals the
attraction of the particles without the surface.
For K L : I H :*: P L : P I : : L N : I Q.
.*. annulus described by I H : annulus described by K L
::IQ.IH: K L. L N : : P I^ : P L«
.*. attraction on the annulus I H : attraction on the annulus K L
PP PL '
* • P 1*' PL«* '
and so on for every other annulus, and one set of annuli equals the part
within the surface, and the other set equals the part without.
506. Prop. LXXII. To find the attraction on a particle placed with"
out a solid sphei'e, force oc^. r. 5.
Let the sphere be supposed to be made up of spherical surfaces, and
the attraction of these surfaces upon P will oc yv ;, and therefore
^ distance *
the whole attractions
number of surfaces content of sphere diameter ^
^ P"S^ * P"S^ " PS*^
and if P S bear a given ratio to the diameter, then
the whole attraction on P cc r 7 — ; a diameter.
diameter ^
507. Prop. LXXIIL To find the attraction on the particle placed
toithin
Let P be the particle ; with rad. S P describe
the interior sphere P Q ; then by Prop. LXX.
(considering the sphere to be made of spherical
surfaces,) the attraction of all the particles con
tained between the circumferences of the two
circles on P will be nothing, inasmuch as they
are equal on each side of P, and the attraction
p g3
of the other part by the last Prop, oc p^ a P S.
400
A COMMENTARY ON
[Sect. XII.
508. Prop. LXXIV. If the attractions of the particles of a spliere
(X rr— — 5—5 , and two similar spheres attract each other, then the spheres
will attract with a force «« as
distance ^
of their centers.
For the attraction of each, particle a v= 5 from the center of the
' *^ distance*
attracting sphere (A), and therefore with respect to the attracted particle
the attracting sphere is the same as if all its particles were concentrated
in its center. Hence the attraction of each particle in (A) upon the
whole of (B) will a j.— ^ of each particle in B from the center of P,
and if all the particles in B were concentrated in the center, the attraction
would be the same ; and hence the attractions of A and B upon each other
will be the same as if each of them were concentrated in its center, and
therefore a
distance
t '
509. Prop. LXXVI. Let the spheres attract each other, and let
them not be homogeneous, but let them be homogeneous at correspond
ing distances from the center, then they attract each other with forces
I
aB.
distance '
Suppose any number of spheres C D and E F, I K and L M, &c, to
be concentric with the spheres A B, G H, respectively; and let C D and
I K, E F and L M be homogeneous respectively ; then each of these
spheres will attract each other with forces a^. 7: . Now suppase
distance * '^^
the original spheres to be made up by the addition and subtraction of
similar and homogeneous spheres, each of these spheres attracting each
Book I.]
other with a force a s.
NEWTON'S PRINCIPIA.
1
401
 ; then the sum or diiFerences will attract
distance ^
each other in the same ratio.
510. Prop. LXXVII. Let the force oc distance, to find the attraction
of a sphere on a particle placed without or within it.
Let P be the particle, S the center, draw two planes E F, e f, equally
distant from S ; let H be a particle in the plane E F, then the attraction
of H on P a HP, and therefore the attraction in the direction S P a
P G, and the attraction of the sum of the particles in E F on P towards
S oc circle E F . P G, and the attraction of the sum of the particles in
(e f) on P towards S cc circle e f . P g, therefore the whole attraction of
E F, e f, a circle EF(PG+Pg) cc circle E F . 2 P S, therefore the
whole attraction of the sphere a sphere X P S.
When P is within the sphere, the attraction of the circle E F on P to
wards S oc circle E F . P G, and the attraction of the circle (e f ) towards
S oc circle e f . P g, and the difference of these attractions on the whole
attraction to S a circle EF(Pg— PG) oc circle E F . 2 P S. There
fore the whole attraction of the sphere on P a sphere X P S.
511. Lemma XXIX. If any arc be described with the center S, rad.
S B, and with the center P, two circles be described very near each other
Vol. I. C c
402
A COMMENTARY ON
[Sect. XII.
cutting, first, the circle in E, e, and P S in F, f ; and E D, e d, be drawn
perpendicular to P S, then ultimately,
Dd: Ff::PE: PS.
For
and
Dd:Ee::DT:ET::DE:ES
E e
Ee: Ff
Dd: Ff
e r ::SE :SG
— ::DE: SO:
P E : P S.
612. Prop. LXXIX. Let a solid be generated by the revolutions of a«.
evanescent lamina E F f e round the axis P S, then the force with which
the solid attracts Pa DE'. Ffx force of each particle.
Draw E D, e d perpendiculars upon P S ; let e d intersect E F in r ;
draw r n perpendicular upon E D. Then E r : n r : : P E : ED, .*.
Er.ED = nr.PS = Dd.PE, .. the annular surface generated by
the revolution of Era Er.EDa Dd.PE, and (P E remaining the
same) a D d. But the attraction of this annular surface on P a D d .
P E, and the attraction in the direction P E : the attraction in the direc
tion P S : : P E : P D,
PD
.*. the attraction in the direction P S a
PE
.Dd.PE a PD.Dd
and the whole attraction on P of the surface described by E F a sum of
the PD.Dd.
Let P E = r, D F = X,
.♦. P D = r — X,
•. PD.Dd=rdx — xdx.
.'. sum ofPD.DdS=yrdx — xdx =
2rx— x» D E'
a DE%
2. 2
and therefore the attraction of lamina a D E '. F f X force of each particle.
Book I.]
NEWTON^S PRINCIPIA.
403
513. Prop. LXXX. Take D N proportional to p^p — X force
of each particle at the distance P E, or if ^^ represent that force, let D N
"T) fr 2 PS
a ' , then the area traced out by D N will be proportional to
JL ill. V
the whole attraction of the sphere.
a
For the attraction of lamina EFfeaDE*. F fx force of each parti
i a (Lemm;
DE'^. PS
J) ]g2 p g
cle a (Lemma XXIII) ^^^ . D d x force of each particle, or
PE
p ^ ^ D d, .*. D N . D d a attraction of lamina E F f e, and the
sum of these areas or area A N B will represent the whole attraction of
the sphere on P.
514. Prop. LXXXI. To find the area A N B.
Draw the tangent P H and H I perpendicular on P S, and bisect P I
in L ; then
Cc2
404 A COMMENTARY ON [Sect. XII.
PE«= PS' + SE« + 2PS.SD
But
SE* = SH' = PS. SI,
PE'^rrPS^ + PS.SI + 2PS.SD
= PSJPS + SI + 2SD}
= P S J{P I + I S) + S I + 2 S D}
= PS^2LI + 2SI + 2SD
= 2PSJLI + SI + SDJ = 2PS.LD
DE«=SE= — SD'zrSE'^ — (LD — LS)«
= SE2 — LD« — LS2 + 2LD.LS
= 2LD.LS — LD^ — (LS+ SE)(LS — SE)
= 2LD.LS — LD* — LB.LA,
^., DE^.PS 2LD.LS.PS
PE.V V2SD.PS.V
LD^PS LB. LA. PS
V2LD.PS.V V2LD.PS.V
and hence if V be given, D N may be represented in terms of L D and
known quantities.
515. Ex. 1. Let the force a j. : to find the area A N B.
distance
si"^^^.«di;^*v'"^'"P^'
2LS.LD.PS LD^PS AL.LB.PS
..DJNa 2LD.PS 2LD.PS 2LD.PS
Tc LD AL.LB
a LSg WTTDT*
^XTT^^ A T c rk 1 LD.Dd AL.LB.Dd
.♦. D N . D d, or d . area a L S . D d 2 I Ti '
.'. area AND between the values of L A and L B
= LS.(LBLA) LB'LA' _ALJLB ^LB
Now
LB* — LA« = (LB + LA).(LB — LA)
= (LS + AS + LS — AS)AB = 2LS.AB,
Avrn Tc AR 2LS.AB AL. LB ,LB
.'. area AND = LS.AB . ^ 1 =1 — a~
4 2 L A
_ LS.AB AL.LB ,LB
"2 2 UA'
Book I.j
NEWTON'S PRINCIPIA.
405
516. To construct this area.
To the points L, A, B erect L 1, A a, B b,
perpendiculars, and let A a = L B, and B b i
= L A, through the points (a), (b), de
scribe an hyperbola to which L 1, L B are
asymptotes. Then by property of the hy
perbola, AL.Aa = LD.DF,
^^ AL.Aa AL.LB
.. D F =
..DF.Dd =
LD ~ LD
A L.LB.Dd
LD '
..areaAaFD =/DF.Dd = AL.LB/LD,
T B
.♦.hyperboUcarea AafbB= Ah.liBfj—^.
The area AaBb = Bb.AB + ^ ^^ ^ "
Bb.AB , an + Bb .^ Aa + Bb.^
= 2 + § ^^ 2 ^^
LB + L A
. A B = L S . A B,
.*. area a f b a = area A a B b — area A a f b B
= LS.AB — AL.LB/i^.
517. Ex. 2. Let the force a ,._ . ; to find the area A N B.
distance ^ "
but
LetV =
.. D N =
V.PE =
.. D N =
PE^
2 A S^'
2LS.LD.PS
LD^PS
PE.V
PE.V
AL.LB. PS
PE.V
PE _ 4PS^LD' _ PS j^,
2AS*~ 2AS^ _4ro^gj. A.1^ ,
SI.LS SI AL.LB. SI
LD 2
../DN.x' = Si.LS/LD
SI.LD AL.LB. SI
2 ' 2LD
.*. area between the values of L A and L B
c.TTc/'LB SI. (LB — LA) /LB. SI AL.SL
= SI.LSy j^ 2 \~2 2 .
LB
izSI.LS/^ — SLAB.
Cc3
406
A COMMENTARY ON
[Sect. XIL
To construct this area.
1 a
S Dd
Take S I = S s, and describe a hyperbola passing through a, s, b, to which
L 1, L B are asymptotes ; then as in the former case, the area A a n b B
•LB
.LB
= AL.SB./^ = LS.Ss/^ = SI.LS/^
.. the area A N B = S I . L S/]^ — SLAB.
518. Prop. LXXXII. Let I be a particle within the sphere, and P
the same particle without the sphere, and take
S P : S A : : S A : S I,
then will the attracting power of the sphere on I : attracting power of the
sphere on P
: : V S I. V force on I : V S P. V force on P.
D N force on the point P : D' N' force on the point I
DE' PS D E' IS
PE.V '' lE.V
PS.IE.V'rIS.PE.V.
I^t
V : V :: PE" : IE",
Book L]
NEWTON'S PRINCIPIA.
407
then
but
DN:D'N':: PS.IE.IE°:IS.PE.:eE°,
P S : S E : : S E : S I,
and the angle at S is common,
.'. triangles P S E, I S E are similar,
.. P E : 1 E
.. D N : D' N'
P S : S E ; : S E : S I,
PS.SE.IE : PS.SI.PE,
SE.IE" : SI.PE"
VS P.IE" : VSI.PE*
VSP : SI^ VS I.PS2.
519. Prop. LXXXIII. To find the attraction of a segment of aspheie
upon a corpuscle placed within its centre.
Draw the circle F E G with the
center P, let R B S be the segment of
the sphere, and let the attraction of the
spherical lamina E F G upon P be
proportional to F N, then the area de
scribed by F N a whole attraction of """"
the segment to P.
Now the surface of the segment
E F G a P F D F, and the content
of the lamina whose thickness is O x
PF DF O.
Let F (X jv and the attraction on P of the particle in that
distance "
1~) F * O
spherical lamina, oc ( Prop. LXXIII.) p^^
a
r2PF FD — FD^) O
2FDO FD^ O
PF'i
2 F D F D *
.. if F N be taken proportional to p ^ „_j — p^^ , the area traced
out by F N will be the whole attraction on P.
520. Prop. LXXXIV. To find the attraction when the body is placed
ia the axis of the segment, but not in the center of the sphere.
Pel
408
A COMMENTARY ON
[Sect. XIII.
Describe a circle with the radius P E, and the segment cut ofF by the
revolution of this circle E F K round P B, will have P in its center, and
the attraction on P of this part may be found by the preceding Proposi
tion, and of the other part by Prop. LXXXI. and the sum of these at
tractions will be the whole attraction on P.
SECTION XIII.
621. Prop. LXXXV. If the attraction of a body on a particle placed
iu contact with it, be much greater than if the particle were removed at
any the least distance from contact, the force of the attraction of the par
ticles a in a higher ratio than that of p , .
° distance *
For if the force a tt— ^ , and the particle be placed at any distance
from the sphere, then the attraction a t: •„ from the center of the
^ distance*
sphere, and .*. is not sensibly increased by being placed in contact with
the sphere, and it is still less increased when the force a in a less ratio
than that of r^ r» and it is indifferent whether the sphere be homo
distance ^
geneous or not ; if it be homogeneous at equal distances, or whether the
body be placed within or without the sphere, the attraction still varying in
the same ratio, or whether any parts of this orbit remote from the point of
contact be taken away, and be supplied by other parts, whether attractive
or not, .*. so far as attraction is concerned, the attracting power of this
sphere, and of any other body will not sensibly differ ; .*. if the pheno
Book I.] NEWTON'S PRINCIPIA. 409
mena stated in the Proposition be observed, the force must vary in a higher
ratio than that of p •„ .
distance*
522. Prop. LXXXVI. If the attraction of the particles a in a higher
ratio than t . » or a r. , then the attraction of a body placed
distance ^ distance "
in contact with any body, is much greater than if they were separated
even by an evanescent distance.
For if the force of each particle of the sphere oc in a higher ratio than
that of T 5 , the attraction of the sphere on the particle is indefinitely
Cll S l3.ll X^C
increased by their being placed in contact, and the same is the case for
any meniscus of a sphere ; and by the addition and subtraction of attrac
tive particles to a sphere, the body may assume any given figure, and
.*. the increase or decrease of the attraction of this body will not be sensi
bly different from the attraction of a sphere, if the body be placed in con
tact with it.
523. Prop. LXXXVII. Let two similar bodies, composed of particles
equally attractive, be placed at proportional distances from two particles
which are also proportional to the bodies themselves, then the accelerat
ing attractions of corpuscles to the attracting bodies will be proportional
to the whole bodies of which they are a part, and in which they are simi
larly situated.
For if the bodies be supposed to consist of particles which are propor
tional to the bodies themselves, then the attraction of each particle in one
body : the attraction of each particle in the other body, : : the attraction
of all the particles in the first body : the attraction of all the particles in
the second body, which is the Proposition.
CoR. Let the attracting forces a tt— , then the attraction of a
° distance °
particle in a body whose side is A : — B
A^ B^
distance ^ from A ' distance " from R
A^ 21
A° • B^
1 1
' • A°3 ' B°3'
if the distances oc as A and B.
410
A COMMENTARY ON
[Sect. XiII.
524. Prop. LXXXVIII. If the particles of any body attract with a
force a distance, then the whole body will be acted upon by a particle
without it, in the same manner as if all the particles of which the body is
composed, were concentrated in its center of gravity.
Let R S T V be the body, Z the par
ticle without it, let A and B be any
two particles of the body, G their cen
ter of gravity, then A A G = B B G,
and then the forces of Z of these parti
cles Qc A A Z, B B Z, and these
forces may be resolved into A A G +
A G Z, B B G + B G Z, and A A G
being = B B G and acting in opposite
directions, they will destroy each other,
and .*. force of Z upon A and B will be
proportional to A Z G } B Z G, or to (A + B) Z G, .*. particles A
and B will be equally acted upon by Z, whether they be at A and B, or
collected in their center of gravity. And if there be three bodies A, B,
C, the same may be proved of the center of gravity of A and B (G) and
C, and .*. of A, B, and C, and so on for all the particles of which the
body is composed, or for the body itself.
525. Prop. LXXXIX. The same applies to any number of bodies
acting upon a particle, the force of each body being the same as if it
were collected in its center of gravity, and the force of the whole system
of bodies being the same as if the several centers of gravity were collected
in the common center of the whole.
526. Prop. XC. Let a body be placed in a perpendicular to the plane
of a given circle drawn from its center ; to find the attraction of the circu
lar area upon the body.
With the center A, radius = A D, let
a circle be supposed to be described, to
whose plane A P is perpendicular. From
any point E in this circle draw P E, in
P A or it produced take P F = P E, and
draw F K perpendicular to P F, and let
F K oc attracting force at E on P. Let
i K L be the curve described by the point
K, and let I K L meet A D in L, take
P H = P D, and draw H I perpendicular
Book I.] NEWTON'S PRINCIPIA. 411
to P H meeting this curve in I, then the attraction on P of the circle
a A P the area A H I L.
For take E e an evanescent part of A D, and join P e, draw e C per
pendicular upon P E, .. E e : E C : : P E : A E, .♦. E e . A E = E C x
P E a annulus described by A E, and the attraction of that annulus in
PA
the direction P A cc E C . P E . p^ x force of each particle at E oc E C X
P A X force of each particle at E, but E C = F f, .. F K . F f <x E C x
the force of each particle at E, .*. attraction of the annulus in the direction
PA a P A . F f . F K, and .. P A x sum of the areas F K . Ff or P A
the area A H I L is proportional to the attraction of the whole part de
scribed by the revolution of A E.
527. Cor. 1. Let the force of each particle a r it at P F =r x,
let b = force at the distance a,
ba«
.*. F K the force at the distance x = — 5 ,
X*
.. FK.Ff =
X
badx
528. Cor. 2. Letthe force a j^ — r , then T K = —  ,
..attraction = PA. FK.Ff= PAy5^^
aPA — Qc A — ^p,
and between the values of P A and P H, the attraction
cr PA ^ ' « 1 ^^
^^ PA~"PH "* *~PH
I ., ^.,. ba«
distance "
.^ .■ r. * /b a° , PA 1 , /.
.'. attraction = P A / — —d x a r X r — r + t>or.,
•/x" n — 1 x"~*
and between the values of P A and P H,
attraction = ^^ {^^^^ — FTT^}
1 PA
^ PA"i~PH"i •
529. CoR. 3. Let the diameter of a circle become infinite, or P H
oc cc, then the attraction gc p > .._i ! •
530. Prop. XCL To find the attraction on a particle placed in the
axis produced of a regular solid.
413
A COMMENTARY ON
• R E
[Sect. XIII.
Let P be a body situated in the axis A B of the curve D E C G, by
the revolution of whicli the solid is generated. Let any circle II F S
perpendicular to the axis, cut the solid, and in the semidiameter F S of
the solid, take F K proportional to the attraction of the circle on P, then
F K . F f QC attraction of the solid w^hose base = circle R F S, and depth
= F f, let I K L be the curve traced out by F K, .*. A L K F a at
traction of the solid.
Cor. 1. Let the solid be a cylinder, the force varying as y— „ ,
Then the attraction of the circle R F S, or F K which is proportional
to that attraction a 1 — ^^ .
Let P F = X, F R = b,
.. F K a 1 —
.. FK. Ff ex dx —
Vx^ + b«'
X x'
Vx^ + b*'
.. area a — x v'x '^ + b * .
Book I.]
NEWTON'S PRINCIPIA.
413
Now if P A = X, attraction = 0,
.. Cor. = PD — P A,
.. whole attraction = PB — PE + PD — PA
= AB — PE + PD.
LetAB=a)=PE = PD,
.'. atraction = A B.
531. CoR. 3. Let the body P be placed
within a spheroid, let a spheroidical shell
be included between the two similar
spheroids DOG, K N I, and let the
spheroid be described round S which
will pass through P, and which is simi
lar to the original spheroid, draw D P E,
F P G, very near each other. Now P D
= BE, PF = CG, PH = BI, PK
= CL.
.. F K = L G, and D H = I E,
and the parts of the spheroidical shell which are intercepted between these
lines, are of equal thickness, as also the conical frustums intercepted by
the revolution of these lines, and
.*. attraction on P by the part D K : . . . . G I
number of particles in D K _ ... G*
• • WW' '• "Fg~«
PD^ . PG' . . I . ,
'•• PD« ' P G^^ •• '
and the same may be proved of every other part of a spheroidical shell, and
.•. body is not at all attracted by it; and the same may be proved of all the
other spheroidical shells which are included between the spheroids, A O G,
and C P M, and .*. P is not affected by the parts external to C P M, and
,. (Prop. LXXIL),
attraction on P : attraction on A : : PS: AS.
532. Prop. XCIIl. To find the attraction of a body placed without an
infinite solid, the force of each particle varying as y. ^ , where n is
greater than 3.
Let C be the body, and let G L, H M, K O, &c. be the attractions
at the several infinite planes of which a solid is composed on the
414
A COMMENTARY ON
[Sect XIII.
body Cj then the area G L O K equals the whole attraction of a solid
onC.
L
"^
 N
O
G
H
I
K
1
m
n
Now if the force a y. „ 
distance"
Then
H M a Qi^n2 (Cor. 3. Prop, XC)
.../HM.dx a/^, « r^ + Cor.
a
and if H C = oo
then the area G L O K oc
C G"3 C H»3'
1
G C»3*
Case 2. Let a body be placed within the solid.
6
N
C
I
K
1
Let C be the place of the body, and take C K = C G ; the part of
the solid between G and K will have no effect on the body C, and there
fore it is attracted to remain as if it were placed without it at the distance
CK.
1 1
.*. attraction x
QC
CK^a  CG»3*
Book I.]
NEWTON'S PRINCIPIA.
415
SECTION XIV.
534. Prop. XCIV. Let a body move through a similar medium, ter
minated by parallel plane surfaces, and let the body, in its passage through
this medium, be attracted by a force varying according to any law of its
distance from the plane of incidence. Then will the sine of inclination be
to the sine of refraction in a given ratio.
a\h ^ —
^\ K
a
^^
^. o
N
\
1
B
v^
b
Q\
\>
^K
M
Let A a, B b be the planes which terminate the medium, and G H be
the direction of the body's incidence, and I R that of its emergence.
Case 1. Let the force to the plane A a be constant, then the body will
describe a parabola, the force acting parallel to I R, which will be a diameter
of the parabola described. H M will be a tangent to the parabola, and if
K I be produced I L will also be a tangent to the parabola at I. Let K I
produced meet G M in L ^ith the center L, and distance L I describe
a circle cutting I R in N, and draw L O perpendicular to I R. Now by a
property of the parabola M I =. I v,
.. M L = H L, .. M O = O R, and .. M N = I R.
The angle L M I=the angle of incidence, and the angle MIL = sup
plement of M I K r= supplemental angle of emergence.
Now
L.MI = MH« = 4ML^
416
but
A COMMENTARY ON
[Sect. XIV.
B
1/
b
C
K/
13
R/
d
MN.MI = MI.IR = MQ.MP=ML+LQ.ML — LQ
= ML« — LQ'
.. L : I R : : 4 M L« : M L^ — L Q«
but L and I R are given
.. 4ML« a ML« — LQ«
.. ML'^ aLQ« a LI^
,*. M L a L I or sin. refraction : sin. inclination in a given ratio.
Case 2. Let the force vary according to ^ Gy
any law of distance from A a.
Divide the medium by parallel planes A a,
B b, C c, D d, &c. and let the planes be at
evanescent distances from each other, and
let the force in passing from A a to B b,
from B b to C c, from C c to D d, &c. be
uniform.
.*. sin. I at H : sin. R at H : : a : b
sin. R or I at I : sin. R at K : : c : d
sin. R or I at K : sin. R at R : : e : f, and so on.
.'. sm. I at H : sin. RatR::a.c.e:b.d.f and in a constant pro
portion.
535. Prop. XCV. The velocity of a particle before incidence : velocity
after emergence : : sin. emergence : sin. incidence.
G
Take A H = I d, and draw A G, d K perpendicular upon A a, D d,
meeting the directions of incidence and emergence in G, K. Let the
motion of the body be resolved into the two G A, A H, Id, d k, the ve
Book L] NEWTON'S PRINCIPIA. 417
locity perpendicular to A a cannot alter the motion in the direction A a ;
therefore the body will describe G H, I K in the same time as the spaces
A H, I d are described, that is, it will describe G H, I K in equal times
before the incidence and after the emergence.
Velocity before incidence : velocity after emergence : : G H : I K
A H . Id
sin. incidence ' sin. emergence
: : sin. emergence : sin. incidence.
536. Prop. XCVI. Let the velocity before incidence be greater than
the velocity after emergence, then, by inclining the direction of the inci
dent particle perpetually, the ray will be refracted back again in a similar
curve, and the angle of reflection will equal the angle of incidence.
A
xH
h.
yk
B \p
P/
b
c \q
./
c
\
^^R^
d
E
e
Let the medium be separated by parallel planes A a, B b, C c, D d,
E e, &c. and since the velocity before incidence is greater than the
velocity after emergence. .*. sin. of emergence is greater than sin. of in
cidence. .'. H P, P Q, Q R, &c. will continually make a less angle with
H a, P b, Q c, R d, &c. till at last it coincides with it as at R ; and after
this it will be reflected back again and describe the curve R q p h g simi
lar to R Q P H G, and the angle of emergence at h will equal the angle
of incidence at H.
537. Prop. XCVIL Let sin. incidence : sin. refraction in a given ra
•tio, and let the rays diverge from a given point ; to find the surface of
medium so that they may be refracted to another given point.
Let A be the focus of incident, B of refracted rays, and let C D E
be the surface which it is requued to determine. Take D E a small arc,
Vol. T. D d
418 A COMMENTARY ON [Sect. XIV.
and draw E F, E G perpendiculars upon A D and D B ; then D P\ D G
are the sines of incidence and refraction ; or increment of A D : decrement
of B D : : sin. incidence : sin. refraction. Take .*. a point C in the axis
through which the curve ought to pass, and let C M : C N : : sin. inci
dence : sin. refraction, and points where the circles described with radii
A M, B N intersect each other will trace out the curve.
538. Cor. 1. If A and B be either of them at an infinite distance or at
any assigned situation, all the curves, which are the loci of D in different
situations of A and B with respect to C, will be traced out by t'lis
process.
AC B
539. Cor. 2. Describe circles with radii A C and C B, meeting A D,
B D in P and Q ; then P D : D Q : : sin. incidence : sin. refraction, since
P D, D Q are the increments of B C and A C.
BOOK II.
SECTION I.
1. Prop. I. Suppose the resistance oc velocity, and supposing the whole
time to be divided into equal portions, the motion lost will « velocity, and
oc space described. Therefore by composition, the whole decrement of the
velocity cc space described.
Cob. Hence the whole velocity at the beginning of motion : that part
which is lost : : the whole space which the velocity can describe : space
already described.
2. Prop. II. Suppose the resistance oc velocity.
Case 1. Suppose the whole time to be divided into equal portions, and
at the beginning of each portion, the force of resistance to make a single
impulse which will a velocity, and the decrement of the velocity
a resistance in a given time, a velocity. Therefore the velocities
at the beginning of the respective portions of time will be in a con
tinued progression. Now suppose the portions of time to be diminished
sine limited and then the number increased ad infinitum, then the force of
resistance will act constantly, and the velocity at the beginning of equal
successive portions of time will be in geometric progression.
Case 2. The spaces described will be as the decrements of the velocity
oc velocity.
3. CoR. 1. Hence if the time be represented by any line and be divid
ed into equal portions, and ordinates be drawn perpendicular to this
line in geometric progression, the ordinates will represent the velocities,
and the area of the curve which is the logarithmic curve, will be as the
spaces described.
Dd2
420
A COMMENTARY ON
[Sect. 1.
Suppose L S T to be the logarithmic curve to the asymptote A Z.
A L, the velocity of the body at the beginning of the motion.
P Q
K Z
The space described in the time A H with the first velocity continued
uniform : space described in the resisting medium, in the same time : :
A H P L : area A L S H : : rect. A L X P L : rect. A L X PS*
: : P L : P S (if A L = subtan. of the curve).
Also since H S, K T representing the velocities in the times A H, A K ;
P S, Q T are the velocities lost, and therefore cc spaces described.
4. Cor. 1. Suppose the resistance as well as the velocity at the begin
ning of the motion to be represented by the line C A, and after any time by
the line C D. The area A B G D will be as the time, and A D as the
space described.
For if A B G D increase in arithmetical progression the areas being
the hyperbolic logarithms of the abscissas, the abscissa will decrease in
geometrical progression, and therefore A D will increase in the same
proportion.
5. Prop. III. Let the force of gravity be represented by the rectangle
• Let the subtaogent = M. Then the whole area of the curve = M X A L.
.. the area ALSH = MXAL — MXHS=MXPS=ALXPS.
Book II.]
NEWTON'S PRINCIPIA.
421
BACH, and the force of resistance at the beginning of the motion by
the rectangle B A D E on the other side of A B.
D d A I i
Describe the hyperbola G B K between the asymptotes A C and C H
cutting the perpendiculars D E, d e, in G and g.
Then if the body ascend in the time represented by the area D G g d,
the body will describe a space proportional to the area E G g e, and the
whole space through which' it can ascend will be proportional to the area
EGB.
If tlie body descend in the time A B K I, the area described is B F K.
For suppose the whole area of the parallelogram B A C H to be di
r «Jr
I
F
k 1 mn
A I
I K L M N
[
H
vided into portions, which shall be as the increments of the velocity in
equal times, therefore A k, A 1, A m, A n, &c. will oc velocity, and there
fore a resistances at the beginning of the respective times.
Let A C : A K : : force of gravity : resistance at the beginning of the
second portion of time, then the parallelograms B A C H, k K C H, &c.
will represent the absolute forces on the body, and will decrease in geome
trical progression. Hence if the lines K k, L 1, &c. be produced to meet
Dd3
422 A COMMENTARY ON [Sect. I.
the curve in q, r, &c. these hyperbolic areas being all equal will repre
sent the times, and also the force of gravity which is constant. But the
area B A K q : area Bqk::Kq:4kq::AC:^AK;: force of
gravity : resistance in the middle of the first portion of time.
In the same way, the areas q K L r, r L M s, &c. are to the areas
q k 1 r, r 1 m s, &c. as the force of gravity to the force of resistance in the mid
dle of the second, third, &c. portions of time. And since the first term is
constant and proportional to the third, the second is proportional to the
fourth, similarly as to the velocities, and therefore to the spaces described.
.*. by composition B k q, B r 1, B s m, &c. will be as the whole spaces
described, Q. e. d.
The same may be proved of the ascent of the body in the same way.
6. Cor. 1. The greatest velocity which the body can acquire : the velo
city acquired in any given time : : force of gravity : force of resistance
at the end of the given time.
7. Cor. 2. The times are logarithms of the velocities.
8. Cor. 4. The space described by the body is the difference of the space
representing the time, and the area representing the velocity, which at the
beginning of the motion are mutually equal to each other.
* Suppose the resistance to oc velocity.
rv
c' : v' : : r : — j =retardingforcecorresponding with the velocity (v)
c
r v
.*. v d v = — g X —J X d X,
J c* dv
.♦. d X = — X —
g V
.*. X = — b X 1 V + C,
.'. X = b X 1 —
V
__dx __ bdv
~ V ~~ v^ '
.. t = — b X + Cor.
v
_ , J 1_ _l 1_
~~ V c v c
.*. the times being in geometiical progression, the velocities C, d, E, &c.
will be in the same inverse geometrical progression.
Also the spaces will be in arithmetical progression.
Book II.]
NEWTON'S PRINCIPIA.
423
9. Prop. IV. Let D P be the direction of the projectile, and let it
represent the initial velocity ; draw C P perpendicular to C D, and
N
let D A : A C : : resistance : gravity. Also DP: C P : : resistance :
gravity, .. DAxDP:CPxCA::R:G. Between D C, C P de
scribe a hyperbola cutting D G and A B perpendicular to D C in G and B,
from R draw R V pei*pendicular cutting D P in V and the hyperbola in T,
complete the paraUelogram G K C D and make N : Q B : : C D : C P.
Take
,, G T t „ G T E I
V r = — .i^f — or R r =
N
N
for ^ince
R V =
and
GTEI
N : Q B : ; C D : C P :
DR X QB
N
D R X QB — GTt
D R : R V,
= Rr
N ~ N
in the time represented by D R T G the body will be at (v), and the great
est altitude = a, and the velocity ex r L.
For the motion may be resolved into two, ascending and lateral. The
lateral motion is represented by D R, and the motion in ascent by R r,
which
aDRxQB — GTt,
or
DRxAB— DG.RT
N'
Ddl
^»i A COMMENTARY ON [Sect. II.
or
D R X A B— D R X AQ
N '
D R : R r : : N : A B — A Q, or Q B
: : C D : C P,
: : lateral motion . ascending motion at the beginning,
(r) will be the place of the body required.
SECTION II.
10. Prop. V. Suppose the resistance to vary as the velocity ^
Then as before, the decrement of velocity a resistance cc velocity
A KI.M T D
Let the whole time A D be divided into a great number of equal por
tions, and draw the ordinates A B, K k, L 1, M m, &c. to the hyperbola
described between the two rectangular asymptotes, C H, CD; then by the
property of the hyperbola,
A B : K k : : C K : C A,
.. AB— Kk:Kk::AK:CA
::ABxAK:ABxCA.
.. AB — KkaABxKk.
In the same way
Kk — LI a KkS &c.
or
A B S K k S L P, &c.
are proportional to their differences.
.*. velocities will decrease in the same proportion. Also the spaces de
scribed are represented by the areas described by the ordinates ; hence in
Book II.]
NEWTON'S PRINCIPIA.
425
the time A M the space described may be represented by the whole area
A M mB.
Now suppose the lines C A, C K, &c. and similarly A K, K L, &c. in
geometrical progression, then the ordinates will decrease in the inverse
geometrical progression, and the spaces will be all equal to each other.
Q. e. d.
] 1. Cor. 1. The space described in the resisting medium : the space de
scribed with the first velocity continued uniform for the time AD:: the
hyperbolic area A D G B : rectangle A B X AD.
12. Cor. 3. The first resistance equals the centripetal force which would
generate the first velocity in the time A C, for if the tangent B T be drawn
to the hyperbola at B, since the hyperbola is rectangular A T = A C, and
with the first resistance continued uniform for the time A C the whole
velocity A B would be destroyed, which is the time in which the same ve
locity would be generated by a force equal the first resistance. For the
first decrement is A B — K k, and in equal times there would be equal de
crements of velocity.
13. Cor. 4. The first resistance : force of gravity : : velocity generated
by the force equal the first resistance in the time A C : velocity generated
by the force of gravity in the same time.
14. CoR. 5. P^ice versd^ if this ratio is given, every thing else may be
found.
C Q P L K I A
15. Prop. VIII. Let C A represent the force of gravity, A K the resis
tance, .*. C K represents the absolute force at any time (if the body de
scend) ; A P, a mean proportional to A C and A K, represents the velo
city ; K L, P Q are contemporaneous increments of the resistance and
the velocity.
Then since
AP^aAK, KLa2APxPQxAPxKC,
426
A COMMENTARY ON
[Sect. II.
tlie increment of velocity a force when the time is given,
..KLxKNaAPxKCxKN,
.*. ultimately K L O N (equal the increment of the hyperbolic area)
oc A P a velocity, a space described, and the whole hyperbolic area =
the sura of all the K L O Ns which are proportional to the velocity, and
.*. space desci'ibed. .*. If the whole hyperbolic area be divided into equal
portions the absolute force C A, C I, C K, &c. are in geometrical pro
gression. Q. e. d.
16. Cor. 1. Hence if the space described be represented by a hyper
bolic area, the force of gravity, velocity, and resistance, may be repre
sented by lines which are in continued proportion.
17. Cor. 2. The greatest velocity = A C.
18. Cor. 3. If the resistance is known for a given velocity, the greatest
velocity : given velocity : : V force of gravity : v^ given resistance.
1 9. Prop. IX. Let A C represent the greatest velocity, and A D be per
pendicular and equal to it. With the center D and radius A D describe
the quadrant A t E and the hyperbola A V Z. Draw the radii D P, D p.
Then
Case 1. If the body ascend ; draw D v q near to D p, .*. since the sector
and the triangle are small,
Dvt:Dpq::Dt*:Dp^
Dqp
.'. D V t a
Dp^
Book II.] NEWTON'S PRINCIPIA. 427
^ A D X p q p q
"^ AD* + ADxAK "^ C~K
cc increment of the time.
••. by composition, the whole sector oc whole time till the whole
V= 0.'
Case 2. If the body descend; as before
D VT:DPQ::DT«: DP^
:: DX*: D A": : TX*: AP*
::DX^ — TX«:DA'' — AP«
:: AD^: AD^ — AD X A K
 : : A D : C K.
By the property of the hyperbola,
TX^ = DX^ — D A*
.. D A^ = DX^ — TX*
••^^^ «AD X CK« CTT
oc increment of the time.
,*. by composition, the whole time of descent till the body acquire its
greatest V = the whole hyperbolic sector DAT.
20. Cor. 1. If A B = i A C.
The space which the descending body describes in any time : space
which it would describe in a nonresisting medium to acquire the greatest
velocity : : area ABNK:aATD, which represents the time. For
since AC:AP::AP:AK
KL:iPQ::AP:iAC
and
KN: AC ::AB:CK
.. KLON:DTV::AP:AC
: : vel. of the body at any time : the greatest vei.
Hence the increments of the areas oc velocity gc spaces described.
.*. by composition the whole A B N K : sector A T D : : space described
to acquire any velocity : space described in a nonresisting medium 'for
the same time.
21. Cor. 2. In the same way, if the body ascend, the space described
till the velocity = A p : space through which a body would move : :
A B n k : A D t.
22. Cor. 3. Also, the velocity of a body falling for the time A T D :
velocity which a body would acquire in a nonresisting medium in the
same time : : A A D P : sector T D A ; for since the force is constant,
428 A COMMENTARY ON [Sect. II.
the velocity in a nonresisting medium a time, and the force in a resist
ing medium aAPaAADP.
23. Cor. 4. In the same wa)', the velocity in the ascent : velocity with which
a body should move, to lose its whole motion in the same time : : A A p D
: sector A t D : : A p : arc A t.
For let A Y be any other velocity acquired in a nonresisting medium
in the same time with A P.
.. A P : A C : : A P D : this area
and
AP:AC::APD:AeD.
Therefore the area which represents the time of acquiring the greatest
velocity in a nonresisting medium = A C D.
In the same way, let Ay be velocity lost in a nonresisting medium in
the same time as A p in a resisting medium.
.*. Ap:Ay::AApD: area which represents the time of losing the
velocity A p.
.*. time of losing the velocity A y = A A p D.
24. Cor. 5. Hence the time in which a failing body would acquire the
velocity A P ; time in which, in a nonresisting medium, it would acquire
the greatest velocity : : sector A D T : A C A D.
Also the time in which it would lose the velocity A p : time in which,
in a nonresisting medium, it would lose the same velocity : : arc A t :
tangent A p.
25. CoR. 6. Hence the time being given, the space described in ascent
or descent may be known, for the greatest velocity which the body can
acquire is constant, therefore the time in which a body falling in a non
resisting medium, would acquire that velocity is also known. Then the
sector ADTorADtcAADC:: given time : time just found; there
fore tho velocity A P is known or A p.
Then the area ABNKorABnk:ADTorADt:: space sought
for : space which the body would describe uniformly with its greatest
velocity.
26. Cor. 7. Hence vice versa, if the space be given, the time will be
known.
Book II.] NEWTON'S PRINCIPIA. 429
27. Prop. X. Let P F Q be the curve meeting the plane P Q. Let
T
L M
B C D E Q
G, H, I, K be the points in the curve, draw the ordinates ; let B C = C D
= D E, &c.
Draw H N, G L tangents at H and G, meeting the ordinates produced
in L and N, complete the parallelogram C H M D. Then tlie times
(X. V Li hi and V N I, and the velocities cc G H and H I, and the times
G H TT T
Qc ; let T and t = times, and the velocities cc — rp— and — — , therefore
the decrement of the velocity arising from the retardation of resistance and
G H H T
the acceleration of gravity oc —^p — , also the accelerating force of
gravity would cause a body to describe 2 I N in the same time, therefore
the increment of the velocity from G =
2NI
, again the arc is increased
M I X N I
by tlie space = HI — HN= RI=: jj^ , therefore the de
crement from tlie resistance alone =
HI
GH_Hl 2 M I X N I
T t "^ t X H I
GHxt uT,2MIxNI _., T
resistance : gravity : : rp W 1 + rirr — : 2 IN 1.
HI
Again, let
and
A B, C D, C E, &c. be — o + o, 2o, 3o, &c.
C H = P
MI = Qo+ Ro=^+So^ + &c.
.. D I = P — Q o + &c.
EK = P_2Qo — 4Ro« — &c.
BG=P + Qo + &c.
480 A COMMENTARY ON [Sect. II.
(BG — CH)» + BC«(= GH*) = o^+ Q^o*+ 3QRo' + &c.
.. G H« = 1 + Q'^ X o« + 3 Q II o^
.. G H = ^/ 1 + Q» X o + ^^'''
V 1 4 Q*
and
H T = o V~T+~Q' + Si2l=.
Subtract from C H ^ the sum G B and D I, and R o* and R o '^ +
3 S o ^ will be the remainder, equal to the sagittaa of the arcs, and which
are proportional to L H and N I, and therefore, in the subtracted num
ber of the times,
t / R + 3 S o R + So , .3So
•••T^x/ R ^ 2R °^^+TR'
... _^_ = o V 1 + Q« + :;^YTW "" "^ "SR
Q Ro« , 3So^ Vl + Q' , 3So QRo*
= ^l + Q' + vT + Q^+ 2R +2RXvT+Q^
QRo«
Mix NI _ Ro' X Qo + Ro' + &c.
HT "■ o. V iTTQ* QRo«
vi + Q^
G H X t „ . , 2MI X NI „ ., ,
.*. resistance ; gravity : : Fp H J H rr — — : 2 JN 1
3S0« V l+Q\ gR .
2R "^"^
: :3 S V 1 + Q=: 4 R«.
Tiie velocity is equal to that in the parabola whose diameter = H C,
H N* 1 f Q'
and the lat. rect. =  „ „ ■ or n — • The resistance « density x V S
, » , J . resistance 3 S V 1 + Q« . , R
therefore the density « — a T~WT directly «
J 1 s
directly oc
R V 1 + Q«
28. Ex. 1. Let it be a circular arc, CH = e, AQ = n, AC = a,
CD = o,
..DP = n»— (a+o)« = n' — a«— 2a© — o'=e*— Sao— o*,
Book II.]
and therefore
NEWTON'S PRINCIPIA.
DT ao n^o* an'o^
e 2e' 2e' '
P = e,Q = i,R = ^,S = ^„
.'. density «
S
a
a n^ 2 e
R V 1 + Q^ 2e
n
a a sm. ^ .
a — oc — a a tangent.
n e e cos. °
3 a n^ " n* «
The resistance : gravity : : ,, , ■ X r = r^ J : 3 a : 2 n.
2e
e e
29. Ex. 2. Of the hyperbola.
P A CD y
P I X b = P D S
.. put P C = a, C D = o, Q P = c,
.*. a + o X c — a — o = ac — a* — 2ao + co — o'
.. DI
2a + c
. o
b b •" b'
and since there is no fourth term,
S = 0,
.*. draw y = 0.
30. Prop. XIII. Suppose the resistance to a V + V*.
431
I> F
Case I. Suppose the body to ascend ; with the center D and rad. D B,
432
A COMMENTARY ON
[Sect. II.
describe the quadrant B T F; draw B P an indefinite line perpendicular
to B D, and parallel to D F. Let A P represent the velocity ; join D P,
D A, and draw D Q near D P.
.*. resistance «AP* + 2BAxAP, suppose gravity « D A%
.*. decrement of V « gravity + resistance ocAD'^+AP'^+2BAxAP.
oc D P^
D P Q (a P Q) : D T V : : D P* : D T*,
.. D T V a D T = oc 1,
therefore the whole sector E T D, is proportional to the time.
Case 2. Suppose the force of gravity proportional to a less quantity
than DAS draw B D perpendicular to B P, and let the force of gravity
P Q
a A B « — B D 2. Draw D F parallel to P B and = D B and widi the
center D — ^ axismajor = ^ axisminor = D B, describe a hyperbola
from the vertex F, cutting A D produced in E, and D P, D Q in T, V.
Now since the body is supposed to ascend.
The decrement of the velocity o:AP==f2AB x AP+AB« —
BD« a BP« — BDHB P'' = A P*f A B*^ + 2 A B x B P).
Also, DTV:DPQ::DT'':DP2(by similar triangles)
: : T G* : B D '^ (T G perpendicular to G)
: : D F*: P B'^ — D B^.
Now D P Q a decrement of velocity a P B '^ — D B ',
.*. DTVaDF*al a increment of the time, since the time flows uni
formly.
Book II.] NEWTON'S PRINCIPIA.
Case 3. If the body descend ; let gravity oc B D * — A B *.
483
With center D and vertex B, describe the rectangular hyperbola B T V,
cutting the lines D A, D P, D Q produced in E, T, V.
The increment of V « B D '
a BD*
DTV:DPQ(« PQ) ;
AB'^ — 2ABxAP — AP«
:(AB + AP)*a BD — BP«
D T*: D P*
GT':BP«::GD^— BD*:BP«
GD':BD*::BD'':BD«— BPS
.. DT Va BD*oc 1,
.♦. the whole sector E D T a time.
81. CoR. With the center C and distance D A describe an arc similar
toBT.
Then the velocity A P : the velocity which in the time E D t a body
would lose or acquire in a nonresisting medium : : a D A P : sector
ADt.
For V in a non resisting medium a time.
32. In the case of the ascent,
Let the force of gravity <x I. Resistance a 2 a v f v *
.. d va 1 + 2 a V + v«
d V
•'• T — r~:3 ; — ^2 oc time.
lj2avfv'
.". by Demoivre's first formula,
f. or time =
when
f. ; ;r — ; = — ^ X cir. arc. rad. = g and
1 + 2a Y4 V* g* ^
tangent = v } a
Vol. I.
434 A COIVIMENTARY ON [Sect. III.
The whole time .*. when v = = ^ x cir. arc rad. = g
and tangent = a f C.
.♦. coi^ time = — x cir. arc rad. = g and tangent v + a — cir. arc rad.
= g and tangent a.
.. the time of ascent = sector EDT — g' = l — a*.
33. In the case of descent,
dval — 2a V — v*
let
V  a = X
••. d V = d X
.. v*}2avia'' = x2
.. l+a^ — x2= l_2av — v'
•••f=ix/f^^+C,(g'= ! + ■>')
2g ^ g
Time = 0, V =r 0,
/. X = a,
2g ^ g — a
.. Cor^ time = 1 X f^^^  f^^^ .
o o to
34. Prop. XIV. Take A C proportional to gravity, and A K to the
resistance on contrary sides if the body ascend, and vice versa.
Between the asymptotes describe a hyperbola, &c. &c.
Draw A b perpendicular to C A, and
Ab:DB::DB«:4BA X A C.
The area A b N K increases or decreases in arithmetic progression it
the forces be taken in geometric progression.
Now
A K Qc resistance a2BAP + AP*.
Let
2BAP + AP*
AK =
.•.KL =
Z
2B A X PQ+2APX PQ
Book II.]
NEWTON'S PRINCIPIA.
435
B
D
A
N
KQ P
B
II
V
b
\
/Qp
"" AJ^
■^ LK
/^T
^
/f
E
KL =
2 B P Q
Now
..KLON = iMAP«xLO.
Ab:LO::CK:CA
DB:Ab::4BAx CA:DB*
LO= ^^'
.•.KLON =
4BA X CK
2PBxPQxBD^
4BAx CK X Z •
Case 1. Suppose the body to ascend,
gravity a: AB' + BD^ = K^'^^Ti
Ee2
•J.S6 A COMMENTARY ON [Sect. IV.
. „ AP» + 2BAP
A K = 2
.•.DP« = CKx Z.
.•.DT*:DP*::DB»:CK x Z
and in the other two cases the same result will obtain.
Make
DTV = DBx m.
.•.DBxm:iDBxPQ::DB':CKxZ
.. BD'xPQ=:2BDxmxCKxZ.
.'.AbNK = ^^'x BDxm
A B
.■.AbNKDTV= ^^'^fS^Px i^aAP.« velocity.
AB
.*. it will represent the space.
SECTION IV.
35, Prop. XV. Lemma. The
/I. O P Q = a rectangle = ^i O Q R
and
^ S P Q = £. of the spiral = ^ S Q R
.. ^ O P S = z. O Q S.
.'. the circle which passes through the points P, S, O, also passes
rcl
2"
Circle
through Q. Also when Q coincides with P, this — ^ — touches the spiral.
.'. ^ P S O z. in a — r — whose diameter = P O.
Book II.] NEWTON'S PRINCIPIA. i^t
Also
T Q : P Q : : P Q : 2 P S.
r. PQ=^ = 2PS X TO
which also follows from the general property of every curve.
PQ"= P V X QR.
36. Hence the resistance « density X square of the velocity.
37. Density a i^ j centripetal force « density ^ « tt 5 .
•^ distance ^ '' distance^
Then produce S Q to V so that S V = S P, and let P Q be an arc
described in a small time, P R described in twice that time, .♦. the decre
ments of the arcs from what would be described in a nonresisting me
dium a T^
.*. decrement of the arc P Q =  decrement of the arc P R
.'. decrement of the arc PQ = ^Rr(ifQSr = area P S Q).
For let P q, q v be arcs described (in the same time as P Q, Q R) in a
nonresisting medium,
PSq— PSQ = QSq = qSv — QSr
= rSv — QSq
.. 2QSq = rSv
.♦. if S T ultimately = S t be the perpendicular on the tangents
STxQqrr^StXrv
.. 2 Q q = r v
and
R v = 4 Q q.
.. 2 Q q = R r.
Hence
Resistance : centripetal force : :  R r : T Q,
Also
T Q X S P^ a time", (Newt. Sect. II.)
.. P Q 2 X S P a time 
.*. time a P Q X VHP
also
VatQ a
V SQ
Ee3
P Q X V S P V/ S P
1
438 A COMMENTARY ON [Sect. IV.
P Q : Q R
PQ: Q r
V SQ: V S P
SQ: V SQ X SP
SQ: S P
since the areas are equal, and the angles at P and Q are equal.
.. PQ: Rr::SQ:SP— V SQ x SP
: : S Q : ^ V Q
For
SQ = SP — VQ
.•.SQxSP = SP* — VQx SP
.. v/SQxSp=SPiVQX^_&c.
.. ^ V Q ultimately = S P — V S P x S Q
T» • ^ decrement of V R r
Resistance « _,,_ a PQ^xSP
. hJQ
PQxSQxSP
^VQ:PQ::OS:PO
and
^OS
S Q = S P oc
O P X SP*
O s
.. density X square of the velocity oc resistance a Tyjj o~pi
• • ^^"^'^y ^ OPXSP
O S
and in the logarithmic spiral jYn ^^ constant
.. density cc ^g . Q. e. d.
38. Cor. 1. V in spiral = V in the circle in a non resisting medium at
(.he same distance.
39. Cor. 3. Resistance : centripetal force : : ^ R r : T Q
..iVQx PQJPQ^
SQ • SP
::iVQ:PQ
: : ^ O S : O P.
.*. the ratio of resistance to the centripettJ force is known if the spiral be
given, and vice versa.
40. Cor. 4. If the resistance exceed I the centripetal force, the body
cannot move in this spiral. For if the resistance equal I the centripetal
Book II.] NEWTON'S PRINCIPIA. 43U
force, O S = O P, .*. the body will descend to the center in a straiglit
line PS.
V of descent in a straight line : V in a nonresisting medium of de
scent in an evanescent parabola : : 1 : V 2; for V in the spiral = V in the
circle at the same distance, V in the parabola = V in the circle at
^ distance.
Hence since time ex ^ ,
time of descent in the 1st case : that in 2d : : V 2 : 1.
41. Cor. 5. V in the spiral P Q R = V in the line P S at the same
distance. Also
PQR: PS in a given ratio:: PS: PT:: OP: OS
.. time of descending PQR: that of P S : : O P : O S.*
Length of the spiral = T P = sector of the /l T P S.
a:b::b:c::c:d::d:e
a + b I c + &c. : b + c + d + &c. : : a : b
.. a  b ( c }■ &c. : a : : a : a — b.
42. Cor. 6. If with the center S and any two given radii, two
circles be described, the number of revolutions which the body makes
between the two circumferences in the different spirals oc tangent of the
P S
angle of the spiral a yt^ .
The time of describing the revolution : time down the difference of ilie
radii : : length of the revolution : that difference.
2d a 4th,
.'. time a length of the revolution cc secant of the angle of the spinsl
OP
QC
o s*
• pq: pt:
: S
P:
Sy
d w
p d X
: X
: p.
Vr* — p
.'. A w
X <1 X
.'. TV
X*
"i '
2^ r» — p
Et
4
440 A COMMENTARY ON [Sect. IV
43. Con. 7. Suppose a body to revolve as in the proposition, and to cut
the radius in the points A, B, C, D, the intersections by the nature of the
spiral are in continued proportion.
,,,. (. , . perimeters described
1 unes ot revohition a ^
and velocity a
1
V distance
a A S^ B S^ CS^,
5 5 5
.*. the whole time : lime of one revolution ::AS2fBS*+ &c. : A S ■
:: A S^: AS^
1
BSI
44. Prop. XVI. Suppose the centripetal force x
S P " + ' '
time a P Q X S P 2
and velocity cc ~
S P 2
PQ : Q R
Qr : PQ
Qr : QR
.. Q r : R r
S Q<f : SP2
SP : SQ
SQ2
SPs
SQ2* : SQ2' — SP2'
For
S Q : 1— i n . V Q.
SP = SQ+ VQ,
Book II.] NEWTON'S PRINCIPIA. 441
.. SP^i = SQ^i +  — 1. VQ X SQ^2 + &c.
... SQ2» — SP^'i = i_'^ X VQ xSQ^^.
Then as before it may be proved, if the spiral be given, that the density
CO ^p . Q. e. d.
45. Cor. 1.
Resistance : centripetal force : : 1 — g n . O S : O P,
for the resistance : centripetal force : :  II r : T Q
:: (ll) X VQx PQ PQ'
2 8 Q 2 S P
l~X VQ:PQ
:: 1— x OS: OP.
46. CoR. 2. If n + 1 = 3, 1 — ^ = 0,
.*. resistance = 0.
Cor. 3. If n + 1 be greater than 3, the resistance is propelling,
SECTION VI.
47. Prop. XXIV. The distances of any bodies' centers of oscillation from
the axis of motion being the same, the quantities of matter oo weight
X squares of the times of oscillation in vacuo.
force X time
For the velocity jjenerated qd ~ t — • Force on bodies at
•' ° quantities or matter
e(]ual distances from the lowest points go weights, times of describing
corresponding parts of the motion x whole time of oscillation,
t, , force X time of oscil.
.*. quantities or matter oc , — .
' velocities
00 weights X squares of the times,
since the velocities genei'ated x : for equal spaces.
° times ^ '■
48. CoR. 1. Hence the times being the same, the quantities of mattei*
00 weights.
Colt. 2. If the weights be the same, the quantities of matter co tiuic^
Cor. 3. If the quantities of matter be the same, the wciglits cc : j ..
442 A COMMENTARY ON [Sect. VJ
49. Coil. 4. Generally the acceleratinff force oc ^.r— of matter
quantities '
and L 00 T T*,
. J WxT«
L '
.'. if W and Q be given L oo T 2.
If T and Q be given L oo W.
.^ ^ K 11 ^, ^.^ r ,, weightx time* of oscillation
50. Cor. 5. generally the quantity of matter qd — j —. .
51 Prop. XXV. Let A B be the arc which a body would describe in a
nonresisting medium in any time. Then the accelerating force at ajiy
point D 00 C D ; let C D represent it, and since the resistance oo time,
it may be represented by the arc C o.
.'. the accelerating force in a resisting medium of any body d,  o d.
Take ;
o d : C D : : e B : C B.
Therefore at the beginning of motion, the accelerating force will be in
this ratio, .*. the initial velocities and spaces described will be in the same
ratio, .*. the spaces to be described will also be in the same ratio, and
vanish together, .•. the bodies will arrive at the same time at the points
C and o.
In the same way when the bodies ascend, it may be proved that they
will arrive at their highest points at the same time. .'.If A B : a B in
the ratio C B : o B, the oscillations in a nonresisting and resisting me
dium will be isochronous. Q. e. d.
Book II.]
NEWTON'S PRINCIPIA.
448
Cor. The greatest velocity in a resisting medium is at the point o.
The expression for the ^ time of an oscillation in vacuo, or time of de
scent down to the lowest point a quadrant whose radius = 1. Now
B
\ /
R
/
«\
N
y's
M\ I
^
suppose the body to move in a resisting medium when the resistance
: force of gravity : : r : 1 .
Then vdv = — gFdx + grdz = — gd^x + grdz. Now by
a property of the cycloid, if ^ be the axis, dx:dz::x:::z:a,
.. d x =
z d
.♦. V d V = ~ xzdz + grdz — —
a /6
= ^ X z'' + g r z,
Now
Xz2 + 2grz+C.
z = d, V = o,
v^ = ^ X d
a
2 g r X d — z
a
X d« — 4ard + 2adrz — z*,'
.. V =y— a— xVd'' — 2ard + 2arz — zS
— dz /. a . — d z
..dt ^ ~J g ^ Vd« — 2ard + 2arz— zl
Assume
z — a r = y,
.•.,z ' — 2arz+a*r^ = yS
.'. 2arz — z* = a^v' — yS
d' — 2 a r d + 2 a r z — z^ = (d — ar)^ — y^ = (b' — y'.)
444 A COMMENTARY ON CSkct. VI.
aiid
(1 z = d y
a — dy
and
.*. t = /" — X circular arc, radius = 1,
z — a r
COS. = J + C and C = o.
d — a r
— X circular arc
whose COS. = J , .*. time in vacuo : time in resisting medi
d — ar' *" .w...w..^
a r
mm
: : quadrant : arc whose cos. = j .
Cor. 1. Time of descent to the point of greatest acceleration is constant,
for in that case z = a r,
••. t = /* — X quadrant, for d v = 0,
.. V d V = 0,
.'. — gzdz + garz = 0,
.'. z = a r,
.'. z : r : : a : 1.
Cor. 2. To find the excess of arc in descent above that in ascent.
vdv= +gTc(xfgrdz,
I ff z d z ,
. .. V d V = — ^ ff r d z
a
o
V* mz' , ^
..^ grz + C,
.. v«= ^ (d — z») — (z — d) X 2 a r
a
= ^ X (d » — 2 a r d) — (2 a r z— z ')
which when the body arrives to the highest point = 0,
d" — 2a rd — 2arz — z* = 0,
.. d ' — 2 a r d = z * + 2 a r z,
.. z + a r = d — a r,
.'. z = d — 2 a r,
.. d — z = 2 a r,
Book II.]
NEWTON'S PRINCIPIA.
445
52. Prop. XXVI. Since V oc arc, and iesistance a V, resistance a arc.
.'. Accelerating force in the resisting medium a arcs.
Also the increments or decrements of V a accelerating force.
.*. the V will always a arc.
But in the beginning of the motion, the forces which oo arcs will generate
velocities which are proportional to the arcs to be described. .. the velo
cities will always co arcs to be described.
.*. the times of oscillation will be constant.
53. Prop. XXVIII. Let C B be the arc described in the descent, C a
in the ascent.
.. A a = the difference (if A C = C B)
Force of gravity at D : resistance : : C D : C O.
C A = CB
Oa = O B
.. CA — OaorAa — eO = CBOB = CO
.. CO = i Aa
.*. Force of gravity at D : resistance : : C D : 1 A a
.\ At the beginning of the motion,
Force of gravity : resistance : •• 2 C B : A a
: : 2 length of pendulum : A a.
54. Prob. To find the resistance on a thread of a sensible thickness.
Resistance go V * X D ^ of suspended globe.
.*. resistance on the whole thread : resistance on the globe C
446 A COMMENTARY ON [Sect. VI.
2a'b*. (a— b)* : a'r«c — r«c^ (a — 2b)\ c = a + r.
: a^b«. (a— b)* : 3 a«r*c« b— ba b*r*c» + 4 b='r»c*,
a'b . (a — b)« : 3 a'r *c» — ba b r »c' + 4b « r« c«,
.*. resistance on the thread : whole resistance
::a'b. (a— b)« : r*c« . (3 a' — b ab + 4 b«).
Cor. If the thickness (b) be small when conipared with the length (a)
8a« — bab4 4b*=3a' — bab + 3 b ' (nearly) = 3. (a — b) ^
.*. Resistance on the whole thread : resistance on the globe
: : a^ b: 3r2c«
and
Resistance on the thread : whole resistance to the pendulum
: : a ' b : a 3 b + 3 r * c ^
Suppose, instead of a globe, a cylinder be suspended whose ax. = 2 r.
Now by differentials
die resistance on the circumference : resistance on the base : : 2 : 3.
'%
By composition the resistance to the cylinder : resistance on the square
= 2 r : : 2 : 3.
Resistance a x * x',
.*. resistance ax',
.'. resistance to the whole thread oc x\
Resistance on A E a (a — 2 b) » if 2 b = E D.
.'. Resistance on the thread : resistance of the globe
:: 16.a'b». (a — b) ^ : 3 p . a ' — (a — 2 b; ^ xr^ (a + i)'.
55. Prop. XXIX. B a is the whole arc of oscillation. In the line O Q
take four points S, P, Q, R, so that if O K, S T, P I, Q E be erected
Book II.]
NEWTON'S PRINCIPIA.
447
perpendiculars to O Q meeting a rectangular hyperbola between the
asymptotes O Q, OK in T, I, G, E, and through I, K F be drawn
O S P rRQ M
parallel to O Q, meeting Q E produced in F. The area P I E Q may
be : area P I S T : : C B : C a. Also IEF:ILT::OR:OS.
Draw M N perpendicular to O Q meeting the hyperbola in N, so that
P L M N may be proportional to C Z, and P 1 G R to C D.
Then the resistance : giavity : : ^^ xTEF — IGHiPINM.
Now since the force oc distance, the arcs and forces are as the hyper
bolic areas. .*. D d is proportional to R r G g.
(O T?
fYn '^ ^ ^ — ^ ^ ^)
= Gllgh — ?^^A4^:RrxGR::HG — ^J^:GR::0RX
OQ
OR
OQ
HG — lEZ X ^:OP xPI(ORxHG = ORxHR —
OPxPI = PIHR=PIRG+IGH)::PIRG+lGH —
^xIEF.OPxPI.
Now if Y = ^ X I E F — I G H, the increment Y a P I G R — Y.
Let V = the whole from gravity. .*. V — R = actual accelerating
force. .". Increment of the velocity a V — R X increment of the time.
As the resistance oc V ' the increment of resistance a V X increment of
,, , . J . , .^ increment of the space ^ . r
the velocity, and the velocity a : 7^, — . . .*. Increment 01
''  •' mcrement 01 the tune
resistance cc V — R if the space be given, cc P I G R — Z, if Z be the
area which represents the resistance R e.
Since the increment Y a P I G R — Y, and the increment of Z
44>d
A COMMENTARY ON
[Skct. VIII.
ooPIGR — Z. IfY and Z be equal at the beginning of the motion and
begin at the same time by the addition of equal increments, they will still
remain equal, and vanish at the same time.
Now both Z and Y begin and end when resistance = 0, i. e. when
O R
.lEF — IGH =
or
OQ
ILT
OS
O R X I E F
xOR — IGH = 0.
IG H = Z
OQ
O R
.. Resistance : gravity : : ^^ .lEF — IGH:PMNI.
SECTION VIll.
56. Prop. XLIV. The friction not being considered, suppose the mean
K
M
N
altitude of the water in the two arms of the vessel to be A B, C D. Then
when the water in the arm K L has ascended to E F, the water in the arm
M N will descend to G H, and the moving force of the water equals the
excess of the water in one arm above the water in the other, equals twice
A E F B. Let V P be a pendulum, R S a cycloid = ^ length of the
canal, and P Q = A E. The accelerating force of the water : whole
weight : : A E or P Q : P R.
Book IL] NEWTON'S PRINCIPIA. 449
Also, the accelerating force of P through the arc P Q : whole weight
of P : : P Q : P R; therefore the accelerating force of the water and P
cc the weights. Therefore if P equal the weight of the water in the canal,
the vibration of the water in the canal will be similar and cotemporaneous
with the oscillations of P in the cycloid.
Cor. 1. Hence the vibrations of the water are isochronous.
CoR. 2. If the length of the canal equal twice the length of the
pendulum which oscillates in seconds; the vibrations will also be performed
in seconds.
Cor. 3. The time of a vibration will « V L.
Let the length = L, A E = a,
then the accelerating force : whole weight : : 2 a : L,
2 a
.*. accelerating force = y ;
2 A
.'. when the surface is at 0, the accelerating force = — j — .
Put E = X,
A = a — X,
.'. accelerating force = "^ ,
, g . 2 a d X — 2 X d X
.. V d V = ^2 ,
_ 2g
V ^ = ys X 2 a X — X 2,
V
= ^ j^ X a/ 2a X — X*
dt— — = ^ ^ ^^^^
V V 2ga'' V 2ax
=J
X cir. arc rad. = a, and vers. = x
2ga
+ cor", and cor". = 0,
♦.• t = 0, X = 0,
.. if p = 3. 14159, &c.
' = V 2li; X l" Swhen (x) = i.)l= j'^ X f
.*. time of one entire vibration = p x ^ / rr — = time of one entire vi
^ V 2g
bration of a pendulum whose length = — .
Voi I. Ff
450
A COMMENTARY ON
[Sect. VIII.
D
67. Cor. 1. Since the distance (a) above the quiescent surface does
not enter into the expression. The time will be the same, wiiatevev be
the value of A E.
58. Cor. 2. The greatest velocity is at A = /,J ^ X «> a y'~'^iJ~i
I AE»
69. Prop. XLVII. Let E, F, G be three physical points in the line
B C, which are equally distant ; E e, F f,
G g the spaces through which they move
during the time of one vibration. Let s, p, y
be their place at any time. Make P S =
E e, and bisect it in O, and with center O
and radius O P = O S, describe a circle.
Let the circumference of this circle repre
sent the time of one vibration, so that in
the time P H or P H S h, if H L or h 1
be drawn perpendicular to P S and E £ be
taken = P L or P 1, E « may be found in
E ; suppose this the nature of the medium.
Take in the circumference P H S h, the arcs
HI, IK, hi, i k which may bear the
same ratio to the circumference of the circle as E F or F G to
B C. Draw I M, K N or i m, k n perpendicular to P S. Hence
PI, or P H S i will represent the motion of F . and P K or
P H S k that of G . E «, F<p, G y = P L, P M, P N or P 1,
P m, P n respectively.
Hence eyorEG+Gy — Ei = GE — LN = expan
sion at £ 7 ; or = E G + 1 n.
.*. in going, expansion : mean expansion : : G E — L N : E G
In returning, : : : E G + In : E G
Now join I O, and draw K r perpendicular to H L, H K r,
I O M are similar triangles, since the iLKHr = ^KOk=^
I O i = z I O P and A at r and M = 90°,
.. L N : K H : : I M : I O or O P, and by supposition K H :
EG:: circumference PSLP:BC::OP:V = radius of
the circle whose circumference = B C.
.•. by composition LN:GE::IM:V.
.'. expansion : mean expansion : : V — I M : V,
G
F
E 
B
Book II.] NEWTON'S PRINCIPIA. 451
.♦. elasticity : mean elasticity : : y j j^ : y. In the same way, for the
points E and G, the ratio will be y _^^ ^^ : ~ a y^K N * ^
: : excess of elasticity of E : mean elasticity
H L— KN 1
' • V '— H Lx V— K NxV + HLxKN'T
: : H L — K N : V.
Now J
V a 1.
.*. the excess of E's elasticity cc H L — K N, and since H L — K N
= H r : H K : : O M : O P,
.. H L — K N a O M,
••. excess of E's elasticity oc O M.
Since E and G exert themselves in opposite directions by the arc's ten
dency to dilate, this excess is the acceleratinsr force of e y, .•. accelerating
force 00 O M.*
ON THE HARMONIC CURVE.
Since the ordinates in the harmonic curve drawn perpendicular to the
axis are in a constant ratio, the subtenses of the angle of contact will be
in the same given ratio. Now the subtenses a — j — t^ , and when
° rad. oi curv.
the curve performs very small vibrations, the arcs are nearly equal.
Now the curv. oc — , , .*. subtense a curvature,
rad.
Hence the accelerating force on any point of the string a curvature at
that point.
• Now bisect F f in n,
. •. O M = n ^
For
OM=OP— PM=nF— F^=:fi(p
i. e. the accelerating force a distance from il the middle point. Q. e. d.
Ff.?
452 A COMMENTARY ON [Sect. VIII.
To fijid the equation to the harmonic curve.
O S
Let A C be the axis of the harmonic curve C B A, D the middle point,
draw B D peipendicular cutting the curve in B; draw P M perpendi
cular to B D cutting the curve in P, and cutting the quadrant described
with the center D and radius D B in N. Draw P S perpendicular to A C.
Put
BD = a, PM = y, BM = x,
.. D M = a — X = P S.
r = rad. of curv. at B, B P = z,
, d z cl X
.'. rad. of curv. = ,~^ (if d e be constant).
Now
B D : P S : : curvature at B : curvature at P
: : rad. of cur. at P : rad. at B
or
a : a
— d z d X
X : : —J — : r,
d'y
Now
.*. rad*y + adzdx — xdxdz = 0,
X ' d z
.'. rady + adzx — = + C.
X = 0, d y = d X,
radz = + C = C,
X ^ d z
rady + axdz ^ —
= r a d z.
Put
a X
x*
.•. r a d y = ra — b' d z,
.. r^i*dy»= (ra — b*)« X dx* + r*a^dy*— 2rabdy'+ b^dyS
Rook II.] NEWTON'S PRINCIPIA. 453
.•. (ra — b2)2xdx2= 2 r a b ''dy ^— b* d y 2,
.. r« aMx*= 2 r ab^dy*
if (b) be small compared to (a j,
.•.dy* =
r ad X*
2b« *
.••dy =
V r a
V 2 ax
Xdx
— x^
 >V a
X
adx
V 2
ax —
X*
.♦. y z= ^ / — X circular arc whose rad. = a, and vers. = x
•^ 'V a
I C, and cor". = 0,
because when y = 0, x = 0,
.*. arc = 0.
.. C D = J^ X quadrant B N E,
*and therefore
CD
V a "■
B N E»
B N X ,^ Z
BNE
60. Prop. XLIX. Put A = attraction of a homogeneous atmosphere
when the weight and density equal the weight and density of the medium
through which the physical line E G is supposed to vibrate. Then every
thing remaining as in Prop. XLVII. the vibration of the line E G will
be performed in the same times as the vibrations in a cycloid, whose
length = P S, since in each case they would move according to the same
law, and through the same space. Also, if A be the length of a pendulum,
since T a V L
The time of a vibration : time of oscillation of a pendulum A
: : V~FO : V^A.
Also (Prop. XLVII.), the accelerating force of EG in medium : ac
celerating force in cycloid
:: A X HK: Vx EG;
since H K : G E : : P O : V.
:: PC X A : V«.
F f 3
454 A COMMENTARY ON [Sect VIIT.
Now
T cc ^ ffT when L is given.
.*. the lime of vibration : time of oscillation of the pendulum A
: : V : A
: : B C : circumference of a circle rad. = A.
Now B C = space described in the time of one vibration, therefore
the circumference of the circle of radius A = space described in the time
of the oscillation of a pendulum whose length =r A.
Since the time of vibration : time of describing a space =r circum
ference of the circle whose rad. = A : : B C : that circumference.
Cor. 1. The velocity equals that acquired down half the altitude of
A. For in the same time, with this velocity uniform, the body would de
scribe A ; and since the time down half A : time of an oscillation : : r :
circumference. In the time of an oscillation the body would describe the
circumference.
Cor. 2. Since the comparative force or weight oc density X attraction
elastic forcG
of a homogeneous atmosphere, A go —^ r , and the velocity <xi V A.
V elastic force
oc , .^^^^ .
V density
SCHOLIUM.
61. Prop. XLIX. Sound is produced by the pulses of air, which
theory is confirmed, 1st, from the vibrations of solid bodies opposed to it.
2d. from the coincidence of theory with experiment, with respect to the
velocity of sound.
The specific gravity of air : that of mercury : : 1 : 11890.
Now since the alt. a — ^ , .*. 1 : 11890 : : 30 inches : 29725 feet = ,
sp. gr.
altitude of the homogeneous atmosphere. Hence a pendulum whose
length = 29725, will perform an oscillation in 190'', in which time by
Prop. XLIX, sound will move over 186768 feet, therefore in 1'' sound
will describe 979 feet. This computation does not take into considera
tion the solidity of the particles of air, through which sound is pro
pagated instantly. Now suppose the particles of air to have the same
density as the particles of water, then the diameter of each particle : dis
Book I.] NEWTON'S PRINCIPIA. 455
tance between their centers : : 1 : 9, or 1 : 10 nearly. (For if there are
two cubes of air and water equal to each other, 1) the diameter of the par
ticles, S the interval between them, S + D = the side of the cube, and if
N = N°. N S + N D z= N". in the side of the cube, N". in the cube
30 N \ Also, if M be the N°. in the cube of water, M D the side of the
cube and the N°. in the cube a M ^.
Put 1 : A : : N 3 : M ^
.. M = A ^ N,
By Proposition
NS + ND = MD = NA^D,
.. S = D X A*_i,
.. S:D:: A^ — 1 : 1,
.. S + D : D : : A^ : 1 : : 9 : 1 if A = 870
or 10 : 1 if A = 1000).
Now the space described by sound : space which the air occupies : : 9 : II,
m
9
' 979
.'. space to be added = ^ = 108 or the velocity of sound is 1088
feet per 1".
Again, also the elasticity of air is increased by vapours. • Hence since
the velocity a — ■ . ^ ; if the density remain the same the velocity
V density
a V elasticity. Hence if the air be supposed to consist of 11 feet, 10 of
air, and I of vapour, the elasticity will be increased in the ratio of 11 : 10,
therefore the velocity will be increased in the ratio of 11 : 10 or 21 : 20,
therefore the velocity of sound will altogether be 1142 feet per 1'', which
is the same as found by experiment.
In summer the air being more elastic than in winter, sound will be
propagated with a greater velocity than in winter. The above calculation
relates to the mean elasticity of the air which is in spring and autumn.
Hence may be found the intervals of pulses of the air.
By experiment, a tube whose length is five Paris feet, was observed to
give the same sound as a chord which vibrated 100 times in 1", and in
the same time sound moves through 1070 feet, therefore the interval of
the pulses of air = 10.7 or about twice the length of the pipe.
Ff4
456
A COMMENTARY ON
[Sect; VJIT.
62. On the vibrations of a harmonic string.
The force with which a string tends to the center of the curve : force
which stretches the string : : length : radius of curvature. Let P p be a
small portion of the string, O the center of the curve ; join O P, O p, and
draw P t, p t, tangents at P and p meeting in t, complete the parallelo
gram P t p r. Join t r, then P t, p t represent the stretching force of
the string, which may be resolved into P x, t x and p x, t x of which
P X, p X destroy each other, and 2 t x = force with which the string
tends to the center O. Now the AtPr= ^ /lF O p, .'. z. tV x =z
P O p, .*. t r : P t : : P p : O P, i. e. the force with which any particle
moves towards the center of the curve : force which stretches it : : length
: radius.
63. To find the times of vibration of a harmonic string.
D
^/""^^
o\
B
C
A
P
Let w = weight of the string. L = length.
Dd:L
weight D d : w
weight of D d =
D d X w
Book II.]. NEWTON'S PRINCIPIA. 4^57
Also
D d : — ^ —  = rad. of curve : : the moving force of D d : P
. 4.U ' r £T\ J PxDdxap*
.'. the movinff force of D d = —  — = —
° L, w
.*. accelerating force = ^r^ — X .p— ^
^ L* Dd X w
 P X ap*
"" Lw.
if D O = X, D C = a, O C = a — X,
,'. the accelerating force at O = — ^—4
... V d s = ^: P X a (1 X — X d X
I Ji w
... V* =SPpl xTax — z^
L w
.. v = . / ^T ^* X V2ax — x«.
'V L w
•. C and 1 = 0,
d X / L w d X
.. d t =  = ^
V VgPp* v'2ax — X
.•.t=J — o—i X cir. arc rad. = I
^ e P P*
and
X
vers, sine = — ,
a
when X = a,
t = 0.
Lw .. , / Lw ^ P
••• J apZy^ ^ quadrant = J
YPp^ 4—  V YFP" ^ 2
/ L w
V ff P«
.*. time of a vibration r= ^ / — =r 1"
^ gP
.'. number of vibrations in 1" = ^ / ^ — .
V L w
CoR. Time of vibration = time of the oscillation of a pendulum whose
1 1 L w
length =p^.
458
For this time
A COMMENTARY, &c.
[Sect. IX.
=v
gp ■
64. Prop. LI. Let A F be a cylinder moving in a fluid round a
fixed axis in S, and suppose the fluid divided into a great number of solid
I H G
orbs of the same thickness. Then the disturbing force a translation of
parts X surfaces. Now the disturbing forces are constant. .*. Transla
tion of parts, from the defect of lubricity a r • Now the diffcr
^ distance
. On A Q draw
f.^, 1 ^. translation
ence oi the angular motions a — n— a
distance d stance*'
A a, B b, C c, &c. : : r: ^ j then the sum of the differences will
a hyperbolic area.
.*. periodic time x
a
a distance.
angular motion hyperbolic area
In the same way, if they were globes or spheres, the periodic time
would vary as the distance *.
END OF THE FIRST VOLUME.
COMMENTARY
NEWTON'S PRINCIPIA.
A SUPPLEMENTARY VOLUME.
DESIGNED FOR THE USE OF STUDENTS AT THE UNIVERSITIES.
BY
J. M. F. WRIGHT, A. B.
LATE SCHOLAR OF TRIXITV COLLEGE, CAMBRIDGE, AUTHOR OF SOLUTIONS
OF THE CAMBRIDGE PROBLEMS, &C. &C.
IN TWO VOLUMES.
VOL. II.
' LONDON:
PRINTED FOR T. T. & J. TEGG, '73, CHEAPSIDEj
AND RICHARD GRIFFIN & CO., GLASGOW.
MDCCCXXXIII.
c\\^
GLASGOW :
GEORGE EROOKMAX, PlUNTKn, TlLLAl'JtLO.
INTRODUCTION
VOLUME II.
'AND TO TU£
MECANIQUE CELESTE.
ANALYTICAL GEOMETRY
1. To determine the position of a point injixed space.
Assume any point A in fixed space as known and immoveable, and let
Z' z
three fixed planes of indefinite extent, be taken at right angles to one
another and passing through A. Then shall their intersections A X',
A Y', A 7J pass through A and be at right angles to one another.
U INTRODUCTION.
This being premised, let P be any point in fixed space; from P draw
]* z parallel to A Z, and from z where it meets the plane X A Yi draw
z X, z y parallel to A Y, AX respectively. Make
A X = X, Ay = y, P z = z.
Then it is evident that if x, y, z are given, the point P can be found
■practically by taking A x = x, A y = y, drawing x z, y z parallel to
AY, AX; lastly, from their intersection, making z P parallel to A Z
and equal to z. Hence x, y, z determine the position of the point P.
The lines x, y, z are called the rectangular coordinates of the point P ;
the point A the origin of coordinates ; the lines AX, AY, A Z the axes
of coordinates, A X being further designated the axis of x, AY the axis
of y, and A Z the axis of z; and the planes X A Y, Z A X, Z A Y co
ordinate planes.
These coordinate planes are respectively denoted by
plane (x, y), plane (x, z), plane (y, z) ;
and in like manner, any point whose coordinates are x, y, z is denoted
briefly by
point (x, y, e).
If the coordinates x, y, z when measured along AX, AY, A Z be
always considered positive ; when measured in the opposite directions,
viz. along A X' A Y', A Z', they must be taken negatively. Thus ac
cordingly as P is in the spaces
Z A X Y, Z A Y X', Z A X' Y', Z A Y' X;
Z'AXY, Z'AYX', Z'AX'Y', Z'AY'X,
the point P will be denoted by
point (x, y, z), point ( — x, y, z), point ( — x, — y, z), point (x, — y, z)\
point (x, y,  z), point ( x, y,  z), point ( x,  y,  z), point (x,  y, • z)
respectively.
2. Given the position of iivo points (a, /3, 7), («', /3', 7') in Jixed space,
tojind the distance bet'isoeen them.
The distance P P' is evidently the diagonal of a rectangular parallelo
piped whose three edges are parallel to A X, A Y, A Z and equal to
as a', i3s/3', 7s/.
Hence
P F = V (aaO*+ (/3^')*+ iy — yV .... (1)
the distance required.
Hence if P' coincides with A or a', /3', 7' equal zero,
P A = VV* h /32 + 7« (2)
AxNALYTICAL GEOMETRY. iii
3. Calling the distance of any point P (x, y, z) from the origin A of
coordinates the radiusvector, and denoting it by g, suppose it inclined to
the axes A X, A Y, A Z or to the planes (y, '/,), (x, z), (x, y), by the
angles X, Y, Z.
Then it is easily seen that
X = f cos. X, y = f COS. Y, z = ^ cos. Z (3)
Hence (see 2)
COS. X rr 7— — Y, r~, — iT > COS. Y = , , , , , , — ^,
V(x2+y* + z*)' V(x*+y* + z')' ,
'°^^= V(x'+V + 3') • <*^
SO that when the coordinates of a point are given, the angles 'which the ra
diusvector makes 'with each of the axes may hence be found.
Again, adding together the squares of equations (3), we have
(x» + y« + z«) = ^2 (COS.2X + cos.2 Y + cos.'Z).
But
^2= x^ + y« + z^ (see 2),
.. cos. 2 X + cos. 2 Y + COS. * Z = 1 (5)
which shows that when two of these angles are given the other may be
found.
4. Given two points in space, viz. (a, jS, y), (a* (3', y'), and one of the
coordinates of any othei^ 'point (x, y, z) in the straight line that passes
through them, to determine this other point ; that is, required the equations
to a straight line given in space.
The distances of the point (a, j8, y) from the points («', /3', /), and
(x, y, z) are respectively, (see 2)
P F = V (a_a')^+ 0/3')'+ (7— ^')%
and
P Q = V (« — x) ^ + — y) « + (7 — z) ^
But from similar triangles, we get
(yz)^: (PQ)«:: (7/)(PF)*
whence «
which gives
"H« — «')'+(^/301(7 — z)*=(7/)'.U« — x)' + (/3 — y)^}
* But since a, a' are independent of /3, /3' and vice versa, the two first
terms of the eqnation,
(a_a )\ (y_z)« (y/)* («_x)^  {yyj (/3_y)' + (/3/3')^ (yz)' =
a2
IV
INTRODUCTION.
are essentially different from the last. Consequently by (6 vol. 1.)
(a — a')^(7z)* = (y_/)=!(a_x)«
0—^') ' (yz) 2 = (y— /) ^ (,8— y) 2 ^
■which give
z — 7 = +?^^^(a — x'))
";z;/ [ (6)
These results may be otherwise obtained ; thus, p g p',is the projection
of the given line on the plane (x, y) &c. as in fig.
p q p'
Hence
Also
z — y : / — y : : p q : p p'
: : m n : m p'
• : : y — /3 : ^'— ^
iV
z — "y": / — 7::pq:pp'::pr:pm
: : a — X : a — a.
Hence the general forms of the equations to a straight line given in
space, not considering signs, are
z = a X + bl
z = a' y + b' f
To find where the straight line meets the planes, (x, y), (x, z), (y, z),
we make
z = 0, y =i: 0, X = 0,
which give
Jb^
hii^Si^l
ANALYTICAL GEOMETRY,
b
z = b'
b' — b ^
X =
a
z = b
b — b'
y =
=:;} («)
a
which determine the points required.
To find when the straight Une is parallel to the planes, (x, y), (x, z),
(y, z), we must make z, y, x, respectively constant, and the equations be •
come of the form
," = " ^ ,1 (8)
ay = ax + b — bj ^
To find when the straight line is perpendicular to the planes, (x, y),
(x, z) (y, z), or parallel to the axes of z, y, x, we must assume x, y;
X, z; y, z; respectively constant, and z, y, x, will be any whatever.
To find the equations to a straight line passing through the origin of
coordinates ; we have, since x = 0, and y = 0, when z = 0,
z =
z = a y.
5. To Jind the conditions that two straight lines in Jixed space may inter
sect one another ; and also their point of intersection.
Let their equations be
z = ax + A )
z = by + Bj
z = a' X + A' \ ,
z=b'y+ B'f
from which eliminating x, y, z, we get the equation of condition
a'A — aA^ _ b' B — b B^
a' — a "" b' — b
Also when this condition is fulfilled, the point is found from
A — A' B — B' a' A — n A' ,,^x
X = , , y = <, r, Z = ; . . . (10)
a' — a ' ^ b — b a' — a ^
6. To Jind the angle /, at inhich these lines intersect.
Take an isosceles triangle, whose equal sides measured along these
lines equal 1, and let the side opposite the angle required be called i ;
then it is evident that
cos. I = 1 — w i '
vi INTRODUCTION.
But if at the extremities of the line i, the points in the intersecting lines
be (x', y', z'), (x", y", z"), then (see 2)
i = = (x' — x'O ' + (y— y ') ' + {z' — z") «
.. 2 COS. I = 2 — J(x' — x") * + (y' — y") * + (z' — z") ^
But by the equations to the straight lines, we have (5)
z' = a x' f A ")
z'=by' + Bj"
z" = a' x" + A' >
z"=b'y" + B'/
and by the construction, and Art. 2, if (x, y, z) be the point of intereec
tion,
(X _ X')* + (y — y) ' + (z — z)« = 1 I
— x")* + (y — y")^ + {z — z'T = ij
Also at the point of intersection,
z = ax+A = a'x + A'"i
z = by + B = b'y + B'j
From these several equations we easily get
z — z' = a (x — a')
y — y'=^ ('' — ^')
z — z" = a' (X — x'O
yy"=p(xx)
whence by substitution,
. (X — x')' + aMxxO* + ^[(x — xO*= 1
(X x")^ + a'» (X  X)' + ^ (xx'O^ = 1
which give
1
X X =
X" =
1
Hence
(X'— x'0'= !— 5 +
l + a.+g l + a''+^, ^'(l + '+p)V(> + ''''+e^)
Also, since
ANALYTICAL GEOMETRY. vfi
yv'= ^(xx')
y/'=^(xx'o
and
ue have
z — z' = a (x — x')
z — z" = a' (x — x'O
iy^y=^—L^.^' 1 '
(z'E)'= ^— ,+
a* ' « a« f /■ . aS , /, „ a'2>
Hence by adding these squares together we get
2(l + aa'+^;) ^
{■
2 COS. 1=2 — (1 + 1—
which gives
II / • a a
+ a a' + r^,
COS. 1= ^ (II;
Tliis result may be obtained with less trouble by drawing straight lines
from the origin of coordinates, parallel to the intersecting lines ; and then
finding the cosine of the angle formed by these new lines. For the new
angle is equal to the one sought, and the equations simplify into
z' = ax' = b y', z" = a' x" = b' y'H
z=ax = by, z=a'x =b'y !
x'2 + /2+z'2 = 1 f
x"* + y"2 + z"' = 1 J
From the above general expression for the angle formed by two inter
secting lines, many particular consequences may be deduced.
For instance, required the conditions requisite that t'iXO straight lines
given in space may intersect at right angles.
That they intersect at all, this equation must be fulfilled, (see 5)
a' A — a A'  b'B — b B' ;
a' — a "" b' — b
«4
▼iii INTRODUCTION.
and that being the case, in order for them to intersect at right angles,
we have
1 = — , COS. 1 =
and therefore
1 + aa' + ^, = (12)
7. In the preceding No. the angle between two intersecting lines is
expressed in a function of the rectangular coordinates, which determine
the positions of those lines. But since the lines themselves would be
given in parallel position, if their inclinations to the planes, (x, y), (x, z),
(y, z), were given, it may be required, from other data, to find the same
angle.
Hence denoting generally the complements of the inclinations of a
straight line to the planes, (x, y), (x, z), (y, z), by Z, Y, X, the problem
may be stated and resolved, as follows : i
Required the angle made by the two straight lines, whose angles qfinclina
fton are Z,Y,X; Z', Y,', X'.
X«et two lines be drawn, from the origin of the coordinates, parallel
to given lines. These make the same angles with the coordinate planes,
and with one another, as the given lines. Moreover, making an isosceles
triangle, whose vertex is the origin, and equal sides equal unity, we have^
as in (6),
COS. I = 1— li'^ = 1 — ij(x — xO' + (yy')' + (z — zO^l *
the points in the straight lines equally distant from the origin being
(x, y, z), (x', y', z').
But in this case,
x« + y« + z'^ = 1
x'2 + y'*+ z'2= 1
and
X = cos. X, y = cos. Y, z = cos. Z
x' = COS. X', y' = COS. Y', z' = cos. Z'
\ COS. I = X x' + y y' + z z'
= cos. X. cos. X' + COS. Y. cos. Y' + cos. Z. cos. Z'. . (13)
Hence when the lines pass through the origin of coordinates, the same
expression for their mutual inclination holds good ; but at the same time
X, Y, Z ; X', Y', Z', not only mean the complements of the inclinations
to the planes as above described, but also the inclinatio7is of the lines to
the axes of coordinates of x, y, z, resjiectively.
ANALYTICAL GEOMETRY. ix
8. Given the length (L) of a straight line and the complements of its in
clinations to the planes (x, y), (x, z), (y z), viz. Z, Y, X, tojind the lengths
of its projections upon those planes.
By the figure in (4) it is easily seen that
L projected on the plane (x, y) = L. sin. Z"\
(X, z) = L. sin. Y V . . . (U)
(y, z) = L. sin. Xj
9. Instead of determining the parallelism or direction of a straight line
in space by the angles Z, Y, X, it is more concise to do it by means of
Z (for instance) and the angle ^ which its projection on the plane (x, y)
makes with the axis of x.
For, drawing a line parallel to the given hne from the origin of the co
ordinates, the projection of this line is parallel to that of the given line,
and letting fall from any point (x, y, z) of the new line, perpendiculars
upon the plane (x, y) and upon the axes of x and of y, it easily appears,
that
X = L cos. X = (L sin. Z) cos. 6 (see No. 8)
y ~= L. cos. Y = (L. sin. Z) sin. d
which give
cos. X = sin. Z. cos. 6'\ . .
cos. Y = sin. Z . sin. 6} ^
Hence the expression (13) assumes this form,
COS. I = sin. Z . sin. TJ (cos. & cos. d' + sin. 6 sin. ^) + cos. Z cos. Z'
= sin. Z . sin. Z' cos. (^ — ^0 + cos. Z . cos. Z' . . . . (16)
which may easily be adapted to logarithmic computation.
The expression (5) is. merely verified by the substitution.
10. Given the angle of intersection (I) hetvoeen two lines in space and
their inclinations to the plane (x, y), to Jind the angle at tsohich their pno
jections upon that plane intersect one another.
If, as above, Z, 7/ be the complements of the inclinations of the lines
upon the plane, and 6, (/ the inclinations of the projections to the axis of
X, we have from (16)
,. ... COS. I — cos. Z. COSv TI ,.„.
^o'i'O^ .in.Z.sin.Z' '">
This result indicates that I, Z, Z' are sides of a spherical triangle
(radius = 1), ^ — ^ being the angle subtended by I. The form may at
once indeed be obtained by taking the origin of coordinates as the center
of the sphere, and radii to pass through the angles of the spherical tri
angle, measured along the axis of z, and along lines parallel to the
giveu lines.
X INTRODUCTION.
Having considered at some length the mode of determining the posi
tion and properties of points and straight lines in fixed space, we proceed
to treat, in like manner, of planes.
It is evident that the position of a plane is fixed or determinate in posi
tion when three of its points are known. Hence is suggested the follow
ing problem.
11. Having given the three points (a, jS, y), (a', Q', /), (a", /3", /') in
space, tojind the eqtmtion to the plane passing through them ; that is, to
Jind the relation between the coordinates of any othei' point in the plane.
Suppose the plane to make with the planes (z, y), (z, x) the intersec
tions or traces B D, B C respectively, and let P be any point whatever
in the plane ; then through P draw P Q in that plane parallel to B D,
&c, as above. Then
z — QN = PQ' = QQ' cot. D B Z
= y cot. D B Z.
But
QN = AB — NA. cot. C B A
= A B + X cot. C B Z,
.\ z = A B + X cot. C B Z + y cot. D B Z.
Consequently if we put A B = g, and denote by (X, Z), (Y, Z) the
inclinations to A Z of the traces in the planes of (x, z), (y, z) respectively,
we have
z = g + X cot. (X, Z) + y cot. (Y, Z) . , . . (18)
Hence the form of the equation to the plane is generally
z=Ax+By+C ri9)
ANALYTICAL GEOMETRY. xi
NoMTto find these constants there are given the coordinates of three
points of the plane ; that is
y— Aa +B/3 +C
/ = Aa'+B/3'+C
/' = A a" + B ^" + C
from which we get
B  y«^ — /« + /«^^/^«^ + /^« — y«^^ _ ^^, ,v 7>
(21)
(22)
p _ ^''(y «' — /«) + g(/a^^ — /^gQ +/3^(/^« — yg^O
^ a/3' — a'3 + a'/3'' — a"/3' + a''/3 — «/3''
Hence when the trace coincides with the axis of x, we have
C = 0,
and
A = cot. 5=0
^" (ya'ya) + 13 (/«"_/'«') + /S'(/'a7a'0 = >
7 ^' — / /3 + / /3" — /' ^' + /' ^ _ y /3" = ) ' • • ^ ^
R_i i^n . (/ c^" 7" cc') + {S'  ^") . {y" a  y a")
/3" • a^' — of ^ + of /3" — a" p' + a" ^ — a ^" •"" ^"^^^
and the equation to the plane becomes
z = By (25)
When the plane is parallel to the plane (x, y),
A = 0, B = 0, '
and.
z = C (26)
from which, by means of A = 0, B = 0, any two of the quantities /, y\ y"
being eliminated, the value of C will be somewhat simplified.
Hence also will easily be deduced a number of other particular results
connected with the theory of the plane, the point, and the straight line, of
which the following are some.
To find the projections on the planes (x, y), (x, z), (y, z) of the intersec
tion of the planes,
z=Ax + By+C,
z = A'x+ B'y+ C
Eliminating z, we have
(A — A')x + (B— BOy + C— C = .... (27)
which is the equation to the projection on (x, y).
xii INTRODUCTION.
Eliminating x, we get
(A' — A)z + (AB' — A'B)y + AC'A'C = .... (28)
which is the equation to the projection on the plane (y, z).
And in like manner, we obtain
(B'— .B)z + (A'B — AB')x+BC'B'C = . . . . (29)
for the projection on the plane (x, z).
To find the conditions requisite that a plane atid straight line shall be
parallel or coincide.
Let the equations to the straight line and plane be
X = a z + A"i
y =bz + B/
z = A' X + B' y + C.
Then by substitution in the latter, we get
z(A'a+ B'b — 1) + A'A+ B'B + C'=0.
Now if the straight line and plane have only one point common, we
should thus at once have the coordinates to that point.
Also if the straight line coincide with the plane in the above equation,
z is indeterminate, and (Art. 6. vol. I.)
A' a + B' b — 1 = 0, A' A + B' B + C = . . . (27)
But finally if the straight line is merely to be parallel to the plane, the
above conditions ought to be fulfilled even when the plane and line are
moved parallelly up to the origin or when A, B, C are zero. The only
condition in this case is
A' a + B' b = I (28)
To find the conditions that a straight line be perpefidicular to a plane
z = Ax + By + C.
Since the straight line is to be perpendicular to the given plane, the
plane which projects it upon (x, y) is at right angles both to the plane
(x, y) and to the given plane. The intersection, therefore, of the plane
(x, y) and the given plane is perpendicular to the projecting plane. Hence
the trace of the given plane upon (x, y) is perpendicular to the projec
tion on (x, y) of the given straight line. But the equations of the traces
of the plane on (x, z), (y, z), are
z= Ax+ C, z = By + C^
**" ' ^ (29)
z=Ax\^, z = r)yt i_^^
1 C 1 Cf
^=A"A'y = B"B)
and if those of the perpendicular be
x = a z + A/1
y = bz + B,J
ANALYTICAL GEOMETRY. xiii
it is easily seen from (11) or at once, that the condition of these traces
being at right angles to the projections, are
A + az=0, A + b = 0.
To draw a straight line passing through a given point (a, iS, y) at right
angles to a given plane.
The equations to the straight line, are clearly
X — a + A (z — 7) = 0, y — /3 + B (z — 7) = 0. . . . (30)
To find the distance of a given point (a, /3, y)from a given plane.
The distance is (30) evidently, when (x, y, z) is the common point hi
the plane and perpendicular
V {z—yy + (y_^)2 4 (X — «)« = (z — 7) V' 1 + A^ + B*.
But the equation to the plane then also subsists, viz.
z = Ax + By + C
from which, and the equations to the perpendicular, we have
z — 7=C — 7+Aa + B/?,
therefore the distance required is
C — 7 + Ao + B 3
(31)
A* + B«
To Jind the angle I formed by iiw planes
z = Ax + By+C,
z = A' X + B' y + C.
If from the origin perpendiculars be let fall upon the planes, the angle
which they make is equal to that of the planes themselves. Hence, if
generally, the equations to a line passing through the origin be
X = a z )
y = bzJ
the conditions that it shall be perpendicular to the first plane are '
A + a = 0,
B + b = 0,
and for the other plane
A' + a = 0,
B' + b = 0.
Hence the equations to these perpendiculars are
X + A z
y + Bz
X + A'z
y + B'z
'. = 0/
X + A'z = \
= 0,j
*• ^ ~ V(l + A» + B^)
_ A (
'•"  V(l + A'' 4 B»))
»v INTRODUCTION.
which may also be deduced from the forms (30).
Hence from (11) we get
J __ 1 + A A^ + B B^
~ V(l+A2+B*) V(l + A'» + J^ 1 • • • ^^^)
Hence to find the incli7iatioti (s) of a plane taith the plane (x, y).
We make the second plane coincident witli (x, y), which gives
A' = 0, B' = 0,
and therefore
"^•'= V(l+A' + b«) (*3)
In like manner may the inclinations (^), (»j) of a plane
z=Ax+By+C
to the planes (x, z), (y, z) be expressed by
y B
COS. L :=
(34>
COS.
Hence
COS. 2 s + COS. 2 ^ + COS. * »j = 1 (35)
Hence also, if /, ^', ;;' be the inclinations of another plane to (x, y)j
(x, z), (y, z).
COS. I = cos. i COS. l' + COS. ^ COS. (^' + cos. n COS. r! . , . (36)
To find the inclination v of a straight line x = a z + A', y = b z + B',
to the plane z = Ax + By+C.
The angle required is that which it makes with its projection upon the
plane. If we let fall from any part of the straight line a perpendicular
upon the plane, the angle of these two lines will be the complement of y.
From the origin, draw any straight line whatever, viz. x = a' z, y = b'z.
Then in order that it may be perpendicular to the plane, we must have
a' = — A, b' = — B.
The angle which this makes with the given line can be found from (11);
consequently by that expression
1 — A a — B b „^v
''"• "  V (1 +a*+ b*) V (1 + A«+ B^) • • • ^^^^
Hence we easily find that the angles made by this line and the coor
dinate planes (x, y), (x, z), (y, z), viz. Z, Y, X are found from
rj _ 1
<^s ^  V(i + a^ + b«) '
COS. Y =
ANALYTICAL GEOMETRY. xv
b
V {1 + a» + b'')'
a
V (1 4. a^ + b
which agrees with what is done in (3).
•='^^= V(l+\'+b') (3«)
TRANSFORMATION OF COORDINATES.
12. To transfer the origin of coordinates to the point (a, /3, y) ^without
changing their direction.
Let it be premised that instead of supposing the coordinate planes at
right angles to one another, we shall here suppose them to make any
angles whatever with each other. In this case the axes cease to be rec
tangular, but the coordinates x, y, z are still drawn parallel to the axes.
This being understood, assume
X = x' + «, y = / + /3, z = z' + 7 (39)
and substitute in the expression involving x, y, z. The result will contain
x', y', z' the coordinates referred to the origin (a, 3, 7).
When the substitution is made, the signs of a, jS, y as explained in (1),
must be attended to.
13. To change the direction of the axes from rectangular, >withoui
qffecti7ig the origin.
Conceive three new axes A x', A y', A z', the first axes being supposed
rectangular, and these having any given arbitrary direction whatever.
Take any point ; draw the coordinates x', y', z' of this point, and project
them upon the axis A X. The abscissa x will equal the sum, taken with
their proper signs, of these three projections, (as is easily seen by drawing
the figure) ; but if (x x')> (y, yOj (z> z') denote the angles between the
axes A X, A x' ; A y, A y' ; A z, A z' respectively ; these projections
are
x' COS. (x' x), y' cos. (y' x), z' cos. (z' x).
In like manner we proceed with the other axes, and therefore get
X = x' cos. (x' x) + y' COS. (y' x) + z' cos. {z' x) ^
y = y' COS. iy y) + z' cos. (z' y) + x' cos. (x' y) > . . . (40)
z = z' COS. (z' z) f y' cos. (y' z) + x' cos. (x' z) J
XVI
INTRODUCTION.
s. * (x' x) + COS. ' (x' y) + COS. * x' z = 1 "^
5' (y'x) + cos.«(y'y) +cos.2(y'z)= 1 > . .
s. * (z' x) + COS. * (z' y) 4. COS. * (z' x) = 1 )
}
(41;
(42)
Since (x' x), (x' y), (x' z) are the angles which the staight line A x',
makes with the rectangular axes of x, y, z, we have (5)
COS. * (x' x) + COS. ' (x' y) + cos.
COS.
cos.
We also have from (13) p.
cos.(xy)=cos.(x'x)cos.(y'x)+cos.(x'y)cos.(y'y)+cos.(x'z)cos.(y'z)
cos.(xV) = cos.(x'x)cos.(z'x) + cos.(x'y)cos.(z'y) + cos.(x'z)cos.(z'z)
cos.(y'z') =cos.(y'x)cos.(z'x)4cos.)y'y)cos.(z'y)4cos.(y'z)cos.(z'z)
The equations (40) and (41), contain the nine angles which the axes of
x', y', z' make with the axes of x, y, z.
Since the equations (41) determine three of these angles only, six of
them remain arbitrary. Also when the new system is likewise rectangu
lar, or COS. (x'yO = cos. (x' z') — cos. (y'z') = 1, three others of the
arbitraries are determined by equations (42). Hence in that case there
remain but three arbitrary angles.
14. Required to transform the rectangular axe of coordinates to other
rectangular axeSt having the same origin, but ixeo of "which shall be situated
in a given plane.
Let the given plane be Y' A C, of which the trace in the plane (z, x) is
A C. At the distance A C describe the arcs C Y', C x, x x' in the planes
C A Y', (z, x), and X' A X. Then if one of the new axes of the coordi
nates be A X', its position and that of the other two, A Y', A Z', will be
determined by C x' = f>, C x = vj/, and the spherical angle x C x' = ^ =
inclination of the given plane to the plane (z, x).
Hence to transform the axes, it only remains to express the angles
• (x'x), (y'x), &c. which enter the equations (40) in terms of ^ 4/ and p.
ANALYTICAL GEOMETRY. xvii
By spherics
COS. (x'x) ^ COS. vj/ COS. + sin. ^ sin. p cos. S,
Tn like manner forming other spherical triangles, we get
COS. (/ x) r= COS. (90<' + (p) COS. ^ + sin. ^ sin. (GO*' + p) cos. ^
COS. (x' y) = cos. (96« + 4/) cos. <p + sin. (90° + 4) sin. p cos. 6
cos. (y'y) = cos. (90°+vl/)cos.(90^f) + sin.(90° + '4/)sin.(90<' + p)cos.d
So that we obtain these four equations
cos. (x' x) = COS. v}^ COS. p + sin. vj/ sin. p cos. tf
cos. (y' x) = — sin. ^z sin. p + sin. %J/ cos,
) COS. f^
5. f COS. ^ f
n. ^.^t)s. ^C
1 ro<?. 4 ^
}
COS. (x' y) = — sin. 4' cos. <p + cos. ^ sin. p<:t)r " '^ * • . /
COS. (y' y) = sin. vj/ sin. p + cos. vl^ cos. p cos. 4
Again by spherics, (since A Z' is perpendicular to A C, and die inclin
ation of the planes Z' A C, (x, y) is 90° — 6) we have
cos (z' x) = sin. vJ/ sin. ^ i
cos. (z y) = cos. v]/ sin. d f ''
And by considering that the angle between the planes Z A C, Z A X', =
90° + 6, by sphericsj'we also get
cos. (x'z) = — sin. (p sin. 6
COS. (y'z) = — cos. (p sin. 6 ^ (45)
cos. (z'z) = cos. 6
which give the nine coefficients of equations (40).
Equations (41), (42) will also hereby be satisfied when the systems are
rectangular.
15. To find the sedion of a surface made hy a plane.
The last transformation of axes is of great use in determining the na
ture of the section of a surface, made by a plane, or of the section made
by any two surfaces, plane or not, provided the section lies in one plane ;
for having transformed the axes to others, A Z', A X', A Y', the two lat
ter lying in the plane of the section, by the equations (40), and the de
terminations of the last article, by putting z' = in the equation to the
surface, we have that of the section at once. It is better, however, to
make z := in the equations (40), and to seek directly the values of
cos. (x'x), COS.. (yx), &c. The equations (40) thus become
X = x^ cos. •vl' + y' sin. 4 cos. 6 x
y = x' sin. ^ — y' cos. •4' cos. 6 K. ..... (46)
z = y sin. & J
16. To determine the nature and position of all surfaces of the second
order , or to discuss the general equation of the second order^ viz.
ax* + by* + cz ^ 2dxy + 2exz + 2fyz + gx J hy +iz = k . . (a)
First simplify itby such a transformation of coordinates as shall destroj'
b
xviii INTRODUCTION.
the terms in x y, x z, y z ; the axes from rectangular will become oblique,
by substituting the values (40), and the nine angles which enter these,
being subjected to the conditions (41), there will remain six of them
arbitrary, which we may dispose of in an infinity of ways. Equate to
zero the coefficients of the terms in x' y', x' z', y' t.
But if it be required that the new axes shall be also rectangular, as this
condition will be expressed by putting each of the equations (42) equal
zero, the six arbitrary angles will be reduced to ihree^ which the three
coefficients to be destroyed will make known, and the problem will thus
be determined.
This investigation will be rendered easier by the following process :
Let x= az, y=/3zbe [the equations of the axis of x' ; then for
brevity making
1 = L_:
V (1 + a^ + ^^)
we find that (3)
cos. (x'x = a 1, cos. (x'y) = /SI, Cos. x'z = 1.
Reasoning thus also as to the equations x = a'z, y = j8' z of the axis
of y', and the same for the axis of z', we get
cos. (y'x) = a' I', cos. (y'y) = /3'1', cos. (y'z) = 1'
COS. (z' x) = a" 1", COS. \tI y) = /3" \', cos. (z' z) = V.
Hence by substitution the equations (40) become
X = 1 a x' + F a' y' + 1" o!
y = l/3x' +
z = 1 x' + 1' y'
The nine angles of the problem are replaced by the six unknowns a,
a', o!\ 13, /3', (3", provided the equations (41) are thereby also satisfied.
Substitute therefore these values of x, y, z, in the general equation of
the 2d degree, and equate to zero the coefficients of x' y', x' z', y' z', and
we get
(aa + diS + e) a' + (da + b/3 + f)/3' + e a + f /S + c = 0"
(a a + d iS + e) a" + (d a + b /3 + f) ^" + e a + f /3
(a«" + d/3" + e) a' + (da" + b/3"+ f) /3' +e a" + f/3'
One of these equations may be found without the others, and by making
the substitution only in part. Moreover from the symmetry of the pro
cess the other two equations may be found from this one. Eliminate a',
B' from the first of them, and the equations x = a' z, y = ^' z, of the
axis of y'; the resulting equation
(a a + d ^ + e) X + (d a + b ^ + f) y + (e a + f S + c) 2 = . . (b)
is that of a plane (19).
F a' y' + 1" a" z' ^
1' iS' y + Y' ^" l' K
\ y' + Y' z'. J
3 + c =0\
3 + c = l
r + c = J
ANALYTICax. geometry. xix
But the first equation is the condition which destroys the term x'y't
since if we only consider it, a, /3, a\ /3', may be any whatever that will
satisfy it ; it suffices therefore that the axis of y' be traced in the plane
above alluded to, in order that the transformed equations may not contain
any term in x' y.
In the same manner eliminating a", j3", from the 2d equation by means
of the equations of the axis of z', viz. x = a" z, y = jS" z, we shall have
a .'lane such, that if we take for the axis of z every straight line which it
will there trace out, the transformed equation will not contain the term in
X' z',. But, from the form of the two first equations, it is evident that this
second plane is the same as the first : therefore, if we there trace the axes
of y and z* at pleasure, this plane will be that of y' and z', and the
transformed equation will have no terms involving x' y or x z'. The
direction of these axes in the plane being any whatever, we have an in
finity of systems which will serve this purpose ; the equation (b) will be
that of a plane parallel to the plane which bisects all the parallels to x,
and which is therefore called the Diametrical Plane,
Again, if we wish to make the term in y' z' disappear, the third equa
tion will give a', /?, and there are an infinity of oblique axes which will
answer the three required conditions. But make x', y', a', rectangular.
The axis^of x' must be perpendicular to the plane (y z') whose equa
tion we have just found ; and that x = a z, y = /3 z, may be the equa
tions (see equations b) we must have
a a + d /3 + e = (e « + f /^ + c) a . . , . (c)
d a + b /3 + f = (e a + f /3 + c) ^ . . . , (d)
Substituting in (c) the value of a found from (d) we get
{ (a — b)fe + (f= — e^) d J/S^
+ J (a_b)(c — b)e+ (2d^— f2 — e*)e + (2c — a — b)fd} /3*
+ J (c— a)(c — b) d+ (2e2 — f2 — dO d + (2b — a— c) fe ] ^
+ (a — c) fd + (f^ — d^) e = 0.
This equation of the 3d degree gives for /3 at least one real root; hence
the equation (d) gives one for a; so that the axis of x' is determined so as
to be perpendicular to the plane (y, z*,) and to be free from terms in
X' z", and y' z'. It remains to make in this plane (y*, z',) the axes at right
angles and such that the term x' y' may also disappear. But it is evident
that we shall find at the same time a plane (x, z',) such that the axis of y'
is perpendicular to it, and also that the terms in x' y, t ^ are not involved.
But it happens that the conditions for making the axis of y' perpendicular
to this plane are both (c) and (d) so that the same equation of the 3d de
1%
»c . INTRODUCTION.
gree must give also ^. Tlie same holds good for the axis of z. Conse
quently the tlnee roots of the equation (3 are all real, and are the values
of /3, ^, j8". Therefore a, a', a", are given by the equation (d). Hence,
T'here is only 07ie system of rectangular axes tsohich eliminates x' y', x' z',
x'y'; and there exists one in all cases. These axes are called the Prijici
vol axes of the Suiface.
Let us analyze the case which the cubic in /3 presents.
1. If we make
(a_b)fe + (f2_e'^)d =
the equation is deprived' of its first term. This shows that then one of
the roots of /3 is infinite, as well as that a derived from equation (d) be
comes e a + f /3 = 0. The corresponding angles are right angles. One
of the aKes, thai of z' for instance, falls upon the plane (x, y), and we
obtain its equation by eliminating a and jS from the equations x = a z,
y = /3 z, which equation is
ex + fy =
The directions of y', z' are given by the equation in /3 reduced to a
Quadrature.
2ndly. If besides this first coefficient the second is also = 0, by substi
tuting b, found from the above equation, in the factor of /3 ^^ it reduces to
the last term of the equation in /3, viz.
(a— c) fd + (f2_d=) e = 0.
These two equations express the condition required. But the coeffi
cient of B is deduced from that of /3 ^ by changing b into c and d into e,
and the same holds for the first and last term of the equation in jS.
Therefore the cubic equation is hIso thus satisfied. There exists therefore
an infinity of rectangular systems, which destroy the terms in x' y', x' z',
y' z'. Eliminating a and b from equations (c) and (d) by aid of the above
two equations of condition, we find that they are the product of fa — d
and e3 — d by the common factor eda + fdjS + fe. These factors
are therefore nothing ; and eliminating a and /3, we find
fx = dz, ey =r dz, edx + fdy + fez = 0.
The two first are the equations of one of the axes. The third that ol
a plane which is perpendicular to it, and in which are traced the two
other axes under arbitrary directions. This plane will cut the surface in
a curve vherein all the rectangular axes are principal, which curve is
therefore a circle, the only one of curves of the second order which has
that property. The surface is one then of revolution round the axig
whose equations we have just given.
ANALYTICAL GEOMETRY. xxi
The equation once freed from the three rectangles, becomes of the
form
k z * + m y * 4 n X * + q X 4 q' y + q'' z = h . . . . (e)
The terms of the first dimension are evidently destroyed by removing
the origin (39). It is clear this can be effected, except in the case
where one of the squares x % y ^, z * is deficient. We shall examine these
cases separately. First, let us discuss the equation
kz* + my* f nx* = h (f)
Every straight line passing through the origin, cuts the surface in two ,
points at equal distances on both sides, since the equation remains the same
after having changed the signs of x, y, z. The origin being in the middle
of all the chords drawn through this point is a center. The surface therefore
has the property of possessing a center 'whe7iever the transformed equation
has the squares of all the variables.
We shall always take n positive : it remains to examine the cases where
k and m are both positive, both negative, or of different signs.
If in the equation (f) k, m, and n, aie all positive, h is also positive ;
and if h is nothing, we have x = 0, y = 0, z = 0, and the surface has
but one point.
But when h is positive by making x, y, or z, separately equal zero, we
find the equations to an ellipse, curves which result from the section of
the surface in question by the three coordinate planes. Every plane
parallel to them gives also an ellipse, and it will be easy to show tlie
same of all plane sections. Hence the surface is termed an Ellip
soid.
The lengthy A, B, C, of the three principal axes are obtained by find
ing the sections of the surface through the axes of x, y, and z. Th :e
give
kC* = h, mB*= h, nA= = h.
from which eliminating k, m and n, and substituting in equation (f) we get
C« t B*^ A« I (47)
A*B«z2 + A»C2y^ + B'C'x'^ = A*B'C0
which is the equation to an Ellipsoid referred to its center a7id principal
axes.
We may conceive this surface to be generated by an ellipse, triiced in
the plane (x, y), moving parallel to itself, whilst its two axes vary, the *
curve sliding along another ellipse, traced in the plane (x, z) as a direct
b 3
3ncit INTRODUCTION.
rix. If two of tlie quantities A, B, C, are equal, we liave an ellipsoid of
revolution. If all three are equal,, we have a sphere.
Now suppose k negative, and m and h positive or
kz — my* — ax^=: — h.
Making x or y equal zero, we perceive that the sections by the planes
(y z) and (x z), are hyperbolas, whose axis of z is the second axis. All
planes passing through the axis of z, give this same curve. Hence the
surface is called an hyperholoid. Sections parallel to the plane (x y) are
always real ellipses. A, B, C V — 1 designating the lengths upon the
axes from the origin, the equation is the same as the above equation ex
cepting the first term becoming negative.
Lastly, when k and h are negative
k z2 + my2 + nx^ = — h;
all the planes which pass through the axis of z cut the surface in hyper
bolas, whose axis of z is the first axis. The plane (x y) does not meet
the surface and its parallels passing through the opposite limits, give
ellipses. This is a hyperholoid also, but having two vertexes about the
axis of z. The equation in A, B, C is still the same as above, excepting
that the term in z' is the only positive one.
When h = 0, we have, in these two cases,
k2'= my2 + nx* . . (48)
the equation to a cone, which cone is the same to these hyperboloids that
an asymptote is to hyperbolas.
It remains to consider the case of k and m being negative. But by a sim
ple inversion in the axes, this is referred to the two preceding ones. The
hyperholoid in this case has one or two vertexes about the axis of x ac
cording as h is negative or positive.
When the equation (e) is deprived of one of the squares, of x * for in
stance, by transferring the origin, we may disengage that equation from
the constant term and from those in y and z ; thus it becomes
kz^ + my^'rrhx (49)
The sections due to the planes (x z), (x y) are parabolas in the same
or opposite directions according to the signs of k, m, h ; the planes par
allel to these give also parabolas. The planes parallel to that of (y z)
give ellipses or parabolas according to the sign of m. Tne surface is an
elliptic paraboloid in the one case, and a hyperbolic paraboloid in the
other case. When k = m, it is a paraboloid of revolution.
When h = 0, the equation takes the form
a'z^ + b=y2 =
ANALYTICAL GEOMETRY. xxiii
according to the signs of k and m. In the one case we have
z = 0, y =
whatever may be the value of x, and the surface reduces to the axis of x.
In the other case.
(a z + b y) (a z — b y) =
shows that we make another factor equal zero; thus we have the system
of two planes which increase along the axis of x.
When the equation (e) is deprived of two squares, for instance of x *,
y % by transferring the origin parallelly to z, we reduce the equation to
kz« + py + qx = (50)
*rhe sections made by the planes drawn according to the axis of z, are
parabolas. The plane (x y) and its parallels give straight lines par
allel to them. The surface is, therefore, a cylinder whose base is a para
bola, or a parabolic cylinder.
If the three squares in (e) are wanting, it will be that of a plane.
It is easy to recognise the case where the proposed equation is decom
posable into rational factors ; the case where it is formed of positive
squares, which resolve into two equations representing the sector of two
planes.
PARTIAL DIFFERENCES.
17. If u =r f (x, y, z, &c.) denote any function of the variable x, y, z,
&c. d u be taken on the supposition that y, z, &c. are constant, then is the
result termed the partial difference of u relative to x, and is thus written
a
X.
^d x/
Similarly
denote the partial differences of u relatively to y, z, &c. respectively.
Now since the total difference of u arises from the increase or decrease
of its variables, it is evident that
d„=(^)dK+(«)dy+(^")dz+&C.. ..(«•)
"iv  INTRODUCTION.
But, by the general principle laid down in (6) Vol. I, this result may
be demonstrated as follows ; Let
u + du = A + Adx+Bdy+Cdz+&c.
A'dx«+ B'dy^+ C'dz* + &c.i
+ Mdx.dy + Ndx.dz+&c.j
+ A" dy.^ + &c.
Then equating quantities of the same nature, v.e have
du = Adx+Bdy+Cdz + &c.
Hence when d y, d z, &c. = 0, or when y, z, &c. are considered con
stant
d u = A d X
or according to the above notation
In the same manner it is shown, that
&c
Hence
du = (^) dx + (41) cly + (^i) d. + &c. as before.
Ex. 1. u = X y Zj &c.
/du\ /du\ /du\
Car) = y^' (dj) =''^' (^) = "J'
.'. du = yzdx + xzdy + xydz
d u _ d X d y d z
u ~~ X "y" z
Ex. 2. u = X y z, &c. Here as above
du dx.dy.dz.o
— = H — ^ H + &c.
u X y z
And in like manner the total difference of any function of any number
of variables may be found, viz. by first taking the partial differences, as in
the rules laid down in the Comments upon the first section of the first
book of the Principia.
But this is not the only use of partial differences. They are frequently
used to abbreviate expressions.* Thus, in p. 13, and 14, Vol. II. we
ANALYTICAL GEOMETRY. xxv
learn that the actions of M, /*, jm.", &c. upon ^ resolved parallel to x,
amount to
d»(^ + x) (^'{^'—^) 5 fi" (x'—x)
d t«  [(X'— x)'+(y'~y)H (^— z)^^"*"[x'x)*+(y"— y)V (z"z)^]l
. ^"' (x'"x) 4. &c — M^ ^
^[(x'"— x)« + (y"'_y)«+ (z"'z)^]^ + ^''' [(x^ + y^+ z«) ]
retaining the notation there adopted.
But if we make
and generally
V(xx)« + (y— y)' + (z'— z)^ = §
0.1
<V/(x"°— X""")* + (y"n_y"'n») 2 ^ ^z""— z"») * = ^
n, m,
and then assume
 X = ^' + ^' + &c (A)
Si
0, 1 0, 2
+^+^ + &0 .(B)
8 1,3
+jq_ + tt+s,,. (C)
2. 2,4
&C
we get
S S'
0,1 . 0,2
\dy/ ^dyy g' f'
0, 1 0, 2
We also get
VdxJ "■ p ■*" VdxV
0,1
/d^N __ ^/."(x"— x ) /.y (k" x) , /^N
Vdx7  ■" f' f' Wx'7
0, 2 1, 2
/d^\ _^ f x, fi"'{x"' — ^x) ■ <ttV"(x"' — x') ^y(x'" — x") / d D \
VdFv "" P P s^ '*'Vdx'"/
0,3 1,3 8,3
XXVI INTRODUCTIOxV.
Hence since (B) has one term less than (A) ; (C) one term less than
^B) ; and so on ; it is evident that
(d4) + (u4) + i'^) +^— (K)(^.)(rx)^
+C^) + (d4) +«''••
and therefore that
See p. 15, Vol. II.
Hence then X is so assumed that the sum of its partial differences re*
lative to x, x', x" &c. shall equal zero, and at the same time abbreviate
the expression for the forces upon fi along x from the above complex
formula into
dt* ~ /I \dx) s' '
and the same may be said relatively to the forces resolved parallel to
y, z, &c. &c.
Another consequence of this assumption is
/d\\ /dx\
^•^^•Mdy) = ^yCd^)
For • '
/ d X X ^;t'(x— x)y ^f/'(x"—'K)y ^^'"(x"'— x)y , „
y V dir)  — p — + — p — + — p + ^''
0,1 0,2 0.3
/d X V ^VY x:— x')y' ^VV — xQy ' . ^/^'(x— x)y
y te; = — p — + p + ^''' ~ e
,., (^ '^\ _/«'V"(x"'x") y" , A^V"'(x"x') y" /./."(x"x)y' (/r^"('x"x')y''
^ te; ^^— — + — p +^*^ p ~ f '
2,3 2,4 0,2 1,2
&C.
Hence it is evident that
3 V (^\  A^/*'(x— x)(y— y') /^/(x— x) (y— y") g^^^
'•'^ Vdx / f ' ^ f '
_^ A^V'(x"x')(yy") ^ ^>"(x'W) (yy" ) _^ g^^
^ /*V(Vx") (y y") _^ ^V"(x"xj)(yy"0 ^ g^^^
2, 3 2. *
&C.
ANALYTICAL GEOMETRY. xxvii
In like manner it is found that
0, 1 t,2
^>"(y"— y') (x— X") /xV"(y"— y') (x'—x'") , _
1. 2 i; 3
&C.
which is also perceptible from the substitution in the above equation of
y for X, X for y; y' for X', x' for y' ; and so on.
But >
, (y'y) (x— x) = (x— x) (y—y)
{y"— y) (x— x") = (x"— x) (y—y")
&c.
consequently
See p. 16. For similar uses of partial differences, see also pp. 22, and
105.
CHANGE OF THE INDEPENDENT VARIABLE.
When an expression is given containing diiFerential coefficients, such
ns
dx' dx^^""
in which the first differential only of x and its powers are to be found, it
shows that the differential had been taken on the supposition that dx is
constant, or that d ^ x = 0, d ' x = 0, and so on. But it may be re
quired to transform this expression to another in which d * x, d ' x shall
appear, and in which d y shall be constant, or from which d * y, &c. shall
be excluded. This is performed as follows :
For instance if we have the expression
d v^
1 4 y
^ dx' dy
d* y dx
dx« .
the differential coefficients
dy dfy
dx' dx"
xxviii INTRODUCTION.
may be eliminated by means of the equation of the curve to which we
mean to apply that expression. For instance, from the equation to a
parabola y = a x *, we derive the values of
IZ and — y
dx ^""^ dx*
which being substituted in the above formula, these differential coefficients
will disappear. If we consider
dy d^y
d^' dx*
unknown, we must in general have two equations to eliminate them from
one formula, and these equations will be given by twice differentiating the
equation to the curve.
When by algebriacal operations, d x ceases to be placed underneath
d y, as in this form
y(<^^' + dyO ^52j
dx* + dy* — yd*y
the substitution is effected by regarding d x, d y, d * y as unknown ; but
then in order to eliminate them, there must be in general the same
number of equations as of unknowns, and consequently it would seem the
elimination cannot be accomplished, because by means of the equation to
the curve, only two of the equations between d x, d y, d * y can be ob
tained. It must be remarked, however, that when by means of these two
equations we shall have eliminated d y and d * y, there will remain a com
mon factor d X *, which will also vanish. For example, if the curve is
always a parabola represented by the equation y = ax , by differentiat
ing twice we obtain
dy = 2axdx0d2y = 2adx*
and these being substituted in the formula immediately above, we shall
obtain, after suppressing the common factor d x *,
y(l+4a»x')
4, a^ x'^ — 2ay*
The reason why d x * becomes a common factor is perceptible at once,
for when from a formula which primitively contained
d'y dy
dx** dx'
d * V
we have taken away the denominator of . — f all the terms independent
of r—^ and j^ must acquire the factor d x * ; then the terms which
dx^ d X ^
were affected by r%> do not contain d x, whilst those affected by t^
ANALYTICAL GEOMETRY. xxix
contain d x. When we afterwards differentiate the equation of the curve,
and obtain results of the form dy = Mdx, d^y = Ndx^ these values
being substituted in the terms in d^y, and in dy dx, will change them,
as likewise the other terms, into products of d x ^. ,
What has been said of a formula containing differentials of the two first
orders applying equally to those in which these differentials rise to supe
rior orders, it thence follows that by differentiating the equation of the
curve as often as is necessary, we can always make disappear from the
expression proposed, the differentials therein contained.
The same will also hold good if, beside these differentials which we have
just been considering, the formula contain terms in d * x, in d ' x, &c. ;
for suppose that there enter the formula these differentials d x, d y, d ' x,
d ' y and that by twice differentiating the equation represented by y = f x,
we obtain these equations
F (x, y, d y, d x) =
F(x, y, dx, dy, d'x, d»y) = 0,
we can only find two of the three differentials d y, d ^ x, d  y, and we see
it will be impossible to eliminate all the differentials of the formula ; there
is therefore a condition tacitly expressed by the differential d '^ x ; it is
that the variable x is itself considered a function of a third variable which
does not enter the formula, and which we call the independent variable.
This will become manifest if we observe, that the equation y = f x may
be derived from the system of two equations
X = F t, y = (pt
from which we may eliminate t. Thus the equation
(X — c)'
is derived from the system of two equations
X = b t + c, y = a t%
and we see that x and y must vary by virtue of the variation which t may
undergo. But this hypothesis that x and y vary as t alters, supposes that
there are relations between x and t, and between y and t. One of these
relations is arbitrary, for the equation which we represent generally by
y = f X, for example
y = ^' (x — c) s
if we substitute between x and t, the arbitrary relation.
XXX INTRODUCTION,
this value being put in the equation
will change it to
(f — c')*
an equation which, being combined with this,
_ t'
^  r« '
ought to reproduce by elimination,
(X — c) '
the only condition which we ought to regard in the selection of the varia
ble t.
We may therefore determine the independent variable t at pleasure.
For example, we may take the chord, the arc, the abscissa or ordinate
for this independent variable ; if t represent the arc of the curve, we
have
t = V (dx« + dy^);
if t denote the chord and the origin be at the vertex of the curve, we
have
t = V(x* + y^);
lastly, if t be the abscissa or ordinate of the curve, we shall have
t = X, or t = y.
The choice of one of the three hypotheses or of any other, becoming in
dispensible in order that the formula which contains the differentials, may
be delivered from them, if we do not always adopt it, it is even then tacitly
supposed that the independent variable has been determined. For ex
ample, in the usual case where a formula contains only the differentials
d X, d y, d* y, d' y, &c. the hypothesis is that the independent variable
t has been taken for the abscissa, for then it results that
dx ,
t = X, j^ = 1,
d^x
and we see that the formula does not contain the seconJ, third, &c. dif
ferentials.
ANALYTICAL GEOMETRY. xxxi
To establish this formula, in all its generality, we must, as above, sup
pose X and y to be functions of a third variable t, and then we have
^ d^ __ d y tl X
d t ~ cTx • dT '
from which we get
djr
^ = ^ (53)
dx d X ^
dT
taking the second differential of y and operating upon the second membei
as if a fraction, we shall get
d X d* y d y d' x
d'y _ d~t ' dl dT* d t
d X ~~ d X*
TP
In this expression, d t has two uses ; the one is to indicate that it is
the independent variable, and the other to enter as a sign of algebra.
In the second relation only will it be considered, if we keep in view that
t is the independent variable. Then supposing d t' the common factor,
the above expression simplifies into
d' y __ dxd'y — dyd'x
dx ■" dx* *
and dividing by d x, it will become
d* y _ dxd'y — dyd'x
dx^ ~ dx^
Operating in the same way upon the equation (53), we see that in
taking t as the independent variable, the second member of the equation
ought to become identical with the first ; consequently we have only one
change to make in the formula which contains the differential coefiicients
^,^„ VIZ. to replace J/, by
d X d* y — d y d*x .^v
To apply these considerations to the radius of curvature which is given
by the equation See p. 6L vol. I.)
^ = — 57 — '
dx«
xxxii INTRODUCTION.
if we wish to have the value of R, in the case where t shall be the inde
pendent variable, we must change the equation to
R =
dx d* y — d y d' X *
dx'
and observing that the numerator amounts to
(dx' + dy') ^
dx^
^we shall have
1 (dx' + dy')i
" " dxd*y — dy»d*x ^^^^
This value of R supposes therefore that x and y are functions of a third
independent variable. But if x be considered this variable, that is to say,
if t = X, we shall have d * x =0, and the expression again reverts to the
common one
^ _ (dx'+dy«)t ^
(> + 1&
d x d* y d* y
dT*
But if, instead of x for the independent variable, we wish to have the
ordinate y, this condition is expressed by y = t ; and differentiating this
equation twice, we have
d t  ^' d t '  "•
The first of these equations merely indicates that y is the independent
variable, which effects no change in the formula ; but the second shows
us that d • y ought to be zero, and then the equation (55) becomes
_(dxM:_dyV
^  dyd^x ^^^^
We next remark, that when x is the independent variable, and
consequently d ' x = 0, this equation indicates that d x is constant.
Whence it follows, that generally the independent variable has always
a constant differential.
Lastly, if we take the arc for the independent variable, we shall have
dt = V (dx' + dy«);
Hence, we easily deduce
dx« dy_* _ , .
dt* + dt« "" '
ANALYTICAL GEOMETRY. xxxiii
dillerentiating this equation, we shall regard d t as constant, since t is the
independent variable ; we get
2d xd'x 2dy d^y _
dt^ + dt« ~ '
which gives
dxd^x = — dyd^y
Consequently, if we substitute the value of d ' x, or that of d * y, in the
equation (55), we shall have in the first case
R = .'f'l:w'"f,. dx= ^"'''' + ''^ )dx. . (67)
(d x + d y *) d ■* y d " y ^ '
and in the second case,
(dx + dy)^ V(dx' + dy)
^ (dx +dy*)d''x'^y d^ "y • ^^^^
In what precedes, we have only considered the two differential coeffi
cients
d y d ** y
dl^'dlP'
but if the formula contain coefficients of a higher order, we must, by
means analogous to those here used, determine the values of
d^y t.d*y ^
T— ^3 of j— ^ &c.
dx^ d x^
which will belong to the case where x and y are functions of a third in
dependent variable. "'
PROPERTIES OF HOMOGENEOUS FUNCTIONS.
IfMdx + Ndy+ Pdt4 &i.\ = dz, be a homogeneous Junction of
any lutmber of variables^ x, y, t, &c. in which the dimension of each tei'm is
n, then is
Mx + Ny + Pt + &c. = nz.
For let M d x + N d y be the differential of a homogeneous function
z between two variables x and y ; if we represent by n the sum of the
exponents of the variables, in one of the terms which compose this func
tion, we shall have therefore the equation
Mdx + Ndy = dz.
« Making ^ = q, we shall find (vol. I.)
F(q) X x» = z;
c
xxxiv INTRODUCTION.
and replacing, in the above equation, y by its value q x, and calling M'
N', what M and N then become, that equation transforms to
M' d X + N' d. q X = d z ;
and substituting the value of z, we shall have
M' d X + N' d (q z) = d (x" F. q.)
But d (q z) =: q d x + X d q. Therefore
(M' + N'q) dx + N'xdq = d (x" F. q).
But, (M* + N' q) d X being the differential of x " F q relatively to x, we
have (Art 6. vol. 1.)
M' + N'q = nx"' X F.q.
If in this equation y be put for q x, it will become
M + N^ = nx"'F. q,
X ^
or
Mx 4 Ny = nz.
This theorem is applicable to homogeneous functions of any number of
variables ; for if we have, for example, the equation
Mdx+ Ndy+ Pdt = dz,
in which the dimension is n in every term, it will suffice to make
s^= q, — = r
X ^' X
to prove, by reasoning analogous to the above, that we get z = x" F (q, r),
and, consequendy, that
Mx + Ny + Pt = nz (59)
and so on for more variables.
THEORY OF ARBITRARY CONSTANTS.
An equation V = between x, y, and constants, may be considered as
the complete integral of a certain differential equation, of which the order
depends on the number of constants contained in V = 0. These constants
are named arbitrary constants, because if one of them is represented by ff,
and V or one of its differentials is put under the form f (x, y) = a, we see
that a will be nothing else than the arbitrary constant given by the integra
tion of d f (x, y). Hence, if the differential equation in question is of the
order n, each integration introducing an arbitrary constant, we have
V = 0, which is given by the last of three integrations, and contains, at
ANALYTICAL GEOMETRY. xxxv
least, n arbitrary constants more than the given differential equation. Let
therefore
F(x,y) = 0, F (x,y,5^) = 0, F (x,y,^,^4^.) = &c. (a)
be the primitive equation of a differential equation of the second order
and its immediate differentials.
Hence we may eliminate from the two first of these three equations,
the constants a and b, and obtain I
^ ('^'^''dl'^) = ^'^ (""'^'di'^ = ^ . . . . (b)
If, without knowing F (x, y) = 0, we find these equations, it will be
sufficient tQeliminate from them ^ , to obtain F (x, y) = 0, which will
be the complete integral, since it will contain the arbitrary constants a, b.
If, on the contrary, we eliminate these two constants between the
above three equations, we shall arrive at an equation which, containing*
the same differential coefficients, may be denoted by
''('''J'Jx'3F>) = « (<=)
But each of the equations (b) will give the same. In fact, by eliminating
the constant contained in one of these equations and its immediate differ
ential, we shall obtain separately two equations of the second order,
which do not differ from equation (c) otherwise than the values of x and
of y are not the same in both. Hence it follows, that a differential equa
tion of the second order may result from two equations of the first order
which are necessarily different, since the arbitrary constant of the one is
different from that of the other. The equations (b) are what we call the
first integrals of the equation (c), which is independent, and the equation
F (x, y) = is the second integral of it.
Take, for example, the equation y = a x + b, which, because of its
two constants, may be regarded as the primitive equation of an equation
of the second order. Hence, by differentiation, and then by elimination
of a, we get
^y ^y , u
T^ = a,y = xr^ + b.
dx "^ dx
These two first integrals of the equation of the second order which we
are seeking, being differentiated each in particular, conduct equally, by
d ^\
the elimmation of a, b, to the independent equation r— ^, = 0. In the
c3
xxxvi INTRODUCTION.
case where the number of constants exceeds that of the required arbitrary
constants, the surplus constants, being connected with the same equations,
do not acquire any new relation. Required, for instance, the equation of
the second order, whose primitive is
y = ^ax' + bx + c = 0;
differentiating we get
^' = ax + b.
dx
The elimination of a, and then that of b, from these equations, give
separately these two first integrals
^ = ax + b, y = xj — i ax^ + c . . . (d)
Combining them each with their immediate differentials, we arrive,
d ^ V ,. .
by two different ways, at t — ^ = a. If, on the contrary, we had elimi
nated the third constant a between the primitive equation and its imme
diate differential, that would not have produced a different result; for
we should have arrived at the same result as that which would lead to
the elimination of a from the equations (d), and we should then have
fallen upon the equation x r— ^ = r^ — b, an equation which reduces
d*v
to j — ^ =r a by combining it with the first of the equations (d).
Let us apply these considerations to a differential equation of the third
order : differentiating three times successively the equation F (x, y) = 0,
we shall have
F fx V ^^  F/^x V ^ ^^  F^x V ^ ^ '^^ 
These equations admitting the same values for each of the arbitrary
constants contained by F (x, y) = 0, we may generally eliminate these
constants between this latter equation and the three preceding ones, and
obtain a result which we shall denote by
/ dyd*yd^y\ ^ , .
This will be the differential equation of the third order of F (x, y) = 0,
and whose three arbitrary constants are eliminated ; reciprocally,
F (x, 3") = 0, will be the third integral of the equation (e).
If we eliminate successively each of the arbitrary constants from the
 ANALYTICAL GEOMETRY. xxxvii
equation F (x, y) = 0, and that which we have derived by differentiation,
we shall obtain three equations of the first order, which will be the second
integrals of the equation (e).
Finally, if we eliminate two of the three arbitrary constants by means
of the equation F (x, y) = 0, and the equations which we deduce by two
successive differentiations, that is to say, if we eliminate these constants
from the equations
F (x,y) = 0. F (x,y, pj = 0, F (x. y, ^I, d^;) = „ . . (f)
we shall get, successively, in the equation which arises from the elimina
tion, one of the three arbitrary constants ; consequently, we shall have as
many equations as arbitrary constants. Let a, b, c, be these arbitraiy
constants. Then the equations in question, considered only with regard
to the arbitrary constants which they contain, may be represented by
f c =. 0, p b c= 0, p a = (g)
Since the equations (f) all aid in the elimination which gives us one of
these last equations, it thence follows that the equations (g) will each be
of the second order ; we call them the first integrals of the equation (e).
Generally, a differential equation of an order w will have a number n
of first integrals, which will contain therefore the differential coefficients
dvd"~'v..
from 5^ to , — ^i inclusively; that is to say, a number „_i of differential
coefiicients ; and we see that then, when these equations are all known,
to obtain the primitive equation it will suffice to eliminate from these equa
tions the several differential coefficients.
PARTICULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS.
It is easily seen that a particular integral may always be deduced from
the complete integral, by giving a suitable value to the arbitrary con
stant.
For example, if we have given the equation
xdxjydy = dyVx* + y^ — a*,
whose complete integral is
y + c = V (x^ + y2 — a*),
whence (for convenience, by rationalizing,) we get
(^'^'>ai^ + ^''y^ + ''' = ^ .... no
«2
xxxviii INTRODUCTION.
and the complete, integral becomes
2 cy + 0*— x« + a*^ = .... (i)
Hence, in taking for c an arbitrary constant value c = 2 a, we shall
obtain this particular integral
2cy + 5 a« — x2 = 0,
which will have the property of satisfying the proposed equation (h) as
well also as the complete integral In fact, we shall derive from tliis
particular integral
— x' — 5 a* d y __ 3C
^ "" 2 c ' cfx ~ "c '
these values reduce the proposed to
(x^a«)^' = ^(x« + c'^5a^),
an equation which becomes identical, by substituting in the second mem
ber, the value of c *, which gives the relation c = 2 a. Let
Mdx + Ndy = 0,
be a differential equation of the first order of a function of two variables
X and y ; we may conceive this equation as derived by the elimination of
a constant c from a certain equation of the same order, which we shall
represent by
mdx4ndy = 0,
and the complete integral
F (X, y, c) = 0,
which we shall designate by u. But, since every thing is reduced to
taking the constant c such, that the equation
Mdx+Ndy=rO,
may be the result of elimination, we perceive that is at the same time
permitted to vary the constant c, provided the equation
Mdx + Ndy = 0,
holds good ; in this case, the complete integral
F {X, y, c) =
will take a greater generality, and will represent an infinity of curves of
the same kind, differing from one another by a parameter, that is, by a
constant.
Suppose therefore that the complete integral being differentiated, by
considering c as the variable, we have obtained
ANALYTICAL GEOMETRY. xxxix
an equation which, for brevity, we shall write
d y = p d X + q d c (k)
Hence it is clear, that if p remaining finite, q d c is nothing, the result
of the elimination of c as a variable from
F (X, y, c) = 0,
and the equation (k), will be the same as that arising from c considered
constant, from
F (x, y, c) = 0,
and the equation
d y = p d X
(this result is on the hypothesis
Mdx+Ndy = 0),
for the equation (k), since
q d c = 0,
does not diifer from
d y = p dx;
but in order to have
q d c = 0,
one of the factors of this equation = constant, that is to say, that we
have
d c = 0, or q = 0.
In the first case, d c = 0, gives c = constant ^ since that takes place
for particular integrals; the second case, only therefore conducts to a par
ticular solution. But, q being the coefficient of d c of the equation (k),
we see that q = 0, gives
dx
This equation will contain c or be independent of it. If it contain c,
there will be two cases ; either the equation q = 0, will contain only c
and constants, or this equation will contain c with variables. In the first
case, the equation q = 0, will still give c =r constant, and in the second case,
it will give c = f (x, y) ; this value being substituted in the equation
F (x, y, c) = 0, will change it into another function of x, y, which will
satisfy the proposed, without being comprised in its complete integral,
and consequently will be a singular solution ; but we shall have a parti
cular integral if the equation c = f (x, y), by means of the complete ^'a
tegral, is reduced to a constant.
c 1
xl INTRODUCTION.
Wlien the factor q = from the equation q d c = not containing
the arbitrary constant c, we shall perceive whether the equation q =
gives rise to a particular solution, by combining this equation with the
complete integral. For example, if from q = 0, we get x = M, and put
this value in the complete integral F (x, y, c) = 0, we shall obtain
c = constant = B or c = f y ;
In the first case, q = 0, gives a particular integral ; for by changing c
into B in the complete integral, we only give a particular value to the
constant, which is the same as when we pass from the complete integral
to a particular integral. In the second case, on the contrary, the value
f y introduced instead of c in the complete integral, will establish between
X and y a relation different from that which was found by merely re
placing c by an arbitrary constant. In this case, therefore, we shall have
a particular solution. What has been said of y, applies equally to x.
It happens sometimes that the value of c presents itself under the form
— : this indicates a factor common to the equations u and U which is ex
traneous to them, and which must be made to disappear.
Let us apply this theory to the research of particular solutions, when
the complete integral is given.
Let the equation be
y dx — x'dy = av'(dx* + dy')
of which the complete integral is thus found.
Dividing the equation by d x, and making
dy
df =P
we obtain
y — px = a V{1 + p«).
Then differentiating relatively to x and to p, we get
J J 1 apdp
dy — pdxxdp= ^^/^^p.) ;
observing that
d y = p d X,
this equation reduces to
J . apdp _
xdp H TJT^, — Sx =
^ ^ V(l + p*)
and this is satisfied by making d p = 0. This hypothesis gives p = con
stant = c, a value which being put in the above equation gives
ANALYTICAL GEOMETRY. xU
y — ex = a V(l + c*) (1)
This equadon containing an arbitrary constant c, which is not to be
found in the proposed equation, is the complete integral of it.
This being accomplished, the part q d c of the equation d y = p d x f
q d c will be obtained by differentiating the last equation relatively to c
regarded as the only variable. Operating thus we shall have
1 , a c d c „
consequently the coefficients of d c, equated to zero, will give us
ac , ,
^ =  V(l + C) ^"^^
To find the value of c, we have
(1 + c')^x2 = a'c^,
which gives
C'= T^^T. 1 +C' = ^
and
^(' + =■> = V(a'x) =
by means of this last equation, eliminating the radical of the equation (m)
we shall thus obtain
c =  V(a»x^) ("5
This value and that of ^(1 + c^) being substituted in the equation (\)
will give us
x' __ a^
y+ V(a« — X*) ■" V(a2 — X*)
whence is derived
y = V(a^x^),
an equation which, being squared, will give us
y^ = a« — x^;
and we see that this equation is a particular solution, for by differentiating
it we obtain
J xd X
d y = ;
this value and that of V (x '^ + y ^), being substituted in the equation
originally proposed, reduce it to
a^ = a«.
In the application which we have just given, we have determined the
xhi INTRODUCTION.
value of c by equating to zero the differential coefficient \r^j This
process may sometimes prove insufficient. In fact, the equation
dy = pdx + qdc
being put under this form
Adx + Bdy + Cdc =
where A, B, C, are functions of x and y, we shall thence obtain
tly = — gdx — g dc (o)
dx = — ^dy — ^dc (p)
and we perceive that if all that has been said of y considered a function of
X, is applied to x considered a function of y, the value of the coefficient of
d c will not be the same, and that it will suffice merely that any factor of B
destroys in C another factor than that which may destroy a factor of A,
in order that the value of the coefficient of d c, on both hypotheses, may
appear entirely different. Thus although very often the equations
§ = 0, c =
give for c the same value, that will not always happen ; the reason of
which is, that when we shall have determined c by means of the equation
dc  "'
dx
It will not be useless to see whether the hypothesis of 5 — gives the same
result.
Clairaut was the first to remark a general class of equations susceptible
of a particular solution ; these equations are contained in the form
dy . n dy
y = ^x + F. ^
"" dx ; dx
on equation which we shall represent by
y = px + Fp (r)
By differentiating it, we shall find
dy = pdx + xdp + (j^) dp;
this equation, since d y = pdx, becomes
ANALYTICAL GEOMETRY. xliii
and since d p is a common factor, it may be thus written :
We satisfy this equation by making d p = 0, which gives p = const.
= c; consequently, by substituting this value in the equation (r) we
shall find
y = ex + F c.
This equation is the complete integral of the equation proposed, since
an arbitrary constant c has been introduced by integration. If we differ
entiate relatively to c we shall get
Consequently, by equating to zero the coefficients of d c, we have
■ dFc 
which being substituted in the complete integral, will give the particular
solution.
THE INTEGRATION OF EQUATIONS OF PARTIAL DIFFERENCES.
An equation which subsists between the differential coefficients, com
bined with variables and constants, is, in general, a partial differential
equation, or an equation of partial differences. These equations are thus
named, because the notation of the differential coefficients which they
contain indicates that the differentiation can only be eflPected partially ;
that is to say, by regarding certain variables as constant. This supposes,
therefore, that the function proposed contains only one variable.
The first equation which we shall integrate is this ; viz.
'dz>
Vdx/
f If contrary to the hypothesis, z instead of being a function of two vari
ables X, y, contains only x, we shall have an ordinary differential equation,
which, being integrated, will give
z = a X + c
bnt, in the present case, z being a function of x and of y, the i/s con
tained in z have been made to disappear by differentiation, since differen
xIiT INTRODUCTION.
tiating relatively to x, we have considered y as constant We ought,
therefore, when integrating, to J3reserve the same hypothesis, and suppose
that the arbitrary constant is in general a function of y ; consequently, we
shall have for the integral of the proposed equation
z = ax + fy.
Required to integrate the equation
in which % is any function of x. Multiplying by d x, and integrating,
we get
z =/Xdx + py.
For example, if the function X were x ^ + a ", the integral would be
x^
z = — +a2x + ?)y.
In like manner, it is found that the integral of
is "
z = X Y + p y .
Similarly, we shall integrate every equation in which (t— ) is equal to
a function of two variables x, y. If, for example,
/d z\ __ X
Vdx/ ~ V ay + x^'
considering y as constant, we integrate by the ordinary rules, making the
arbitrary constant a function of y. This gives
z = v'Cay + x'^) + <py.
Finally, if we wish to integrate the equation
('^\  ^
Vdx/ ~ V(y« — X*}
regarding y as constant, we get
z = sin.' + py.
Generally to integrate the equation
(^)dx=F(x,y)dx,
we shall take the integral relatively to x, and adding to it an arbitrary
function of y, as the constant, to complete it, we shall find
z = /"FCx, y) dx + f y.
ANALYTICAL GEOMETRY. xlv
Now let us consider the equations of partial differences which contain
two differential coefficients of the first order ; and let the equation be
'd Z\ . ^T /d z^
in which M and N represent given functions of x, y. Hence
'd^\ M /d
substituting this value in the formula
isent given functions of x,
/d z\ _ M /d z\
which has no other meaning than to express the condition that z is a
function of x and of y, we obtain
or
, /dzxNdx — Mdy
Let X be the factor proper to make Ndx — Mdya complete differ
ential d s ; we shall have
X (N d x — M d j) = d s.
By means of this equation, we shall eliminate Ndx — M d y from the
preceding equation, and we shall obtain
^" = rN'(§l)^^
Finally, if we remark that the value of ( y ) is indeterminate, we may
take it such that — ^ . (^ — \ d s may be integrable, which would make it
a function of s ; for we know that the differential of every given function
of s must be of the form F s . d s. It therefore follows, that we may
assume
an equation which will change the preceding one into
d z = F s . d s
which gives
z = © s.
xlvi INTRODUCTION.
Integrating by this method the equation
we have in this case
M = y,
and
N = x;
consequently
d s = X (x d X + y d y).
It is evident that the factor necessary to make this integrable is z.
Substituting this for X and integrating, we get
s = x'^ + y*.
Hence the integral of the proposed equation is
z = f (x= + y').
Now let us consider the equation
H^) + ^0 + ^ = o^
in which P, Q, R are functions of the variables x, y, z ; dividing it by P
and making
I = M, ^ = N,
we shall put it under this form :
and again making
/d z\
and
'd z
it becomes
i^) = ^'
p + Mq + N = (a)
This equation establishes a relation between the coefficients p and q ot
the general formula
= pdx + qdy;
without which relation p and q would be perfectly arbitrary, for as it has
been already observed, this formula has no other meaning than to indicate
that z is a function of two variables x, y, and that function may be any
ANALYTICAL GEOMETRY. xlvii
whatever ; so that we ought to regard p and q as mdeterminate m Jiis last
equation. Eliminating p from it, we shall obtain
dz + Ndx = q(dy — Mdx)
and q will remain always indeterminate. Hence the two members of this
equation aie heterogeneous (See Art. 6. vol. 1), and consequently
dz + Ndx = 0, dy — Mdx = (b)
If P, Q, R do not contain the variable z, it will be the same of M and
N; so that the second of these equations will be an equation of two varia
bles X and y, and may become a complete diflferential by means of a factor
X. This gives
X (d y — M d x) = 0.
The integral of this equation will be a function of x and of y, to wlncn
we must add an arbitrary constant s ; so that we shall have
F(x, y) = s;
whence we derive
y = f(x, s).
Such will be the value of y given us by the second of the above equa
tions; and to show that they subsist simultaneously we must substitute
this value in the first of them. But although the variable y is not shown,
it is contained in N. This substitution of the value of y just found,
amounts to considering y in the first equation as a function of x and of
the arbitrary constant s. Integrating therefore this first equation on that
hypothesis we find
z = — yN d X + p s.
To give an example of this integration, take the equation
and comparing it with the general equation (a), we have
M=^, N = A V(x^ + y^).
These values being substituted in the equations (b) will change them to
d z — — V (x» } y2) d x = 0, d y — ^ d X = (cj
Let X be the factor necessary to make the last of these integrable, and
we have
or rather
x(dy_Idx) =0,
xlviu INTRODUCTION.
1 V
which is integrable when X = — ; for then the integral is ^ = constant.
Pat therefore
X
and consequently
y = s X.
By means of this value of y, we change the first of the equations
(c) into
1 V X* — s* X* J _
d z — a . d X = 0,
X
or rather into
dz = adx V(l + s*).
Integrating on the supposition that s is constant, we shall obtain
z = a/dx V (1 + s') + ?)s
and consequently
z = a X V (1 + s*) + p s.
Substituting for s its value we get
= a ^/ (X' + y') + f (i).
x
In the more general case where the coefficients P, Q, R of the equation
contain the three variables x, y, z it may happen that the equations
(b) contain only the variables which are visible, and which consequently
we may put under the forms
d z = f (x, z) d X = 0, d y = F (x, y) d X.
These equations may be treated distinctly, by writing them as above,
z =/f(x,z)dx + z, y =/F(x,y)dx + *y
for then we see we may make z constant in the first equation and y in
the second ; contradictory hypotheses, since one of three coordinates
X, y, z cannot be supposed constant in the first equation without its being
not constant in the second.
Let us now see in what way the equations (b) may be integrated in the
case where they only contain the variables which are seen in them.
Let /» and X be the factors which make the equations (b) integrable.
If their integrals thus obtained be denoted by U and by V, we have
A(dz + Ndx) = dU, /x(dy — Mdx) = d V.
ANALYTICAL GEOMETRY. xlix
By means of these values the above equation vvrill become
dUrrq^dV (d) .
Since the first member of this equation is a complete differential the
second is also a complete differential, which requires q — to be a function
of V. Represent this function by f V. Then the equation (d) will
become
dU = pV.dV
which gives, by integrating,
U = «i»V.
Take, for example, the equation ^
"yCjl) +'''(dy) = y^:
which being written thus, viz.
/d z\ , X /d z\ z
we compare it with the equation
(ai) + M0 + N = « .
and obtain
M = '^, N = — 
y X
By means of these values the equations (b) becomes
dz .dx=0,dy dx = 0;
X y
which reduce to
xdz — zdx = 0,ydy — xdx = 0.
Tiic factors necessary to make these integrable are evidently — j and 2.
Substituting which and integrating, we find — and y * — x ' for the in
tegrals. Putting, therefore, these values for U and V in the equation
U = * V, we shall obtain, for the integral of the proposed equation,
^=a.(y^x^)
It must be remarked, that, if we had eliminated q instead of p, the equa
tions (b) would have been replaced by these
Mdz+Ndy = 0,dy— Mdx = 0. . . . (e)
and since all that has been said of equations (b) applies equally to these,
d
I INTRODUCTION.
It follows that, in the case where the first of equations (b) was not in
te<»rable, we may replace those equations by the system of equations (e),
which amounts to employing the first of the equations (e) instead of the
first of the equations (b).
For instance, if we had
this equation being divided by a z and compared with
will give us
M = ^,N=^y
a a z
and the equations (b) will become •
dz + '^^dxrzOjdy+^dxrrO;
a z •'a
which reduce to
azdz + xydx = 0,ady + xdx,= . . . (f)
The first of these equations, which, containing three variables, is not
immediately integrable, we replace by the first of the equations (e), and
we shall have, instead of the equations (f), these
— ^dz + ^dy = 0,ady + xdx = 0;
which reduce to
2ydy — 2zdz = 0,2ady + 2xdx = 0;
equations, whose integrals are
y* — z*,2ayfx*'
These values being substituted for U and V, will give us
y* — z» = p (2 ay + x«).
It may be remarked, that the first of equations (e) is nothing else than
the result of the elimination of d x from the equations (b) .
Generally we may eliminate every variable contained in the coefficients
M, N, and in a word, combine these equations after any manner what
ever ; if after having performed these operations, and we obtain two in
tegrals, represented by U = a, V = b, a and b being arbitrary constants,
we can always conclude that the integral is U = * V. In fact, since
a and b are two arbitrary constants, having taken b at pleasure, we may
compose a in terms of b in any way whatsoever ; which is tantamount to
saying that we may take for a an arbitrary function of b. This condition
will be expressed by the equations a = f (b). Coiisequently, we shall
ANALYTICAL GEOMETRY. If
have the equations U = p b, V = b, in which x, y, z represent the same
coordinates. If we eliminate (b) from these equations, we shall obtain
U = p V.
This equation also shows us that in making V = b, we ought to have
U = p b = constant ; that is to say, that U and V are at the same time
constant; without which a and b would depend upon one another, where
as the function <p is arbitrary. But this is precisely the condition expressed
by the equations U = a, V = b.
To give an application of this theorem, let
Dividing by z x and comparing it with the general equation we
have
M = — ^, N = — ^;
X zx
and the equations (b) give us
d z — ^ d X = 0, d y + ^ d X =
zx "^ X
or
zxdz — y*dx = 0, xdyfydz= 0.
The first of these equations containing three variables we shall not at
tempt its integration in that state; but if we substitute in it for y d x its
vahie derived from the second equation, it will acquire a common factor
x, which being suppressed, the equation becomes
zdzfydy = 0,
and we perceive that by multiplying by 2 it becomes integrable. 1 he
other equation is already integrable, and by integrating we find
z* + y* = a, xy = b,
whence we conclude that
z' + y' = ?xy.
We shall conclude what we have to say upon equations of partial differ
ences of the first order, by the solution of this problem.
Given an equation "which contains an arbitraiy function of one or more
variables^ tofnd the equation of partial dijfhences which produced it.
Suppose we have
z = F(x'' + y»).
Make
X* + y^ =u (0
and the equation becomes
z = F u.
d2
lii INTRODUCTION.
The difler^iitial of F u must be of the form ^ u . d u. Conse
d z = d u . 9 u
If we tal?e tlie differential of z relatively to x only, that is to say, in
regarding y as constant, we ought to take also d u on the same
hypothesis. Consequently, diriding the preceding equation by d x,
we get
/d z\ /d u\ *
fc) = (dl^)^"
Again, considering x as constant and y as variable, we shall similarly
find
'd z\ /d u
(d^) = (d^)^"
y/ \dy.
But the values of these coefficients are found from the equation (f).
which gives
G
(^D=.(^)=^y
Hence our equations become
and eliminating <p u from these, we get the equation required j viz.
/d Zy. /d Zx
As another example, take this equation
z« + 2 ax = F (x ^ y).
Making
x — y = u,
It becomes
z'42ax=Fu
ar.d differ ntiating, we get
d (z * ■+ 2 a x) = d u ^ u .
Then taking the differential relatively to x, we have
rd to y, we get
and similarly, with regard to y, we get
'd z\ /d u
ANALYTICAL GEOMETRY. Ini
But since
X — y = u
which, being substituted in the above equation, gives us
2z(^) +2a = pu,2z(^) =fU
and eliminating <p u from these, we have the equation required ; viz.
We now come to
EQUATiaNS OP PARTIAL DIFFERKNCKS OF TH^ SECOND ORDER.
An equation of Partial Differences of the second ojxleif in which z is a
function of two variables x, y ought always to contain one or more of the
differential coefficients
/d * z\ /d * Z\ / d * z \
VdlTV* Vjp/' Id^Hdy/
independently of the differential coefficients which enter equations of the
first order.
We shall merely integrate the simplest equations of this kind, and shall
begin with this, viz.
il^d = »•
Multiplying by d x and integrating relatively to x we add to the inte
gral an arbitrary function of y ; and we shall thus get
/dz\
(d^) = py
Again multiplying by d x and integrating, the integral will be com
pleted when we add another arbitrary function of y, viz. v}/ y. We thus
obtain
z = xpy + ^'y.
Now let us integrate the equation.
d8
Uv INTRODUCTION.
in which P is any function of x, y. Operating as before we first obtain
and the second integration gives us
z =/{/Pdx + fy]dx + ^}.y.
In the same manner we integrate
&^
and find
2 =/ipx +/Pdy} dy + ^^x.
The equation
\djdk) " ^
must be integrated first relatively to one of the variables, and then rela
tively to the other, which will give
z =/{py +/Pdx} dy + px.
In general, similarly may be treated the several equations
idf) = ^' (dxriy""*') = ^' (dx^dy"*) = ^' ^'''
in which P, Q, R, &c. are functions of x, y, which gives place to a series
of integrations, introducing for each of them an arbitrary function.
One of the next easiest equations to integrate is this :
Cdp) + P (^) = Q=
in which P and Q will always denote two functions of x and y.
Make
and the proposed will transform to
0+Pu = Q.
To integrate this, we consider x constant, and then it contains only
two variables y and u, and it will be of the same form as the equation
dy + Py dx = Qdx
whose integral (see Vol. 1. p. 109) is
y = e/P'"' J/Qe/P^Mx + C}.
Hence our equation gives
u = c/"M/Qe^^,dy + ^xj.
ANALYTICAL GEOMETRY. Jv
But
Hence by integration we get
z =/{e^P<Jy(/Qe^P'iydy) + px} dy + 4x.
By the same method we may integrate
(d*z \ Ti /d z\ _ d'z . T» /d z\ ^
JTdy) + P (dx) = Q' dTd^ + P (37) = Q'
in which P, Q represent functions of x, and because of the divisor d x d y,
we perceive that the value of z will not contain arbitrary functions of the
same variable.
THE DETERMINATION OF THE ARBITRARY FUNCTIONS WHICH ENTER
THE INTEGRALS OF EgUATIONS OF PARTIAL DIFFERENCES O.Y
THE FIRST ORDEK.
The arbitrary functions which complete the integrals of equations of
partial differences, ought to be given by the conditions arising from the
nature of the problems from which originated these equations ; problems
generally belonging to the physical branches of the Mathematics.
But in order to keep in view the subject we are discussing, we shall
limit ourselves to considerations purely analytical, and we shall first seek
what are the conditions contained in the equation
/d z\
Since z is a function of x, y, this equation may be ^msidered as that of
a surface. This surface, from the nature of its equation, has the following
property, that (r—) must always be constant. Hence it follows that
every section of this surface made by a plane parallel to that of x, y is a
straight line. In fact, whatever may be the nature of this section, if we
divide it into an infinity of patts, these, to a small extent, may be con
sidered straight lines, and will represent the elements of the section, one
of these elements making with a parallel to the axis of abscissae, an angle
rdz
all
whose tangent is (7—) • Since this angle is constant, it follows that
the angles formed in like manner by the elements of the curve, with par
Ivi INTRODUCTION.
allels to the axis of abscissae will be equal. Wluch proves that the sec
tion in question is a straight line.
We might arrive at the same result by considering the integral of the
equation
'dz>
(H) =
which we know to be
z = ax + ^y,
since for all the points of the surface which in the cutting plane, the or
dinate is equal to a constant c. Replacing therefore f y by f c, and
making p c = C, the above equation becomes
z = ax + C;
this equation being that of a straight line, shows that the section is a
straight line.
The same holding good relatively to other cutting planes which may be
drawn parallel to that of x, 2, we conclude that all these planes will cut the
surface in straight lines, which will be parallel, since they will each form
with a parallel to the axis of x, an angle whose tangent is a.
If, however, we make x = 0, the equation z = a x + f y reduces to
z = f y, and will be that of a curve traced upon the plane of y, z; this
curve containing all the points of the surface whose coordinates are x = 0,
will meet the plane in a point whose coordinate is x =0; and since we
have also y = c, the third coordinate by means of the equation
z = ax + C
will be '
z = C.
What has been said of this one plane, applies equally to all others
which are parallel to it, and it thence results that through all the points
of the curve whose equation is z = p y, and which is traced in the plane
of y, z, will pass straight lines parallel to the axis of x. This is ex
pressed by the equations
'd zy
(D =
and
z = ax + f y;
and since this condition is always fulfilled, whatever may be the figure of
the curve whose equation is z =r p y, we see that this curve is arbi
trary.
From what precedes, it follows that the curve whose equation is z = py,
ANALYTICAL GEOMETRY.
Ivii
may be composed of arcs of different curves, which unite at their extre
mities, as in this diagram
A C
or which have a break off in their course, as in this figure.
N
In the first case the curve is discontinuous, and in the second it is dis
contiguous. We may remark that in this last case, two different ordinates
P M, P N corresponding to the same abscissa A P; finally, it is possible,
that without being discontiguous, the curve may be composed of an in
finite series of arcs indefinitely small, which belong each of them to
different curves ; in this case, the curve is irregular, as will be, for
instance, the flourishes of the pen made at random ; but in whatever way
it is formed, the curve whose equation is z = p y, it will suffice, to con
struct the surface, to make a straight line move parallelly with this condi
tion, that its general point shall trace out the curve whose equation is
z = py,
and vhich is traced at random upon the plane of y, z.
If instead of the equation
(ffs) = "•
we had
in which X was a function of x, then in drawing a plane parallel to the
plane (x, z), the surface will be cut by it no longer in a straight line, as
in the preceding case. In fact, for every point taken in this section, the
tangent of the angle formed by the element produced of the section, with
a parallel to the axis of x, will be equal to a function X of the abscissa x
of this point ; and since the abscissa x is different for every point it foJ
Iviii INTRODUCTION.
lows that this angle will be different at each point of the section, which
section, therefore, is no longer, as before, a straight line. The surface
will be constructed, as before, by moving the section parallelly, so that its
point may ride continually in the curve whose equation is z = ^ y.
Suppose now that in the preceding equation, instead of X we have a
function, P of x, and of y. The equation
(p.) = P.
containing three variables will belong still to a curve surface. If we cut
tliis surface by a plane parallel to that of x, z, we shall have a section in
which y will be cwistant ; and since in all its points (j— ) will be equal
to a function of the variable x, this section must be a curve, as in the pre
ceding case. The equation
'dz>
(n)
P
being integrated, we shall have for that of the surface
z =/Pdx + py;
if in this equation we give successively to y the increasing values y', y%
y"', &c. and make P', P", P'", &c. what the function P becomes in these
cases, we shall have the equations
z=/PMx + <py, z=/P"dx + ?y" ■»
z =/P"'dx + f y'", z =/F'''dx + ^y"" &c. /
and we see that these equations will belong to curves of the same nature,
but different in form, since the values of the constant y will not be the
same. These curves are nothing else than the sections of the surface
made by planes parallel to the plane (x, z) ; and in meeting the plane
(y, z) they will form a curve whose equation will be obtained by equating
to zero, the value of x in that of the surface. Call the value of/Pdx,
in this case, Y, and we shall have
z = Y + py;
and we perceive that by reason of <p y, the curve determined by this equa
tion must be arbitrary. Thus, having traced at pleasure a curve, Q R S,
upon the plane (y, z), if we represent by R L the section whose equation
Q
L
IS t =yp'd x 4 py', we shall move this section, always keeping the ex
ANALYTICAL GEOMETRY. Hx
tremity R applied to the curve Q R S ; but so that this section as it
movTes, may assume the successive forms determined by the above group
of equations, and we shall thus construct the surface to which will belong
the equation
Finally let us consider the general equation
whose integral is U = 9 V. Since U = a, V = b, each of these equa^
tions subsisting between three coordinates, we may regard them as be
longing to two surfaces ; and since the coordinates are common, they
ought to belong to the curve of intersection of the two surfaces. This
being shown, a and b being arbitrary constants, if in U = a, we give to
X and y the values x', y' we shall obtain for z, a function of x', of y' and
of a, which will determine a point of the surface whose equation is U = a.
This point, which is any whatever, will vary in position if we give succes
sively different values to the arbitrary constant a, which amounts to say
ing that by making a vary, we shall pass the surface whose equation is
U = a, through a new system of points. This applies equally to V = b,
and we conclude that the curve of intersection of the two surfaces will
change continually in position, and consequently will describe a curved
surface in which a, b may be considered as two coordinates ; and since
the relation a = p b which connects these two coordinates, is arbitrary
we perceive that the determination of the function p amounts to making
a surface pass through a curve traced arbitrarily.
To show how this sort of problems may conduct to analytical condi
tions, let us examine what is the surface whose equation is
We have seen that this equation being integrated gives
z = f (x* + y*).
Reciprocally we hence derive
x*f.y' = *z.
If we cut the surface by a plane parallel to the plane (x, y) the equation
of the section will be
x^ f y« = * c;
and representing by a * the constant * c, we shall have
X* + y* = a^
This equation belongs to the circle. Consequently the surface will
Ix INTRODUCTION.
hare this property, viz. that every sectjoo maJe by a plane iMrallel to the
plane (x, y) will be a circle.
This property is also indicated by the equation
y O = " (dy)
for this equation gives
dy
^ dx
This equation shows us that the subnormal ought to be always equal to
the abscissa which is the property of the circle.
The equation z = P (x* + y*) showing merely that all the sections
parallel to the plane (x, y) are circles, it follows thence that the law ac
cording to which the radii of these sections ought to increase, is not
comprised in this equation, and that consequently, every surface of revo
lution will satisfy the problem ; for we know tliat in this sort of surfaces,
the sections parallel to the plane (x, y) are always circles, and it is need
less to say that the generatrix which, during a revolution, describes the
surface, may be a curve discontinued, discontiguous, regular or irregular.
' Let us therefore investigate the surface for which this generatrix will
be a parabola A N, and suppose that, in this hypothesis, the surface is
cut by a plane A B, which shall pass through the axis of z ; the trace of
L
this plane upon the plane (x, y) will be a straight line A L, which, being
drawn through the origin, will have the equation y = a x ; if we repre
sent by t the hypothenuse of the right angled triangle A P Q, constructed
upon the plane (x, y) we shall have
t« = X* + y'i
but t being the abscissa of the parabola A M, of which Q M = z is the
ordinate, we have, by the nature of the curve,
t * = b z.
Putting for t * its value x * + y *, we get
z = \j(y* + x«),orz = ~(a«x«fx') = ^x'(l + fl')r
ANALYTICAL GEOMETRY. Ixi
and making
i (a + 1) = m,
we shall obtain
% = mx*;
so that the condition prescribed in the hypothesis, where the generatrix
is a parabola, is that we ought to have
z = m X % when y = a x.
Let us now investigate, by means of these conditions, the arbitrary
function which enters the equation z = p (x * 4 y *). For that pur
pose, we shall represent by U the quantity x* + y '» which is effected by
the symbol p, and the equation then becomes
z = pU;
^nd we shall have the three equations
x* + y'z= U, y = ax, z = mx*.
By means of the two first we eliminate y and obtain the value of x '
which being put into the third, will give
2: = m.5p^^
an equation which reduces to
th^ value of z being substituted in the equation z = p U, will change
it to
and putting the value of U in this equation, we shall find that
and we see that the function is determined. Substituting this value of
f (x *  y •) in the equation z = p (x * + y ^)> we get
z=l(x« + y^),
for the integral sought, an equation which has the property required,
since the hypothesis of y = ax gives
z = m x '.
This process is general ; for, supposing the conditions which determine
the arbitrary constant to be that the integral gives F (x, y, z) = 0, when
we have f (x, y, z) =0, we shall obtain a third equation by equating to
ixu
INTRODUCTION.
U the quantity which follows f, and then by eliminating, successively,
two of the variables x, y, z, we shall obtain each of these variables in a
function of U ; putting these values in the integral, we shall get an equa
tion whose first member is <p U, and whose second member is a compound
expression in terms of U ; restoring the value of U in terms of the vairi
bles, the arbitrary function will be determined.
THE ARBITRARY FUNCTIONS WHICH ENTER THE INTEGRALS OF THE
E2UATIONS OF PARTIAL DIFFERENCES OF THE SECOND ORDER.
Equations of partial differences of the second order conduct to integrals
which contain two arbitrary functions ; the determination of these func
tions amounts to making the surface pass through two curves which may
be discontinuous or discontiguous. For example, take the equation
(d * z\ ^
•whose integral has been found to be
z = xpy4N/y
Let A X, A y, A z, be the axis of coordinates; if we draw a plane
K L parallel to the plane (x, z), the section of the surface by this plane
will be a straight line ; since, for all the points of this section, y being
equal to A p, if we represent A p by a constant c, the quantities ?y, •4' y
will become p c, ^ c, and, consequently, may be replaced by two con
stants, a, b, so that tlie equation
z = xfy + 4y
ANALYTICAL GEOMETRY. Ixifi
will become
z = a X f b,
and this is the equation to the section made by the plane K L.
To find the point where this section meets the plane (y, z) make
X = 0, and the equation above gives z = vj/ y, which indicates a curve
a m b, traced upon the plane (y, z). It will be easy to show that the
section meets the curve a m b in a pomt m ; and since this section is a
straight line, it is only requisite, to find the position of it, to find a second
point. For that purpose, observe that when x = 0, the first equation
reduces to
whilst, when x = 1, the same equation reduces to
z = p y + ^ y.
Making, as above, y = Ap = c, these two values of z will become
z = b, z = a 4 b,
and determining two points m and r, taken upon the same section, m r
we know to be in a straight line. To construct these points we thus pro
ceed : we shall arbitrarily trace upon the plane (y, z) the curve a m b,
and through the point p, where the cutting plane K L meets the axis of
y, raise the perpendicular pm = b, which will be an ordinate to the
curve ; we shall then take at the intersection H L of the cutting plane,
and the plane (x, y), the part p p' equal to unity, and through the point
p', we shall draw a plane parallel to the plane (y, z), and in this plane
construct the curve a' m' b', after the modulus of the curve a m b, and so
as to be similarly disposed ; then the ordinate m' p' will be equal to m p ;
and if we produce m' p' by m' r, which will represent a, we shall deter
mine the point r of the section.
If, by a second process, we then pi'oduce all the ordinates of the curve
a' m'b', we shall construct a new curve a' r' b', which will' be such, that
drawing through this curve and through a m b, a plane parallel to the
plane (x, z), the two points where the curves meet, will belong to the
same section of the surface.
From what precedes, it follows that the surface may be constructed, by
moving the straight line m r so as continually to touch the two curves,
a m b, a' m' b'.
This example suffices to show that the determination of the arbitral'^
functions which complete the integrals of equations of partial differences
of the second order, is the same as making the surface pass through two
curves, which, as well as the functions themselves, may be discontinuous,
discontiguous, regular or irregular.
lar INTRODUCTION
CALCULUS OF VARIATIONS.
If we have given a function Z = F, (x, y, j/, y"), wherein y', y" mean
/dyx /d^yx
y itself being a function of x, it may be required to make L have certain
properties, (such as that of being a maximum, for instance) whether by
assigning to x, y numerical values, or by establishing relations between
these variables, and connecting them by equations. When the equation
y = p X is given, we may then deduce y, y', y" ... in terms of x and sub
stituting, we have the form
Z = f X.
By the known rules of the differential calculus, we may assign the values
of X, when we make of x a maximum or minimum. Thus we determine what
are the points of a given curve, for which the proposed function Z, is
greater or less than for every other point of the same curve.
But if the equation y = p x is not given, then taking successively for
px different forms, the function Z = f x will, at the same time, assume
different functions of x. It may be proposed to assign to ^ x such a
form as shall make Z greater or less than every other form of p x,yor the
same numerical value of x 'whatever it may he in other respects. This latter
species of problem belongs to the calculus of variations. This theory
relates not to maxima and minima only; but we shall confine our
selves to these considerations, because it will suffice to make known all
the rules of the calculus. We must always bear in mind, that the varia
bles X, y are not independent, but that the equation y = p x is unknown,
and that we only suppose it given to facilitate the resolution of the prob
lem. We must consider x as any quantity whatever which remains the same
for all the differential forms of f x ; the forms of f , p', f " .... are therefore
variable, whilst x is constant.
In Z = F (x, y, y', y". . .) put y + k for y, y' + k', for y'. . . , k being
an arbitrary function of x, and k', k," . . . the quantities
dk^ dMc
dx' dx»"*
But, Z will become
Z, = F(x,y + k,y + k', y + V'...)
ANALYTICAL GEOMETRY. Ixv
Taylor's theorem holds good whether the quantities x, y, k be depen
dent or independent. Hence we have
so that we may consider x, y, y', y" . . . as so many independent variables.
The nature of the question requires that the equation y = 9 x should
be determined, so that for the same value of x, we may have always
Z^ > Z, or Z^ •< Z : reasoning as in the ordinary maxima and minima,
we perceive that the terms of the first order must equal zero, or that we
have
''(dT) + >''(d) + ''"(P) + ^^ = »
Since k is arbitraiy for every value of x, and it is not necessary that its
value or its form should remain the same, when x varies or is constant,
k', k" . . . is as well arbitrary as k. For we may suppose for any value
X = X that k = a + b (x — X) + ^ c (x — j^) « + &c., X, a, b, c . . .
being taken at pleasure ; and since this equation, and its differentials,
ought to hold good, whatever is x, they ought also to subsist when
x = X, which gives k = a, k' = b, k" = c, &c. Hence the equation
Z, = Z f . . . cannot be satisfied when a, b, c . . . are considered inde
pendent, unless (see 6, vol. I.)
(af) = »'(dT) = '''(^) = «(^».)=''>
n being the highest order of y in Z. These different equations subsist
simultaneously, whatever may be the value of x ; and if so, there ought
to be a maximum or minimum ; and the relation which then subsists be
tween X, y will be the equation sought, viz. y = f x, which will have the
property of making Z greater or less than every other relation between
X and y can make it. We can distinguish the maximum from the mini
mum from the signs of the terms of the second order, as in vol. L
p. (3L)
But if all these equations give different relations between x, y, the
problem will be impossible in the state of generality which we have
ascribed to it ; and if it happen that some only of these equations subsist
mutually, then the function Z will have maxima and minima, relative to
some of the quantities y, y', y" . •. without their being common to them
all. The equations which thus subsist, will give the relative maxima and
minima. And if we wish to make X a maximum or minimum only relatively
iatvi INTRODUCTION.
' lo one of the quantities y, y', y'' . . . , since then we have only one equa
tion to satisfy, the problem will be always possible.
From the preceding considerations it follows, that first, the quantities
X, y depend upon one another, and that, nevertheless, we ought to make
them vary, as if they were independent, for this is but an artifice to get
the more readily at the result.
Secondly, that these variations are not indefinitely small ; and if we em
ploy the differential calculus to obtain them, it is only an expeditious
means of getting the second term of the developement, the only one
which is here necessary.
Let us apply these general notions to some examples.
Ex. 1. Take, upon the axis of x of a curve, two abscissas m, n; and
draw indefinite parallels to the axis of y. Let y =• p x be the equation
of this curve: if through any point whatever, we draw a tangent, it will
cut the parallels in points whose ordinates are
1 = y + y' ("^ — x), h = y + y' (n — x) .
If the form of p is given, every thing else is known ; but if it is not
« given, it may be asked, what is the curve which has the property of
having for each point of tangency, the product of these two ordinates less
than for every other curve.
Here we have 1 X h ; or
Z= {yx (ra — x)y'] + Jy + (nx)y'}.
From the enunciation of the problem, the curves 'which pass through the
same point (x, y) have tangents taking different directions, and that which
is required, ought to have a tangent, such that the condition Z = maximum
is fulfilled. We may consider X and y constant; whence
/d^\ _ 2/ 2 X — m — n _ 1 1
\ d y'/ ~ * y (x — m) (x — n) ~x — mx — n'
Then integrating we get
y' = C(x — m) (x — n).
The curve is an ellipse or a hyperbola, according as C is positive or
negative ; the vertexes are given by x = m, x = n ; in the first case, the
prod uct h X 1 or Z is a maximum^ because y" is negative ; in the second,
Z is a minimum or rather a negative maximum ; this product is moreover
constant, and 1 h = — i C (m — n)*, the square of the semiaxis.
Ex. 2. What is the curve for which, in each of its points, the square of
the subnormal added to the abscissa is a minimum P
We have in this case
Z = (y/ + x)«
ANALYTICAL GEOMETRY. Ixvii
whence we get two equations subsisting mutually by making
y y' + X =
and thence
. X * + y 2 = r '.
Therefore all the circles described f''om the origin as a center wf J alone
satisfy the equation.
The theory just expounded has not been greatly extended ; but it serves
as a preliminary developement of great use for the'comprshension of a
far more interesting problem which remains to be considered. This re
quires all the preceding reasonings to be applied to a function of the form
/ Z: the sign y indicates the function Z to be a differential and that after
having integrated it between prescribed limits it is required to endow it
with the preceding properties. The difficulty here to fae overcome is that
of resolving the problem without integrating.
When a body is in motion, we may coujpare together either the differ
ent points of the body in one of its positions or the plane occupied suc
cessively by a given point. In the first case, the body is considered fixed,
and the symbol d will relate to the change of the coordinates of its surface ;
in the second, we must express by a convenient symbol, variations alto
gether independent of the first, which shall be denoted by 3. When we
consider a curve immoveable, or even variable, but taken in one of its po
sitions, d X, d y . . . announce a comparison between its coordinates ; but
to consider the different planes which the same point of a curve occupies,
the curve varying in form according to any law whatever, we shall write 3
X, 3 y . . . which denote the increments considered under this point of view,
and are functions of x, y ... In like manner, d x becoming d (x f 3 x)
will increase by d 5 x ; d * x will increase by d ' 3 x, &c.
Observe that the variations indicated by the symbol b are finite, and
wholly independent of those which d represents ; the operations to which
these symbols relate being equally independent, the order in which they
are used must be equally a matter of indifference as to the result. So
that we have
3 . d X = d . 3 x
d^3x = 3.d*x
&c.
fhV=^* U.
and so on.
It remains to establish relations between x, y, 7. . . .such thatyZ may
be a maximnm or a minimum letween given limits. That the calculus may
be rendered the more symmetrical, we shall not suppose any differential
Ixviu INTRODUCTION
constant ; moreover we shall only introduce three variables because it will
be easy to generalise the result To abridge the labour of the process,
make
d X = x^, d * X = X/^, &c.
so that
z = F (x, x„ x,„ . . . y^ y„ y,„ . . . z, z„ z,, . . .).
Now X, y and z receiving the arbitrary and finite increments 3 x, 5 y,
3 ^ d X or X, becomes
d (x + 3 x) = d X + a d X or x, + 3 X,.
In the same manner, x,, increases by d x^, and so on ; so that develop
ing Z, by Taylor's theorem, and integrating y Z becomes
/•Z,=/Z+/{(^)^x + (^)ay+(^)az+(.^)a.,
The condition of a maximum or minimum requires the integral of the
terms of the first order to be zero between given limits ivhatever may be
3 X, 3 y, 3 z as we have already seen. Take the differential of the known
function Z considering x, x^, x^^ . . . y, y^, y^, . . . as so many independent
variables; we shall have
dZ=mdx + ndx^ + pdx,,+... Mdy + Ndy,. . .+ (udz + vdz, ...
ni, n . . . M, N . . . /x, v . . . being the coefficients of the partial differences
of Z relatively to x, x^ . . . y, y, . . . z, z^, . . . considered as so many varia
bles ; these are therefore known functions for each proposed value of Z.
Performing this differentiation exactly in the same manner by the symbol
2, we have
3 Z = m ax + n 8d X + p 3d«x + q 3 d'x
+ M3y + N3dy + P3d'y + qad^y + . . . ^(A)
fj«,3z+ v3dz + '33d*z+p/3d^y +
+ q3d'x + . . . ^
+ qad^y + . . . t
+ V 3 d ^ y + . . . )
But this known quantity, whose number of terms is limited, is precisely
that which is under the sign f, in the terms of the first order of the de
velopement : so that the required condition of max. or min. is that
between given limits, whatever may be the variations 3 x, 3 y, 3 z. Ob
serve, that here, as before, the differential calculus is only employed as a
means of obtaining easily the assemblage of terms to be equated to zero;
so that the variations are still any whatever and finite.
ANALYTICAL GEOMETRY. Ixix
We have said that d . 3 x may be put for d . 6 x ; thus the first hne is
equivalent to
m5x + n.d5x + p.d*3x+q.d'5x + &c.
m, n . . . contains differentials, so that the defect of homogeneity is here
only apparent. To integrate this, we shall see that it is necessary to
disengage from the symbol f as often as possible, the terms which con
tain d 3. To eflfect this, we integrate hy 'parts which gives
ynd3x = n. Sx — /dn.3x
/p.d2ax = p d ^x — d p 5x+/d'p3x
/qd^ax=qd'^ax~dq.dax+ d== q. d x — /d' q . 3 x
&c.
Collecting these results, we have this series, the law of which is easily
recognised ; viz.
/{m — d n + d « p — d 3 q 4 d * r — . . .) 3 X
+ (n— dp + d*q — d^r + d^s — ...)3x
+ (p— dq + d«r— d^s + dn — ...)dax
+ (q —  d r + . . .) d * 3 x
+ &c.
The integral of (A) ory . 3 z = , becomes therefore
(B).../{(m d n + d^ p...)3x+(Md N+d 2 P...)3 y+ (/^d i^...)3z}=0
C (ndp + d2q...)3xf(NdP+d2Q_...)3y + (»d*...)dz
(C)...^ +(pdq + d==r...)d3x+(PdQ + ...)d3y + (^d;^...)d3z
(.+(qdr...) d^3x...+ K =
K being the arbitrary constant. The equation has been split into two,
because the terms which remain under the sign y cannot be integrated, at
least whilst 3 x, 3 y, 3 z are arbitrary. In the same manner, if the nature
of the question does not establish some relation between 3 x, 3 y, 3 z, the
independence of these variations requires also that equation (B) shall again
make three others ; viz.
0= m — d n + d * p — d » q +d * r ■— . . . '),
0=M — dN+d=P — d3Q+d*R— ... V. . (D)
= /i— dy + d«ff — d';^/ + d^g — . . . J
Consequently, to find the relations between x, y, z, which make y Z a
maximum, we must take the differential of the given function Z by con
sidering x, y, z, d X, d y, d z, d ^ x, ... as so many independent vari
ables, and use the letter 3 to signify their increase ; this is what is termed
taking the variation of Z. Comparing the result with the equation (A),
we shall observe the values of m, M, /tt, n, N . . . in terms of y, y, z, and
e3
Ixx INTRODUCTION.
their differences expressed by d. We must then substitute these in the
equations (C), (D) ; the first refers to the limits between which the
maximum should subsist; the equations (D) constitute the relations re
quired; they are the differentials of x, y, z, and, excepting a case of
absurdity, may form distinct conditions, since they will determine nume
rical values for the variables. If the question proposed relate to Geo
metry, these^ equations are those of a curve or of a surface, to which
belongs the required property.
As the integration is effected and should be taken between given limits,
the terms which remain and compose the equation (C) belong to these
limits: il is become of the form K + L = 0, L being a function of
X, y, z, 3 X, 3 y, 3 z . . . Mark with one and two accents the numerical
values of these variables at the first and second limit. Then, since the
integral is to be taken between these limits, we must mark the different
terms of L which compose the equation C, first with one, and then with
two accents ; take the first result from the second and equate the differ
ence to zero ; so that the equation
L// — L, =
contains no variables, because x, d x . . . will have taken the values
x^ 3 x^ . . . x^^ 8 x^^ . . . assigned by the limits of the integration. We
must remember that these accents merely belong to the limits of the
integral.
There are to be considered four separate cases.
1. Jf the limits a7e given andjixed^ that is to say, if the extreme values
of X, y, z are constant, since 3 x^, d 3 x^ . . . d x^^, d 3 x^^, &c. are zero, all
the terms of L^ and L^, are zero, and the equation (C) is satisfied. Thus
we determine the constants which integration introduces into the equations
(D), by the conditions conferred by the limits.
2. If the limits are arbitrary and independent^ then each of the coeflfi
cients 3 x, , 3 x^^ . . . in the equation (C) is zero in particular.
3. If there exist equations of condition^ (which signifies geometrically
that thecurve required is terminated at points which are not fixed, but
which are situated upon two given curves or surfaces,) for the limits, that
is to say, if the nature of the question connects together by equations,
some of the quantities x^, y^, z,, x,^, y^^, z,, we use the differentials of these
equations to obtain more variations 3 x^, 3 y^ 3 z^, d x,^, &c. in functions
of the others ; substituting in L,, — L, =0, these variations will be re
duced to the least number possible : the last being absolutely independent,
the equation will split again into many others by equating separately their
coeflRcients to zero.
ANALYTICAL GEOMETRY. Ixxi
Instead of this process, we may adopt the following one, which is more
elegant. Let
u = 0, V = 0, &c.
be the given equations of condition; we shall multiply their variations
3 u, 3 V ... by the indeterminates X, >/. . . This will give X3u4.X'3v + .,,
a known function of 3 x^ d x^^, d y^ . . . Adding this sum to h„ — h,, we
shall get
L,, — L, + X 3 u + X' d V + . . . = . . . . (E).
Consider all the variations 3 x,, 3 x,,, ... as independent, and equate
their coefficients separately to zero. Then we shall eliminate the inde
terminates X, X'. . . from these equations. By this process, we shall arrive
at the same result as by the former one ; for we have only made legiti
mate operations, and we shall obtain the same number of final equations.
It must be observed, that we are not to conclude from u = 0, v = 0,
that at the limits we have du=0, dv = 0; these conditions are inde
pendent, and may easily not coexist. In the contrary case, we must
consider d u = 0, d v = 0, as new conditions, and besides X 3 u, we
must also take X' 3 d u . . .
4. Nothinjr need be said as to the case where one of the limits is fixed
and the other subject to certain conditions, or even altogether arbitrary,
because it is included in the three preceding ones.
It may happen also that the nature of the question subjects the varia
tions 3 X, 3 y, 3 z, to certain conditions, given by the equations
s = 0, ^ = 0,
and independently of limits; thus, for example, when the required curve
is to be traced upon a given curve surface. Then the equation (B) will
not split into three equations, and the equations ( D) will not subsist. We
must first reduce, as follows, the variations to the smallest number possi
ble in the formula (B), by means of the equations of condition, and equate
to zero the coefficients of the variations that remain ; or, which is tanta
mount, add to (B) the terms XB e + W 8 6 + . . .; then split this equation
into others by considering 3 x, 3 y, 3 z as independent ; and finally elimi
nate X, X' . . . I
It must be observed, that, in particular cases, it is often preferable to
make, upon the given function Z, all the operations which have produced
the equations (B), (C) instead of comparing each particular case with the
general formulae above given.
Such are the general principles of the calculus of variations : let us
illustrate it with examples.
«4
Ixxii
INTRODUCTION.
Ex. 1. What is the curve C M K o/' ivhich the length M K, comprised
hetAeen the given radiivectors A M, A K is the least possible^
We have, (vol. I, p. 000)> >f ^ be the radiusvector,
s =/(i;«d<J* + d^) = Z
it is required to find the relation r = 9 <l, which^renders Z a minimuoi
the variation is
A 7 — rd^'^ + r'd^.ad^ + dr.od r
^  V (r « d ^ « + d r *) *
Comparing with equation (A); where we suppose x = r, y = ^, we
have
ds
the equations (D) are
■' rd^« ' dr ,, . ., r«d«
m = — i — , n = 1—, M = 0, N =
ds
ds
r d 6
* , /d r\ r * d ^
Eliminating d ^, and then d s, from these equations, and ds*=: r*d^;
4. d r S we perceive that they subsist mutually or agree ; so that it is
sufficient to integrate one of them. But the perpendicular A I let fall
from the origin A upon any tangent whatever. T M is
A J = A M 4 sin. A M T = r sin, /3,
which is equivalent, as we easily find, to
r tan. /3
ivhich gives
V (1 I tan. * /3)
r'd tf
(\d
ds
= c;
V (r « d tf « + d r *)
and since this perpendicular is here constant, the required line is a
straight line. The limits M and K being indeterminate, the equations
(C) are unnecessary.
Ex. 2. Tojind the shortest line hetvceen two given points^ or two given
curves..
ANALYTICAL GEOMETRY. Ixxiil
The length s of the line is
/Z =/V(dx* + dy« + dz^).
It is required to make this quantity a minimum ; we have
as d s ^ d s
md comparing with the formula (A), we find
m = 0,M = 0,A^ = 0,n =4^,N=^,v=4^
d s d s d s
the other coefficients P, p, cr . . , are zero. The equations (D) become,
therefore, in this case,
whence, by integrating .
dx = ads,dy =:bds,dz = cds.
Squaring and adding, we get
a«+ b* + c« = 1,
a condition that the constants a, b, c must fulfil in order that these equa
tions may simultaneously subsist. By division, we find
dy__b dz__£
dx~"a'dx'~a*
whence
b X = a y + a', c x = a z + b';
the projections of the line required are therefore straight lines — the line is
therefore itself a straight line.
To find the position of it, we must know the five constants a, b, c,
a', b'. If it be required to find the shortest distance between two given
fixed points (x , y,, zj, (x^ , y^^, z^J, it is evident that 3, x, d x,,, 5 y^ . . . are
zero, and that the equation (C) then holds good. Subjecting our two
equations to the condition of being satisfied when we substitute therein
x^, x^ , y, . . . for X, y, z, we shall obtain four equations, which, with
a' + b^ + c'z::!, determine the five necessary constants.
Suppose that the second limit is a fixed point (x^^, y,,, z^J, in the plane
(x, y), and the first a curve passing through the point (x^, y^ z,), and also
situated in this plane ; the ec(uation
b X = a y + a'
then suffices. Let y^ = f x^ be the equation of the curve ; hence
dy, =: Adx,;
the equation (C) becomes ,
^ = (ds) >>^ + (af) 'r.
Ixxiv INTRODUCTION.
and since
equations
and since the second limit is fixed it is sufficient to combine together the
3y, = A8x,
dxjx, + dy,3y, = 0.
Eliminating d y, we get
dx, + Ady, = 0.
We might also have multiplied the equation of condition
dy, — A 8 X, =
by the indeterminate X, and have added the result to L^, which would
have given
(df).*"' + ft) '^' + ''^'''^''' = "'
whence
ft)^A = o. (i4)+x = o.
Eliminating X we get
dx, + Ady, = 0.
But then the point (x,, yj is upon the straight line passing through the
points (x„ y„ z,), (x,,, y,,, z„), and we have also
b d x^ = a d y„
whence
and
a = — b A
iy 1 _ b_;
dx A a
which shows the straight line is a normal to the curve of condition. The
constant a' is determined by the consideration of the second limit which is
given and fixed.
It would be easy to apply the preceding reasoning to three dimensions,
and we should arrive at similar conclusions ; we may, therefore, infer
generally that the shortest distance between two curves is the straight
line which is a normal to them.
If the shortest line required were to be traced upon a curve surface
whose equation is u = 0, then the equation (B) would not decompose into
three others. We must add to it the term X 3 u ; then regarding 5 x, 3 y,
i z as independent^ we shall find the relations
d.fe) + 4:'") = o.
^d s ' d X
ANALYTICAL GEOMETRY. Ixxt
From these eliminating X, we have the two equations
which are those of the curve required.
Take for example, the least distance measured upon the surface of a
sphere, whose center is at the origin of coordinates : hence
u = X* 4 y* + z* — r* = 0,
Our equations give, making d s constant,
z^' X z= X d' z^ zd*y = yd*z,
whence
yd*x=:xd*y.
Integrating we have
zdx — xdz = ads, zdy — ydz = bds, ydx — xdy = cds.
Multiplying the first of these equations by — y, the second by x, the
third by z, and adding them, we get
aynbx + cz
the equation of a plane passing through the origin of coordinates. Hencf
the curve required is a great circle which passes through the points A'
C, or which is normal to the two curves A' B and C D which are lirai ts
and are given upon the. spherical surface.
When a body moves in a fluid it encounters a resistance which ceteris
Ixxvi INTRODUCTION.
paribus depends on its form (see vol. I.) : if the body be one of revolu
tion and moves in the direction of its axis, we can show by mechanics
that the resistance is the least possible when the equation of the gener
ating curve fulfils the condition
y d d y ' __
d
or
y ■ / . " ''^ — i =5 minimum.
d X* 4 dy
2 — • iitumiifUiiif
z = P
1 + y"
Let us determine the generating curve of the solid of least resistance
(see Principia, vol. II.).
Taking the variation of the above expression, we get
 — 2ydy'dx — 2yy'^ ^ .
'^=^>" = (dx'+dy')« = (T+f^'>P = Q>^^
M  ^y'  /^^^ N  yy''(^ + y^') &c •
"•^""dx^ + dy*" 1 +y"" ^^ " (1 + y'^)' '
the second equation (D) is
M — dN = 0;
and it follows from what we have done relatively to Z, that
^(M^«) = ^ai + ^^y' = y'^N+Ndy',
because
M *= d N.
Thus integrating, we have
^ + 1 + y'«  ^^ y  (1 + y'*)* •
Therefore
a(l + y")' = 2yy".
Observe that the first of the equations (D) or m — d n = 0, would
have given the same result — n = a ; so that these two equations conduct
to the same result. We have
y ~ 2y"
•^T" y'^^ y'* '
substituting for y its value, this integral may easily be obtained ; it remains
to eliminate y' from these values of x and y, and we shall obtain the
equation of the required curve, containing two constants which we shall
determine from the given conditions.
ANALYTICAL GEOMETRY. Ixxvii
Ex. 3. fVhat is the curve A B M in tsofiich the area B O D M comprised
befaoeen the arc B M the radii of curvature B O, D M and the arc O D
of the evolute, is a minimum ?
The element of the arc A M is
dsrrdxVl +y ;
the radius of curvature M D is
y— '
and their product is the element of the proposed area, or
;^ _ (l+y'')dx _ (dx +dy^) ^
■* "" y" d X d y *
It is required to find the equation y =r f x, which makes yZ, a mini
mum.
Take the variation 8 N, and consider only the second of the equations
(D), vi^hich is sufficient for our object, and we get
M = 0, N — dP=4a,
d X d^y
(i.+ y'*)*
, , d X * + d y 2 . , 1 + y'
N = — , ■„ ^ . 4 d y = X,/ 4. y',
P = —
/'»dx •
But
ii±yv\^
d (ii±XL) = Nd/+Pdy"dx
= 4ady+dPdy' + Pd/'dx,
putting 4 a + P for N. Moreover y'' d x = d y', changes the last
terms into
^ (y" d P + P d y") d X =" d (P /').. d X = — d :(il±^').
Ixxviii INTRODUCTION.
Integrating, therefore,
y "2(a/ + b)"clx' (l + j/«)« '
finally,
^ = c+ ^^~f 4btan.'/;
On the other side we have
y =// d X = y' X — /x d /
or
y = y'x — cy'— /y^^^dy'—Zbdy tan.'y';
this last term integrates by parts^ and we have
y = y' X — c y' — (by — a) tan.'y + f.
Eliminating the tangent from these values of x and y, we get
by = a(xc) + ^\y'~y^r + bf,
V(by — ax+g}_ ^^ ,ds ^ ^by — ax + g)'
finally,
s = 2 V (b y — a X + g) + h.
This equation shows that the curve required is a cycloid, whose four
constants will be determined from this same number of conditions.
Ex. 4. What is the curve of a given length s, between two jixed point Sj
for which fy d s z^ a maximum ?
We easily find
(y + ^) (ji) = ^' ^^^^"^^ ^ ^ = V ny+'^^rc'j '
and it will be found that the curve required is a catenaiy.
.yds.
Sincey^^ is the vertical ordinate of the center of gravity of an arc
whose length is s, we see that the center of gravity of any arc whatever of
the catenary is lower than that of any other curve terminated by the
same points.
Ex. 5. Reasoning in the same way for y y * d x = minimum, and
y y d X = const, we find y * + X y = c, or rather y = c. We have
here a straight;.line parallel to x. Since "^^ — = — is the vertical ordinate
of the center of gravity of every plane area, that of a rectangle, whose
side is horizontal, is the lowest possible ; so that every mass of water
ANALYTICAL GEOMETRY. Ixxix
whose upper surface is horizontal, has its center of gravity the lowest
possible, ■ ,
FINITE DIFFERENCES.
If we have given a series a, b, c, d, . . . take each term of it from that
which immediately follows it, and we shall form the^r^if differences^ viz.
a' = b — a, b' = c — b, c' = d — c, &c.
In the same manner we find that this series a', b', c', d' . . . gives the
second differences
a." = b' — a', b^' = c' — b', c" = d' — c', &c.
which again give the third differences
a!" = h" — a", b'" = c" — b", c'" = d" — c", &c.
These differences are indicated by A, and an exponent being given to
it will denote the order of differences. Thus A "^ is a ferm of the series
of nth differences. Moreover we give to each difference the sign which
belongs to it ; this is — , when we take it from a decreasing series.
For example, the function
y = x' — 9x + 6
in making x successively equal to 0, 1, 2, 3, 4 . . . gives a series of
numbers of which y is the general term, and from which we get the
following differences,
for x = 0, 1, 2, 3, 4, 5, 6, 7 . . .
series y = 6,2,4, 6, 34, 86, 168, 286...
first diff. A y = — . 8, — 2, 10, 28, 52, 82, 118 .. .
second diff. A « y = 6, 12, 18, 24, 30, 36 . . .
third diff. A ' y = 6, 6, 6, 6, 6, . . .
We perceive that the third differences are here constant, and that the
second difference is an arithmetic progression : we shall always arrive at
constant differences, whenever y is a rational and integer function of x ;
which we now demonstrate.
In the monomial k x "^ make x = a, jS, y, . . . ^, z, X (these numbers
having h for a constant difference), and we get the series
.k a », k /3 "",... k 5 •", k X «, k X «».
Since x = X — h, by developing k x "» =: k (X — h) ", and designating
Dy m. A', A'' . . . the coefficients of the binomial, we find, that
k (X"* — X «") = k m h X >" 1 — k A' h * X ""^ + k A" ' h. . .
Ixxx INTRODUCTION.
Such is the first difference of any two terms whatever of the series
k a », k ^ "» . . . k X % &c.
The difference which precedes it, or k (x " — 6^) is deduced by
changing X into x and x into 6' and since x = X — h, we must put
X — h for X in the second member:
k m h(Xh) »»kA' h « (Xh '^) ...=k m h X'"ijA'+m(ml)}kh'X'»8 ^,,
Subtracting these differences, the two first terms will disappear, and
we get for the second difference of an arbitrary rank
km (m— 1) h«X»8 + k B'h'X^'s + . . . ^
In like manner, changing X into X — h, in this last developement, and
subtracting, the two first terms disappear, and we have for the third
difierence
km(m— 1) (m — S) h3xn>3 + kB"h*X»*. ..,
and so on continually.
Each of these differences has one term at least, in its developement,
like the one «bove ; the first has m terms ; the second has m — 1 terms ;
third, m — 2 terms ; and so on. From the form of the first term, which
ends by remaining alone in the mth difference, we see this is reduced to
the constant
1.2.3...mkh'n.
If in the functions M and N we take for x two numbers which give the
results m, n ; then M + N becomes m + n. In the same manner, let
m', n' be the results given by two other values of x ; the first difference,
arising from M + N, is evidently
(m — m') + (n — n').
that is, the difference of the sum is the sum of the differences. The same
may be shown of the 3d and 4th . . . differences.
Therefore, if we make
X = a, /3, y . . .
in
k X "> + p X "i + . . .
the mth difference will be the same as if these were only the first term
k X "», for that of p x '»*, q x ""^ ... is nothing. Therefore the mth
difference is constant, "when for x 'me substitute numbers in arithmetic pro
gression, in a rational and iritegcr Junction ofn.
We perceive, therefore, that if it be required to substitute numbers in
arithmetic progression, as is the case in the resolution of numerical equa
tions, according to Newton's Method of Divisors, it will suffice to find
the (m + 1) first results, to form the first, second, &c. differences. The
X = 0.
1
2.
3
Series 1 .
1
.1.
13
1st. . . .
2
2.
12
2nd ..
3,
10
3d ...
6
6. 6
. 6. 6
6. 6 .. .
10 . 16 .
22 . 2^ .
34 . 40 . . .
2. 12
28 . 50 .
78 . 112. . .
— 1.1.
13 . 41 .
91 . 169 . . .
1. 2
. 3. 4
.5. 6...
ANALYTICAL GEOMETRY. Ixxxi
mlh difference will have but one term ; as we know it is constant and
= 1 . 2 . 3 . . . m k h "^, we can extend the series at pleasure. That of
the (m — l)th differences will then be extended to that of two known
terms, since it is an arithmetic procession ; that of the (m — 2)th differ
ences will, in its turn, be extended j and so on of the rest.
This is perceptible in the preceding example, and also in this ; viz.
3d Diff. 6
2nd . . 4
1st . —2
Results 1
For X
These series are deduced from that which is constant
6.6.6.6...
and from the initial term already found for each of them : any term is
derived by adding the ttSjO terms on the left tichich immediately ^precede it.
They may also be continued in the contrary direction, in order to obtain
the results of x = — 1, — 2, — 3, &c.
In resolving an equation it is not necessary to make the series of results
extend farther than the term where we ought only to meet with numbers
of the same sign, which is the case when all the terms of any column are
positive on the right, and alternate in the opposite direction; for the
additions and subtractions by which the series are extended as required,
preserve constantly the same signs in the results. We learn, therefore,
by this method, the limits of the roots of an equation, whether they be
positive or negative.
Let y^ denote the function of x which is the general term, viz. the
X + 1th, of a proposed series
yo + yz + yi + . . . yx + yx+i+ . • .
which is formed by making
X = 0, 1, 2, 3 . . .
For example, yg will designate that x has been made = 5, or, with re
gard to the place of the terms, that there are 5 before it (in the last ex
ample this is 91). Then
ji — yo = ^ yo J y2 — yi = ^ yi » ya — ys = y2 • • •
A yl _ Ayo = A^yo, Ay2 — AV, = ASyj , A yg _ A yg = A^y^ . . .
A2y, — A^yo = A^yo , A^yg — A^y, = A^yi , A^yg — A^y^ = A^yg . . .
&c. /
kxxii INTRODUCTION,
and generally we have
yx— yxi = Ay ,_i
Ay x — Ay,_i = A«y»_,
A*yx — A«y, _ I = A^y, _ I
&c.
Now let us form the differences of any series a, b, c, d . . . in this
manner. Make
b = c + a'
c = b + b'
d = c + c'
&c.
b' = a' + a"
c' = b' + b"
d' = c + c"
&c.
b'' = a" + a"'
c'' = b" + b'"
d" = c" + d"
&c.
and so on continually. Then eliminating b, b', c, c', &c. from the first
set of equations, we get
b = a + a'
c = a + 2 a' 4 a"
d = a + 3 a' + 3 a" + a"'
e = a + 4 a' + 6 a'' + 4 a!" + a!'"
f = a + 5 a' + 10 a" + &c.
&c.
Also we have
a' = b — a
a" = c — 2 b + a
a!" =d — 3c + 3b — a
&c.
But the letters a', a'', a'", &c. are nothing else than A y^, aVq, A^yo . • •
a, b, c . . . being yc, yi, y? • • • j consequently
y, = yo + A yo
72 = yo + 2Ayo+ A?y
yg = yo + 3 A yo + 3 A'^yo + A^y^
&c.
.ANALYTICAL GEOMETRY. Ixxxiu
And
^ yo = yi — yo
^^yo = ya — • 2 yi + yo
^^yo = ys — 3 ya + 3 yi — Vo
■^*yo = y* — 4. ya +6 y.2 + 4 yi 4 yc
&c.
Hence, generally, we have
y, = yo + xAyo+ x ^^ . A^yo + ^ "^^ ' ^^^ '^'^ +.(A)
n — 1 n — In — 2 . ,„.
Anyo = yn — n y + n .— ^— . y — n . —  — . —j— y + . . . (B^
n— 1 ^ n— 2 <* «' n — 3
These equations, which are of great importance, give the general term
of any series, from knowing its first term and the first term of all the
orders of differences ; and also the first term of the series of nth differ
ences, from knowing all the terms of the series yo, yi, y2 • • •
To apply the former to the example in p. (81), we have
yo= 1
Ayo = — 2
A'y„ = 4 .
AVo = 6
A*y, =
whence
y, = 1 — 2x+2x(x— l) + x(x— l)(x — 2) = x' — x^ — 2x+l
The equations (A), (B) will Jje better remembered by observing that
y, = (I + Ay„)%
A"yo = (yi)% .
provided that in the developements of these powers, we mean by the
exponents of A y^, the orders of differences, and by those of y the place
in the series.
It has been shown that a, b, c, d . . . may be the values of yx, when
tliose of X are the progression al numbers
ra, m + h, m + 2 h . . . m + i h
that is
a = y^ , b = ym+ h j'C = &c.
In the equation (A), we may, therefore, put ym+s h for y„ y^ fory,,. A y^
for A yo, &c. and, finally, the coefficients of the i'** power. Make i h = z,
and write A, A » ... for A y^, A»y„ . . . and we shall get
zA. . z.(zh)A^ , z(z h)(z2h)A^ , ^.
y»+. = ym + J + 21? — + 2Y^ + ••• t»i
/2
Ixxxiv INTRODUCTION.
This equation will give y^ when x = m + z, z being either integer or
fractional. We get from the proposed series the differences of all orders,
and the initial terms represented by A, A^, &c.
But in order to apply this formula, so that it may be limited, we must
arrive at constant differences ; or, at least, this must be the case if we
would have A, A* . . . decreasing in . value so as to form a converging
series : the developeraent then gives an approximate value of a term cor
responding to
X = m + z;
it being understood that the factors of A do not increase so as to destroy
this convergency, a circumstance which prevents z from surpassing a
certain limit.
For example, if the radius of a circle is 1000,
the arc of 60° has a chord 1000,0 . ^ ^, r.
65« 1074,6 ^^.'A'z: 2,0
70" 1147,2 *^»"
75° 1217,5 '2,3
Since the difference is nearly constant from 60"* to 73°, to this extent
of the arc we may employ the equation (C); making h = 5, we get for
the quantity to be added to y = 1090, this
m
}, 74,6. z — 3% z (z — 5) = 15,12. z — 0,04. z«
So that, by taking z = 1, 2, 3.. . then adding 1000, we shall obtain the
chords of 61°, 62°, 63° ; in the same manner, making z the necessary
Jraction, we shall get the chord of any arc whatever, that is intermediate
to those, and to the limits 60° and 75°. It will be better, however, when
it is necessary thus to employ great numbers for z, to change these limits.
Let us now take
log. 3100
=
y
= 4913617
I6g. 3110
log. 3120
log. 3130
m
= 4927604
= 4941346
= 4955443
A.
= 13987
13942
13897
A.« = — 45
—.45
We shall here consider the decimal part of the logarithm as being an
integer. By making h = 10, we get, for the part to be added to log.
3100, this
1400,95 X z — 0, 2 25 X z^
To get the logarithms of 3101, 3102, 3103, &c. we make
z = 1,2, 3....;
and in like manner, if we wish for the log. 3107, 58, we must make
ANALYTICAL GEOMETRY. Ixxxv
z — 7, 58, whence the quantity to be added to the logarithm of 3100 is
10606. Hence
log. 310768 = 5,4924223.
The preceding methods may be usefully employed to abridge the
labour of calculating tables of logarithms, tables of sines, chords, &c.
Another use which we shall now consider, is that of inserting the inter
mediate terms in a given series, of which two distant terms are given.
This is called
INTERPOLATION.
It is completely resolved by the equation (C).
When it happens that A^ = 0, or is very small, the series reduces to
z yA
whence we learn that the results have a difference which increases propor
tionally to z.
When A * is constant, which happens more frequently, by changing z
into z + 1 in (C), and subtracting, we have the general value of the first
difference of the new interpolated series ; viz.
First difference a' = ^ + liZzil+L^.s
A.*
Second difference a" == ^.
If we wish to insert u terms between those of a given series, we must
make
h = n + 1 ;
then making z = 0, we get the initial term *of the differences
A''
(n+1)*
A' = ^ — i n A'' ;
n + 1 ^
we calculate first A", then A' ; the initial term A' will serve to compose
the series of first differences of the interpolated series, (A" is the constant
difference of it) ; and then finally the other terms are obtained by simple
additions.
If we wish in the preceding example to find the log. of 3101,
/3
IxxKvi INTRODUCTION.
3102, 3103 ... we shall interpolate 9 numbers between those which are
given: whence
u = 9
A"= — 0,45
A' = 1400,725.
We first form the arithmetical progression whose first term is A', and
— 0,45 for the constant. The first differences are
1400,725; 1400,725; 1399,375; 1398,925, &c.
Successive additions, beginning with log. 3100, will give the consecutive
logarithms required.
Suppose we have observed a physical j:)henomenon every twelve holirs,
and that the results ascertained by such observations have been
For hours ... 78 __ oo
12 ... 300 ^  ^^*^ ^2 _ 144
24 ... 666 S6^
36 ... 1176 510 H4.
&c.
»
If we are desirous of knowing the state corresponding to 4'', S^ 12 '',
&c., we must interpolate two terms; whence
11 = z. A" = 16, a' = 58
composing the arithmetic progression whose first term is 58, and common
difference 16, we shall have the first differences of the new series, and
then what follow
First differences 68, 74, 90, 106, 122, 138 .. .
Series 78, 136, 210, 300, 406, 528, 646 , . .
A 0^ 4^, 8^ 16" 20h, 24".
The supposition of the second differences being constant, applies almos;
to all cases, because we may choose intervals of time which shall favour
such an hypothesis. This, method is of great use in astronomy; and
even when observation or calculation gives results whose second differ
ences are irregular, we impute the defect to errors which we coiTect by
establishing a greater degree of regularity.
Astronomical, and geodesical tables are formed on these principles.
We calculate directly different terms, which we take so near that their
first or second differences may be constant; then we interpolate to obtain
the intermediate numbers.
Thus, when a converging series gives the value of y by aid of that of a
variable x ; instead of calculating y for each known value of x, when the
formula is of frequent use, we determine the results y for the continually
ANALYTICAL GEOMETRY. Ixxxvii
increasing values of x, in such a manner that y shall always be nearly of
the same value : we then write in the form of a table every value by the
side of that of x, which we call the argumeni of this table. For the
numbers x which are intermediate to them, y is given by simple proposi
tions, and by inspection alone we then find the results lequired.
When the series has two variables, or arguments x and z, the values
of y are disposed in a table by a sort of double entry s taking for coordi
nates X and z, the result is thus obtained. For example, having made
z = 1, we range upon the first line all the values of y corresponding to
X = 1, V, 3...;
we then put upon the second line which z = z gives ; in a third line those
which z = 3 gives, and so on. To obtain the result which corresponds to
X = 3, z = 5
we slop at the case which, in the third column, occupies the fifth place.
The intermediate values are found analogously to what has been already
shown.
So far we have supposed x to increase continually by the same differ
ence. If this is not the case and we know the results
y = a, b, c, d . . .
which are due to any suppositions
X = a, /3, 7, . . .
we may either use the theory which makes a parabolic curve pass through
a series of given points, or we may adopt the following:
By means of the known corresponding values
a, a ; b jS ; &c.
we form the consecutive functions
b — a
A =
A.=
&c.
B =
B,=
B,=
&c.
c— b
7— ^
d — c
7
A
y — a
A,— A
A,— A
• — 7
/4
bcxxviii
INTRODUCTION.
C = ^1 — ^
d — a
&c.
D =
c, — c
and so on.
By elimination we easily get
b = a + A (3 — a)
c = a + A(7 — a) + B(7— a) (7~/3)
d = a + A(a a) + B(3— a) (a— ^) + C(3 — a)(a — 13)(3~7)
&c.
and generally
y^= a+A(x — t*) + B(x — a)(x^) + C (x — «) (x— /3) (x— 7)+&c.
We must seek therefore the first differences amongst the results
a, b, c . . .
and divide by the differences of
a, /3, 7 . . .
which will give
"•"A, Ai, A2, &c.
proceeding in the same manner with these numbers, we get
B, B„ B2, &c.
which in like manner give
C, Ci) QZi &c.
and, finally substituting, we get the general term required.
By actually multiplying, the expression assumes the form
a + a'x + a'x'^ + ...
of every rational and integer polynomial, which is the same as when w€
neglect the superior differences.
The chord of 60" = rad. = 1000
= 1033
35
= 1077
42
56
A=li
Ai= 14,82
A2= 14,61
— 0,18
— 0,21
B =—0,035
Bi=— 0,031
620.20'
65M0'
69°. 0' =1133
We have 
« = 0, /3 = 21, 7 = 5^, a = 9.
We may neglect the third differences and put
y, = 100 + 15,082 x — 0,035 x«.
Considering every function of x, y^, as being the general term of the
series which gives
X = m, m + b, m + 2 h, &c.
ANALYTICAL GEOMETRY. boa.ix
if we take the differences of these results, to obtain a new series, the
general iettti will be what is called the Jirst difference of the proposed
fiinction y^ which is represented by A y^. Thus we obtain this difference
by changing x into x + h in y^ and taking y^ from the result ; the re
mainder will give the series of first differences by making
X = m, m + h, m f 2 h, &c.
Thus if
yx =
Ay X =
a + X
(x+h)^
a + x+h a + x*
It will remain to reduce this expression, or to develope it according to
the increasing powers of h.
Taylor's theorem gives generally (vol. I.)
dy,,d2yh«„
'' d X d x^ L2
To obtain the second difference we must operate upon a y^ as upon U?e
proposed y^, and so on for the third, fourth, &c. differences.
INTEGRATION OF FINITE DIFFERENCES.
Integration here means the method of finding the quantity whose dif
ference is the proposed quantity ; that is to say the general term y^ of a
ym) ym + hj ym + 2 h> &C.
from knowing that of the series of a difference of any known order. Tliis
operation is indicated by the symbol 2.
For example
2 (3 x2 + X — 2)
ought to indicate that here
h = L
A function yx generates a series by making
X = 0, 1, 2, 3 . . .
the first differences which here ensue, form another series of which
3 x^ 4 X — 2
is the general term, and it is
— 2, 2, 12, 28 . . .
By integrating we here propose to find yx such, that putting x + 1 for
X, and subtracting, the remainder shall be
3 X « + X — 2.
xc INTRODUCTION.
It is easy to perceive tbat, first the symbols 2 and A destroy one another
ttsdojfandd; thus
2 A fx = f x<
Secondly, that
A (a y) = a A y
gives
2 a y = a 2 y.
Thirdly, that as
A(AlBu) = AAt — Bau
so is
S (A t — B u) = A 2 t ~ B 2 u,
t and u being the functions of ;c.
The problem of determining yx by its first difference does not contain
data sufficient completely to resolve it; for in order to recompose the
series derived from y^ in beginning with
— 2, 2, 12, 28, &c.
ive must make the first term
yo = a ■■■' " 
and by successive additions, we shall find
a, a — 2, a + 2, a + 12, &c. <
in which a remains arbitrary.
Every integral may be considered as comprised in the equation (A)
p. 83 ; for by taking
X = 0, 1, 2, 3 . . .
in the first difference given in terms of x, we shall form the series of first ,
diflerences ; subtracting these successively, we shall have the second dif
ferences ; then in like manner, we shall get the third and fourth difJer
ences. The initial term of these series will be
^yo> ^'yo • •
and these values substituted in yx will give y,. Thus, in the example
above, which is only that of page (81) when a = 1, we have
Ayo = —2, A«yo = 4, A^v^ = 6, A ♦ y^ = 0, &c. ;
which give
y, = yo — 2 X — X 2 + X '.
Generally, the first term yo of the equation (A) is an arbitrary constant,
which is to be added to the integral. If the given function is a second
difference, we must by a first integration reascend to the first difference
and thence by another step to y, ; thus we shall have two arbitrary con
stants ; and in fact, the equation (A) still gives y, by finding A", A 3, the
ANALYTICAL GEOMETRY. xci
only difference in the matter being that y^ and A y^ are arbitrary. And
so on for the superior orders.
Let us now find 2 x ™, the exponent m being integer and positive.
Represent this developement by
2x'" = px + qx'' + rx*= + &c.
a, b, c, &C. being decreasing exponents, which as w^ell as the coefficients
p, q, &c. must be determined. Take the first difference, by suppressing
2 in the first member, then changing x into x f h in the second member
and subtracting. Limiting ourselves to the two first terms, we get
x" = pahx«i + ^pa(a— l)h*x^2 4....qbhx''i + ...
But in order that the identity may be established the exponents ought
to give
a — 1 = m
a_2 = b— 1
whence
a = m + 1, b = m.
Moreover tlie coefficients give
l=pah, — ^pa(a — l)h = qb;
whence
P  (m + 1) h ' ^ = — ^•
As to the other terms, it is evident, that the exponents are all integer
and positive ; and we may easily perceive that they fail in the alternate
terms. Make therefore
2x™ = px™ + ^ — ix" +ax'"^ + /3x™3 f yx^^H ...
and determine a, i3, y ... &c.
Take, as before, the first difference by putting x + h for x, and sub
tracting : and first transferrinfj
X 2 ^ >
we find that the first member, by reason of
ph (m + 1) = 1,
reduces to
A' ^' xm2 , A" !^— ^ 3h^ , m— 5 5hg ^_^
^•2:3'' +^ 4 •2.5'' +^ • 6 '^Jf''
To abridge the operation, we omit here the alternate terms of the deve
lopement ; and we designate by
1, m. A', A'', &c.
the coefficients of the binomial.
Making the same calculations upon
ax'"i + /Sx"^ 4. &r.
xcii INTRODUCTION.
we shall have, with the same respective powers of x and of h, '
,. ./ ,xn — 2 m — 3 ,. ,xm — 2 rn — 4
(m— 1) a+ (m— 1). — g— . 3 « + (m — I). — ^— .
+ (m — 3)/3+(m.3).— ^
+
Comparing them term by term, we easily derive
_. "^
" ~ 3l'
• •
5
«+...
m
—
■^/3 4
*
3
P +...
(m
—
4)7+..
^ = r
A'
2.3.4.5 '
A""
7 =
&c.
6.6.7
whence finally we get
sx"" = ■; r—rrr tt + mahx^^ + A"bh^x°»3
(m + 1) h 2
+ A""ch«x»HA'' dh'x™'^+...(D)
This developement has for its coefficients those of the binomial, taken
from two to two, multiplied by certain numerical factors a, b, c . . ., which
are called the numbers of Bernoulli, because James Bernoulli first deter
mined them. These factors are of great and frequent use in the" theory
of series ; we shall give an easy method of finding them presently. These
are their values
a =
1
12
b =
1
120
c =
1
252
d =
1
240
e =
1
132
f —
691
32780
g 
1
12
h —
3617
8160
i =
43867
14364
&c.
ANALYTICAL GEOMETRY. xciii
which it will be worth the trouble fully to commit to memory.
From the above we conclude that to obtain 2 x°, m being any number,
mteger and positive, we must besides the two first terms
^m J 1 X '^
(m + 1) h 2~
also take the developement of
(x + h) ™
reject the odd terms, the first, third, fifth, &c. and multiply the retained
terms respectively by
a, b, c . . .
Now X and h have even exponents only *when m is odd and reciprocally ;
so that we must reject the last term h ™ when it falls in a useless situation ;
the number of terms is I m + 2 when m is even, and it is ^ (m + 3) when
ni is odd ; that is to say, it is the same for two consecutive values of m.
Required the integral of x ^°.
Besides
x^
11 h ^
we must develope (x + h) ""j retaining the second, fourth, sixth, &c. terms
and we shall have
lOx^ah + 120x''bh3 + 252x*ch5 + &c
Therefore
2xi° = 3^— ix»+ lxMi — x'h^ + x^h^ — Jx'h'+ ^xh»
lino DO
In the same manner we obtain
^ 1 X* X
'^ =2h~2
^ 3h"~'2"^"6~'
« , x* x^ . h X*
4 h a ^ 4
^^4_ x'' X* hx^ h'x
5 h 4. ' 3 30
5 __ X ^ x^ 5 h x*^ h ' X '
^ ^ Qh~~'2'^ ^~Y2 12~
, x'' x^ h. x^ h^x*.
7h 2^2 6^ 42'
x« xV 7 hx6 Th^x* . h
5 V ^ — 4 4
8 h 2 "^ 12 24 ^ 12
xciv
INTRODUCTION.
''  yh
X » 2 h X ' 7 li 3 X ^ 2 h ^ X '
~T+ 3 15 * 9
h'x
30
2 X >> = 
10"
x9 3 h x" 7 h^x^ h^x*
" 2 "^ 4 10 ' 2
3h'x«
20
2 x" = TTi &c. as before,
llh
&c.
We shall now give an easy method of dcteimining die Number of
Bernoulli a, b, c. . . In the equation (D) make
x= h = 1;
2 X " is the general term of the series whose first difference is x ". We
shall here consider 2. x° = 1, and the corresponding series which is that
of the natural numbers
0, 1, 2, 3 . . .
Take zero for the first member and transpose
_i 1
m + 1 ^
which equals
2 (m + 1)
""l— m •
Then we get
oT^L^n = a m + b A" + c A '' + <1 A '1 + . . . + k m.
d (m + I )
By making m = 2, the second member is reduced to am, which gives
1
^ "" 12*
Making m = 4, we get
3
Whence
j^ = 4 a + b A''
m — 1 m — 2 ,
= 4a + m.— ^.^jb
= 4 a + 4 b
= 1 + 4 b.
, 1
^ = 120
Again, making m = 6, we get
■^ = 6a + b A'' + c A»*
= 6a+ 20 b +6c
. = i — ^ + 6 c
ANALYTICAL GEOMETRY. xcv
which gives
1
^ ~ 252'
nnd proceeding thus by making
m = 2, 4, 6, 8, &c.
we obtain at each step a new equation which has one term more than the
preceding one, which last terms, viz.
2 a, 4 b, 6 c, . . . m k
will hence successively be found, and consequently,
a, b, c . . . k.
Take the difference of the product
y, = (x — h) X (x + h) (x + 2 h) . . . (x+ i h),
by X + h for x and subtracting ; it gives
A y;, = X (X + h) (x + 2 h) . . . (x + i h) x (i + 2) h;
dividing by the last constant factor, integrating, and substituting for y,
its value, we get
2 X (x + h) (x + 2 h) . . . (x + i h)
"" (iVsJ'h X ^ (^ + ^^) (X + 2 h)...(x + i h)
This equation gives the integral of a product of factors in arithmetic
jprogre&do7i.
, Taking the difference of the second member, we verify the equation
1 —J
^ X (X + h) (x + 2 h) . . .(x + i h) ~ i h x (X + h) . . . [x f (i — 1} h
which gives the integral of any inverse product
Required the integral qfa^.
Let
y^ = a*.
Then
A y^ = a'^ (a'' — 1)
whence
y^ = 2 a'^ (a" — 1) = a*;
consequently
a''
2 a '^ = ■— r ; 4 constant.
a •> — 1
Required the integrals of sin. x, cos. x.
Since
cos. B — cos A = 2 sin. h {^ + B). sin. ^ (A — B)
A cos. X. = cos. (x + h) — cos. X
hs . h • ,
2) ^^" 2
= — 2 sin. (x + g) sin.
xcvi INTRODUCTION.
Integrating and changing x + — into z, we have
2 sm. z = — COS. + constant.
o • h
In the same way we find
sm.
2 COS. z = r 1 constant.
2 sin.
When we wish to integrate the powers of sines and cosines, we trans
form them into sines and cosines of multiple arcs, and we get terms of
the form
A sin. q X, A cos. q x.
Making
q X = X
the integration is perfonned as above.
Required the integral of a froduct^ viz.
Assume
2(uz) = u2z + t
u, z and t being all functions of x, t being the only unknown one. By
changing x into x + h in
u 2 z + t
u becomes u + A u, z becomes z + ^ z, &c. and we have
u2z+UZ + Au2(z + Az) + t+At;
substituting from this the second member
u 2 z + t,
we obtain the difference, or u z ; whence results the equation
= Au2(z + Az) + At
which gives
t = — 2^Au2(z + A z)].
Therefore
2 (U z) = U 2 Z — 2 JA u . 2 (z + A z)}
which is analogous to integrating by parts in differential funcKpns.
There are but few functions of which we can find the finite integral ;
when we cannot integrate them exactly, we must have recourse to series.
Taylor's theorem gives us
dy, , d'^y h« , .
y« = dK'' + dTa +«"
ANALYTICAL GEOMETRY.
= y'h + ^'h^ + &c.
by supposition. Hence
y, = h 2 y' + !^ 2 /' + &C.
Considering y' as a given function of x, \iz. z, we have
y = z
~
r = z'
f" = z"
&c.
and
yx =fy^ X =/zdx
whence
h*
/z d X =: h 2 z + — 2 z' + &c.
i6
which gives
2 z = h*'/z d X — I 2 z' — ^ h » 2 z" — 8cc.
This equation gives 2 z, when we know z', 2 z'', &c. Take the dif
ferentials of the two numbers. That of the first 2 z will give, when di
vided by d X, 2 z\ Hence we get 2 z", then 2 z'", &c. ; and even without
making the calculations, it is easy to see, that the result of the substitution
of these values, will be of the form
2 z = h'/z d X + A z + B h z' 4 C h 2 z" + &c.
It remains to determine the factors A, B, C, &c. But if
2 = X"»
we get
/z d X, z', z", &c.
and substituting, we obtain a series which should be identical with the
equation (D), and consequently defective of the powers m — 2, m — 4,
so that we shall have
/'zdx z ahz' bh^z'" . ch^z"'" . dhV"''' , 
'^ = h ¥+! + IT + 2:3:4r + 27776 + ^^
a, b> c, &c. being the numbers of BernouUu
For example, if
z = 1 X
y* 1 x.dx = xlx»^x
z' = x^
z" = &c
y
xcviii INTRODUCTION
consequently
2lx = C + xlx — X — ^lx + a X' + b x' + c x» + &c.
The series
a, b, c . . . k, 1,
having for first differences
a', b', c' . . . k'
we have
b = a + a'
c = b + b
d = c + c'
&c.
1 = k + k'
equations wliose sum is
1 = a + a' + b' + c' + . . . k'.
If the numbers a', b', c', &c. are known, we may consider them as being
the first differences of another series a, b, c, &c. since it is easy to com
pose the latter by means of the first, and the first term a. By definition
we know that any term whatever 1', taken in the given series a', b', c', &c.
is nothing else than A 1, for T = m — 1 ; integrating
1' = A 1
we have
2 1' = 1
or
2 1' = a' + b' + c' . . . + k',
supposing the initial a is comprised in the constant due to the integra
tion. Consequently
The integral of any term 'whatever of a series^ "ive obtain the sian of all
the terms that precede it, and have
2 yx = yo + yi + yg + • • • y x  1
In order to get the sum of a series, we must add yx to the integral ; or
which is the same, in it must change x into x + Ij before we integrate.
The arbitrary constant is determined by finding the value of the sum y^
when
X = 1.
We kncm therefore ho'w to find the summing term of every series whose
general term is knoivn in a rational and integer function qfn, '
Let
y, = Ax""  Bx«+ C
m and n being positive and integer, and we have
A 2x" — B 2 X* + C sx**
ANALYTICAL GEOMETRY. xcix
for the sum of the terms as far as y, exclusively. This integral being
onco found by equation D, we shall change x into x + 1, and determine
the constant agreeably.
For example, let
ya=x(2x— 1);
changing x into X + 1, and integrating the result, we shall find
4 x3+ 3 X* — X
22x' + 32x+2x'' =
2.3
X + 1 4x— 1
there being no constant, because when x = 0, the sum = 0.
The series
Im O m 9 m
, <& ) ** . • •
of the m^** powers of the natural numbers is found by taking 2 x ™ (equa
tion D); but we must add afterwards the x'** term which is x™; that is to
say, it is sufficient to change — ^ x " , the second terra of the equation
(D), into 1^ X™; it then remains to determine the constant from the term
we commence from.
For example, to find
S°= 1 + 2* + 3« + 4« + ...x*
we find 2 X*, changing the sign of the second term, and we have
x' x* X _ x+ 1 2x + 1
^'3+'2"*"6' """"S^ 2~'
the constant is 0, because the sum is when x = 0. But if we wish to
find the sum
S' = (n + 1)« + (n + 2)« + ...x'
S' = 0, whence x = n — 1, and the constant is
n— 1 2n— 1
"•— 2— 3"'
which of course must be added to the former ; thus giving
S'= (n + 1)*+ (n + 2)'+...x''
_ X + 1 2 X + 1 n — 1 2 n — 1
~ '^ • 3 • 2 " • 2~ • 3
= ^ X {x.(x+ 1). (2x + 1)— n.(n— l)(2n — 1)
= ^X {2 (x^^n^) + 3 (x« + nO + X — n].
This theory applies to the summation o^^gt^rate numbets, of the dif
ferent orders ;«t^
r2 
INTRODUCTION.
First order,
1 . 1 . 1 . 1 . 1 . 1 . 1 , &c.
Second order,
1.2.3.4.5. 6 . 7 , &c.
Third order.
1.3. 6 . 10. 15. 21 . 28, &c.
Fourth order,
1 .4. 10.20.35. 56 . 84, &c.
Fifth order.
1.5. 15.35.70. 126.210, &c.
and so on.
The law which (
every term follows being the sum of the one immediate
'y over it added to
the preceding one. The general terms are
First, 1
Second, x
Th;^.A ^
.(x + 1)
2
Fo„„,, X (X + 1) (X + 2)
Ami
ntb x.(x>> 1) (x+2)...x + p — 2
P 1.2.3...P— 1
To sum the Pyramidal numbers, we have
S = 1 + 4 + 10 + 20 + &c.
Now the general or x^** term in this is
y, = i . X (X + 1) (X + 2).
But we find for the (x — 1)* term of numbers of the next order
^(xl)x(x+ l)(x + 2); ■
finally changing x into x + 1, we have for the required form
S = ^x.(x + l)(x + 2)(x +3).
Since S = 1, when x = 1, we have
1 = 1 + constant, consequently
.'. constant zz 0.
Hence it appears that the sum of x terms of the fourth order, is the
x'*" term or general term of the fifth order, and vice versa ; and in like
manner, it may be shown that the x"^'' term of the (n + l)"^** order is the
sum of X terms of the n^'' order.
Inverse Jigurate numbers are fractions which have 1 for the numerator,
and a figurale series for the denominator. Hence the x^ term of the p"'
order is
1.2.3...(p— _0
x(x+ l)...x + p — 2
ANALYTICAL GEOMETRY. oi
and the integral of this is
^ ^ 1.2.3...(p_l)
(p~2)x(x +l)...(x + p — 3)*
Changing x into x + 1> then determining the constant by makinw
X = 0, which gives the sum = 0, we shall have
c = P
p — 2'
and the sum of the x first terms of this general series is
Pl 1.2.3...'(pl)
p — 2 (p — 2)(x+l)(x+2)...(x + p — 2)
In this formula make
p = 3, 4, 5 . . .
and we shall get
4.2. ij_i4. 1.2 _ 2 2
■*"3'*'6+10"' x(x+l) ~1 x+1
+ i. + i4.i+ 12.3 3 3
'A I lA • an '
4 ^ 10^20^ •*'x{x+ l)(x + 2) 2 (x+l)(x+2)
1.1 i , 1.2.3.4 4 2.4
5 ^ 10 ^35^'"x(x+l)(x+2) (x + 3) 3 (x4 J)...(x + 3)
1 1 J_ 1.2.3.4.5 5 2.3.5
"T 5" « o 1 "r K« T" • • •
6 ^ 21^ 66^ • •x(x+l)...(x+4) 4 (x+1) . . . (x + 4)
and so on. To obtain the whole sum of these series continued to infinity,
we must make
X = 00
which gives for the sum required the general value
P1
p2
which in the above particular cases, becomes '
2 3 4 5
1' 2' 3'4'^'^*
To sum the series '
sin. a + sin. (a + h) + sin. (a + 2 h) + . . . sin. (a + x — 1 h)
we have
cos. (a + h X —  j
2 sin. (a + X h) = C ■ r
2sin.^^
changing x into x + 1, and determining Cby the condition that x = — 1
makes the sum = zero, we find for the summingterm.
hx / . , . h
cos.
. (a — g) — cos. (a + h X + ^ )
2 sm.^
cii INTRODUCTION,
or
. / , h N . h (x + 1)
sin. (^a + g X j sin. ^ ^
"T~ir
.sin.^
In a similar manner, if we wish to sum the series
COS. a + COS. (a + h) 4 cos. (a + 2 h) + • • . cos. (a + x — 1 ii^
we easily find the summingteiia to be
sin. (a g) —sin. (a 4 h x + ^ )
'
2sm.
or
/ ^ h • . h (X + 1)
cos. (a + 2 ^; sin g
. h
^'2
A COMMENTARY
OM
N E W T O N'S PRINCIPIA,
SUPPLEMENT
TO
SECTION XI.
460 Prop. LVIT, depends upon Cor. 4 to the Laws of Motion,
which is
If any number of bodies mutually attract each other, their center of gra
vity will either remain at rest or will move uniformly in a straight line,
. First let us prove this for two bodies.
Let them be referred to a fixed point by the rectangular coordinates
^ X, y ; x^ y',
and let their masses be
/*, fJ''
Also let their distance be g, and f (g) denote the law according to which
they attract each other.
Then
will be their respective actions, and resolving these parallel to the axes of
abscissas and ordinates, we have (46)
d^x ,,,,x'— X
l'~  > <•)
d
Vou II.
8 A COMMKNTARY ON [Sect. XI.
Ji,, .,L„y (2)
dt* > ^
Hence multiplying equations (1) by /«. and those marked (2) by fi' and
adding, &c. we get
Atd'x + fi,'d^x' __
dF " "'
and
/*d^y +A^^d'y' _
dt" "
and integratuig
d X , , d x'
"•di +■"• — = '=
d y , / d y' ,
''•dT + '' dT = <^
Now if the coordinates of the center of gravity be denoted by
X, y,
we have by Statics
/i X + /a' x'
X =
and
d X __ 1 / djc , dx\ _ c
■ * dT ~ /A + ytt' * V^ • "dl "^ ^^ dT/ "(«, + /*'
d t ~ /4 + /a' V'^ d t ^ d t / /!* + /*''
But
d X d y
Tt' Tt
represent the velocity of the center of gravity resolved parallel to the axes
of coordinates, and these resolved parts have been shown to be constant
Hence it easily appears by composition of motion, that the actual velocity
of the center of gravity is uniform, and also that it moves in a straight
line, viz. in that produced which is the diagonal of the rectangidar par
allelogram whose two sides are d x, d y.
If
c = 0, c' =
then the center of gravity remains quiescent
Book l.J NEWTON'S PRINCIPIA. 3
461 The general proposition is similarly demonstrated, thus.
Let the bodies whose masses
^\ l^'\ l^"\ &c.
be referred to three rectangular axes, issuing from a fixed point by the
coordinates
„/// ,jff/ „ni
•^ J y 5 ■^
&c.
Also let
f 1, 2 be the distance of ^', ^'^
r I,' I,'"
gl,3 /tj/"
§2,3 •• Z*}/"
&C. &C.
and suppose the law of attraction to be denoted by
Now resolving the attractions or forces
l^" f (gl.2)
&c.
parallel to the axes, and collecting the parts we get
"dp = ''"^(^'.«) ^i^ + "" f (?^'^) V— + ^^ •
^^ i\,2 fl, 3
IT^ " /*'f (?1.2)^^=^ + /*'"f(f2.3)^^— ^^ + &C.
"^ gl,2 i2,3
" "^ ?1, 3 g2, 3
&C. = &C.
Hence multiplying the first of the above equations by f/, the second by
n", and so on, and adding, we get
/d^x^ + fi,''d^7i" + /j.'"d''yi'" + &c. _
dt'
Again, since it is a matter of perfect indifference whether we collect the
forces parallel to the other axes or this ; or since all the circumstances are
similar with regard to these independent axes, the results arising from
similar operations must be similar, and we therefore have also
fi' d^y ' + fi"d^ y" + fj/" d ^ y'" + &c. _
dT^ • *
fi/ d^z'\ (jJ' d ^ t!' + ul" d ° i!" H &c. _
dt* ~
A2
4 A COMMENTARY ON [Sect. XI.
Hence by integration
, dx' ^ „dx'' ,„dx"' .
'^•dT + ^dr + '* 'dT + ^^ = ^
'^ • dT ^'^ dt ^'^ ^t +^*' "^
, dz' „dz" , ,„dz''' ,  „
'^•rt ^'^ dT + ^ dr+^" = "
But X, y, z denoting the coordinates of the center of gravity, by statics
we have
 _ / X^ + fJ.'^ X." + jil" x'" + &c.
^  ti' + fi." + yl" + &c.
y  ^/ + // + 1^'" + &c.
 _ ^/ z^ + // z" + /^^^^ z^^^ + &c.
^  y;^' + ^" 4 /i"/ + &c.
and hence by taking the differentials, &c. we get
dx c
a t ~ / + it*'' + /*''' + &c.
dy r^
a t" ~ fj/ ■\ IJ'" + /*'" + &c.
di__ d^
d t ~ /*' + It*'' + u/" + &c.
that is, the velocity of the center of gravity resolved parallel to any three
rectangular axes is constant. Hence by composition of motion the actual
velocity of the center of gravity is constant and uniform, and it easQy ap
pears also that its path is a straight line, scil. the diagonal of the rectan
gular parallelopiped whose sides are d x, d y, d z.
462. We will now give another demonstration of Prop. LXI. or that
Of two bodies the motion of each about the center of gravity, is the same
as if that center 'was the center of force, and the law of force the same as
that of their mutual attractions.
Supposing the coordinates of the two bodies referred to the center of
gravity to be
we have
= x + x,l x'=x + X/,
Hence since
d X dy
dT ' dT
Book L] NEWTON'S PRINCIPIA.
are constant as it has been shown, and therefore
dt» "' dt== "
we have
d'x _d'x,
dt^  dt«
d^y _ d^y,
dt^ ~ dt''
and we therefore get (46)
dt^ ^ ^^^ e
dt* ^^^^ e
But by the property of the center of gravity
P = . 6
6 being the distance of [*>' from the center of gravity. We also have
^// ^/ _ ^//
Hence by substitution the equations become
dT^'*H;;r^;y'
Similarly we should find
and
Hence if the force represented by
were placed in the center of gravity, it would cause /*' to move about it as
a fixed point; and if
were there residing, it would cause fi to centripetate in like manner.
Moreover if
Hs)=r
A 3
6 A COMMENTARY ON [Sect. Xi
then these forces vary as
so that the law of force &c. &c.
ANOTHER PROOF OF PROP. LXII.
463. Let jtt, ti! denote the two bodies. Then since /i has no motion
round G (G being the center of gravity), it will descend in a straight
line to G. In like manner yl will fall to G in a straight line.
Also since the accelerating forces on /x, ^' are inversely as /(*, (i! or
directly as G a*, G /«,', the velocities will follow the same law and corre
sponding portions of G ^a, G yl will be described in the same times ; that
is, the whole will be described in the same time. Moreover after tliey
meet at G, the bodies will go on together with the same constant velocity
with which G moved before they met
Since here
tt will move towards G as if a force
ff^ + /*'
"'C^')
or
Hence by the usual methods it will be found that if a be the distance
at which ^ begins to fall, the time to G is
yj ^
and if a' be the original distance of /i*', the time is
(//. + y!) a' ^ nr
I '2 V2'
y.'i
But
a : a' :://.': /A
therefore these times are equal, which has just been otherwise shown.
Book I.] NEWTON'S PRINCIPIA. 7
ANOTHER PROOF OF PROP. LXIH.
464. We know from (461) that the center of gravity moves uniformly
in a straight line; and that (Prop. LVII,) /* and ^J will describe about G
similar figures, (Jj moving as though actuated by the force
and Q as if by
Hence the curves described will be similar ellipses, with the center of
force G in the focus. Also if we knew the original velocities of /«. and /*'
about G, the ellipse would easily be determined.
The velocities of /* and ,«.' at any time are composed of two velocities,
viz. the progressive one of the center of gravity and that of each round G.
Hence having given the whole original velocities required to find the separate
j)arts of them,
is a problem which we will now resolve.
Let
V, V
be the original velocities of /t, /*', and suppose their directions to make
with the straight line /«. yl the angles
a, a!.
Also let the velocity of the center of gravity be
V
and the direction of its motion to make with y, ii! the angle
' a.
Moreover let
V, v'
be the velocities of /t, fjJ around G and the common inclination of their
directions to be
^.
Now V resolved parallel to (i, fjf is
V cos. a.
But since it is composed of v and of v it will also be
v cos. a + V cos. 6
.'. V COS. a = v cos. a + V cos. 6.
In like manner we get
V sin. a = V sin. a + v sin. 6.
A4
COS. a •\ fj/y COS. a' = (jH + /*') V COS. a \
sin. a + /a' V'' sin. a' = (/a + /u,') v sin. a J
8 A COMMENTARY ON [Seci. Xl.
and also
V COS. a' = V COS. a — v' COS. 6
V sin. a' z= \ sin. a — v' sin. 6.
Hence multiplying by /i, yl, adding and putting
liy zz y! \'
we get
Ik V COS. a. if yiy COS. a' = (^a + /«') V COS. a
and
II, V sin.
Squaring these and adding them, we get
^2 V^ + /2 v* + 2/*/*' VV COS. (a — a') = (/!* + /*')'v«
which gives
v= ^f/^'V' + /'V+ 2/(^A*'VV^cos.(a — gQl
y> + Ij/
By division we also have
— f6 V sin. a + /i' V sin. a'
tan. a = — Y^ — j^Tj. . .
fjk V COS. a j fji/\' COS. a'
Again, from the first four equations by subtraction we also have
V cos. a — V cos. a' = (v + v') cos. ^ = v . — X— cos. 6
ft!
V sin. a — V sin. a' = (v + v') sin. ^ = v . — —, — sin. ^
and adding the squares of these
V2 + V — 2VV'cos. (a — aO=v^(^^^^y
whence
V = ^' ,. '/iV^+ V'^ — 2 W cos. (a— a')]
v' = ^^ V^V* + V'« — SVV'cos. (a~a')?
^ + /*
and by division
V sin. a — V sin. a'
tan. ^ = vv ^^7 ; .
V cos. a — V'cos. a'
"Whence are known the velocity and direction of projection of fi about
G and (by Sect. III. or Com.) the conic section can therefore be found ;
and combining the motion in this orbit with that of the center of gravity,
which is given above, we have also that of /et.
465. Hence since the orbit of (j> round /u,' is similar to the orbit of
u. round G, if A be the semiaxis of the ellipse which fi describes round
Book I.] NEWTON'S PRINCIPIA. 9
G, and a that of the ellipse which it describes relatively to /*' which is also
in motion ; we shall have
A '. a. '. '. (jf : (jj •\ fjf .
466. Hence also since an ellipse whose semiaxis is A, is described by
the force
«'3 ]
we shall have (309) the periodic time, viz.
^_ 2AgT _ 2':rA^{li + (J.')
2<r a^
V (At + iJ.') '
467. Hence we easily get Prop. LIX.
For if /u, were to revolve round /*' at rest, its semiaxis would be a, and
periodic time
qi/ <a ?r a
.'. T : T' :: V^' : V{fi + fi').
468. Prop. LX is also hence deducible. For if (a revolve round fjt'' at
rest, in an ellipse whose semiaxis is a', we have
T'/  ^^^
and equating this with T in order to give it the same time about (*>' at rest
as about At' in motion, we have
2 era' 2 2Ta^
.*. a : a' :: (At + z*')^ : fi'^.
ANOTHEB PROOF OF PROP. LXIV.
469. Required the motions of the bodies ivhose masses are
fi, fL', fi", fj,'", &C.
and which mutually/ attract each other with forces varying directly as the
distance.
Let the distance of any two of them as ii, ijf, be f ; then the force of /*'
on (<A is
10 A COMMENTARY ON [Sect. XI.
and the part resolved parallel to x is
li! g . ^^~ = /i' (x — x').
In like manner the force of /«." on /tt, resolved parallel to x, is
Ao" (X — X'O
and so on for the rest of the bodies and for their respective forces resolved
parallel to the other axes of coordinates.
Hence
i^, =/(xxO + /x"(xx'0 + &c.
^ = ^(x'x) + /'(x'— x'O + &c.
^' =^(x''x) + /^'(x'^xO + &c.
which give
&c. = &c.
i^ = (/* + A*' + /.'' + &C.) X— {/.X + /.' X' + &C.)
^ = (^ + /.' + /' + &c.)x'— (/.x + //x' + &c.)
^~ = (^ + /t' + fJ," + &c.)x"— {/IX + ^'x' + &c.)
&c. = &c.
Or since
(0. X 4 it*' x' + &c. = (((A + At' + &c.) x
making the coordinates of the center of gravity
X, y, z,
we have
'^■^ = (^ + ^' + &c.)(xx)
^' = (/* + /*' + &c.)(x'x)
^=(A*+^' + &C.)(x''^x).
&c. = &c.
In like manner, we easily get
^=(^ + ^'+&c.)(y — y)
^=(. + ^' + &c.)(y'y)
Book I.] NEWTON'S PRINCIPI A. II
^y^'=(/* + // + &c.)(/'y)
&c. = &c.
and also
(J t2 = i.^ ^ /* t «c.; (z — z;
^' = (^ + ^' + &c.) (z'~ z)
'j'^^^' = (a. + ^'+&c.)(z''z)
&c. = &c.
Again,
X — x,y — y, z — z
x' — x^ y' — y, z' — z
&c. &c. &c.
are the coordinates of /t*, /tt,', (i!\ &c. when measured from the center of
gravity, and
it has been shown already that
d^(x — x) d'x
dt^ ~dt«
d*(yy)_d«y
dt« ~dt^
d*(z — i) d'z
dt* "dt^
and so on for the other bodies. Hence then it appears, that the motions
of the bodies about the center of gravity, are the same as if there were but
one force, scil.
((«, 4 ,«,' f &c.) X distance
and as if this force were placed in the center of gravity.
Hence the bodies will all describe ellipses about the center of gravity,
as a center ; and their periodic times will all be the same. But their
magnitudes, excentricities, the positions of the planes of their orbits, and
of the major axes, may be of all varieties.
Moreover the motion of any one body relative to any other, will be
governed by the same laws as the motion of a body relative to a center
of force, which force varies directly as the distance ; for if we take the
e(juations
jp = (^ + ^' + &c.) (X — x)
^ = (^ + ^' + &c.)(x'x)
12 A COMMENTARY ON [Sect. XL
and subtract them we get
and similarly
and
^^i^_J^ = (/^ + /^' + &C.) (X X')
dTv v')
—hr^ = (^ + ^' + &c.) (y y')
^11^^ = (/^ + ^' + &c.) (z  z').
Hence by composition and the general expression for force (ttI) it
readily appears that the motion of /m about /w', is such as was asserted.
470. Thus far relates merely to the motions of two bodies ; and these
can be accurately determined. But the operations of Nature are on a
grander scale, and she presents us with Systems composed of Three, and
even more bodies, mutually attracting each other. In these cases the
equations of motion cannot be integrated by any methods hitherto dis
covered, and we must therefore have recourse to methods of approxi
mation.
In this portion of our labours we shall endeavour to lay before the
reader such an exposition of the Lunar, Planetary and Cometary Theories,
as may afford him a complete succedaneum to the discoveries of our
author.
471. Since relative motions are such only as can be observed, we refer
the motions of the Planets and Comets, to the center of the sun, and the
motions of the Satellites to the center of their planets. Thus to compare
theory with observations,
// is required to determine the relative motion of a system ofbodies^ ahotU
a body considered as the center of their motions.
Let M be this last body, /*, tJ', /*", &c. being the other bodies of which
is required the relative motion about M. Also let
I, n, y
be the rectangular coordinates of M ;
^+ X, n + y, 7 + z;
^ + x'n + y',7+z';
&c.
those of /t, ij/, &c. Then it is evident that
X5 yj z;
x',/,z'
&c.
Book 1.1 NEWTON'S PRINCIPIA. 13
will be the coordinates of /t*, /»', &c. referred to M.
Call Si S'y &c.
the distances of/*, /t*', &c. from M; then we have
S, g'j &c. being the diagonals of rectangular parallelopipeds, whose sides
are
X, y, z
x', y', z'
&c.
Now the actions of (i, j«.', /tt", &c. upon M are
Ji ifL /Jl, &c
s s s ^
and these resolved parallel to the axis of x, are
/Ct X At' x' (a" x'
Therefore to determine ^ we have
d^_^x ^' ^^'
dt^ ~ ^2 ^ ^'3 + p"3 + ^^'
da b
/U. X
tlie symbol 2 denoting the sum of such expressions.
In like manner to determine n, / we have
dt^ ^g' '
dt^" *" §''
The action of M upon /*, resolved parallel to the axis of x, and in the
contrary direction, is
__Mx
Also the actions of i"', (<*", &c. upon (i resolved parallel to the axis of x
are, in like manner,
fl' {k' — X) fl" JX" — X) fi,"' {x'" — x) „
3 J j J 3 i o^C*
f 0, 1 f 0,2 f 0,3
fd.ra generally denoting the distance between /J''" ••••" and t^'" •• **
But
f0.1 = V (X^ — X)'' + (/ — y)^ + (2^ — Z)'
go.2= V(x" — x)«+(y" — y)'^+(z"— z)'
&c. = &c.
U A COMMENTARY ON [Sect. XI.
f..9 = '^ {^' — xO « + (/' — y) ' + (z'' ^=^^«
and so on.
Hence if we assume
fO,l f0,2
fif (if' U,' fS.'"
+ ^ — + ":; —
fl,2 fl,3
+ ^^^ + &c.
' ?2, S
&C.
and taking the Partial Difference upon the supposition that x is the only
variable, we have
«. Vdxy g'o.l f'o,2
the parenthesis ( ) denoting the Partial Difference. Hence tlie sum of
all the actions of (i', /jJ', &c. on /* is
IJ. Vd x/
Hence then the whole action upon /a parallel to x is
d.'(^ + x) _ 1 /dx> _ Mx
. d t« ~ ^ * Vdx/ f^ '
But
Hi  t^
dt^^^'
d2x__ 1 /d Xx M X ^ /^x .V
•'•dT^ ~ II UxJ g^ ?' ^^
Similarly, we have
^^yl ri^A_My_,/^y
dt«  ^ Vdy; ^^^ " f^^ ^"^^
^1^ _ ^ /d Xx M z ^ ^ z
dt*'  "7 Vdzy ^^ "'f' ^^
If we change successively in the equations (1), (2), (3) the quantities
(<*, X, y, z into
(jf, x', y', z';
II" Ti" m" f" •
^ > X , y , ^ ,
&c.
and reciprocally ; we shall have all the equations of motion of the bodies
/tt', fi"^ &c. round M.
Book L] NEWTON'S PRINCIPIA. 15
If we multiply the equations involving ^ by M + 2. /^ ; that m x, bv
tt : that in x', by /«.', and so on ; and add them together, we shall have
But since
(fil) = ^^^%^ + ^^•
and so on in pairs, it will easily appear that
(n) + (^') + ^^ = »•
d^ ^ d'^ X
whence by integrating we get
sdl d X
1 y _ c , _2^. ^dx
••^^  M + 2.^ M + 2./.*
and again integrating
1 2. /«. X
a arid b being arbitrary constants.
Similarly, it is found that
/ = ^" + ''"'MV^%
These three equations, therefore, give the absolute motion of M in
space, when the relative motions around itof ^, ^', (/.", &c. are known.
Again, if we multiply the equations in x and y by
s./^y
^y + ^M + 2.^'
and
2. ((A X
in like manner the equations in x' and y' by
^ y t '*• M ^. 2. ag'
16 A COMMENTARY ON [Sect. XL
and
and so on.
And if we add all these results together, obseiving that from the nature
of X, (which is easily shown)
and that (as we already know)
we have
xd*y — yd*x S./tx d*y
^'^' dT^ M + 2^ ^' "' dT^"
S.Aty * d^x
M + 2 .« d t "
and integrating, since
/(xd^y — yd*x) =/xd«y— /yd«x
= X d y — /d X d y — (y d x — /d x d y)
= xdy — ydx,
we have
xdy — ydx ^. 2./ctx „ dy
2 . /* . ji ^^ = const. + ^r^p— .l.fl.^
dt ^M + 2..tt dt
2 . /«. y d X
— —= 2 . /i . ■ — —
M + 2./tt* '^ d t
Hence
KK ^ xdy — ydx xdy — ydx. dx
C=M.2.(*. dt  +2./*X2^. ^—^ + 2.^yX2./*^
dy
d t
= M.2./..^^y=^^+2./*/. I (x^x)(dyd y)yy)(dx^dx)  ^ ^ ^^^
c being an arbitrary constant.
In the same mannel^H'e arrive at these two integrals,
c"= M. .. ^. Xd^^ ,_ ^ ^,(/y)(dz'.iz)(z'^)(dy'.ly) J. _ ^^^
c' and c" being two other arbitrary constants.
BoaK.I.]
NEWTON'S PRINCIPIA.
17
Again, if we multiply the equation in x by
o J « 2. At d X
M + 2./*'
the equation in y by
o J o 2 . (ti d y
the equation in z by
n J c\ 2 . /x d z
M + 2. At
if in like manner we multiply the equations in x', /, z' by
o / J / o / 2 . (W d X
2 A* d x' — 2a. ttt—
M + 2. At
o / 1 / o / 2. At d y
2 At d y — 2 At . Kir . ^ 
2 At' d z' — 2 A^' .
2. Ai d z
M + 2. At'
respectively, and so on for the rest ; and add the several results, observ
ing that
(a;)=o
we get
2 dxd^x + dyd^y + dzd'z _ 22.Atdx ^ Atd'^jx
'^ dt«  M + 2At"" dt^
, 22. /^dy A^d'y 2 2 . a^ d z Vd'^z
■^M + 2a6 dt^ ■^M + 2At* dt^
2 M. 2.
Mdg
+ 2 d X;
and integrating, we have
2 . fi
dx^ + dyHdz _ (2.A^dx)^ + (2.A^dy )^ + (2./^dz)«
+ 2 M 2— 4 2 X,
S
which gives
h = M. .^ dx^+dy^+dz' ^ ,^^^,^ I (dx'dx)H(d/dy)^+(dz'dz) '
dt'
— 2 M 2. ^ + 2 x (M + 2 At)
h being an arbitrary constant.
Vol. II. B
{^)
18 A COMMENTARY ON [Sect. XL
These integrals being the only ones attainable by the present state of
analysis, we are obliged to have recourse to Methods of Approximatioi,
and for this object to take advantage of the facilities afforded us by the
constitution of the system of the World. One of the principal of these
is due to the fact, that the Solar System is composed of Partial Systems,
formed by the Planets and their Satellites : which systems are such, that
the distances of the Satellites from their Planet, are small in comparison
with the distance of the Planet from the Sun : whence it results, that the
action of the Sun being nearly the same upon the Planet as upon its Satel
lites, these latter move nearly the same as if they obeyed no other action
than that of the Planet. Hence we have this remarkable property,
namely,
472. The motion of the Center of Gravity of a Planet and its Satellites,
is very nearly the same as if all the bodies formed one in that Center.
Let the mutual distances of the bodies (i, /*', (/>", &c. be very small
compared with that of their center of gravity from the body M. Let
also
x = x+x,; y = y + y^; z = i + z,.
x' = r 4 x/; y' = y + y/; z' = "z + z/;
&c.
X, y, z being the coordinates of the center of gi'avity of the system of
bodies fi, (if, /«.", &c. ; the origin of these and of the coordinates x, y, z ;
x', y', z', &c. being at the center of M. It is evident that x^, y^, z, ;
x/, y/, z/, &c. are the coordinates of /*, /a', &c. relatively to their center of
gravity ; we will suppose these, compared with x, y, z, as small quanti
ties of the first order. This being done, we shall have, as we know by
Mechanics, the force which sollicits the center of gravity of the system paral
lel to any straight line, by taking the sum of the forces which act upon the
bodies parallel to the given straight line, multiplied respectively by their
masses, and by dividing this sum by the sum of the masses. We also
know (by Mech.) that the mutual action of the bodies upon one another,
does not alter the motion of the center of gravity o£ the system ; nor does
their mutual attraction. It is sufficient, therefore, in estimating the forces
which animate the center of gravity of a system, merely to regard the
action of the body M which forms no part of the system.
The action of M upon ij>, resolved parallel to the axis of x is
^x
Book I.] NEWTON'S PRINCIPIA.
19
the whole force which sollicits the center of gravity parallel to this straight
line is, therefore,
? .
2 (J,
Substituting for x and j their values
X _ X + x^
,3
f Ux + X,) ^ + (y + y,)'^''+ (z + z,) ']^
If we neglect small quantities of the second order, sell, the squares and
products of
X/, y/j z, ; x/, y/, z/ ; &c.
and put
7 = >/ (X 2 + P + z^)
the distance of the center of gravity from M, we have
^ _. J , X, 3x(xx, +yy, + zz,)
for omitting x ^ y ' &c., w§ have
p = (X + xj X J(e)^ + 2 (X X, + y y, + z zj] "^ nearly
= (^+x,) X J(7) ' — 3 (7)  Mx X, + y y, + ^zj nearly
= — ^ — ' = — . (x X, + y y, + z z.) nearly.
Again, marking successively the letters x^, y^, z^, with one, two, three,
&c. dashes or accents, we shall have the values of
X X  
.. . , , &c.
i
But from the nature of the center of gravity
2./^x = 0, S./iy = 0, 2./iz =
we shall therefore have
^— — — = — nearly.
Thus the center of gravity of the system is sollicited parallel to the
axis of X, by the action of the body M, very nearly as if all the bodies of
the system were collected into one at the center. The same result evi
dently takes place relatively to the axes of y and z ; so that the forces, by
B2
20 A COMMENTARY ON [Sect. XI.
which the center of gravity of the system is animated parallel to these
axes, by the action of M, are respectively
My J Mz
"When we consider the relative motion of the center of gravity of the
system about M, the direction of the force which sollicits M must be
changed. This force resulting from the action of /*, (j.', &c. upon M, and
resolved parallel to x, in the contrary direction from the origin, is
if we neglect small quantities of the second order, this function becomes,
after what has "been shown, equal to
X 2./i
In like manner, the forces by which M is actuated arising from the
system, parallel to the axes of y, and of z, in the contrary direction, are
y2./A ,z2
and
iiV is)'
It is thus perceptible, that the action of the system upon the body M,
is very nearly the same as if all the bodies were collected at their common
center of gravity. Transferring to this center, and with a contrary sign,
the three preceding forces; this point will be sollicited parallel to the
axes of X, y and z, in its relative motion about M, by the three following
forces, scil.
(M + 2^)^3,_(M+2/.)i,_(M+2^)4^.
(g) is) ' is) '
These forces are the same as if all the bodies a, /«,', /j/\ &c. were col
lected at their common center of gravity ; which center, therefore, moves
nearly (to small quantities of the second ordei) as if all the bodies were col
lected at that center.
Hence it follows, that if there are many systems, whose centers of gra
vity are very distant from each other, relatively to the respective distances
of the bodies of each system ; these centers will be moved very nearly, as
if the bodies of each system were there collected ; for the action of the
first system upon each body of the second system, is the same very nearly
as if the bodies of the first system were collected at their common center
of gravity ; the action of the first system upon the center of gravity of the
second, will be therefore, by what has preceded, the same as on this hy
pothesis ; whence we may conclude generally that the reciprocal action of
Book I.] NEWTON'S PRINCIPIA. 21
diffhent systems upon their respective centers of gravity ^ is the same as if all
the bodies of each system ivere there collected, and also that these centers
move as on that supposition.
It is clear that this result subsists equally, whether the bodies of each
system be free, or connected together in any way whatever ; for their mu
tual action has no influence upon the motion of their common center
of gravity.
The system of a planet acts, therefore, upon the other bodies of the
Solar system, very nearly the same as if the Planet and its Satellites,
were collected at their common center of gravity ; and this center itself is
attracted by the different bodies of the Solar system, as it would be on
that hypothesis.
Having given the equations of motion of a system of bodies submitted
to their mutual attraction, it remains to integrate them by successive
approximations. In the solar system, the celestial bodies move nearly as
if they obeyed only the principal force which actuates them, and the per
turbing forces are inconsiderable ; we may, therefore, in a first approxi
mation consider only the mutual action of two bodies, scil. that of a planet
or of a comet and of the sun, in the theory of planets and comets ; and
the mutual action of a satellite and of its planet, in the theory of satellites.
We shall begin by giving a rigorous determination of the motion of two
attracting bodies : this first approximation will conduct us to a second in
which we shall include the first powers of small quantities or the perturb
ing forces ; next we shall consider the squares and products of these
forces; and continuing the process, we shall determine the motions of the
heavenly bodies with all the accuracy that observations will admit of.
FIRST APPROXIMATION.
473. We know already that a body attracted towards a fixed point,
by a force varying reciprocally as the square of the distance, de
scribes a conic section ; or in the relative motion of the body /^, round
M, this latter body being considered as fixed, we must transfer in a di
rection contrary to that of fi, the action of j" upon M ; so that in this re
lative motion, /u is sollicited towards M, by a force equal to the sum ol
the masses M, and /« divided by the square of their distance. All this
has been ascertained already. But the importance of the subject in the
Theory of the system of the world, will be a sufficient excuse for repre
senting it under another form,
B3
22 A COMMENTARY ON [Sect. XI.
First transform the variables x, y, z into others more commodious for
astronomical purjioses. f being the distance of the centers of /* and M,
call (v) the angle which the projection of g upon the plane of x, y makes
with the axis of x ; and {6) the inclination of § to the same plane ; we
shall have
X = g cos. 6 COS.
y = S cos. 6 sin. v; ^ (1)
z
Next putting
we liave
= g COS. cos. V ; "\
= § COS. 6 sin. v; >
= f sin. 6. J
^ u + fi _ /j^j^^f + yy+zzr) X
y = z . j^ f 
r • /^
Similarly
dQv _ 1 /dXx M+^ /c^x'
dxJ ~ fi\dx) s' C
[t \d x/ f ^
AtX
M „ ^y
 , — i . — 3
/dQx _ _L/dX\ _
Vdy / (Ct \d y/
/ d Q \ _ 1 fA\\ __ M _ ^
Vdzy' ~ /*Vd z/ f3 ^^ gs •
Hence equations (1), (2), (3) of number 471, become
d'x _ /dQ. d^y __ /dQ. d^ _ / d Q v
dt^~Vdxy* dt^ ~ Vdy;' dt*~ Vdzr
Now multiplying the first of these equations by cos. 6. cos. v ; the
second by cos. &. sin. v ; the third by sin. tf, we get, by adding them
d^^ _njj f_d^^ _ /d Qx .
In like manner, multiplying the first of the above equations by — % cosJ X
sin. v; the second by f cos. 6 cos. v and adding them, &c. we have
d v
COS. =' )
d. =('^) («)
And lastly multiplying the first by — g sin. 6. cos. v ; the second by
• — f sin. 6. cos. V and adding them to the third multiplied by cos. 6. we
have
^ 'dV
dG^^cos.^.)
, u» , jdv« . , . , 2?dedd /dQ\ ...
To render the equations (2), (3), (4), still better adapted for use, let
1
u = r
g cos. e
Book I.]
NEWTON'S PRINCIPIA.
23
and
s = tan. 6 ^
u being unity divided by the projection of the radius f upon the plane
of X, y ; and s the tangent of the latitude of ^ from that same plane.
If we multiply equation (3) by g ^ d v cos.* 6 and integrate, we get
h being the arbitrary constant.
Hence
dt =
d V
dQx dv
"W0" + ^/Or
(S)
If we add equation (2) multiplied by — cos. 6 to equation (4) multi
plied by —  — , we shall have
whence
Substituting for d t, its foregoing value, and making d v constant, we
shall have
/dQ>^ d u Vd Q>^ s ^d Q>
=
d v/ u*d v
/dQ\ s^ /dQ\
V (1 uy' u V d s y
d v'
'•■+^/(^)t=
d Qxdv
In the same way making d v constant, equation (4) will become
ds/dQx /dQ\ 2\/dQ\
d^s. dv(dv)"<T;i)(^+^Hd7)
=
d v'
Now making M + /«. z= m, we have (in this case)
Qm
= — or
(6)
C^)
m u
V(l+s«)
and the equations (5), (6), (7) will become
dv
dt
h.u
= T ; f U
d v^
fj'
h«(l + s=)^'
(8)
B4
24 A COMMENTARY ON [Sect. XI.
(These equations may be more simply deduced directly 124 and Wood
house's Phys. Astron.)
Tlic area described during the element of time d t, by the projection
d V
of the radiusvector is ^ — ^ ; the first of equations (8) show that this area
is proportional to that element, and also that in a finite time it is propor
tional to the time.
Moreover integrating the last of them (by 122) or by multiplying by
2 d s, we get
s = y sin. (v — d) (9)
7 and 6 being two arbitrary constants.
Finally, the second equation gives by integration
" = h'(i+V) ^^^"+^' + ^"^^(^ — ^l = ^^^Y^'' ' • (i»)
e and w being two new arbitraries.
Substituting for s in this expression, its value in terms of v, and then
this expression in the equation
' A «. tlv
d t = r — o ;
h u^
the integral of this equation will give t in terms of v; thus we shall have
v, u and s in functions of the time.
This process may be considerably simplified, by observing that the
value of s indicates the orbit to lie wholly in one plane, the tangent of
whose inclination to a fixed plane is y, the longitude of the node 6 being
reckoned from the origin of the angle v. In referring, therefore, to this
plane the motion of a^ ; we shall have
s = and y = 0,
which give
1 /A
u =  = p {1 + e cos. (v — t^)}.
This equation is that of an ellipse in which the origin of g is at the
focus :
h'
is the semiaxismajor which we shall designate by a ; e is the ratio of
the excentricity to the semiaxismajor ; and lastly w is the longitude of
the perihelion. The equation
d V
h u*
Book L] NEWTON'S PRINCIPIA. 25
lience becomes
d t = ^—, ^ X
dv
V fi [1+ ecos. (V — •=r)p"
Develope the second member of this equation, in a series of the angle
V — w and of Its multiples. For that purpose, we will commence by
developing
__J
1 + e COS. (V zsr)
in a similar series. If we make
X =
1 + >/ (1 — e*)'
we shall have
1 I f 1 X.c(^^)v^^ ) .
1+ecos. (v — 1^)~ VI— eHl+?^c(^«) i l+Xc(^^)Vi j '
c being the number whose hyperbolic is unity. Developing the second
member of this equation, in a series; namely the first term relatively
to powers of c(^—'^)^—^, and the second term relatively to powers of
c — (v — bt) v/_i and then substituting, instead of imaginary exponentials,
their expressions in terms of sine and cosine ; we shall find
I + e cos. (v — w) / 1 — e*
Jl_2Xcos. (v — zir) + 2X2COS. 2(v — w)— 2X3cos.3(v— 17) + &c.J
Calling f the second member of this equation, and making q = — : wc
shall have generally
1 ±e — d~.(^)
fl + ecos. (v— =r)J«+i 1.2.3 m. d q"'
for putting
q q + R
R being = cos. (v — tsr)
''(i)
q >^ _ _ ■ 1
dq  (q + R)
dq« (q + R)»
&c. = &c.
26 A COMMENTARY ON [Sect. XL
dq«» ^ 2.3...m (q+K)"" + ' ^
1
" n+ ecos. (v — t^)l"» + i'
Hence it is easy to conclude that if we make
1 #
= (1— eO "^ X
n + e COS. (v — x^)Y
{I +E (1). COS. (V — «r) + E ^2). COS. 2 (v — «r) + &C.}
we shall have generally whatever be the number (i)
E (0 = + 2e4l+i_ VT^j .
the signs + being used according as i is even or odd ; supposing there
fore that u = a~^ V m, we have
ndt = dv{l + E(i)cos. (v — «r) + E(^)cos.2 (v— «r) &c.i
and integrating
n t +6 = V + E ('^ sin. (v — z^) + ^ E (2) sin. 2 (v — ■=r) + &c.
s being an arbitrary constant. This expression for n t + ^ is very con
vergent when the orbits are of small excentricity, such as are those of the
Planets and of the Satellites ; and by the Reversion of Series we can find
V in terms of t : we shall proceed to this presently.
474. When the Planet comes again to the same point of its orbit, v is
augmented by the circumference 2 sr ; naming therefore T the time of the
whole revolution, we have (see also 159)
T — — — ^ ^^^
~" n ~ V m '
This could be obtained immediately from the expression
yVdjr
^ ~ h
__ 2 area of Ellipse __ 2 t a b
" h  h
But by 157
h*=ma(l — e")
X  '^^^^
"" V m '
KOOK I.]
NEWTON'S PRINCIPIA.
27
If we neglect the masses of the planets relatively to that of the sun we
have
which will be the same for all the planets ; T is therefore proportional in
5.
that hypothesis to a 2, and consequently the squares of the Periods are as
the cubes of the major axes of the orbits. We see also that the
same law holds with regard to the motion of the satellites around their
planet, provided their masses are also deemed inconsiderable compared
with that of the planet.
475. The equations of motion of the two bodies M and /(* may also be
integrated in this manner.
Resuming the equations (1), (2), (3), of 471, and putting M+A«' = ni, we
have for these two bodies
_ d ^ X m x'
"  dT^; + T^
=
=
dt
2 "" „3
}> (0)
d '^ z m z
dT^ + 7^.
The integrals of these equations will give in functions of the time t, the
three coordinates x, y, z of the body /(a referred to the center of M ; we
shall then have (471) the coordinates t,, n, 7 of the body M, referred to a
fixed point by means of the equations
^ = a + b t — ^ — ;
^ m
' H = a' + b't — ^;
m
7 = a'' + b"t— ^^
' m
 Lastly, we shall have the coordinates of /*, referred to the same fixed
point, by adding x to ^, y to n, and z to 7 : We shall also have the rela
tive motion of the bodies M and /«■, and their absolute motion in space.
476. To integrate the equations (0) we shall observe that if amongst
the (n) variables x '•^\ x ^^^ x ^") and the variable t, whose difference
is supposed constant, a number n of equations of the following form
di xW . , d>i x(') . ^ d'^x^*^
=
H
dt*i ' dt^^
in which we suppose s successively equal to 1, 2, 3 n ; A, B H
oeing functions of the variables x ('), x ^^\ &c. and of t symmetrical
28
A COMMENTARY ON
[Sect. XI.
with regard to the variables x ^^\ x '^, &c. that is to say, such that they
remain the same, when we change any one of these variables to any other
and reciprocally ; suppose
xO) = a(')x^°' + '^ + bWx(*i + 2) + h") x(n),
x(2) = a^2) x^'i + J) + b® x^"' + 2) + li(2) xn.
X(ni) _ a (ni) X (•»» + !) _. I3 ^"^ *) X (" ' + '^) ^ h^'5 X ^"^
a^'>, h^^\ h ('^; a^*^, b^% &c. being the arbitraries of which the
number is i (n — i). It is clear that these values satisfy the proposed
system of equations : Moreover these equations are thereby reduced to i
equations involving the i variables x ^"■' + ^) x^"\ Their integrals
will introduce i * new arbitraries, which together with the i (n — i) pre
ceding ones will form i n arbitraries which ought to give the integration
of the equations proposed.
477. To apply the above Theorem to equations (0) ; we have
z = a X + b y
a and b being two arbitrary constants, this equation being that of a plane
passing through the origin of coordinates ; also the orbit of /» is wholly in
one plane.
The equations (0) give
= d(f'.^)+mdj4; (0')
= i{s'.^^)+m,i.
Also since
and
j« = x« +y^ + z«
.*. fdg = xdx + ydy + zdz
and differentiating twice more, we have
gd^g + 3 df d^g = X d'x + y d^y + zd'z
+ 3(dxd^x + dyd2y + dzd*z),
and consequently
"• V dt'J ^ (. ilf + ^dt« + dt'f
iQ.fi d'x, , d'y.j d'zl
Substituting in the second member of this equation for d ^ x, d ^ y, d ^' z
Book I.] NEWTON'S PRINCIPIA. 29
their values given by equations (0'), and for d*x, d^y, d*z their values
given by equations (0) ; we shall find
= d(^^i^^+mde).
If we compare this equation with equations (0'), we shall have ia virtue
of the preceding Theorem, by considering r—  , j^ , t— , 3^ , as so many
particular variables x ^^\ x ®, x ^^, x ^*\ and g as a function of the time t;
dgrrXdx+ydy;
X and y being constants ; and integrating
=  + Xx + 7y,
h 2 .
— being a constant. This equation combined with
z = ax + by; g* = x^ + y2 + z^
gives an equation of the second degree in terms of x, y, or in terms of
X, z, or of y, z ; whence it follows that the three projections of the curve
described by fi about M, are lines of the second order, and therefore that
the curve itself (lying in one plane) is a line of the second order or a conic
section. It is easy to perceive from the nature of conic sections that, the
radiusvector § being expressed by a linear function of x, y, the origin of
X, y ought to be in the focus. But the equation
i =  + ^^ + 7y
gives by means of equations (0)
. d'S ^ ""m)
^ = dF + ^— P
Multiplying this by d f and integrating we get
a' being an arbitrary constant. Hence
d t = e ''f ■
which will give g in terms of t ; and since x, y, z are given above in terms
of ^, we shall have the coordinates of /u. in functions of the times.
478. We can obtain these results by the following method, which has
the advantage of giving the arbitrary constants in terms of the coordinates
X, y, z and of their first differences ; which will presently be of great use
to us.
30 A COMMENTARY ON [Sect. XL'
r.et V = constant, be aa integral of the first order of equations (0), V
being a function of x, y, z, ,— , t^ , ~ . Call the three last quantities
x', y'j z'. Then V = constant will give, by taking the differential,
^ •_ /d V\ tl X /d V\ ^ y 1 z*^^ ^\ J 2
"  U xV • dT "^ vay; * ar + vot ' • at
+ VdxV* dt"^Vdy7' dt'+Vdz'y'* dt
But equations (0) give
dx'_ mx dy'__ my dz'__ mz
Tt ~ p~' "dT ~ p' dT ~ P"'
we have therefore the equation of Partial Differences
, /d Vn , , /d Vx , ,d Vx
« = ^ (dir) + y (ay) + ^ ( dr)
in r /'d Vx ^ /<i Vx , /d Vx
It is evident that every function of x, y, z, x', y', z' which, when sub
stituted for V in this equation, satisfies it, becomes, by putting it equal to
an arbitrary constant, an integral of the first order of the equations (0).
Suppose
V = U + U' + U" + &c.
U being a function of x, y, z ; U' a function of x, y, z, x', y', z' but of the
first order relatively to x', y', zf ; U^' a function of x, y, z, x', y', z' and of
the second order relatively to x', y', z', and so on. Substitute this value
of V in tlie equation (I) and compare separately 1. the terms without
x', y', z' ; 2. those which contain their first powers ; 3. those involving their
squares and products, and so on ; and we shall have
/d U\ /d U\ /d U'x
^ = ^(di?) + y(d7) + ^(dF)'
, /d Ux , , /d Ua . , /d Ux m f /d U'\ ^ /d U'\ ^ /d U"x
^(dT)+y(dr)+<^)=piHd^)+y(ay)+<dz'))
&c.
which four equations call (I').
The integral of the first of them is
U' = funct. [x y' — y x', x z' — z x', y z' — z y', x, y, z]
Book I.] NEWTON'S PRINCIPIA. 31
The value of U' is linear with regard to x', y^ z' \ suppose it of this
form
U' = A (x y' — y x') + B (x z' — z x') + C (y z' — z y') ;
A, B, C being arbitrary constants. Make
U"', &c. = ; •
then the third of the equations (F) will become
The preceding value of \5' satisfies also this equation.
Again, the fourth of the equations (F) becomes
of which the integral is
U" = funct. Jx y' — y x', x z' — z x', y z' — z y', x', y ', z'\ .
This function ought to satisfy the second of equations (F), and the first
member of this equation multiplied by d t is evidently equal to d U. The
second member ought therefore to be an exact differential of a function of
X, y, z ; and it is easy to perceive that we shall satisfy at once this condi
tion, the nature of the function U", and the supposition that this function
ought to be of the second order, by making
U" = (D y' — E X') . (X / — y x') + (D z' — F x') (x z' — z x')
+ (E z — F y) (y z' — z y'} + G (x'^ + y 2 + z' ')^
D, E, F, G being arbitrary constants ; and then g being = Vx My^+z^,
we have
U = — (Dx + Ey + Fz + 2G);
Thus we have the values of
U, U', U" ;
and the equation V = constant will become
const.=— Px+Ey+Fz+2G} + (A + Dy' — Ex') (x y — y x')
+ (B + Dz' — Fx') (xz— zx') + (C+Ez' — Fy') (yz'— zy)
This equation satisfies equation (I) and consequently the equations (0)
whatever may be the arbitrary Constants A, B, C, D, E, F, G. Sup
posing all these = 0, 1. except A, 2. except B, 3. except C, &c. and
putting
d X d y d z ^ , ,
T— , 1*^ , ,— tor X , y , z ,
d t ' d t (1 t ' ^ '
32 A COMMENTARY ON [Sect. XT.
we shall have the integrals *
fp xdy — > d X ^,_ xdz — zd x »_ydzz d y
dt '"" dt ''^ dTt
nf _L ^ J°^ dys+dzn y dy.dx zdz . d x
• 17 ar^ — / + dt* + dt^
(V) ^ Of X v/™ dx'+ dz^ ( xdx.dy zdz. dy
nf'a.^/"^ dx' + d y') , xdx.dz , ydy.dz
0_t +z dT^~f + dt^ + dt«
. m 2m . dx« + dy*+dz*
.«=TT"*'"~ dt^
c, c', c'', f, f, f" and a being arbitrary constants.
The equations (0) can have but six distinct integrals of the first order,
by means of which, if we eliminate d x, d y, d z, we shall have the three
variables x, y, z in functions of the time t ; we must therefore have at least
one of the seven integrals {P) contained in the six others. We also per
ceive a priori f that two of these integrals ought to enter into the five
others. In fact, since it is the element only of the time which enters
these integrals, they cannot give the variables x, y, z in functions of the
time, and therefore are insufficient to determine completely the motion of
a about M. Let us examine bow it is that these integrals make but five
distinct integrals.
If we multiply the fourth of the equations (P) by ^ . ^ , and
X U Z "^"^ Z (1 X
add the product to the fifth multiplied by t— , we shall have
n_f z dy— ydz ^, xdz— zdx xdy— ydx/m dx^dy^ )
"*• dl +*• dl +^ dt \g dT^~/
+
xdy — ydxfxdx.dz ydy.dz
dt
fx dx. d z y d y . d z \
t tUt'""*" dt^ )'
„, . . t. xdy — ydx xdz — zdx ydz — zdy,.
Substitutmg for (\^ ^ ' jl » "^ ' '"^"'
values given by the three first of the equations (P), we shall have
f ' cf — f c" ( m d x'+ d y g "> xdx.dz y dy.d z
^ = I "^""iT dl^ j+ TF~+ dt^ *
This equation enters into the sixth of the integrals P, by making
f" = f ' c' — f c^ or = f c" — f c' + f" c. Also the sixth of these
c
integrals results from the five first, and the six arbitraries c, c', c", f, f, f
are connected by the preceding equation.
Book L] NEWTON'S PRINCIPIA. 33
If we take the squares off, f, i" given by the equations (P), then add
them together, and make f '^ + f * + F' * = 1 % we shall have
,„ 2__ / I dx^+dy^+dz ^ (l^i\^ 1 f dx^+dy^+dz' 2 mi
^"""'"t^ d~P Vdt^ \'\ dt° yV
but if we square the values of c, c', c", given by tlie same equations, and
make c* + c'* 4 c"* = h'; we get
%
dx'^ + dy^ + dz' /L^
dt
the equation above thus becomes
__ d x^+ dy^+ dz^ 2 m m» — P
"" r~dT^ g ■*" h^~*
Comparing this equation with the last of equations (P), we shall have
the equation of condition,
m' — V _ jn
h« ~ a *
The last of equations (P) consequently enters the six first, which are
themselves equivalent only to five distinct integrals, the seven arbiti'ary
constants, c, c', c", f, f, f", and a being connected by the two preceding
equations of condition. "Whence it results that we shall have the most
general expression of V, which will satisfy equation (I) by taking for this
expression an arbitrary function of the values of c, c', c", fj and T, given
by the five first of the equations (P).
479. Although these integrals are insufficient for the determination of
X, y, z in functions of the time ; yet they determine the nature of the
curve described by /i about M. In fact, if we multiply the first of the
equations (P) by z, the second by — y, and the third by x, and add the
results, we shall have
= c z — c' y + c'' X,
the equation to a plane whose position depends upon the constants
c, c', d'.
If we multiply the fourth of the equations (P) by x, the fifth by y, and
the sixth by z, we shaU have
n <• I f / , f // . 2 dx«+dy*+dz2 , g^d^^
= fx+f'y + P'z+m^ — f^ ^ 1^7 + Vt*'
but by the preceding number
, dx^4 dy'+ dz' ?IAi!l,»
^ * dt« " dt« "" *
..0 = m g — h» + f X + f y + f" z.
This equation combined with
= c" X — c' y + c z
Vot. II. C
34 A COMMENTARY ON [Sect. XI.
and
g* = x» + y' + z*
gives the equation to conic sections, the origin of f being at the focus.
The planets and comets describe therefore round the sun very nearly
conic sections, the sun being in one of the foci ; and these stars so move
that their radiusvectors describe areas proportional to the times. In fact,
if d V denote the elemental angle included by ^, g + d f, we have
dx* + dy" + dz* = g*d v^ + d g*
and the equation
, dx« + dy*+ dz' g'd g' _ ^^,
dt* dt
becomes
f*d v» = h«d t';
hdt
.'. d V = r— .
Hence we see that the elemental area  f * d v, described by g, is propor
tional to the element of time d t ; and the area described in a finite time is
therefore also proportional to that time. We see also that the angular
motion of/* about M, is at every point of the orbit, as — , ; and since without
sensible error we may take very short times for those indefinitely smally we
shall havef by means of the above equation, the horary motions of the planets
and comets, in the different points of their orbits.
The elements of the section described by fi, are the arbitrary constants
of its motion ; these are functions of the arbitraries c, c', c", f, T, f", and
— . Let us determine these functions,
a
Let 6 be the angle which the intersection of the planes of the orbit and
of (x, y) makes with the axis of x, this intersection being called the li?ie
of the nodes ; also let <p be the inclination of the planes. If x', y' be the
coordinates of fi referred to the line of the nodes as the axis of abscissas,
then we have
x' = X COS. ^ + y sin. 6
y' = y COS. 6 — x sin. 6.
Moreover
z = y' tan. f
.*. z = y COS. 6 tan. f — x sin. 6 tan. p.
Comparing this equation with the following one
= c" X — c' y + c z
Book I.] NEWTON'S PRINCIPIA. 35
we shall have
c' = c COS. 6. tan. p
c" zz. c sin. 6 tan. <p
whence
and
c
tan. 6 zz r
c
tan. p = — ^ ' '
c
Thus are determined the position of the nodes and the inclination of the
orbit, in functions of the arbitrary constants c, c', c".
At the perihelion, we have
gdf = 0, orxdx + ydy + zdz = 0.
Let X, Y, Z be the coordinates of the planet at this point ; the fourth
and the fifth of the equations (P) will give
_Y __ f
X  f
But if I be called the longitude of the projection of the perihelion upon
the plane of x, y this longitude being reckoned from the axis of x, we have
Y
■^ = tan. 1 ;
T f'
.. tan. I = ,
which determines the position of the major axis of the conic section.
If from the equation
2 clx'+ dy' + dz« _ g'dg' _ , ,
^ • dt^ dt'^ ~
we
eliminate a\^ » ^^ means of the last of the equa
tions (P), we shall have
me* f *d e* 1 ,
but d g is at the extremities of the axis major ; we therefore have at these
points
n h *
= p«_2ap+ ?^.
* * m
The sum of the two values of % in this equation, is the axis major, and
their difference is double the excentricity ; thus a is the semiaxis major of
the orbit, or the mean distance of ^t from M ; and
VOi^a)
30 A COMMENTARY ON [Sect. XI.
is the ratio of the excentricity to the semiaxis major. Let
h«x
= J(
1
m sn
and having by tlie above
m _ m' — 1'
we shall get
m 6 = 1.
Thus we know all the elements which determine the nature of the conic
section and its position in space.
480. The three finite equations found above between x, y, z and f give
X, y, z in functions of g ; and to get these coordinates in functions of the
time it is sufficient to obtain f in a similar function ; which will require a
new integration. For that purpose take the equation
m g* ^f ' "
2 m — —2 *_— s_ — us
» a d t«
But we have above
h' = — (m« — 1«) = am(l — e«);
.. d t =
V m /2f — ^" — a(l — e«)
whose integral (237) is
a^
t + T = ^^^ (u — e sin. u) (S)
u being = cos. ^ ( —j, and T an arbitrary constant.
This equation gives u and therefore § in terms of t; and since x, y, z
are given in functions of f, we shall have the values of the coordinates for
any instants whatever.
We have therefore completely integrated the equations (0) of 475, and
thereby introduced the six arbitrary constants a, e, I, 6, <p, and T. The
two first depend upon the nature of the orbit ; the three next depend upon
its position in space, and the last relates to the position of the body u.
at any given epoch ; or which amounts to the same, depends upon the
instant of its passing the perihelion.
Referring the coordinates of the body fi, to such as are more commodious
for astronomical uses, and for that, naming v the angle which the radius
Book I.] NEWTOM'S PRINCIPlA. 37
vector makes with the major axis setting out from the perihelion, the
equation to the ellipse is
^ ~" 1 + e cos. V *
The equation
g = a ( 1 — e COS. u)
indicates that u is at the perihelion, so that this point is tlie origin of two
angles u and v ; and it is easy hence to conclude that the angle u is formed by
the axis major, and by the radius drawn from its center to the point where
the circumference described upon the axis major as a diameter, is met by
tlie ordinate passing through the body /* at right angles to the axis major.
Hence as in (237) we have
V . 1 + e ^ u
tan. ^ = jj 1 . tan. ^ .
2 ^1 — e 2
We therefore have (making T = 0, &c.)
n t = u — e sin.
j= a(l — e
and
1. u ~j
cos. u) I
V /I + e u
(0
n t being the Mean Anomaly,
n the Excentric Anomaly,
V the Time Anomaly.
The first of these equations gives u in terms of t, and the two others
will give f and v when u shall be determined. The equation between u
and t is transcendental, and can only be resolved by approximation.
Happily the circumstances attending the motions of the heavenly bodies
present us with rapid approximations. In fact the orbits of the stars are
either nearly circular or nearly parabolical, and in both cases, we can de
termine u in terms of t by series very convergent, which we now proceed
to develope. For this purpose we shall give some general Theorems
upon the reduction of functions into series, which will be found very use
ful hereafter.
481. Let u be any function whatever of a, which we propose to deve
lope into a series proceeding by the powers of a. Representing this
series by
U = «'+ a.q,+ a*. q2+ «°. qn+ «" + ^ qD+ + &c.
C3
38 A COMMENTARY ON [Sect. XI.
«j qi> q2j &c. being quantities independent of a, it is evident that?^ is what
u will become when we suppose a = ; and that whatever n may be
(j^) = 1.2....n.q„ + 2.3....(n+l).a.q„ + j + &c.
(d ** u\ • •
j — ) being taken on the supposition that every thing in
u varies with a. Hence if we suppose after the differentiations, that a = 0,
in the expression T j — A we have
_/d''u\ 1
^" ~\da°>' ^ 1.2 n*
This is Maclaurin's Theorem (see 32) for one variable.
Again, if u be a function of two quantities a, «', let it be put
u = M 4 a . qi + a 2 . q^^ + &c.
+ «'• qo.i +««'• qi.i + &c.
+ a' *. qo,2 + &C.
the general term being
«''a'°'qn.n
Then if generally
/ d " + °' u \
Vd a " . d a' «'/
denotes the (n + n')''' difference of u, the operation being performed (n)
times, on the supposition that a is the only variable, and then n' times on
that of a' being the only variable, we have
(dli) ~ *^''° + ^ " • ^2.0 + ^ " '• ^3.0 + 4, a' q^^o + 5 a*. q5,o + &c.
+ «' qi.l +2a.«'q2,i +3a'^a'q3^i +4aVq4^i + &c.
+ a' * qi,2 +2aa'^q2,2 +3a«a^q3,2 + &C.
+ "'^ qi,3 + 2aa''q2^3 + &c.
+ a'* qi.4 + &c.
(d^) ~ ^ ^*'' + ^ 2 " 93.0 + 4. 3 a 2 q4 + 5. 4 a' qj^o + &c.
+ 2a qj,! + 3. 2aaq3,i + 4. 3o«c£q4^i 4. &c.
+ 2a« q2.2 + 3.2aa2q3^2 + &c.
+ 2a3q +&C.
(d^«)=2^^> + ^^"^^^ + ^*^
+ 2 a q2_2 + &c.
and continuing the process it will be found that
Book L] NEWTON'S PRINCIPIA. 39
so that when a, a both equal 0, we have
/ d » + "' u \
_ \d a n . d a'"n7
^"•"'~2. 3....n X 2. 3....n' ^^^
A nd generally, if u be a function of a, a, a", &c. and in developing it
into a series, if the coeflScient of a °. a «'. a." ^'. &c. be denoted by q„, ^., a"> &c
we shall have, in making «, 6.^ a", &c. all equal 0,
/ d n + n' + n" + &c. ^ .
0,^ _ Vda" .da»'.da^^°",&C.) .
qn.n'.n". &c. 2.3....n X 2. 3 . . . . n' X 2. 3 . . . . n" X &c. " '^ '
This is Maclaurin's Theorem made general.
482. Again let u be any function of t + a, t' + a, t" + a", &c. and
put
u z= p (t + a, f + <i, t'' + a", &c.)
then since t and a are similarly involved it is evident that
/ (J n + n' + n" + &c. y . , ^J n + n' + n" + &c. y .
Vd a", d a."', d «"n"&c./ ~ Vd t". d t'°'. d t'">". &c.^
and making
«, a, a'', &c. = 0,
or
u = ^ (t, t', t", &c.)
by (2) of the preceding article we have
, dn + n' + n".&c. ^ ^ ^^^ ^/^ ^i ^ g^C,)
__ V d t°. d t^^' . d t^"°"&c /
q n.n'.n'.&c. " 2. 3 .... n X 2. 3 n' X 2. 3 n" X &c. * * * ^*^
which gives Taylor's Theorem in all its generality (see 32).
Hence when
u = P . (t + «)
^" ~ 2. 3....n.dt»
and we thence get
,(t + .) = Mt) + «^ + "^'.^ + &c. (i)
483. Generally, suppose that u is a function of «, a, a", &c. and of
t, t', t", &c. Then, if by the nature of the function or by an equation of
P.. ftial Differences which represents it, we can obtain
/ d» + "' + &°.U N
Vda". da"'. &c./
in a function of u, and of its Diffeiences taken with regard to t, t', Sec
C4
40 A COMMENTARY ON [Sect. XI.
calling it F when for u we put tt or make a, d, a", &c. = ; it is evident
we have
_ F
qn.n'.n.te. ~ 2. 3 ... n X 2. 3 ... n' X 2. 3 . . . n", x &c.
and therefore the law of the series into which u is developed.
For instance, let u, instead of being given immediately in terms of a,
and t, be a function of x, x itself being deducible firom the equation of
Partial Differences
= ^0
in which X is any function whatever of x. That is
Given
u = function (x)
to develope u into a series ascending by the powers of a.
Fii'st, since
•••(rJ = ('^^) w
Hence
/d'uv /d'/Xdux
VdaV~ V da.dt )*
But by equation (k), changing u into y X d u
/ d./Xdu >, _ /d^/X^dux
V d^i )\ dt )'
/dj_ux _ / d'/X'du x
•*• V d aV ~ \ d t « )'
Again
/d^uv _/ dVX«du x
Vda^JV da.dt* )'
But by equation k, and changing u intoyX' d u
/ d/X'du x _ / d/X'du v
V da J~V dt /
/d^ux _ /d»./Xld_ux
•'•VdT'jv dt» ;•
Thus proceeding we easily conclude generally that
• ,d°». xd"/x■_du>y ''°•x°(rt) ^ ,,.
Now, when a = 0, let
X = function of t = T
Book L] NEWTON'S PRINCIPIA. 41
and substitute this value of x in X and u ; and let these then become X
and u respectively. Then we shall have
d'.Xn.^"
/ d°u \ _. d_t
and
d u
d»». X«.
•'• ^° ■" 2.3 ndt^» ^^^
which gives
, ^ dw , a* V d t/ , a^ v d t^ , . , .
" = " + «^Tt + 2 dl +2:3 dT^ +&c....(p)
which is Lagrange's Theorem.
To determine the value of x in terms of t and a, we must integrate
In order to accomplish this object, we have
and substituting
we shall have
dx=(^)Sdt + Xd.!
= ^^{d(t + «X)«(^)d.},
(^).d.(t+aX)
dx =
dXx /dxx"
>+«(d^)rdi)
which by integration, gives
X = 9 (t +.a X) ' . '. (2)
f denoting an arbitrary function.
Hence whenever we have an equation reducible to this form x =
9 (t + a X), the value of u will be given by the formula (p), in a series of
the powers of a.
By an extension of the process, the Theorem may be generalized to the
case, when
u = function (x, x', x'', &c.)
4a A COMMENTARY ON [Sect. XI.
and
X = p (t + a X)
x' =: f {tf + a' X')
x" = ^' (f ' + a" X'O
&c. = &c.
484. Given (237)
u = n t 4 e sin. u
required to develope u or any function of it according to the powers ofe.
Comparing the above form with
X = p (t + « X)
X, t, a, X become respectively
u, n t, e, sin. u.
Hence the formula (p) 483. gives
. / N . , X .// % • . e' d JvI/' (n t) sin. * n t J
■^{n) = ^Knt) + e^^' (n t) sm. n t +  . '^^^^^ i
+ £! d'{4/(nt)sin.3ntl
4,(„t)be,„g=14^.
To farther develope this formula we have generally (see Woodhouse's
Trig.)
6m.'(nt) = [ 2 V — 1 ) ' cos.>(nt) = ( X j;
c being the hyperbolic base, and i any number whatever. Developing the
second members of these equations, and then substituting
cos. r n t + V — I sin. r n t, and cos. r n t — V — 1 sin. rn t
for c'"* •^~', and c"'" * V — ?, r being any number whatever, we shall
have the powers i of sin. n t, and of cos. n t expressed in sines and cosines
of n t and its multiples ; hence we jfind
6 e ^ .
P =r sin. nt+^sin'nt + 55 sin. ' n t f &c.
r= sin. n t — 5^ . {cos. 2 n t — 1 }
e
2. 3. 4. 2 »
. {sin. 3 n t — 3 sin. n t]
( 1 4. 3 )
. < cos. 4 n t — 4 cos. 2 n t + ^ . ^j— ^ >
2.3.2*
1 4. 3
+ 2. 3. 4.5. 2 ^ {'^" ^ " '""^ '^" ^ " ^+172 '^" " *}
Book I.] NEWTON'S PRINCIPIA. 4»v
23:4^»{'=°^'«'"«'°^*°'+tI~^2"'it:II}
— &c.
Now multiply this function by ■^' (n t), and differentiate each of its
terms relatively to t a number of times indicated by the power of e which
multiplies it, d t beuig supposed constant; and divide tliese differentials
by the corresponding power of n d t. Then if P' be the sum of the
quotients, the formula (q) will become
4 (u) = v^n t) + e P^
By this method it is easy to obtain the values of the angle u, and of
the sine and cosine of its multiples. Supposing for example, that
•vp u = sin. i u
we have .
■^ (n t) = i cos. int.
Multiply therefore the preceding value of P, by i. cos. i n t, and deve
lope the product into sines and cosines of n t and its multiples. The
terms multiplied by the even powers of e, are sines, and those multiplied
by the odd powers of e, are cosines. We change therefore any term of
the form, K e '^ ■" sin. s n t, into + K e * •■ s '^^ sin. s n t, + or — obtaining
according as r is even or odd. In like manner, we change any term
of the form, K e^"" + ' cos. s n t, into + K e^'" + '. s'^' + '. sin. s n t, — or
f obtaining according as r is even or odd. The sum of all these terms
will be P' and we shall have
sin. i u = sin., i n t + e P'*
But if we suppose
•4' (u) = u;
then
^^ (n t) = 1
and we find by the same process
e*
u = n t + e sin. n t 4 ^^ . 2 sm. 2 n t
+ £22[3*sin. 3nt — 3sin. nt] .
e*
+ Q 3 .{4»sin. 4n t — 4. 2 'sin. 2 n t]
iS» «S. 4. a
e * f 5 4)
+ g g ^ g gv 5*sin.5nt — 5.3*sin.3nt+j^sin.nt
+ &c.
44 A COMMENTARY ON [Sect. XI.
a formula johich expresses the Excentric Anomaly in terms of the Mean
Anomaly.
This series is very convergent for the Planets. Having thus determin
ed u for any instant, we could thence obtain by means of (237), the cor
responding values of f and v. But these may be found directly as fol
lows, also in convergent series.
485. Required to express g in terms of the Mean Anomaly.
By (237) we have
^ = a (1 — e cos. u).
Therefore if in formula (q) we put
4' (u) = 1 — e cos. u
we have
•vj/' (n t) = e sin. n t,
and consequently
. • o . e' d. sin.' n t , .
1 — e cos. u = 1 — e cos. n t + e * sm. * n t + — . • r — + &c.
A not
Hence, by the above process, we shall find
p e ' e *
i = 1 + ^ — e cos. n t — COS. 2 n t
_ a
,.{3 cos. 3 n t — 3 cos. n t]
2. 2
,4
2. 3. 2 3
® .J4*cos. 4nt — 4. 2*. cos. 2n t
e ' r , « 6. 4 "J
__ . j 5 ' cos. 5 n t5. 3 ^ cos. 3 n t + y^. cos. n t J
2. 3. 4. A (. 1. ^ J
— ostW* {6*cos.6nt— 6. 4* cos. 4 n t+^. 2*cos.2nt
— &c.
486. To express the True Anomaly in terms of the Mean.
First we have (237)
V . u
sin. ^ 1 + e «'"• 2
Vl— e* u
cos. Y ^^' "2
/. substituting the imaginary expressions
and making
X =
»— 1 _ /1 + e c"^^"^— I ,
1 + 1  sj\ — e c"Vi+ 1'
1 + V (l—e*)
Book I.] NEWTON'S PRINCIPIA. 45
we shall have
*^ _ c V X i_xc"Vi '
and therefore
— log.(l^c — "V— ») — log.(l~Xc"Vi)
v_uf \^'Z:^
whence expanding the logarithms into series (see p. 28), and putting
sines and cosines for their imaginary values, we have
2 X ^ 2 X ^ .
V = u + 2 X sin. u \ — ^— sin. 2 u j ~ sin. 3 u + &c.
But by the foregoing process we have u, sin. u, sin. 2 u, &c. in series
ordered by the powers of e, and developed into sines and cosines of n t
and its multiples. There is nothing else then to be done, in order to
express v in a similar series, but to expand X into a like series.
The equation, (putting u=l + Vl — e*)
e*
u = 2— 
u
will give by the formula (p) of No. (483)
1 __ 1 ie« . i(i + 3) e^ i(i + 3)(i + 5) _e^
u'~2'"*" 2>+2 ■*■ 2 •2'+*"^ 2.3 '2^ + ^
and since
u = 1 + V 1 — e*
we have
These operations being performed we shall find
v=nt + 2e — — e' + g^ e^j sin. n t
fl03 . 451 si . . ,
+ 960 "^^"•^"*
. 1223 6 . « ^
the approximation being carried on to quantities of the order e* in
clusively.
46 A COMMENTARY ON [Sect. XL
487. The nngles v and n t are here reckoned from the Perihelion ; but
if we wish to compute from the Aphelion, we have only to make e nega
tive. It would, tlierefore, be sufficient to augment the angle n t by «r, in
order to render negative the sines and cosines of the odd multiples of n t ;
then to make the results of these two methods identical ; we have only in
tlie expressions for § and v, to multiply the sines and cosines of odd
multiples of n t by odd powers of e; and the even multiples by the even
powers. This is confirmed, in fact, by the process, a posteriori.
488. Suppose that instead of reckoning v from the perihelion, we fix
its origin at any point whatever ; then it is evident that this angle will be
augmented by a constant, which we shall call w, and which will express
the Longitude of the Perihelion. If instead of fixing the origin of t at
the instant of the passage over the perihelion, we make it begin at any
point, the angle n t will be augmented by a constant which we will call
e — w ; and then the foregoing expressions for — and v, will become
a
^ = 1 + ^e?—(e— I e)cos.{nt+s—z^)—{ ]:^—l e*)cos.2(nt +«—»')+ &c.
v=nt+t+(2e— e^)sin.(nt +«—»)+( e«—27e*)sin. 2 (nt + s—«')+&c.
where v is the true longitude of the planet and n t + g its mean longi
tude, these being measured on the plane of the orbit.
Let, however, the motion of the planet be referred to a fixed plane a
little inclined to that of the orbit, and <p be the mutual inclination of the
two planes, and 6 the longitude of the Ascending Node of the orbit, mea
sured upon the fixed plane ; also let jS be this longitude measured upon
the plane of the orbit, so that 6 is the projection of jS, and lastly let v^ be
the projection of v upon the fixed plane. Then we shall have
V, — ^, V — /3,
making the two sides of a right angled spherical triangle, v — /3 being
opposite the right angle, and p the angle included between them, and
therefore by Napier's Rules
tan. (v, — d) = COS. 9 tan. (v — /S) (1)
TTiis equation gives v, in terms of v and reciprocally ; but we can ex
press cither of them in terms of the other by a series very convergent
after this manner.
By what has preceded, we have the series
11 X* X'
 v = — u t X sin. u + ^ sin. 2 u + ~ sin. 3 u { &c.
Book L] NEWTON'S PRINCIPIA 4t
from
1 /l + e ^ 1
tan. 2 v=^p_^.tan.u.
by making
^  1 + e
into
e
V^ + i
If we change  v into v, — d, and  u into v — jS, and ^=
COS. f , we have
COS. p 1 .0
X = Zl_— = _ tan.«^; (1)
COS. <p + 1 2 ^ '
The equation between x v and  u will change into the equation be
tween v^ — 6 and v — /3, and the above series will give
V, — ^ = v — jS — tan "  ?>. sin. 2 (v — S) +  tan. '* 5 p. sin. 4 (v — 18)
2 "^ ^ ' ' 2 2
3 2
^Uxn.^ I <p sin. 6 (v — 18) + &c (2)
v u 1 .
If in the equation between  and  , we change ^ v into v — /3 and
1 . . 1 / 1 + e • 1 1111
_ u into V, — 6, and ^ / ^i into , we shall have
2 ' ^1 — e cos. <p
X = tan.^f, (3)
and
V — jS = v^ — ^+tan. ' g (p. sin. 2 (v, — 6)
+ 2 tan. ^ 2 ^ sin. 4 (v, — S)
+ I tan.^! ^. sin. 6(v, — (4)
Thus we see that the two preceding series reciprocally interchange,
l.y changing the sign of tan. * ^ ?'> and by changing v, — ^, v — jS the one
for the other. We shall have v, — 6\n terms of the sine and cosine of
n't and its multiples, by observing that we have, by what precedes
v = nt + « + eQ,
Q being a function of the sine of the angle n t + « — w, and its multi
ples; and that the formula (i) of number (482) gives, whatever is i,
sin. i (v — /3) = sin. i (n t + 1 — /3 + e Q)
♦8 A COMMENTARY ON [Sect. XI.
Lastly, s being the tangent of the latitude of the planet above the fixed
plane, we have
s = tan. <p sin. (v, — 6) ;
and if we call f^ the radiusvector projected upon the fixed plane, we
shall have
f,=f(i+s«r*=f{i.is« + s*&c.},
we shall therefore be able to determine v^, s and ^, in converging series
of the sines and cosines of the angle n t and of its multiples.
489. Let us now consider very excentric orbits or such as are those of
the Comets.
For this purpose resume the equations of No. (237), scil.
 a( l — e')
* ~ 1 + e cos. V
n t = u — e sin. u
1 + e
tan.i V = ^ J— — ^ . tan. i u.
In this case e differs very little from unity; we shall therefore suppose
1 — e = a
a being very small compared with unity.
Calling D the perihelion distance of the Comet, we shall have
D= u(l — e) = aa;
and the expression for ^ will become
(2  g) D D
s 1 — 1 r a ~i r
2 cos. *  v — a COS. V cos. '^2^1"*" 2 — « *^"' 2 ^f
which gives, by reduction into a series
cos.* 2 V
To get the relation of v to the time t, we shall observe that the expres
sion of the arc in terms of the tangent gives
u = 2tan. iujl— I tan.' g " + 5 ^^'* 2 ^ "^ ^^'\
But
1 / a ^ 1
tan.u=,^^:^tan. ^u;
Book L] NEWTON'S PRINCIPIA. 49
we therefore have
Next we have
2 tan. ~ u
sin. u =
1 + tan. 2 — u
— = 2 tan. g u 1 —tan. ^  + tan.* ^_ &c. 
Whence we get
esin.u = 2(l»)^^^tan.iv.{l_^^tan.'Av
+ {2^)' ■''''■' h^}
Substituting these values of u, and e sin. u in the equation n t = u —
e sin. u, we shall have the time t in a function of the anomaly v, by a series
very convergent ; but before we make this substitution, we shall observe
that (237)
n
=:
a
s^
V
m,
D
=
a
a>
1
n
=
a
m
•
_5.
n = a
and since
we have
Hence we find
If the orbit is parabolic
a =
and consequently
D
1
COS. — V
'J V . I , 1 1
tan.  + 3tan.3_ v]^
~~ v' m
which expression may also be got at once from (237).
The time t, the distance D and sum ra of the masses of the sun and
comet, are heterogeneous quantities, to compare which, we must divide
each by the units of their species. We shall suppose therefore t'nat the
mean distance of the sun from the Earth is the unit of distance, so that D
is expressed in parts of that distance. We may next observe that if T
Vol. II. D
50 A COMMENTARY ON [Sect. XI.
represent the time of a sidereal revolution of the Earth, setting off from
the perihelion ; we shall have iii the equation
n t = u — e sin. u
u = at the beginning of the revolution, and u = 2 t at the end of it.
Hence
n T = 2 ff.
But we have
n = a "" * V m = y/ m,
, 2^
.. V m = ^ .
The value of m is not exactly the same for the Earth as for the Comet,
for in the first case it expresses the sum of the masses of the sun and
earth ; whereas in the second it implies the sum of the masses of the sun
and comet : but the masses of the Earth and Comet being: much smaller
than that of the sun, we may neglect them, and suppose that m is the
same foi: all Planets and all Comets and that it expresses the mass of the
2 <K
sun merely. Substituting therefore for V m its value ttt in the preced
ing expression for t ; we shall have
, D^.T /^ 1 ^1 3 1 1
/=VV2l^2^+ 3^" 2M*
This equation contains none but quantities comparable with each other ;
it will give t very readily when v is known ; but to obtain v by means of
t, we must resolve a Cubic Equation, which contains only one real root.
We may dispense with this resolution, by making a table of the values of
V corresponding to those of t, in a parabola of which the perihelion dis
tance is unity, or equal to the mean distance of the earth from the sun.
This table will give the time corresponding to the anomaly v, in any par
abola of which D is the perihelion distance, by multiplying by D 2' , the
time which corresponds to the same anomaly in the Table. We also get
5
the anomaly v corresponding to the time t, by dividing t by D * , and
seeking in the table, the anomaly which corresponds to the quotient
arising from this division.
490. Let us now investigate the anomaly, corresponding to' the time t,
in an ellipse of great excentricity.
If we neglect quantities of the order a ^, and put 1 — e for a, the above
expression of t in terms of v in an ellipse, will give
D « V 2 f tan. ^ v + ^ tan.^ \ v )
Vm t+ (1 — e) tan.«^ V ^ — ^tan. *^ V ^tan. ♦iv}/ *
Then, find by the table of the motions of the comets, the anomaly cor
3
1)
t =
Book I.] NEWTON'S PRINCIPIA. 51
responding to the time t, in a parabola of which D is the perihelion dis
tance. Let U be this anomaly and U + x the true anomaly in an ellipse
corresponding to the same time, x being a very small angle. Then if we
substitute in the above equation U + x for v, and then transform the
second member into a series of powers of x, we shall have, neglecting the
square of x, and the product of x by 1 — e,
^^Dl^2j^^"U+itan.'iU + ^^^^ )
^ "" (+ ^—^ tan. i U {1 — tan.2 i U — I tan. ' ^ U})
But by supposition
D^ V 2
t = ^^ {tan. i U + ^ tan.= i V].
Therefore, substituting for x its sine and substituting for sin. * ^ U its
value (1 — COS. 2 i U) *, &c.
sin. X = y'^ (1 — e) tan. I U {4 — 3 cos. ^ i U — 6 cos. '^ ^ U] .
Hence, in forming a table of logarithms of the quantity
jL tan. 1 U {4 — 3 cos. ^  U — 6 cos. * i U]
it will be sufficient to add the logarithm of 1 — e, in order to have that of
sin. X ; consequently we have the correction of the anomaly U, estimated
from the parabola, to obtain the corresponding anomaly in a very excen
tric ellipse.
491. To find the masses of such planets as have satellites.
The equation
^ _ 2cra^
V m
gives a very simple method of comparing the mass of a planet, having sa
tellites, with that of the sun. In fact, M representing the mass of the sun,
if fi the mass of the planet be neglected, we have
T  ^'L'^^
V M
If we next consider a satellite of any planet /«.', and call its mass p, and
mean distance from the center of /a', h, and Tits periodic time, we shall
have
T
=
2 *r h 2
V/i' + p
. ^' + P
•• M
=
^3 T^
^3 ^ 2^2'
This equation gives the ratio of the sum of the masses of the planet fi
and its satellite to that of the sun. Neglecting therefore the mass of the
52 A COMMENTARY ON [Sect. XI.
satellite, as small compared with that of the planet, ov supposing their ra
tio known, we have the ratio of the mass of the planet to that of the sun.
492. To determine the Elements of Elliptical Motion.
After having exposed the General Theory of Elliptical Motion and
Method of Calculating by converging series, in the two cases of nature,
that of orbits almost circular, and the case of orbits greatly excentric, it
remains to determine the Elements of those orbits. In fact if we call V
the velocity of/* in its relative motion about M, we have
_ dx^ + dy' + dz'
dt^
and the last of the equations (P) of No. 478, gives
I ^ a J
To make m disappear from this expression, we shaU designate by U
the velocity which At would have, if it described about M, a circle whose
radius is equal to the unity of distance. In this hypothesis, we have
I = a = 1,
and consequently
Hence
U''= m.
If a J
This equation will give the semiaxis major a of the orbit, by means of
the primitive velocity of /* and of its primitive distance from M. But a is
positive in the ellipse, and infinite in the parabola, and negative in the
hypei'bola. Thus the orbit described by (t, is an ellipset a parabola, or %
perbola, according as Y is <C. = or 'P' than U ^  . It is remarkable
that the direction of primitive motion has no influence upon the species of
conic section.
To find the excentricity of the orbit, we shall observe that if* repre
sents the angle made by the direction of the relative motion of/* with the
radiusvector, we have
d p'
^ii_ — V^ cos ' g
dt'  ^ cos. E.
Substituting for V ^ its value m \ f » we have
d e' / 2 1 \ ,
Book L] NEWTON'S PRINCIPIA. 5^
But by 480
a dt'' • '
.•.a(le)^=.^sin.= s(^_i);
whence we know the excentricity a e of the orbit.
To find V or the true anomaly, we have
 a(l — e'')
" ~ 1 + e COS. V
a (1 — e) — ?
.•. COS. V = — ^ ? .
This gives the position of the Perihelion. Equations (f ) of No. 480 will
then give u and by its means the instant of the Planet's passing its peri
helion.
To gpt the position of the orbit, referred to a fixed plane passing
through the center of M, supposed immoveable, let <f> be the inclination of
the orbit to this plane, and ^ the angle which the radius f makes with the
Line of the Nodes. Let, Moreover, z be the primitive elevation of /«.
above the fixed plane, supposed known. Then we
shall have, CAD being the fixed plane, A D the
line of the nodes, A B = f , &c. &c.
z = B D . sin. <p ■= g sin. /3 sin. p ;
so that the inclination of the orbit will be known
when we shall have determined /3. For this pur
pose, let X be the known angle which the primitive
direction of the relative motion of /x makes with the fixed plane; then if
we consider the triangle formed by this direction produced to meet the
line of the nodes, by this last line and by the radius f, calling 1 the side
of the triangle opposite to /3, we have
e sin. Q
 sin. (/3 + i) ■
Next we have
2 • ,
J = sm. X
s
consequently
^ _ z sm. s
g sm. X — z COS. s
The elements of the Planetary Orbit being determined by these formu
las, in terms of g and z, of the velocity of the planet, and of the direction
of its motion, we can find the variation of these elements corresponding
D3
54 A COMMENTARY ON [Sect. XI.
to the supposed variations in the velocity and its direction ; and it will be
easy, by methods about to be explained, from hence to obtain the differ
ential variations of the Elements, due to the action of perturbing forces.
Taking the equation
V« = U»{ — ^ }.
In tlie circle a = g and .•.
so that the velocities of the planets in different circles are reciprocally as
the squares of their radii (see Prop. IV of Princip.)
In the parabola, a = oo ,
•■•^ = uvf
the velocities in the different points of the orbit, are therefore in this case
reciprocally as the squares of the radius vectors ; and the velocity at each
point, is to that which the body would have if it described a circle whose
radius = the radiusvector g, as V 2 : 1 (see 160)
An ellipse indefinitely diminished in breadth becomes a straight line,
and in this case V expresses the velocity of /t*, supposing it to descend in
a straight line towards M. Let i^ fall from rest, and its primitive dis
tance be g ; also let its velocity at the distance g' be \' ; the above expres
sion will give
g a I ^ a J
whence
Many other results, which have already been determined afler another
manner, may likewise be obtained from the above formula.
493. The equation
= d'^'+d y' + d"' _„,(£_!)
d t* ^ ^ a /
is remarkable from its giving the velocity independently of the excentricily.
It is also shown from a more general equation which subsists between the
axismajor of the orbit, the chord of the elliptic arc, the sum of the ex
treme radiusvectors, and the time of describing this arc.
To obtain this equation, we have
 a(l — e^)
^ "" 1 + e cos. v
Book I.] NEWTON'S PRINCIPIA. ' 55
g = a (1 — ie COS. u)
t = a '^ (u — e sm. a) ;
in which suppose j, v, u, and t to correspond to the first extremity of the
elliptic arc, and that ^', v', u', t' belong to the other extremity ; so that we
also have
a(le^)
^ 1 + e cos. v'
f' = a ( 1 — e COS. u')
5
t' = a* (u' — e sin. u').
Let now
then, if we take the expression of t from that of t', and observe thiit
sin. u' — sin. u = 2 sin. /S cos. /3'
we shall have
T = 2 a^ {13 — e sin. /3 cos. ^].
If we add them together takine notice that
cos. u' + COS. u = 2 cos. /3. cos. /3'
we shall get
R = 2 a (1 — e cos. ^ cos. /3').
Again, if c be the chord of the elliptic arc, we have
c^ = g ' + / 2 _ 2 J g' COS. (v — v')
but the two equations
a (1— e^) ., • .
= T— ^ ; e = a (1 — e cos. u)
1 + e COS. V *> ^
give these
COS. u — e . aVl — e^ sin. u
cos. V = a : sm. v =
and in like manner we have
cos. u' — e . , a VI — e^ sin. u'
COS. v' = a . ; : sm. v = . ;
§ i
whence, we get
gf'cos. (v — v') = a^(e — cos. u) (e — cos. u') + a'(l — e *) sin. u sin. u ;
and consequently
c' = 2a2(l — e'^)^! — sin. u sin. u' — cos. u cos. u'}
+ a "^ e * (cos. u — cos. u') * ;
D 4.
56 A COMMENTARY ON [Sect. XI.
But
sin. u sin. u' + cos. u cos. u' = 2 cos. ' ^ — 1
COS. u — COS. u' = 2 sin. jS sin. /S'
.. c« = 4 a« sin.  ^ (1 — e* COS.2 jS').
We therefore have these three equations, scil.
R = 2 a { 1 — e cos. /3 cos. /3' ; [
r=2a^Ji8 — e sin. ^ cos. /?'},' !
c« = 4 a^ sin. 2/3 (1 ~ e* cos. *8).
The first of them gives
, 2 a — R
e cos. /3' = ^r '
2 a COS. p
and substituting this value of e cos. /3' in tlie two others, we shall have
= 4a«tan.«/3cos.2(8_ ( ^ ^ ^ ^  .
These two equations do not involve the excentricity e, and if in the
first we substitute for S its value given by the second, we shall get Z* in a
function c, R, and a. Thus we see that the time T depends only on the
semiaxis major, the chord c and the sum R of the extreme ra(^us
vectors.
If we make
_ 2 a — R + c , _ 2 a — R — c
z_ ^ ;z — ;
the last of the preceding equations will give
cos. 2 /3 z= z z' 4 \/ (1 — z^).(l —z'^) ;
whence
2 /3 = cos.  ' z' — cos. ~ ' z
(for cos. (A — B) = cos. A cos. B + sin. A sin. B).
Consequently
sin. (cos.  ' zO — sin. (cos.  ' z)
tan.^ = — i ,'^^ ^ i
we have also
2a~ R
z + z' = •
^ a
Hence the expression of T will become, observing that if T is the du
ration of the sidereal revolution, whose mean distance from the sun is
taken for unity, we have
Book I.] NEWTON'S PRINCIPIA. 57
T = 2 cr,
T = K— {cos.' z' — COS.* z — sin. (cos.' z') + sin.(cos'z) ... (a)
Since the same cosines may belong to many arcs, this expression is
ambiguous, and we must take care to distinguish the arcs which corre
spond to z, z'.
In the parabola, the semiaxis major is infinite, and we have
cos.  ' z' — sin. (cos.  ' z') =  ( ■ — \ ^ .
And making c negative we shall have the value of
cos. — ' z — sin. (cos. — ' z) ;
hence the formula (a) will give the time T employed to describe the arc
subtending the chord c, scil.
^ = ^^^' + / + ^=P(f + / c) ^ ;
the sign — being taken, when the two extremities of the parabolic arc are
situated on the same side of the axis of the parabola.
Now T being = 365.25638 days, we have
^r = 9. 688754 days.
12 ^ •'
The formula (a) gives the time of a body's descent in a straight line to
wards the focus, beginning from a given distance; for this, it is suffi
cient to suppose the axisminor of the ellipse indefinitely diminished. If
we suppose, for example, that the body falls from rest at the distance 2 a
from the focus and that it is required to find the time (T) of falling to
the distance c, we shall have
R = 2aff, P = 2a — c
whence
/ 1 c — a
z' = — 1, z =
a
and the formula gives
^ a«Tf iCja ; 2 a c — c \
T = —T — i T — COS.  ^ f ', 2 — \ .
There is, however, an essential difference between elliptical motion to
wards the focus, and the motion in an ellipse whose breadth is indefinite
ly small. In the first case, the body having arrived at the focus, passes
beyond it, and again returns to the same distance at which it departed ;
but in the second case, the body having arrived at the focus immediately
returns to the point of departure. A tangential velocity at the aphelion,
58 A COMMENTARY ON [Sect. XI.
liowever small, suffices to produce this difference which has no influence
upon the time of the body's descent to the center, nor upon the ve
locity resolved parallel to the axismajor. Hence the principles of the
7th Section of Newton give accurately the Times and Velocities, although
they do not explain all tlie circumstances of motion. For it is clear that
if there be absolutely no tangential velocity, the body having reached the
center of force, will proceed beyond it to the same distance from which it
commenced its motion, and then return to the center, pass through it,
and proceed to its first point of departure, the whole being performed in
just double the time as would be required to return by moving in the in
definitely small ellipse.
494. Observations not conducting us to the circumstances of the pri
mitive motion of the heavenly bodies ; by the formulas of No. 492 we
cannot determine the elements of their orbits. It is necessary for this
end to compare together their respective positions observed at different
epochs, which is the more difficult from not observing them from the
center of their motions. Relatively to the planets, we can obtain, by
means of their oppositions and conjunctions, their Heliocentric Longitude.
This consideration, togetlier with that of the smallness of the excentricity
and inclination of their orbits to the ecliptic, affords a very simple method
of determining their elements. But in the present state of astronomy,
the elements of these orbits need but very slight corrections ; and as the
variations of the distances of the planets from the earth are never so great
as to elude observation, we can rectify, by a great number of observations,
the elements of their orbits, and even the errors of which the observa
tions themselves are susceptible. But with regard to the Comets, this is
not feasible ; we see them only near their perihelion : if the observations
we make on their appearance prove insufficient for the determination of
their elements, we have then no means of pursuing them, even by thought,
through the immensity of space, and when after the lapse of ages, they
again approach the sun, it is impossible for us to recognise them. It be
comes therefore important to find a method of determining, by observa
tions alone during the appearance of one Comet, the elements of its orbit.
But this problem considered rigorously surpasses the powers of analysis,
and we are obliged to have recourse to approximations, in order to obtain
the first values of the elements, these being afterwards to be corrected to
any degree of accuracy which the observations permit.
If we use observations made at remote intervals, the eliminations will
lead to impracticable calculations ; we must therefore be content to con
Book lO NEWTON'S PRINCIPIA. 59
sidei only near observations ; and with this restriction, the problem is abun
dantly difficult.
It appears, that instead of directly making use of observations, it is
better to get from them the data which conduct to exact and simple re
sults. Those in the present instance, which best fulfil that condition, are
the geocentric longitude and latitude of the Comet at a given instant, and
their first and second differences divided by the corresponding powers of
the element of time ; for by means of these data, we can determine rigo
rously and with ease, the elements, without having recourse to a single
integration, and by the sole consideration of the differential equations of
the orbit. This way of viewing the problem, permits us moreover, to
employ a great number of near observations, and to comprise also a con
siderable interval between the extreme observations, which will be found
of great use in diminishing the influence of such errors, as are due to ob
servations from the nebulosity by which Comets are enveloped. Let us
first present the formulas necessary to obtain the first differences, of the
longitude and latitude of any number of near observations ; and then de
termine the elements of the orbit of a Comet by means of these differences ;
and lastly expose the method which appears the simplest, of correcting
these elements by three observations made at remote intervals.
495. At a given epoch, let a be the geocentric longitude of a Comet,
and 6 its north geocentric latitude, the south latitudes being supposed ne
gative. If we denote by s, the number of days elapsed from this epoch,
the longitude and latitude of the Comet, after that interval, will, by using
Taylor's Theorem (481), be expressed by these two series
/d ttN s ■' /d ^ a\ , o
We must determine the values of
by means of several observed geocentric longitudes and latitudes. To do
this most simply, consider the infinite series which expresses the geocen
tric longitude. The coefficients of the powers of s, in this series, ought to
be determined by the condition, that by it is represented each observed
longitude ; we shall thus have as many equations as observations ; and i(
their number is n, we shall be able to find from them, in series, the n
60 A COMMENTARY ON [Sect. XL
quantities a, (5—) , &c. But it ought to be observed that s being sup
posed very small, we may neglect all terms multiplied by s ", s " + ', &c.
which will reduce the infinite series to its n first terms ; which by n ob
servations we shall be able to determine. These are only approximations,
and their accuracy will depend upon the smallness of the teems which are
omitted. They will be more exact in proportion as s is more diminutive,
and as we employ a greater number of observations. The theory of inter
polations is used therefore To find a rational and intega' function ofs such,
that in substituting therein fiar s the number of days xsohich correspond to each
observation, it shall become the observed longitude.
Let iS, /3', ^", &c. be the observed longitudes of the comet, and by
i, i', i", &c. the corresponding numbers of days from the given epoch, the
numbers of the days prior to the given epoch being supposed negative.
If we make
I" 1
«*/3' — d«
= 338; &c.
1'" — 1
&c. ;
the required functions will be
^+ (s — i).a/3(s — i)(s — i0.3'/3+(s— i)(s — i')(s — i'05^ft&c.
for it is easy to perceive that if we make successively s = i, s = i', s = i", &c.
it will change itself into jS, 13', ^", &c.
Again, if we compare the preceding function with this
" + ^ • (j^) + 8 • (d70 + «"=•
we shall have by equating coefficients of homogeneous terms.
a=iS — ia.3+i.i'.5 2/3 — i.i'. i"a3/34&c.
(^)=a/3— (i+i0 3'/3t(ii'+ii"+i'i'0S'/3 — &c.
i(^,)=a^/3(i+i'+i")a3/3+&c.
The higher differences of a will be useless. The coefHcients of these
expressions are alternately positive and negative ; the coefficient of 5 ' /3
is, disregarding the sign, the product of r and r together of r quantities
i, i', . . . . i ^''5 in the value of « ; it is the sum of the products of the
Book L] NEWTON'S PRINCIPIA. 61
same quantities, r — 1 together in the value of \t—\ ; lastly it is the sum
of the products of these quantities r — 2, together in the value of
2 VdsV*
If 7, 7', 7", &c. be the observed geocentric latitudes, we shall have the
values of 6, ( j— ) > ( j — 2) > &c. by changing in the preceding expressions
for a (t— )j ( i^")' ^^* t^^^ quantities /3, 13' , /3" into 7, /, 7".
These expressions are the more exact, the greater the number of ob
servations and the smaller the intervals between them. We might,
therefore, employ all the near observations made at a given epoch, pro
vided they were accurate ; but the errors of which they are always sus
ceptible will conduct to imperfect results. So that, in order to lessen the
influence of these errors, we must augment the interval between the ex
treme observations, employing in the investigation a greater number of
them. In this way with five observations we may include an interval of
thirtyfive or forty degrees, which would give us very near approximations
to the geocentric longitude and latitude, and to their first and second
differences.
If the epoch selected were such, that there were an equal number of
observations before and after it, so that each successive longitude may
have a corresponding one which succeeds the epoch. This condition will
give values still more correct of a, (r—^ and ( , — ^) , and it easily appears
that new observations taken at equal distances from either side of the epoch,
would only add to these values, quantities which, with regard to their last
A 2
terms, would be as s^ (3 — g^to a. This symmetrical arrangement takes
place, when all the observations being equidistant, we fix the epoch at
the middle of the interval which they comprise. It is therefore advanta
geous to employ observations of this kind.
In general, it will be advantageous to fix the epoch near the middle of
this interval ; because the number of days included between the extreme
observations being less considerable, the approximations will be more con
vergent. We can simplify the calculus still more by fixing the epoch at
the instant of one of the observatiqns ; which gives immediately the values
of a, and 0.
62 A COMMENTARY ON [Sect. XI
When we shall have determined as above tlie values of
(di)' (dsO' (rs)'""Md7^)
we shall then obtain as follows the first and second differences of a, and i^
divided by the corresponding powers of the elements of time. If we neg
lect the masses of the planets and comets, that of the sun being the unit
of mass ; if, moreover, we take the distance of the sun from the earth for
the unit of distance ; the mean motion of the earth round the sun will
be the measure of the time t. Let therefore X be the number of se
conds which the earth describes in a day, by reason of its mean sidereal
motion ; the time t corresponding to the number of days will be X s ; we
shall, therefore, have
(d a\ 1 /d a\
dly ~ T \d~s)
VdTv*" x'^VdTv*
Observations give by the Logarithmic Tables,
log. X = 4. 0394622
and also
log. X 2 = log. X f log.
R
R being the radius of the circle reduced to seconds ; whence
log.X^zr 2. 2750444;
.*. if we reduce to seconds, the values of (t— ) > and of (. — ^j , we shall
have the logarithms of ( i") » and of (g^)by taking from the logarithms
of these values the logarithms of 4. 039422, and 2. 2750444. In like
manner we get the logarithms of ( .) , (gpj) j ^^er subtracting the
same logarithms, from the logarithms of their values reduced to seconds.
On the accuracy of the values of
depends that of the following results ; and since their formation is very
simple, we must select and multiply observations so as to obtain them with
the greatest exactness possible. We shall determine presently, by means
of these values, the elements of the orbit of a Comet, and to generalize
these results, we shall
Book I.] NEWTON'S PRINCIPIA.
63
496. Investigate the motion of a system of bodies sollicited by any forces
•whatever.
Let X, y, z be the rectangular coordinates of the first body ; x', y', z'
those of the second body, and so on. Also let the first body be sollicited
parallel to the axes of x, y, z by the forces X, Y, Z, which we shall sup
pose tend to diminish these variables. In like manner suppose the second
body sollicited parallel to the same axes by the forces X', Y', Z', and so
on. The motions of all the bodies will be given by differential equations
of the second order
d^x
dt^
+
X;
=
 dt^ + ^'
=
d^z
dt^
+
d'x'
 dt^
+
X';
;
d^ v'
 dt^ + ^
>
d^
z'
~ d
t*
&c. = &c.
If the number of the bodies is n, that of the equations will be 3 n ; and
their finite integrals will contain 6 n arbitrary constants, which wUl be the
elements of the orbits of the different bodies.
To determine these elements by observations, we shall transform the
coordinates of each body into others whose origin is at the place of the
observer. Supposing, therefore, a plane to pass through the eye of the
observer, and of which the situation is always parallel to itself, whilst the
observer moves along a given curve, call r, r' r", &c. the distances of
the observer from the different bodies, projected upon the plane ;
a, a', a", &c. the apparent longitudes of the bodies, referred to the same
plane, and 6, ^, ^', &c. their apparent latitudes. The variables x, y, z
will be given in terms of r, a, ^, and of the coordinates of the observer.
In like mannei', x', y', z' will be given in functions of r', a', 6\ and of the
coordinates of the observer, and so on. Moreover, if we suppose that the
forces X, Y, Z ; X^ Y', Z', &c. are due to the reciprocal action of the
bodies of the system, and independent of attractions ; they will be given in
functions of r, r', r", &c. ; a, a', a", &c. ; d, ^', 6", &c. and of known quan
tities. The preceding differential equations will thus involve these new
variables and their first and second differences. But observations make
known, for a given instant, the values of
"' (di)' (arrO' Mdi)' (diO' "'(df)' ^'
There will hence of the unknown quantities only remain r, r', x"j &c.
and their first and second differences. These unknowns are in number
3 n, and since we have 3 n differential equations, we can determine them.
64 A COMMENTARY ON [Sect. XL
At the same time we shall have the advantage of presenting the first and
second differences of r, r', v", &c. under a linear form.
The quantities os, tf, r, a', ^, r', &c. and their first differences divided by
d t, being known ; we shall have, for any given instant, the values of
X, y, z, x', y', z'. Sec. and of their first differences divided by d t. If we
substitute these values in the 3 n finite integrals of the preceding equa
tions, and in the first differences of these integrals ; we shall have 6 n
equations, by means of which we shall be able to determine the 6 n arbi
trary constants of the integrals, or the elements of the orbits of the dif
ferent bodies.
497. To apply this method to the motion of the Comets,
"We first observe that the principal force which actuates them is the
attraction of the sun ; compared with which all other forces may be ne
glected. If, however, the Comet should approach one of the greater
planets so as to experience a sensible perturbation, the preceding method
will still make known its velocity and distance from the earth ; but this
case happening but very seldom, in the following researches, we shall ab
stain from noticing any other than the action of the sun.
If the sun's mass be the unit, and its mean distance from the earth the
unit of distance; if, moreover, we fix the origin of the coordinates
X, y, z of a Comet, whose radiusvector is g ; the equations (0) of No. 475
will become, neglecting the mass of the Comet,
d^ X
dt* ^ ^
" dt^ ^ ^\
^^d^z . z
(k)
dt^ ^ e
Let the plane of x, y be the plane of the ecliptic. Also let the axis of
X be the line drawn from the center of the sun to the first point of aries,
at a given epoch ; the axis of y the line drawn from the center of the sun
to the first point of cancer, at the same epoch ; and finally the positive
values of z be on the same side as the north pole of the ecliptic. Next
call x', y the coordinates of the earth and R its radiusvector. This be
ing supposed, transfer the coordinates x, y, z to others relative to the
observer ; and to do this let a be the geocentric longitude, and r its dis
tance from the center of the earth projected upon the ecliptic ; then we
shall have
X = x' + r COS. a ; y = y + r sin. a ; z = r tan. 6.
Book I.] NEWTON'S PRINCIPIA. 65
If we multiply the first of equations (k) by sin. a, and take from the re
sult tlie second multiplied by cos. a, we shall have
d^x d^y.x sin. a — y cos. a
= sm. « ^  COS. a. ^f + ^ ;
whence we derive, by substituting for x, y their values given above,
d ^ x' d  y' x' sin. a — y cos. a
= sm. a. ^j^ _ COS. «. ^ + ^
d r\ /da\ /d^a>
/a r\ /aa\ /ci^av
^•(dT)(dT)HdT5)
The earth being retained in its orbit like a comet, by the attraction of
the sun, we have
which give
d^' x^ dV 2:
" ~ dt^ ^ R^' dt* ^ R =
d ^ x' d '^ y' y' cos. a — x' sin. a
sm. « j^  COS. «. ^ = I ^^3 ^
We shall, therefore, have
= (/ COS. «  X' sin. «) I ^^ _ ^} _ z (^) . (^^)  r (i^,) .
Let A be the longitude of the earth seen from the sun ; we shall have
x' = R COS. A ; y' = R sin. A ;
therefore
y' COS. a — x' sin. a = R sin. (A — «) ;
and the preceding equation will give
/d'a\
/drx _ Rsin.(A— ;a) fj \\ ^' Vdt V
(dJ= ,(a»^ Ir^ e) 3(d«) ■• • n
Now let us seek a second expression for (t — j . For this purpose we
will multiply the first of equations (k) by tan. 6 . cos. a, the second by
tan. 6 sin. a, and take the third equation from the sum of these two pro
ducts ; we shall thence obtain
f d^ X d^ y)
= tan.^cos.«3P^ + sin.a_^
^ X COS. a 4 y sin. a d ^ z z
+ tan. 6 . \^ .f— „ r .
^ f' dt= f'
This equation will become by substitution for x, y, z
= tan.^(^ +^^)cos.a+(^V+ fO^^"'"/
Vol. II. E
66 A COMMENTARY ON [Sect. XI.
_ ^(<u)Cn) _ .GrT^^ Kr.)^ (^")%a„.4 "
COS.  tf (^ COS. ^ d COS. ^ ^ \a t/ j
But
CdH^ +p;^"^^+Cdt^ + f3)si«.a=(x COS. a +/ sin. a) Q 3  j^,)
= Rcos.(A«){Ljl^^j;
Therefore,
(dl) =i'i 75:^7+ ^31)""''+ TdT. \
v \<i i) U J J
R sin. ^ COS. tf COS. (A — a) f 1 1 )^
"^ Vdi;
If we take this vahie of (fj from the first and suppose
^,_ (dt) (dT')^rt) (at^)+<dt) (?t) "'"•^+(dT) '""•'"°^''
^ j^ sin. ^ COS. 6 cos. (A — ■ ) + ( i .) sin. (A — a)
we shall have
^•{pi[^} (')
The projected distance r of the comet from the earth, being always po
sitive, this equation shows that the distance g of the comet from the sun,
is less or greater than the distance R of the sun from the earth, according
as fi' is positive or negative ; the two distances are equal if /i' = 0.
By inspection alone of a celestial globe, we can determine the sign of
(jif ; and consequently whether the comet is nearer to or farther from the
Earth. For that purpose imagine a great circle which passes through
two Geocentric positions of the Comet infinitely near to one another.
Let y be the inclination of this circle to the ecliptic, and X the longitude
of its ascending node ; we shall have
tan. 7 sin. (a — x) = tan. 6 ;
whence
d 6 sin. (a — x) = a a sin. 6 cos. 6 cos. (a — X).
Book L] NEWTON'S PRINCIPIA. 67
Differentiating, we have, also
» = (.Tt) (Tp)(rt) (dT«) + Hdi^ irJ "•"• '
d «>
/a a\ •
Sin. 6 COS. ^;
d  ^^ being the value of d * 6, which would take place, if the apparent mo
tion of the Comet continued in the great circle. The value of /j/ thus be
comes, by substituting for d 6 its value
d a sin. S cos. 6 cos. (a — X)
sin. (a — X) *
''■=i(g)(g)l!^^.
sin. 6 cos. 6 sin. (A — X)
The function  . ' \ ;:' is constantly positive ; the value of u, is there
sm. 6 cos. 6 J f i f
((J 2 ^ /d* ^ \
TT2) — (tTz)^^^ ^^^ same or
a different sign from that of sin. (A — X). But A — X is equal to two
right angles plus the distance of the sun from the ascending node of the
great circle. Whence it is easy to conclude that (j/ will be positive or
negative, according as in a third geocentric position of the comet, inde
finitely near to the two first, the comet departs from the great circle on
the same or the opposite side on which is the sun. Conceive, therefore,
that we make a great circle of the sphere pass through the two geocentric
positions of the comet ; then according as, in a third consecutive geocen
tric position, the comet departs from this great circle, on the same side as
the sun or on the opposite one, it will be nearer to or farther from llie
sun than the Earth. If it continues to appear in this great circle, it will
be equally distant from both ; so that the different deflections of its ap
parent path points out to us the variations of its distance from the sun.
To eliminate ^ from equation (3), and to reduce this equation so as to
contain no other than the unknown r, we observe that g^ = x^ + y^ + Z*
in substituting for x, y, z, their values in terms of
r, a, and ^;
and we have
f2 = x'^' + y'^+ 2rb'cos. a + y s\n. a} + ^^^ I
but we have
x' = R cos. A, y' = R sin. A ;
.. P' = — ^, + 2 R r cos. (A — a) + R';
* cos. ^ d ^
E2
68 A COMMENTARY ON [Sect. XL
But
x' = R COS. A ; y' = R sin. A
.. f * = ^ , , + 2 R r COS. (A — a) + R ^
* COS. ^6 ^ ' ^
If we square the two members of equation (3) put under this form
f'J/*'R«r + 1]= R3
we shall get, by substituting for g *,
{^d + 2 R r cos. (A  «) + ^'Y'^^' ^' ^ + ^^'= ^' • • • ("^^
an equation in which the only unknown quantity is r, and which will rise
to the seventh degree, because a terra, of the first member being equal to
R ^, the whole equation is divisible by r. Having thence determined r,
we shall have f,—) by means of equations (1) and (2). Substituting, for
example, in equation (1), for j — p, its value ^ , given by equation
(3) ; we shall have
(d:)=,7^{(^")+'''^'(^°)}
The equation (4) is often susceptible of many real and positive roots j
reducing it and dividing by r, its last term will be
2 R * COS. 6 ^[/ct' R' + 3 COS. (A — a)].
Hence the equation in r being of the seventh degree or of an odd de
gree, it will have at least two real positive roots if /i' R ^ + 3 cos. (A — a)
is positive; for it ought always, by the nature of the problem, to have
one positive root, and it cannot then have an odd number of positive
roots. Each real and positive value of r gives a different conic section,
for the orbit of the comet ; we shall, therefore, have as many cun^es
which satisfy three near observations, as r has real and positive values ;
and to determine the true orbit of the comet, we must have recourse to a
new observation.
498. The value of r, derived from equation (4) would be rigorously
exact, if
were exactly known ; but these quantities are only approximate. In fact,
by the method above exposed, we can approximate more and more, mere
ly by making use of a great number of observations, which presents the
advantage of considering intervals sufficiently great, and of making the
errors arising from observations compensate one another. But this
Book I.] NEWTON'S PRINCIPJA. 69
method has the analytical inconvenience of employing more than three
observations, in a problem where three are sufficient. This may be
obviated, and the solution rendered as approximate as can be wished by
' three observations only, after the following manner.
Let a and 6, representing the geocentric longitude and latitude of the
intermediate; if we substitute in the equations (k) of the preceding
No. instead of x, y, z their values x' + r cos. « ; y' + r sin. a ; and
r tan. 6 ; they will give ( jr2) j ( j ^ 2) ^"<1 ( j — "2) ^^ functions of r, a, and
6, of their first differences and known quantities. If we differentiate these,
we shall have f , — 3 ^ , (173) and (^tts) ^ terms of r, a, 6, and of their
first and second differences. Hence by equation (2) of 497 we may eli
minate the second difference of r by means of its value and its first differ
(1^ 3 ^ ^A 3 A
1 — 3 ) ' (tts) '
and eliminating the differences of a, and of 6 superior to second differences,
and all the differences of r, we shall have the values of
(dT'} ' (dT*) ' ^^' ^" ^^^^^ °^
/d a\ /d'^ax ^ /d 6\ /d*rf\
^•'"' (di)' (drO'^'Cdi)' (dT^)'
this being supposed, let
be the three geocentric observed longitudes of the Comet; 6^, 0, ^ its
three corresponding geocentric latitudes ; let i be the number of days
which separate the first from the second observation, and i' the interval
between the second and third observation ; lastly let X be the arc which
the earth describes in a day, by its mean sidereal motion; then by (481)
we have
a, = «z.x(_)+ ^^(_)_^(^3)+ &c.;
a = « +.^x(^^)+ ^ ^ (^)+ i;2;3(dY3)+ &c.,
'' = ' ^ Hdi) +X2(drO 1:2:3(3x0 + ^^•>
^ Vd t/ ^ 1. 2 Vd tV ^ 1.2.3 \d tV ^ ^*^'
70 A COMMENTARY ON [Sect. XL
If we substitute in these series for
their values obtained above, we shall have four equations between the
five unknown quantities
/d ax /d * a\ /d ^\ /cl ' ^\
""'Vdl/' VdTV' Vdl;' VdTV*
These equations will be the more exact in proportion as we consider a
greater number of terms in the series. We shall thus have
KdV' VdTV' ^dl/' ^dTV
in terms of r and known quantities ; and substituting in equation (4) of
the preceding No. it will contain the unknown r only. As to the rest,
this method, which shows how to approximate to r by employing three
observations only, would require in practice, laborious calculations, and
it is a more exact and simple process to consider a greater number of ob
servations by the method of No. 495.
499. When the values of r and (i, ) shall be determined, we shall have
those of
^'>''"'(^)' (aT)^"^(^)'
by means of the equations
X = 11 cos. A + r cos. a
y = R sin. A + r sin. «
z = r tan. 6
and of their differentials divided by d t, viz.
iPd = C!i^) ^  ^^^y ^ + (ai)  «' O 
/d z\ /dr\ ^ . , Vd t/
The values of (,^) and of (i^) are given by the Theory of the
motion of the Earth :
To facilitate the investigation let E be the excentricity of the earth's
Book I.] NEWTON'S PRINCIPIA. 71
orbit, and H the longitude of its perihelion; then by the nature of
elliptical motion we have
/dAx _ V(lE^. 1E'
^dT; R^ ' — rt  1 + j^cos. (A — H)*
These two equations give
/d Rx _ E sin. (A — H)
\dt^  V (1 — E*) *
Let R' be the radius vector of the earth corresponding to the longitude
A of this planet augmented by a right angle ; we shall have
1 E"^
^ " 1 — Esin.(A — H)'
whence is derived
p • / A ux R' — 1 + E 2
Esm. (A — H) = ^^ ;
/d Rn _ R^ + E ^ — 1
•*• Vdt)"" R'— V (1— .E«)*
If we neglect the square of the excentricity of the earth's orbit, which is
very small, we shall have
the preceding values of ( ^ — j and (r^) wiU hence become
/dx\ ,T»/ ,N A sin. A /d r\ /da\ .
/^y\ /T3/ i\ • A . COS. A , /dr\ . , /dax
R, R', and A being given immediately by the tables of the sun, thfe esti
mate of the six quantities x, y, z, (r— ) » (dt ) ' (ht) ^^^^ ^^ ^^^^
when r and (r— ) shall be known. Hence we derive the elements of the
orbit of the comet after this mode.
The indefinitely small sector, which the projection of the radiusvector
and the comet upon the plane of the echptic describes during the element
of time d t, is o — » ^^^ ^* ^^ evident that this sector is posi
live or negative, according as the motion of the comet is direct or retro
grade. Thus in forming the quantity x (r^) — y (r — ), it will indicate
by its sign, the direction of the motion of the comet
£4
72 A COMMENTARY ON [Sect. XI.
To determine the position of the orbit, call (p its inclination to the
ecliptic, and I the longitude of the node, which would be ascending if the
motion of the comet were direct or progressive. We shall have
z = y COS. I tan. p — x sin. I tan. p
These two equations give
tan. I =
tan. f =
■M^^~^iV^)}'
sm
Wherein since (p ought always to be positive and less than a right
angle, the sign of sin. I is known. But the tangent of I and the sign of
its sine being determined, the angle I is found completely. This angle
is the longitude of the ascending node of the orbit, if the motion is pro
gressive ; but to this we must add two right angles, in order to get the
longitude of the node when the motion is retrograde. It would be more
simple to consider only progressive motions, by making vary p, the in
clination of the orbits, from zero to two right angles; for it is evident that
then the retrograde motions correspond to an inclination greater than a
right angle.
In this case, tan. 9 has the same sign as x ( t^) — y (t — ) , which will
determine sin. I, and consequently the angle I, which always expresses
the longitude of the ascending node.
If a, a e be the semiaxis major and the excentricity of the orbit, we
have (by 492) in making m = I,
1 _ 2 /djcx /dyx' /dz\'
a " 7 ~~\i\i) ^dt/ Vd t/ '
,(,_., = . ,_.J_{.() + ,()+.(^^)}=.
The first of these equations gives the semiaxis major, and the second
the excentricity. The sign of the function x ( j^) + y ( j^^ ) + z ( j^)
shows whether the comet has already passed its perihelion ; for it ap
proaches if this function is negative; and in the contrary case, the comet
recedes from that point.
Book I.] NEWTON'S PRINCIPIA. 73
Let T be the interval of time comprised between the epoch and pas
sage of the comet over the perihelion ; the two fii'st of equations (f) (480)
will give, observing that m being supposed unity we have n = a ~2^ ,
f = a (1 — e COS. u)
5.
T = a ^ (u — e COS. u).
The first of these equations gives the angle u, and the second T. This
time added to or subtracted from the epoch, according as the comet ap
proaches or leaves its perihelion, will give the instant of its passage over
this point. The values of x, y, determine the angle which the projection
of the radiusvector § makes with the axis of x ; and since we know the an
gle I, formed by this axis and by the line of the nodes, we shall have the
angle which this last line forms with the projection of g ; whence we derive by
means of the inclination p of the orbit, the angle formed by the line of the
nodes and the radius g. But the angle u being known, we shall have by
means of the third of the equations (f), the angle v which this radius forms
with the line of the apsides ; we shall therefore have the angle comprised
between the two lines of the apsides and of the nodes, and consequently,
the position of the peiihelion. All the elements of the orbit will thus be
determined.
500. These elements are given, by the preceding investigations, in terms
of r, (t:) and known quantities ; and since f r ^ is given in terms of r
by No. 497, the elements of the orbit will be functions of r and known
quantities. If one of them were given, we should have a new equation,
by means of which we might determine r ; this equation would have a
common divisor with equation (4) of No. 497; and seeking this di
visor by the ordinary methods, we shall obtain an equation of the first
degree in terms of r ; we should have, moreover, an equation of condition
between the data of the observations, and this equation would be that
which ought to subsist, in order that the given element may belong to the
orbit of the comet.
Let us apply this consideration to the case of nature. First suppose
that the orbits of the comets are ellipses of great excentricity, and are
nearly parabolas, in the parts of their orbits in which these stars are
visible. We may therefore without sensible error suppose a = oo, and
consequently  = 0; the expression for  of the preceding No. will there
fore give
74 A COMMENTARY ON [Sect. XL
^ _ 2 dx' + dy'^ + tiz''
If we then substitute for (n — \ \XJ ^"^ (tt) ^^^^^' values found in
the same No., vre sliall have after all the reductions and neglecting the
square of 11' — 1,
V. d t COS. '^ 6 J
+ 2(^) {(R'i)«>(A«)^R— '}'5)
Substituting in this equation for (r—) its value
found in No. 497, and then making
+ {u„...(^:)+,.n..si„.(A_«,!%Ji}'
and
C
7?1~\ {\— ^(Rl)cos.(Aa)}
\dtJ
+ ^(ai) {(^'  ') ''"■ (^  "' + ^^'} .
we shall have
= Br'+Cr+ ji,~
and consequently
r{Br»+Cr + i,}'=4.
This equation rising only to the sixth degree, is in that respect, more
Book L] NEWTON'S PRINCIPIA. VS
simple than equation (4) of No. (497) ; but it belongs to the parabola
alone, whereas the equation (4) equally regards every species of conic
section.
501. We perceive by the foregoing investigation, that the determina
tion of the parabolic orbits of the comets, leads to more equations than
unknown quantities; and that, therefore, in combining these equations in
different ways, we can form as many different methods of calculating the
orbits. Let us examine those which appear to give the most exact re
sults, or which seem least susceptible of the errors of observations.
It is principally upon the values of the second differences (^ — ^^ and
(d ^ ^\ . ■ .
T — 2 j, that these errors have a sensible influence. In fact, to deter^iine
them, we must take the finite differences of the geocentric longitudes and
latitudes of the comet, observed during a short interval of time. But
these differences being less than the first differences, the errors of obser
vations are a greater aliquot part of them ; besides this, the formulas of
No. 495 which determine, by the comparison of observations, the values
°*^"' ^' (dl)' (dl)' (dT^) ^"^ (dT*) ^^^'^ ^^^^ greater precision the
four first of these quantities than the two last. It is, therefore, desirable
to rest as little as possible upon the second differences of a and 6; and
since we cannot reject both of them together, the method which employs
the greater, ought to give the more accurate results. This being granted
let us resume the equations found in Nos. 497, &c.
f " = ^ + 2 R r cos. (A — a) + R ^.
* cos. ^6
/d_^\
/drN _ Rsin^j(A^a) JJ l_ ^'WtV n.
^ \Tt) vht;
d «>
Cf?) ,. (^) sin. ^ cos. A
/d vn 1 I ^d t V , ^ /d ^n ^ . Vd t/ f
R sin. 6 cos. 6 cos. (A — «) f 1 1 \
^(^)
rd6y
<'=(a^t)' + a' + {«ai)'' + ^'J}'
76 A COMMENTARY ON [Sect. XI.
42.(^^j{(R'I)sin.(A«)4^^^^iA^^}
+ k:^
1 2
S
d
(rl o\
5 — g J , we consider only the first, second and fourtn
of those equations. Eliminating (tt) from the last by means of the
second, we shall form an equation which cleared of fractions, will contain
a term multiplied by ^ ® r *, and other terms affected with even and odd
powers of r and g. If we put into one side of the equation all the terms
affected with even powers of §, and into the other all those which involve
its odd powers, and square both sides, in order to have none but even
powers of §, the term multiplied by ^ ^ r * will produce one multiplied by
g" r*. Substituting, therefore, instead of ^% its value given by the first
of equations (L), we shall have a final equation of the sixteenth degree in
r. But instead of forming this equation in order afterwards to resolve it,
it will be more simple to satisfy by trial the three preceding ones.
T — jj, we must consider the first, third and fourth
of equations (L). These three equations conduct us also to a final equa
tion of the sixteenth degree in r; and we can easily satisfy by trial.
The two preceding methods appear to be the most exact, which we can
employ in the determination of the parabolic orbits of the comets. It is
at the same time necessary to have recourse to them, if the motion of the
comet in longitude or latitude is insensible, or too small for the errors of
observations sensibly to alter its second difference. In this case, we must
reject that of the equations (L), which contains this difference. But al
though in these methods, we employ only three equations, yet the fourth
is useful to determine amongst all the real and positive values of r, which
satisfy the system of three equations, that which ought to be selected.
502. The elements of the orbit of a comet, determined by the above
process, would be exact, if the values of a, and their first and second
differences, were rigorous ; for we have regarded, after a very simple
manner, the excentricity of the terrestrial orbit, by means of the radius
vector R' of the earth, corresponding to its true anomaly + a right an
gle ; we are therefore permitted only to neglect the square of this excen
Book L] NEWTON'S PRINCIPIA. 77
tricity, as too small a fraction to produce by its omission a sensible influ
ence upon the results. But tf, « and their diflferences, are always suscep
tible of any degree of inaccuracy, both because of the errors of observa
tions, and because these diiferences are only obtained approximately. It
is therefore necessary to correct the elements, by means of three distant
observations, which can be done in many ways ; for if we know nearly,
two quantities relative to the motion of a comet, such that the radiusvec
tor corresponding to two observations, or the position of the node, and
^ the inclination of the orbit ; calculating the observations, first with these
quantities and afterwards with others differing but little from them, the
law of the differences between the results, will easily show the necessary
corrections. But amongst the combinations taken two and two, of the
quantities relative to the motion of comets, there is one which ought to
produce greatest simplicity, and which for that reason should be selected.
It is of importance, in fact, in a problem so intricate, and complicated, to
spare the calculator all superfluous operations. The two elements which
appear to present this advantage, are the perihelion distance, and the
instant when the comet passes this point. They are not only easy to be
derived from the values of r and (t— ) ; but it is very easy to correct them
by observations, without being obliged for every variation which they
undergo, to determine the other corresponding elements of the orbit.
Resuming the equation foimd in No. 492
a{l — e') = 2s—' —
a dt^ '
a (1 — e*) is the semiparameter of the conic section of which a is the
semi axismajor, and a e the excentricity. In the parabola, where a is
infinite, and e equal to unity, a (1 — e^) is double the perihelion dis
tance : let D be this distance : the preceding equation becomes relatively
to this curve
^ ^ 2 Vdtr
^YT^ is equal to  . ^ ^ ; in substituting for g'^its value ^+2RrX
^d Rn , /d A>
cos. (A — a) + R% and for (jr) and (, — \ their values found in
No. 499, we shall have
78 A COMMENTARY ON [Sect. XI.
+ r {(R>  1) COS. (A _ «)  !!MA«)
+ r R (^^) sin. (A  a) + R (R'  1).
Let P represent this quantity ; if it is negative, the radiusvector de
creases, and consequently, the comet tends towards its pei'ihelion. But
it goes off into the distance, if P is negative. We have then
the angular distance v of the comet from its perihelion, will be determined
from the polar equation to the parabola,
cor.^v = _;
and finally we shall have the time employed to describe the angle v, by
the table of the motion of the comets. This time added to or subtracted
from that of the epoch, according as P is negative or positive, will give
the instant when the comet passes its perihelion.
603. Recapitulating these different results, we shall have the following
method to determine the parabolic orbits of the comets.
General method of determining the orbits of the comets.
This method will be divided into two parts ; in the first, we shall give
the means of obtaining approximately, the perihelion distance of the comet
and the instant of its passage over the perihelion ; in the second, we shall
determine all the elements of the orbit on the supposition that the former
are known.
Approximate determination of the Perihelion distance of the comet, and
the instant of its ■passage over the perihelion.
We shall select three, four, five, &c. observations of the comet
equally distant from one another as nearly as possible ; with four obser
vations we shall be able to consider an interval of 30° ; with five, an in
terval of 36°, or 40° and so on for the rest ; but to diminish the in
fluence of their errors, the interval comprised between the observations
must be greater, in proportion as their number is greater. This being
supposed.
Let /3, /3', jS", &c. be the successive geocentric longitudes of the comet,
7, y\ y" the corresponding latitudes, these latitudes being supposed positive
or negative according as they are north or south. We shall divide the dif
ference ^' — /S, by the number of days between the first and second ob
servation ; we shall divide in like manner the difference /3" — /? by the
Book L] NEWTON'S PRINCIPIA. 79
number of days between the second and third observation ; and so on.
Let 3 iS, a /3', d /3", &c. be these quotients.
We next divide the difference 8 &' — 3/3 by the number of days be
tween the first observation and the third ; we divide, in like manner, the
difference 5/3" — d ^' by the number of days between the second and
fourth observations ; similarly we divide the difference 3 /3"' — 8 /S" by the
number of days between the third and fifth observation, and so on. Let
a^ ^, 3 2 ^', 3^/3", &c. denote these quotients.
Again, we divide the difference 3^/3' — 3*j8by the number of days
which separate the first observation from the fourth ; we divide in like
manner 3 * /S" — 3 '^ j8' by the number of days between the second obser
vation and the fifth, and so on. Make 3 ^ jS, 3 ^ j8', &c. these quotients.
Thus proceeding, we shall arrive at 3°— ^ ^, n being the number of obser
vations employed.
This being done, we proceed to take as near as may be a mean epoch
between the instants of the two extreme observations, and calling i, i', i'\
&c. the number of days, distant from each observation, i, i', i'', &c. ought
to be supposed negative for the "observations made prior to this epoch;
the longitude of the comet, after a small number z of days reckoned from
the Epoch will be expressed by the following formula :
3 _ i 3 /3 + i i' 3 2 /3 — i i' i" 3 3 /3 + &c.
\ +z{3 ^— (i + i')3 2/3+ (i i'+i i''+i' i")3^i8— (i i' i"+i i' i"'+ii''i"'+. . (p)
ii'i''i"0 3*/3 + &c.5
'+z2^32/3— {i + i'+i'0 5'/3+(ii' + ii''+ii'''+i'i'"+i"+*OS*^ — &c.
The coefficients of — 3 /3, + 3 * /3, — 3^/3, &c. in the part independent
of z are 1st the numbers i and i', secondly the sum of the products two
and two of the three numbers i, i', i" ; thirdly the sum of the products
three and three, of the four numbers i, i', V\ M", &c.
The coefficients of — 3^/3, + 3 * j8, — 3 ^ j8, &c. in the part multiplied
by z *, are first, the sum of the three numbers i, i', i" ; secondly of the
products two and two of the four numbers i, i', i'', M" \ thirdly the sum of
the products three and three of the five numbers i, i', i", i"', i'"', &c.
Instead of forming these products, it is as simple to develope the func
tion 3 + (z — i) 3 /3 + (z _ i) (z — i') 32 /3 + (z — i) (z — iO (z — i'O
X 3 ^ /3 + &c. rejecting the powers of z superior to the square. This
gives the preceding formula.
If we operate in a similar manner upon the observed geocentric lati
tudes of the comet ; its geocentric latitude, after the number z of days
from the epoch, will be expressed by the formula (p) in changing /3 into
y. Call (q) the equation (p) thus altered. This being done,
80 A COMMENTARY ON [Sect. XI.
a will be the part independent of z in the formula (p) ; and 6 that in the
formula (q).
Reducing into seconds the coefficient of z in the formula (p), and
takino from the tabular logarithm of this number of seconds, the logarithm
4,0394622, we shall have the logarithm of a number which we shall de
note by a.
Reducing into seconds the coefficients of z * in the same formula, and tak
ing from the logarithm of this number of seconds, the logarithm 1.9740144,
we shall have the logarithm of a number, which we shall denote by b.
Reducing in like manner into seconds the coefficients of z and z ^ in
the formula (q) and taking away respectively from the logarithms of these
numbers of seconds, the logarithms, 4,0394622 and 1,9740144, we shall
have the logarithms of two numbers, which we shall name h and 1.
Upon the accuracy of the values of a, b, h, 1, depends that of the
method; and since their formation is very simple, we must select and
multiply observations, so as to obtain them with all the exactness which
the observations will admit of. It is perceptible that these values are only
the quantities (r^) , (jri) j ( j;) > (tTs) ' ^^"*^^ ^^ ^^^^ express
ed more simply by the above letters.
If the number of observations is odd, we can fix the Epoch at the
instant of the mean observation ; which will dispense with calculating the
parts independent of z in the two preceding formulas ; for it is evident,
that then these parts are respectively equal to the longitude and latitude
of the mean observation.
Having thus determined the values of a, a, b, &, h, and 1, we shall de
termine the longitude of the sun, at the instant we have selected for the
epoch, R the corresponding distance of the Earth from the sun, and R'
the distance which answers to E augmented by a right angle. We shall
have the following equations
p« = ^, — 2 Rxcos. (E — a) + R* . . . ^ . . (1)
* cos. ^ &
sin. (E — «) f 1 _Ll_bx ..
y^ 2l IT^— RT'J 2^ ^^^
f , , 1 . a * sin. d . cos. O v
y = xl.tan.^+23;+ 2h i\ ... (3)
R sin. 6 COS. 6 ,„ % f 1 1 1 I *
+ 2Ti COS. (E a) I ^3 pi )
/ h X \* ^ f sin. (E — a)
= y' + a'x'+(ytan.« + j3j^^) + 8 5{4j ■'
Book I.] NEWTON'S PRINCIPIA. 81
— (R' — 1) COS. (E — a)} — 2 a X [(R' — 1) sin. (E — a) +
^^^4=^}+^: w
To derive from these equations the values of the unknown quantities
X, y, f, we must consider, signs being neglected, whether b is greater or
less than 1. In the first case we shall make use of equation (1), (2), and
(4). We shall form a first hypothesis for x, supposing it for instance
equal to unity; and we then derive by means of equations (1), (2), the
values of § and of y. Next we substitute these values in the equation (4) ;
and if the result is 0, this will be a proof that Hi i value of x has been
rightly chosen. But if it be negative we must augment the value of x,
and diminish it if the contrary. We shall thus obtain, by means of a
small number of trials the values of x, y and f. But since these unknown
quantities may be susceptible of many real and positive values, we must
seek that which satisfies exactly or nearly so the equation (3).
In the second case, that is to say, if 1 be greater than b, we shall use
the equations (1), (3), (4), and then equation (2) will give the verifi
cation.
Having thus the values of x, y, f, we shall have the quantity
^ = ^^^y + ^ "^ ^^"' ^^~ ^ y '''''' ^^ ~~ "^
+ X {^^^=^'(R'— l)cos. (E—a)} — Eax.«=in(E«)
+ R.(R'— 1).
The Perihelion distance D of the comet will be
the cosine of its anomaly v will be given by the equation
, 1 D
cos ^ — V = —
2
and hence we obtain, by the table of the motion of the comets, the time
employed to describe the angle v. To obtain the instant when the comet
passes the perihelion, we must add this time to, or subtract it from the
epoch according as P is negative or positive. For in the first case the
comet approaches, and in the second recedes from, the perihelion.
Having thus nearly obtained the perihelion distance of the comet, and
the instant of its passage over the perihelion ; we are enabled to correct
them by the following method, which has the advantage of being inde
pendent of the approximate values of the other elements of the orbit
Vol. ir. F
82 A COMMENTARY ON [Sect. XL
An exact DeietTnination of the elements of the orbit, 'when we know ap
proximate values of the perihelion distance of the comet, and of the instant
of its passage over the perihelion.
We shall first select three distant observations of the comet ; then
taking tlie perilielion distance of the comet, and the instant of its crossing
the perihelion, determined as above, we shall calculate the three anomalies
of the comet and the corresponding radiusvectors corresponding to the
instants of the three observations. Let v, v', v'' be tliese anomalies, those
which precede the passage over the perihelion being supposed negative.
Also let f, g' f " be the corresponding radiusvectors of the comet; then
v' — v, V — v will be the angles comprised by g and ^ and by §, §'\
Let U be the first of these angles, U' the second. Again, call a, a' a" the
three observed geocentric longitudes of the comet, referred to a fixed
equinox ; ^, ^, ^' its three geocentric latitudes, the south latitudes being
negative. Let 3, /3', /3'' be the three corresponding heliocentric longi
tudes and =r, tt, zt", its three heliocentric latitudes. Lastly call E, E', YI'
the three corresponding longitudes of the sun, and R, R', R'' its three
distances to the center of the earth.
Conceive that the letter S indicates the center of the sun, T that of the
eaTth, and C that of the comet, 0/ that of its projection upon the plane
of the ecliptic. The angle S T C is the difference of the geocentric lon
gitudes of the sun and of the comet. Adding the logarithm of the cosine
of this angle, to the logarithm of the cosine of the geocentric latitude of
the comet, we shall have the logarithm of ihe cosine of the angle S T C.
"We know, therefore, in the triangle S T C, the side S T or R, the side
S C or f, and the angle S T C, to find the angle C S T. Next we shall
have tlie heliocentric latitude « of the comet, by means of the equation
_ sin. ^ sin. C S T
sm. C i i>
The angle T S C is the side of a spherical right angled triangle, of
which the hypothenuse is the angle T S C, and of which one of the sides
is the angle »•. Whence we shall easily derive the angle T S C, and con
sequently the heliocentric longitude /3 of the comer.
We shall have after the same manner t/, i3'; J', ^" ; and the values of
/3, ^', jS" will show whether the motion of the comet be direct or retro
grade. '
If we imagine the two arcs of latitude », «', to meet at the pole of die
ecliptic, they would make there an angle equal to ^' — /3 ; and in the
Book I.] NEWTON'S PRINCIPIA. 83
spherical triangle formed by this angle, and by the sides — nr,  — ,/
T being the semicircumference, the side opposite to the angle jS' — S
will be the angle at the sun comprised between the radiusvectors e, and
f'. We shall easily determine this by spherical Trigonometry, or by the
formula
sin. ^ ~ Y = cos. ^ — (w f w') — cos "   (/3' — /3) cos. w cos. »',
in which V represents this angle ; so that if we call A the angle of which
the sine squared is
cos * — (j8' — /3) COS. « . cos. w',
and which we shall easily find by the tables, we shall have
sin.^ i V = COS. (^^+lr.'+ A) COS. (1 ., + 1 «'_A).
If in like manner we call V the angle formed by the two radiusvectors
f, ^', we have
sin.'l V = cos.(l ,+ i .' + A')cos.(l ,+ 'A')
A' being what A becomes, when w', /S' are changed into »'', ^'\
If, however, the perihelion distance and the instant of the comet's
crossing the perihelion, were exactly determined, and if the observations
were rigorously exact, we should have
V = U, V = U';
But since that is hardly ever the case, we shall suppose
m = U — V; m' = U' — v.
We shall here observe that the revolution of the triangle S T C, gives
for the angle C S T two different values : for the most part the nature
of the motion of the comets, will show that which we ought to use, and
the more plainly if the two values are very different ; for then the one will
place the comet more distant from the earth, than the other, and it will
be easy to judge, by the apparent motion of the comet at the instant of
observation, which ought to be preferred. But if there remains any un
certainty, we can always remove it, by selecting the value which renders
V and V least different from U and U'.
We next make a second hypothesis in which, retaining the same pas
sage over the perihelion as before, we shall suppose the perihelion dis
tance to vary by a small quantity ; for instance, by the fiftieth part of
F2
84 A COMMENTARY ON [Sect. XI.
its value, and we shall investigate on this hv^jothesis, the values of U — V,
U' — v. Let then
n = U — V ; n' = U' — v.
Lastly, we shall frame a third hypothesis, in which, retaining the same
periheUon distance as m the first, we shall suppose the instant of the pas
sage over the perihelion to vary by a halfday, or a day more or less. In
this new hypothesis we must find the values of
U — VandofU' — V;
which suppose to be
p = U  V, p' = U' — v.
Again, if we suppose u the number by which we ought to multiply the
supposed variation in the perihelion distance in order to make it the
true one, and t the number by which we ought to multiply the supposed
variation of the instant when the comet passes over the perihehon in
order to make it the true instant, we shall have the two following equa
tions:
(m — n ) u + (m — p ) t = m j
(m' — n') u + (m'— p') t = m';
whence we derive u and t and consequently the perihelion distance cor
rected, and the true instant of the comet's passing its perihelion.
The preceding corrections suppose the elements determined by the
first approximation, to be sufficiently near the truth for their errors to be
regarded as infinitely small. But if the second approximation should
not even suffice, we can have recourse to a third, by operating upon the ele*
ments already corrected as we did upon the first ; provided care be taken to
make them undergo smaller variations. It will also be sufficient to calculate
by these corrected elements the values of U — V, and of U' — V. Call
ing them M, M', we shall substitute them for m, m' in the second mem
bers of the two preceding equations. We shall thus have two new equa
tions which will give the values of u and t, relative to the corrections of
these new elements.
Thus having obtained the true perihelion distance and the true instant
of the comet's passing its perihelion, we obtain the other elements of the
orbit in this manner.
Let j be the longitude of the node which would be ascending if the
motion of the comet were direct, and f the inclination of the orbit. We
shall have by comparison of the first and last observation,
. _ tan. « sin. ^' — tan. V sin. j8 ^
^^"* J  tan. u COS. )S" — tan. ^' cos. ^ '
Book I.] NEWTON'S PRINCIPIA. 86
tan. xf"
tan. (p = . jr^, rr . .
sin. {^" — j)
Since we can compare thus two and two together, the three observa
tions, it will be more correct to select those which give to the above frac
tions, the greatest numerators and the greatest denominators.
Since tan. j may equally belong to j and <? + jj j being the smallest of
the positive angles containing its value, in order to find that which we
ought to fix upon, we shall observe that p is positive and less than a right
angle ; and that sin. (/3" — j) ought to have the same sign as tan. xi".
This condition will determine the angle j, and this will be the position
of the ascending node, if the motion of the comet is direct ; but if retro
grade we must add two right angles to the angle j to get the position of
the node.
The hypothenuse of the spherical triangle whose sides are ^" — j and
w'', is the distance of the comet from its ascending node in the third ob
servation; and the difference between v" and this hypothenuse is the
interval between the node and the perihelion computed along the orbit.
If we wish to give to the theory of a comet all the precision which ob
servations will admit of, we must establish it upon an aggregate of the best
observations ; which may be thus done. Mark with one, two, &c. dashes
or strokes the letters m, n, p relative to the second observation, the third,
&c. all being compared with the first observation. Hence we shaH form
the equations
(m — n)u + (m — p)t = m
(m' — n' ) u + (m' — p' ) t = m'
(m''— n'') u + (m'^ — p") t = m''
&c. = &c.
Again, combining these equations so as to make it easier to determine
u and t, we shall have the corrections of the perihelion distance and of the
instant of the comet's passing its perihelion, founded upon the aggregate
of these observations. We shall have the values of
3, &, &', &C. «r, r>', ^", &C.,
and obtain
. _ tan, zf (sin. & + sin. ^" + &c.) — sin. /3 (tan. «/ + tan, rs" + &c.)
"*•' ~ tan. « (cos. /3' + cos. /3" + &c.) — cos. /3 (tan. zr' + tan. «r" + &c.)
_ tan. w + tan. w'' } &c.
*^° ^  sin. (/3' — j) + sin. ifi" — j) + &c. "
504. There is a case, very rare indeed, in which the orbit of a comet
can be determined rigorously and simply ; it is that where the comet has
been observed in its two nodes. The straight line which joins these
F3
86 A COMMENTARY ON [Sect. XI.
two observed positions, passes through the center of the sun and coincides
with tlie line of the nodes. The length of this straight line is determined
by the time elapsed between the two observations. Calling T this time
reduced into decimals of a day, and denoting by c the straight line in
question, we shall have (No. 493)
3
rp2
~ 2N\
(9.688724) 2*
Let /3 be the heliocentric longitude of the comet, at the moment of tlie
first observation ; g its radius vector ; r its distance from the earth ; and a
its geocentric longitude. Let, moreover, R be the radius of the terrestrial
orbit, at the same instant, and E the corresponding longitude of the sun.
Then we shall have
g sin. /3 = r sin. a — R sin. E ;
g COS. jS = r COS. a — R cos. E.
Now ff + /S will be the heliocentric longitude of the comet at the in
stant of the second observation ; and if we distinguish the quantities ^, «,
r, R, and E relative to this instant by a dash, we shall have
o' sin. B — W sin. E' — r' sin. a/ ;
^' cos. S = R' COS. E' — r' cos. a'.
These four equations give
_ r sin a — R sin. E _ r^ sin, of — R^ sin. E^ ^ ^
^^'^  rcos.a — Rcos.E ~ r' cos. «' — R' cos. E' '
whence we obtain
, _ R R^ sin. (E — EQ — R r sin. (« — EQ
~~ r sin. {a' — a) — R sin. (a' — E)
We have also
{i + g') sin. /3 = r sin. « — x' sin. a' — R sin. E + R' sin. E'
(? + s') COS. /3 = r cos. a — r' cos. a.' — R cos. E + R' cos. E'.
Squaring these two equations, and adding them together, and substitut
ing c for f + f', we shall have
c2 = R2 — 2RR'cos.(E' — E) + R''
+ 2 r JR' cos. (a __ EO — R cos. (a_ E)}
+ 2 r' {R COS. {a' _ E) — R' cos. (a' — E')
+ r*^— 2rr'cos. (a' — a)\v'\
If we substitute in this equation for r' its preceding value in terms of r,
we shall have an equation in r of the fourth degree, which can be resolved
by the usual methods. But it will be more simple to find values of r, r'
by trial such as will satisfy the equation. A few trials will suffice for tliat
puipose. rf.
Book L] NEWTON'S PRINCIPIA. 87
By means of these quantities we shall have /3, § and /. If v be the
angle which the radius j makes with the perihelion distance called D ;
«r — V will be the angle formed by this same distance, and by the radius g'.
"VYe shall thus have by the equation to the parabola
D , D
S = i — '> S =
1 ' * . 1
cos. '■' — V sm. ^ "5 V
which give
tan.
We shall therefore have the anomaly v of the comet, at the instant of
the first observation, and its perihelion distance D, whence it is easy to
find the position of the perihelion, at the instant of the passage of the
comet over that point. Thus, of the five elements of the orbit of the co
met, four are known, namely, the perihelion distance, the position of the
perihelion, the instant of the comet's passing the perihelion, and the posi
tion of the node. It remains to learn the inclination of the orbit; but for
that purpose it will be necessary to have recourse to a third observation,
which will also serve to select from amongst the real and positive roots of
the equation in r, that which we ought to make use of.
505. The supposition of the parabolic motion of comets is not rigorous ;
it is, at the same time, not at all probable, since compared with the cases
that give the parabolic motion, there is an infinity of those which give the
elliptic ot hyperbolic motions. Besides, a comet moving in either a para
bolic or hyperbolic orbit, will only once be visible; thus we may with
reason suppose these bodies, if ever they existed, long since to have dis
appeared ; so that we shall now observe those only which, moving in or
bits returning into themselves, shall, after greater or less incursions into
the regions of space, again approach their center the sun. By the follow
ing method, we shall be able to determine, within a few years, the period
of their revolutions, when we have given a great number of very exact
observations, made before and after the passage over the perihelion.
Let us suppose we have four or a greater number of good observations,
which embrace all the visible part of the orbit, and that we have deter
mined, by the preceding method, the parabola, which nearly satisfies these
observations. Let v, v', v", v"', &c. be the corresponduig anomalies;
Si i'> i"i i"i ^^* ^^® radiusvectors. Let also
v' — v = U, v" — V = U', y'" — v = U", &c.
F4
88 A COMMENTARY ON [Sect. XI.
Then we shall estimate, by the preceding method with the parabola
already found, the values of U, U', U", &c., V, V, V", &c. Make
m = U — V, m' = U' — V, m" = U" — V", &a.
Next, let the perihelion distance in this parabola vary by a very small
quantity, and on this hypothesis suppose
n = U — V; n' = U' — V; n" = U" — \", &c.
We will form a third hypothesis, in which the perihelion distance re
maining the same as in the first, we shall make the instant of the comet's
passing its perihelion vary by a very small quantity ; in this case let
p = U — V; p' = U' — V; p'' = U'^ — M"i &c.
Lastly, we shall calculate the angle v and radius f, witli the perihelion
distance, and instant over the perihelion on the first hypothesis, supposing
the orbit an ellipse, and the difference 1 — e between its excentricity and
unity a very small quantity, for instance jq. To get the angle v, in this
hypothesis, it will suffice (489) to add to the anomaly v, calculated in the
parabola of the first hypothesis, a small angle whose sine is
i (le)tan. iv {4— 3eos.«i v— Gcos.^^ v}.
Substituting afterwards in the equation
D f, l_e „ 1
g =
cos. ^ — v
A'.
for V, this anomaly, as calculated in the ellipse, we shall have the corre
sponding radiusvector ^. After the same manner, we shall obtain v', f ,
v", f", &c. Whence we shall derive the values of U, U', U'', &c. and
(by 503) of V, V, V", &c.
In this case let
q = U — V; q' = U' — V; q" = U" — V^ &c.
Finally, call u the number by which we ought to multiply the supposed
variation in the perihelion distance, to make it the true one ; t the number
by which we ought to multiply the supposed variation in the instant over
the perihelion, to make it the true instant; and s that by which we should
multiply the supposed value of 1 — e, in order to get the true one ; and
we shall obtain these equations :
(m — n) u + (m — p) t + (m — q'; s = m ;
(m' — n') u + (ra' — p') t + (m' — q') s = m ;
(m" — n'O u + (m" — p") t + (m" — q") s = m";
(m'" — n'") u + (m'" — p'") t + {^" — q'") s = m'";
&c.
Book I.] NEWTON'S PRINCIPIA. 89
We shall determine, by means of these equations, the values of u, t, s ;
whence will be derived the true perihelion distance, the true instant over
the perihelion, and the true value of 1 — e. Let D be the periheHon
distance, and a the semiaxis major of the orbit; then we shall have
a = Tj ; the time of a sidereal revolution of the comet, will be expressed
by a number of sidereal years equal to a or to f^j j*, the mean
distance of the sun from the earth being unity. We shall then have
(by 503) the inclination of the orbit and the position of the node.
Whatever accuracy we may attribute to the observations, they will
always leave us in uncertainty as to the periodic times of the comets. To
determine this, the most exact method is that of comparing the observa
tions of a comet in two consecutive revolutions. But this is practicable,
only when the lapse of time shaU bring the comet back towards its peri
heUon.
Thus much for the motions of the planets and comets as caused by the
action of the principal body of the system. We now come to
506. General methods of determining by successive approximatio7is, the
motions of the heavenly bodies.
In the preceding researches we have merely dwelt upon the elliptic
motion of the heavenly bodies, but in what follows we shall estimate them
as deranged by perturbing forces. The action of these forces requires only
to be added to the differential equations of elliptic motion, whose integrals
in finite terms we have already given, certain small terms. We must deter
mine, however, by successive approximations, the integrals of these same
equations when thus augmented. For this purpose here is a general me
thod, let the number and degree of the equations be what they may.
Suppose that we have between the n variables y, y', y", &c. and the
time t whose element d t is constant, the n diflferential equations
d' v'
&c. = &c.
Pj Qj P^ Q'j &c. being functions of t, y, y', &c. and of the differences to
the order i — 1 inclusively, and a being a very small constant coefficient,
which, in the theory of celestial motions, is of the order of the perturb
ing forces. Then let us suppose we have the finite integrals of those
90 A COMMENTARY ON [Sect. XI.
equations when Q, Q', &c. are nothing. Differentiating each i — 1
times successively, we shall form with their differentials i n equations by
means of wliich we shall determine by elimination, the arbitrary constants
c, c', c'', &c. in functions of t, y, y', y'', &c. and of their differences to the
order i — 1. Designating therefore by V, V, V, &c. these functions
we shall have
c = V; e = V; c" = ^"\ &c.
These equations are the i n integrals of the (i — 1)^ order, which the
equations ought to have, and which, by the elimination of the differences
of the variables, give their finite integrals.
But if we differentiate the preceding integrals of the order i — 1, we
shall have
= dV; = dV'; = d V"; &c.
and it is clear that these last equations being differentials of the order i
without arbitrary constants, they can onlv be the sums of the equations
d> v'
= &c.
each multiplied by proper factors, in order to make these sums exact dif
ferences. Calling, therefore, F d t, F' d t', &c. the factors which ought
respectively to multiply them in order to make = d V ; also in like
manner making H d t, H' d i', &c. the factors which would make = d V,
and so on for the rest, we shall have
dV = Fdt{iU+p} + Fdti^tF}+&c.
dV'=Hdt{^ + p} + H'dt{^y/+F}+&c.
&c.
F, F', &c. H, H', &c. are functions of t, y, y', y", &c. and of their dif
ferences to the order i — 1 . It is easy to determine them when V, V, &c.
d ' y
are known. For F is evidently the coefficient of r4 in the differential
d ' v'
of V; F' is the coeiSicient of p^ in the same differential, and so on.
^ d ' V d ' v'
In like manner, H, H', &c. are the coefficients of j— j , , j , &c. in the
differential of V. Thus, since we may suppose V, V, &c. known, by dif
Book I.] NEWTON'S PRINCIPIA. 91
ferentiating with regard to ^ ^._\ , . ._\ , &c. we shall have the
factors by which we ought to multiply the diiFerential equations
= i + p, = ^y; + F, &c.
in order to make them exact diiferences.
Now resume the diiFerential equations
= ^f + P+«.Q; o = i^^y+F + «.Q',
&c.
If we multiply the first by F d t, the second by F' d t, and so on, we
shall have by adding the results
= dV + adtfFQ+FQ' + &C.1,
In the same monner, we shall have
= dV' + adt{HQ+H^Q' + 8cc.
&c.
whence by integration
c — «/d t JF Q + F Q' + &c.} = V;
c' — a/d t {H Q + H' Q' + &c.} = V;
&c.
We shall thus have i n differential equations, which will be of the same
form as in the case when Q, Q', &c. are nothing, with this only differ
ence, that the arbitrary constants c, c', c'', &c. must be changed into
c_a/dtlFQ + FQ'+&c.}, c _a/dt^HQ + H'Q'+&c.]&c.
But if, in the supposition of Q, Q', &c. being equal to zero, we eliminate
from the i n integrals of the order i — 1, the differences of the variables
y, y', &c. we shall have n finite integrals of the proposed equations. We
shall therefore have these same integrals when Q, Q', &c. are not zero, by
changing in the first integrals, c, c\ &c. into
c _ a/d t JF Q + &c.}, c' — a/d t{UQ+ &c.^&c.
507. If the differentials
d t [F Q + F Q' + &c.}, d t JH Q + H' Q' + kc.]kc.
are exact, we shall have, by the preceding method, finite integrals of the
proposed differentials. But this is not so, except in some particular cases,
of which the most extensive and interesting is that in which they are
linear. Thus let P, P', &c. be linear functions of y, y', &c. and of their
differences up to the order i — 1, without any term independent of these
variables, and let us first consider the case in which Q, Q', &c. are no
thing. The differential equations being linear, their successive integrals
92 A COMMENTARY ON [Sect. XI.
are likewise linear, so that c = V, c' = V, &c. being the i n integrals of
the order i — 1, of tlie linear differential equations
 iiy + p  ^' + F &c
V, V, &c. may be supposed linear functions of y y', &c. and of their dif
ferences to the order i — 1. To make this evident, suppose that in the
expressions for y, y', &c. the arbitrary constant c is equal to a determinate
quantity plus an indeterminate 3 c; the arbitrary constant c' equal to a
determinate quantity plus an indeterminate 3 c' &c. ; then reducmg these
expressions according to the powers and products of 5 c, h c', &c. we shall
have by the formulas of No. 487
y = Y + ^c()+ac'(i)+&c.
^ = Y'+*«(^') + ^^(^') + ^^
+ 17^ (d^) + ^^
&c.
Y, Y', {^ — j , &c. being functions oft without arbitrary constants. Sub
stituting those values, in the proposed differential equations, it is evident
that 5 c, 5 c', &c. being indeterminate, the coefficients of the first powers
of such of them ought to be nothing in the several equations. But these
equations being linear, we shall evidently have the terms affected with the
first powers of 3 c, h c', &c. by substituting for y, y', &c. these quantities
respectively
/d Yn , . /d Yn , , . ^
These expressions of y, y', &c, satisfy therefore separately the proposed
equations ; and since they contain the i n arbitraries 3 c, 3 c', &c. they are
complete integrals. Thus we perceive, that the arbitraries are under a
linear form in the expressions of y, y', &c. and consequently also in their
differentials. Whence it is easy to conclude that the variables y, y', &c.
and their differences, may be supposed to be linear in the successive inte
grals of the proposed differential equations.
d ' y d ' y
Hence it follows, that F, F', &c. being the coefficients of yj , TTi »
Book L] NEWTON'S PRINCIPIA. 93
&c. in the differential of V ; H, H', &c. being the coefficients of the same
differences in the differential of V, &c. these quantities are functions ot
variable t only. Therefore, if we suppose Q, Q', &c. functions of t alone,
the diffei'entials
d t {F Q + F Q' + &c.^ ; d t {H Q + H' Q' + &c.} ; &c.
will be exact.
Hence there results a simple means of obtaining the integrals of any
number whatever n of linear differential equations of the order i, and
which contain any terms a Q, a Q', &c. functions of one vai'iable t, having
known the integrals of the same equations in the case where Q, Q', &c.
are supposed nothing. For then if we differentiate their n finite integrals
i — 1 times successively, we shall have i n equations which will give, by
elimination, the values of the i n arbitrary constants c, c\ &c. in functions
of t, y, y', &c. and of their differences to the i — 1'** order. We shall thus
form the i n equations c = V, c' = V, &c. This being done, F, F', &c.
will be the coefficients of t— ; — f , , ^. v> &c. in V: H, H', &c. will
be the coefficients of the same differences in V, and so on. We shall,
therefore, have the finite integrals of the linear differential equations
= ^ + P + «Q; = ^ + F + aQ'; &c.
by changing, in the finite integrals of these equations deprived of their last
terms a Q, a Q', &c. the arbitrary constants c, c', &c. into
c — a/dt JFQ+FQ'+&cl, c — a/dt[HQ + H'Q'+&c.l&c.
Let us take, for example, the linear equation
d^ v
0=^+P^y + aQ. ■
The finite integral of the equation
d" V
is (found by multiplying by cos. a t, and then by parts getting
d^v dy „. dy,^ ^dy,
/ COS. a t . j^ = COS. a t 3^ + a / sm. a t , f . d t = cos. a t . ^f +
'^ dt dt*^ dt at
a sin. a t . y — a.^ f cos. a t . y .'. c = a cos. a t . ^p + a sin. a t . y, &c.)
c . c'
y = — sm. a t + — cos. a t,
•'a a
c, c' being arbitrary constants.
94 A COMMENTARY ON [Sect. XL
This integral gives by differentiation
d y / • » "
Y^ = c COS. at — c sin. a t.
at
If we combine this with tlie integral itself, we shall form two integrals
of the first order
c = a y sin. a t + r^ cos. at;
dy .
c' = a y cos. at r^ sin. a t ;
^ d t
and therefore shall have in this case
F = cos. at; H = — sin. a t,
and the complete integral of the proposed equation will therefore be
c . " c' a sin. a t ^^ ,
y = — sin. a t + — cos. at J Q d t cos, a t
3 3* 2i
a cos. a t ^f^ J * •
H y Q d t sin. a t.
Hence it is easy to conclude that if Q is composed of terms of the form
sin.
K . * (m t + s) each of these terms will produce in the value of y the
corresponding term
a K sin. , ^ , .
m * — a ^ COS. ^
Sill
If m be equal to a, the term K ' (m t + «) will produce in y, 1st. the
term — j — ^ . * (a t + e) which being comprised by the two terms
^ a cos*
c . c' cc "K. t cos.
— sin. a t  COS. at, maybe neglected ; 2dly. the term + — — . . \a.i\ e),
a a ic a sm.
+ or — being used according as the term of Q is a sine or cosine. We
thus perceive how the arc t produces itself in the values of y, y', &c. with
out sines and cosines, by successive integrations, although the differentials
do not contain it in that form. It is evident this will take place when
ever the functions F Q, F', Q', &c. H Q, H' Q', &c. shall contain con
stant terms.
508. If the differences
d t JF Q + &c.}, d t JH Q + &c.}
are not exact, the preceding analysis will not give their rigorous integrals.
But it affords a simple process for obtaining them more and more nearly
by approximation when a is very small, and when we have the values of
Book I.j NEWTON'S PRINCIPIA. 95
y, y', &c. on the supposition of a being zero. Differentiating these values,
i — 1 times successively, we shall form the differential equations of the
order i — 1, viz.
c = V; c'  V^&c.
d i y d ' v'
The coefficients of j— y , ,— "V j &c. in the differentials of V, V'', &c.
d t ^ d t » ' '
being the values of F, F', &c. H, H', &c. we shall substitute them in the
differential functions
d t (F Q + F Q' + &c.) ; d t (H Q + H' Q' + &c) ; &c.
Then, we shall substitute in these functions, for y, y', &c. their first
approximate values, which will make these differences functions of t and of
the arbitrary constants c, c', &c.
Let T d t, T d t, &c. be these functions. If we change in the first
approximate values of y, y', &c. the arbitrary constants c, c', &c. re
spectively into c — a y T d t, c' — a y T d t, &c. we shall have the
second approximate values of those variables.
Again substitute these second values in the differential functions
d t . (F Q + &c.) ; d t (H Q + &c.) &c.
But it is evident that these functions are then what T d t, T' d t, &c.
become when we change the arbitrary constants c, c', &c. into c — ctfT d t,
c' — a/T' d t, &c. Let therefore T,, T/, &c. denote what T, T, &c.
become by these changes. "We shall get the third approximate values of
y, y', Sec. by changing in the first c, c', &c. respectively into c — ^yT, d t,
c — /T; d t, 8s:c.
Calling T/^, T^/, in like manner, what T, T', &c. become when
we change c, c, &c. into c — af T/ d t, c' — «y T/ d t, &c. we shall
have the fourth approximate values of y, y', &c. by changing in the first
approximate values of these variables into c — ^f^i, d t, c' — ay T/ d t,
&c. and so on.
We shall see presently that the determination of the celestial motions,
depends almost always upon differential equations of the form
d 2 V
= 2^?+ a^y + aQ,
Q being a rational and integer function of y, of the sine and cosine of
angles increasing proportionally with the time represented by t. The
following is the easiest way of integrating this equation.
First suppose « nothing, and we shall have by the preceding No. a first
value of y.
Next substitute this value in Q, which will thus become a rational and
90 A COMMENTARY ON [Sect. XL
entire function of sines and cosines of angles proportional to the time.
Then integrating the differential equation, we shall have a second value
ofy approximate up to quantities of the order a inclusively.
Again substitute this value in Q, and, integrating the differential equa
tion, we shall have a third approximation of y, and so on.
This way of integrating by approximation the differential equations of
the celestial motions, although the most simple of all, possesses the dis
advantage of giving in the expressions of the variables y, y', &c. the arcs
of a circle (symbols sine and cosine) in the very case where these arcs
do not enter the rigorous values of these variables. We perceive, in
fact, that if these values contain sines or cosines of angles of the order a t,
these sines or cosines ought to present themselves in the form of series, in
the approximate values found by the preceding method ; for these last
values are ordered according to the powers of a. This developement
into series of the sine and cosine of angles of the order a t, ceases to be
exact when, by lapse of time, the arc a t becomes considerable. The ap
proximate values of y, y', &c. cannot extend to the case of an unlimited
interval of time. It being important to obtain values which include both
past and future ages, the reversion of arcs of a circle contained by the
approximate values, into functions which produce them by their develope
ment into series, is a delicate and interesting problem of analysis. Here
follows a general and very simple method of solution.
609. Let us consider the differential equation of the order i,
= ^+ P + aQ
dy d '~ W
a being very small, and P and Q algebraic functions of y, r^ , . . . . , ^ _x ,
and of smes and cosines of angles increasing proportionally with the time.
Suppose we have the complete integral of this differential, in the case of
a = 0, and that the value of y given by this integral, does not contain the
arc t, without the symbols sine and cosine. Also suppose that in inte
grating this equation by the preceding method of approximation, when a
is not nothing, we have
y = X + t Y 1 t^ Z + t^ S + &c.
X, Y, Z, &c. being periodic functions of t, which contain the i arbitraries
c, c', c", &c. and the powers of t in this expression of y, going on to in
finity by the successive approximations. It is evident the coefficients
of these powers will decrease with the greater rapidity, the less is a.
In the theory of the motions of the heavenly bodies, « expresses the order
of perturbing forces, relative to the principal forces which animate them.
Book I.] NEWTON'S PRINCIPIA. 97
d' y
If we substitute the preceding value of y in the function t— ^ + P+c^Q,
it will take the form k + k' t + k" t + &c., k, k', k'', &c. being perio
dic fimctions of t ; but by the supposition, the value of y satisfies the dif
ferential equation
= ^+ P + aQ;
d t '
we ought therefore to have identically
= k + k' t + k" 1 2 + &c.
If k, k', k", &c. be not zero this equation will give by the reversion of
series, the arc t in functions of sines and cosines of angles proportional to
the time t. Supposing therefore a to be infinitely small, we shall have t
equal to a finite function of sines and cosines of similar angles, which is
impossible. Hence the functions k, k', &c. are identically nothing.
Again, if the arc t is only raised to the first power under tlie symbols
sine and cosine, since that takes place in the theory of celestial motions,
the arc will not be produced by the successive differences of y. Substi
tuting, therefore, the preceding value of y, in the function Ji + P + °' • Q>
the function of k f k' t + &C' ^o which it transforms, will not contain
the arc t out of the symbols sine and cosine, inasmuch as it is already con
tained in y. Thus changing in the expression of y, the arc t, without the
periodic symbols, into t — 6, 6 being any constant whatever, the function
k + k' t + &c. will become k + k' (t — ^) + &c. and since this last
function is identically nothing by reason of the identical equations k =
k' = 0, it results that the expression
y = X + (t — Y + (t — ^)2 Z + &c.
also satisfies the differential equation
d' V
o = ^? + P + «Q
Although this second value of y seems to contain i + 1 arbitrary con
stants, namely, the i arbitraries c, c, c", &c. and tf, yet it can only have i
distinct ones. It is therefore necessary that by a proper change in the
constants c, c', &c. the arbitrary 6 be made to disappear, and thus the
second value of y will coincide with the first This consideration will fur
nish us with the means of making disappear the arc of a circle out of the
periodic sj^mbols.
Give the following form to the second expression for y :
y = X + (t  . R.
Vot. TI. O
»8 A COMMENTARY ON [Sect. XL
Tlien supposing 6 to disappear from y, we have
(rl) = »
and consequently
Differentiating successively this equation we shall have
'dRx /d°Xx . , ,, /d2R>
whence it is easy to obtain, by eliminating R and its differentials, from the
preceding expression of y,
y = X+ (t^)(^) + ^3;^. (^ + 43^.(g^) + &C.
X is a function of t, and of the constants, c, c', c", &c. and since these
constants are functions of 6, X is a function of t and of 6, which we can
represent by <p (t, 6). The expression of y is by Taylor's Theorem
the developement of the function p (t, ^ + t — 6), according to the powers
of t — 6. We have therefore y = ^ (t, t). Whence we shall have y by
changing in X, ^ into t. The problem thus reduces itself to determuie
X in a function of t and 6, and consequently to determine c, c', c", &c.
in functions of ^.
To solve this problem, let us resume the equation
y = X + (t — ^) . Y + (t — ^)^ Z + &c.
Since the constant 6 is supposed to disappear from this expression of y,
we shall have the identical equation
"^OY+c') { (a4H4 +('')'{ (^)«4 +^^ •• '="
Applying to this equation the reasoning which we employed upon
= k + k't + k" t^ + &c.
we perceive that the coefficients of the successive powers of t — 6 ought
to be each zero. The functions X, Y, Z, &c. do not contain ^, inasmuch
as it is contained in c, c', &c. so that to form the partial differences
(5— ^ , (3 — ^ , (rT) > &c. it is sufficient to make c, c', &c. vary in
these functions, which gives
/d Xx _ /d Xx d c , /d Xn d c' . /d Xx d c"
VdT)  \d~c)dJ + Vd~c'>''d7 + VdV'JTT "*■ ^''*
Book L] NEWTON'S PRINCIPIA. 99
/dYx _ /d Yxdc , /d Y^dc'^ /d Yx dc'' „
&C. = &C. !
Again, it may happen that some of the arbitraiy constants c, c', c", &c.
multiply the arc t in the periodic functions X, Y, Z, &c. The differentia
tion of these functions relatively to 6, or, which is the same thing, relatively
to these arbitrary constants, will develope this arc, and bring it from without
the symbols of the periodic functions. The differences (jt)» \rj)^
Cyr ^ , &c. will be then of this form :
(tit) = Y' + ' Y";
&C.
X', X'', Y', Y", Z', Z'', &c. being periodic functions of t, and containing
moreover the arbitrary constants c, c', c", &c. and their first differences
divided by d 6, differences which enter into these functions only under a
linear form ; we shall have therefore
&c.
Substituting these values in the equation (a) we shall have
= X' + ^ X" — Y
+ (t — ^) ^ Y' + ^ Y'' + X'' — 2 ZJ
+ (t ~ ^)MZ' + ^ Z" + Y" — 3^} + &c.;
whence we derive, in equalling separately to zero, the coefficients of the
powers of t — ^,
= X' + ^ X" — Y
= Y' + ^ Y" + X'^ — 2 Z
= Z' + ^Z" + Y" — 3S;
&c.
G2
100 A COMMENTARY ON [Sect. XL
If we differentiate the first of these equations, i — 1 times successively
relatively to t, we shall thence derive as many equations between the
quantities c, c', c'', &c. and their first differences divided by d 6. Then
integrating these new equations relatively to ^, we shall obtain the con
stants in terms of d.
Inspection alone of the first of the above equations will almost always
suffice to get the differential equations in c, c', c", &c. by comparing se
parately the coefficients of the sines and cosines which it contains. For
it is evident that the values of c, c', &c. being independent of t, the dif
ferential equations which determine them, ought, in like manner, to be in
dependent of it The simpUcity which this consideration gives to the pro
cess, is one of its principal advantages. For the most part these equations
will not be integrable except by successive approximations, which will
introduce the arc 6 out of the periodic symbols, in the values of c, c', &c.
at the same time that this arc does not enter the rigorous integrals. But
we can make it disappear by the following method.
It may happen that the first of the preceding equations, and its i — 1
differentials in t, do not give a number i of distinct equations between the
quantities c, c', c'', &c. and their differences. In this case we must have
recourse to the second and following equations.
When we shall have thus determined c, c', c", &c. in functions of 6,
we shall substitute them in X, and changing afterwards 6 into t, we shall
obtain the value of y, without arcs of a circle or free from periodic symbols,
when that is possible.
510. Let us now consider any number n of differential equations.
» = rn + P + "«=
= ^^r + P' + " Q' ;
&c.
P, Q, P', Q' being functions of y, y', &c. of their differentials to the order
i ij and of the sines and cosines of angles increasing proportionally
with the variable t, whose difference is constant. Suppose the approximate
integrals of these equations to be
y = X t t Y + t^ Z + t' S + &c.
y' = X, I t Y, I t' Z, + t^ S, + &c.
X, Y, Z, &c. X,, Y,, Z^, &c. being periodic functions of t and containing
i n arbitrary constants c, c', c", &c. We shall have as in the preceding
No.
Book I.] NEWTON'S PRINCIPIA. 101
= X' + ^X" — Y;
= Y' + ^Y" + X'' — 2Z;
= Z' + Z" + Y" — 3 S ;
&c.
The value of y' will give, in like manner, equations of this form
= X/ + Qx;' Yr,
= Y/ + ^ Y/' + X/' — 2 Z, ;
&c.
The values of y''', y'", &c. will furnish similar equations. We shall
determine by these different equations, selecting the most simple and
approximable, the values of c, c', c", &c. in functions of d. Substituting
these values in X, X', &c. and then changing 6 into t, we shall have the
values of y, y', &c. independent of arcs free from periodic symbols when
that is possible.
511. Let us resume the method already exposed in No. 506. It thence
results that, if instead of supposing the parameters c, c', c", &c. constant,
we make them vary so that we have
d c = — a d t JF Q + F Q' + &cj ;
dc' = — adtJHQ + H'Q' + &c.} ;
we shall always have the i n integrals of the order i — 1,
c = V; c' = V; c"  V"; &c.
as in the case of a = 0. Whence it follows that not only the finite in
tegrals, but also all the equations in which these enter the differences
inferior to the order i, will preserve the same form, in the case of
a = 0, and in that where it is any quantity whatever ; for these equations
may result from the comparison alone of the preceding integrals of the
order i — 1. We can, therefore, in the two cases equally differentiate
i — 1 times successively the finite integrals, without causing c, c', &c. to
vary ; and since we are at liberty to make all vary together, there will
thence result the equations of condition between the parameters c, c', &c.
and their differences.
In the two cases where a = 0, and a = any quantity whatever, the
values of y, y', &c. and of their differences to the order i — 1 inclusively,
are the same functions of t and of the parameters c, c', &c. Let Y be any
function of the variables y, y', y", &c. and of their differentials inferior to
the order i — 1, and call T the function of t, which it becomes, when we
substitute for these variables and their differences their values in t. We
can differentiate the equation Y = T, regarding the parameters c, c', &c.
constant ; we can only, however, take the partial difference of Y relatively
G3
102 A COMMENTARY ON [Sect. XL
to one only or to many of the variables y, y', &c. provided we suppose
what varies with these, to vary also in T. In all these difterentiations, the
parameters c, c', c", Sec. may always be treated as constants ; since by
substituting for y, y', &c. and their differences, their values in t, we shall
have equations identically zero in the two cases of « nothing and of a any
quantity whatever.
When the differential equations are of the order i. — 1, it is no longer
allowed, in differentiating them, to treat the parameters c, c', &c. as con
stants. To differentiate these equations, consider the equation p = 0, p
being a differential function of the order i — 1, and which contains the
parameters c, c', c", &c. Let d f be the difference of this function taken
in regarding c, c', &c. constant, as also the differences d ' ~ ^ y, d ' ~ ^ y', &c.
Let S be the coefficient of j — r^ in the entire difference of <p. Let S'
d t'~'
d • V .
be the coefficient of j — j^ in this same difference, and so on. The c; na
tion f = when differentiated will give
= .,+(^)dc+(^^,)dC + &e.
d ' V . r d ' v' .
Substituting for ^ — r^j its value — d t tP + a Q? ; for j — r^, its value
— d t {P' + a Q'J &c. we shall have
— d t ^S P + S' F + &c.} _ a d t {S Q f S' Q' + &c.} . (t)
In the supposition of a = 0, the parameters c, c', c", &c. are constant.
We have thus
= 3 f> — d t JS P + S' F + &c.}
If we substitute in this equation for c, c', c", &c. their values V, V, V,
&c. we shall have differential equations of the order i — 1, without arbi
traries, which is impossible, at least if this equation is to be identically
nothing. The function
3?) — dt JS P + S' F + &c.]
becoming therefore identically nothing by reason of equations c = V,
cf = V, &c. and since these equations hold still, when the parameters
c, c', c", &c. are variable, it is evident, that in this case, the preceding
Book L] NEWTON'S PRINCIPIA. 103
fiinction is still identically nothing. The equation (t) therefore will be
come
«=(rD''=+(dv)'i '' + «'«•
— a d t JS Q + S' Q' + &c.} (x)
Thus we perceive that to differentiate the equation p = 0, it suffices to
vary the parameters c, c', &c. in p and the differences d ^ ~ ^ y, d ' ~ ^ y',
&c. and to substitute after the differentiations, for — a Q, a Q', &c. tlie
. . d' y" d' y o •
quantities j^ , —^ , &c.
Let vj/ = 0, be a finite equation between y, y', &c. and the variable t. If
we designate by 5 4, 6 ^ vp, &c. the successive differences of ^z, taken in
regarding c, c', &c. as constant, we shall have, by what precedes, in that
case where c, c', &c. are variable, these equations :
^ = 0; b^ = Q; 52v}/ = h'''^ ^ = 0;
changing therefore successively in the equation (x) the function f into v}/,
3 v]/, 6 * vj/, &c. we shall have
rdvI/>
Thus the equations 4/ = 0, vj^' = 0, See. being supposed to be the n
finite integrals of the differential equations
d' v'
' &c.
we shall have i n equations, by means of which we shall be able to de
termine the parameters c, c', c'\ &c. without which it would be necessary
for that purpose to form the equations c = V, c = V, &c. But when
the integrals are under this last form, the determination will be more
simple.
612. This method of making the parameters vary, is one of great utility
G3 ' ,
104 A COMMENTARY ON [Sect. XL
in anal^'sis and in its iipplications. To exhibit a new use of it, let us take
the differential equation
P being a function of t, y, of their differences to the order i — ], and of
the quantities q, q', &c. which are functions of t. Suppose we have the
finite integral of this differential equation of the supposition of q, q', &c.
being constant, and represent by p = 0, this integral, which shall contain
i arbitraries c, c', &c. Designate by d (p, 8^ (p, d^ (p, &c. the successive differ
ences of <p taken in regarding q, q', &c. constant, as also the parameters
c, c', cf', &c. If we suppose all these quantities to vary, the differences of
p will be
^^ + (d!)<''=+(d)'''='+^ + ©O') + 0<"i'+«'
making therefore
3 <p will be still the first difference of (p in the case of c, c', &c. q, q', &c.
being variable. If we make, in like manner,
8' f, d^ (p, 3 ' f will likewise be the second, third, &c. differences of
<p when c, c', &c. q, q', &c. are supposed variable.
Again in the case of c, c', &c. q, q', &c. being constant, the differential
equation
is the result of the elimination of the parameters c, c', &c. by means of
the equations p = 0, d <p = 0, d»p = 0, ....d«f» = 0. Thus, these
last equations still holding good when q, q', &c. are supposed variable, the
equation p = will also satisfy, in this case, the proposed differential
equation, provided the parameters c, c', &c. are determined by means
of the 1 preceding differential equations ; and since their integration
gives i arbitrary constants, the function <p will contain these arbitraries,
and the equation p = will be the complete integral of the proposed
equation.
Book L]
NEWTON'S PRINCIPIA.
105
This method, the variation of parametei's, may be employed with ad
vantage when the quantities q, q', &c. vary very slowly. Because this
consideration renders the integration by approximation of the differential
equations which determine the variables c, c', c", &c. in general much
easier.
513. Second Approximation of Celestial Motions.
Let us apply the preceding method to the perturbations of celestial
motions, in order thence to obtain the most simple expressions of their
periodical and secular inequalities. For that purpose let us resume the
differential equations (1), (2), (3) of No. 471, which determine the relative
motion of ^ about M. If we make
^ __ im' (X x^ + y / + z zQ _^ tJ"{^^"'\'yy" ^z^")
(x'2 + y'2 + z'^)8. (x''*^ + y'"^ + z"^)^
>.
+ &c.
/i
X being by the No. cited equal to
T +
ij' K:
+
x) ^ + (y'y) *+ {2f—z) '} 2 j(x'' _ x) H (y" — y) ^+ (z''z) '\ '
Ui (L
we shall have
{ (x" — ^T + {y" —y'r + (z"  z') '} ^
If, moreover, we suppose M + i" = ni aiid
i = V x*+ y* + z*
g' = Vx'«"+ f + z'^
0113 + ^BJE+fU^V
dt^ + e +Vdx>'
_ d°y my , /dR\
^  dl^ + 1^ + \dj)
m z /d R\
T + &C
=
(P)
dt' ^•
The sum of these three equations multiplied respectively by d x, d y, d z
gives by integration
dx^+dys + dz^ 2m . m
=
cl t'
^^ + ^ + ^fd'R
(Q)
the differential d R being only relative to the coordinates x, y, z of the
body /ti, and a being an arbitrary constant, which, when R = 0, becomes
by No. 499, the semiaxis major of the ellipse described by /* about
M.
106 A COMMENTARY ON [Sect. XI.
The equations (P) multiplied respectively by x, y, z and added to the
integral (Q) will give
_ , d^e* m . m , _ ^ ,r> , ^^^\ , f^^\ , /^Rx
o=^H^y +V+2/^^ + ^(dT) + ndl?) + HdT)' W
We may conceive, however, the perturbing masses /i', /j,", &c. multi
plied by a coefficient a, and then the value of § will be a function of the
time t and of «. If we develope this function according to the powers of a,
and afterwards make a = 1, it will be ordered according to the powers
and products of the perturbing masses. Designate by the characteristic
d when placed before a quantity, this differential of it taken relatively to a,
and divided by d a. When we shall have determined 3 f in a series or
dered according to the powers of «, we shall have the radius f by multi
plying this series by d «, then integrating it relatively to a, and adding to
the integral a function of t independent of «, a function which is evidently
the value of § in the case where the perturbing forces are nothing, and
where the body fi describes a conic section. The determination of § re
duces itself, therefore, to forming and integrating the differential equation
which determines d §.
For that purpose, resume the differential equation (R) and make for the
greater simjjlicity
differentiating this relatively to a, we shall have
0=%l±S + ^' + 2fSdR + i.sn' (S)
Call d v the indefinitely small arc intercepted between the two radius
vectors f apd § + d s; the element of the curve described by fi around M
will be V ds^ + §^d\K We shall thus have
dx2 + dy2 + dz2 = dg2_j.g2dv^
and the equation (Q) will become
Eliminating — from this equation by means of equation (R) we shall
have
dv2 pd
dt^ d t' s
r^+^ + fR'
whence we derive, by differentiating relatively to a,
2g'dv.dav _ gd^ag — agd^g 3m^
Jfe  dT^ p— +f3K — R dg.
Book I.] NEWTON'S PRINCIPIA 107
m p 8 p ,
If we substitute in this equation for — ^j— ^ its value derived from equa
tion (S), we shall have
d3,.^ d(dg3g + 2gciag)+dtq3/a^R+2g3R^+R^ag}
g^ d v ^ ^
By means of the equations (S), (T), we can get as exactly as we wish the
values of 5 g and of 3 v. But we must observe that d v being the angle
intercepted between the radii § and g + d g, the integral v of these angles
is not wholly in one plane. To obtain the value of the angle described
round M, by the projection of the radiusvector f upon a fixed plane, de
note by v^ , this last angle, and name s the tangent of the latitude of /* above
this plane ; then g ( I + s ') '^ will be the expression of the projected ra
diusvector, and the square of the element of the curve described by /«,
will be
g^dv/ • g'ds« ^
1 + s^^ "^ + (1 + s^)2'
But the square of this element is also g^dv'^ + dg*; therefore we have,
by equating these two expressions
d v . = == .
' V 1 + s^
We shall thus determine d v, by means of d v, when s is known.
If we take for the fixed plane, that of the orbit of /* at a given epoch,
d s
s and J— will evidently be of the order of perturbing forces. Neglecting
therefore the squares and the products of these forces, we shall have
V = v^ . In the Theory of the planets and of the comets, we may neglect
these squares and products with the exception of some terms of that
order, which particular circumstances render of sensible magnitude, and
which it will be easy to determine by means of the equations (S) and (T).
These last equations take a very simple form, when we take into account
the first power only of the disturbing forces. In fact, we may then con
sider d g and a v as the parts of g and v due to these forces ; 5 R, 5. g R'
are what R and g R' become, when we substitute for the coordinates of
the bodies their values relative to the elliptic motion : We may designate
them by these last quantities when subjected to that condition. The
equation (S) thus becomes.
108 A COMMENTARY ON [Sect. XL
The fixed plane of x, y being supposed that of the orbit of /«, at a given
epoch, z will be of the order of perturbing forces : and since we may
neglect the square of these forces, we can also neglect the quantity
(d R
1 — j. Moreover, the radius ^ differs only from its projection by quan
tities of the order z '. The angle which this radius makes with the axis
of X, differs only from its projection by quantities of the same order.
This angle may therefore be supposed equal to v and to quantities nearly
of tlie same order
X = g cos. V J y = ^ sin. v ;
whence we get
/d Rx . /dRx /d Rx
'^(dT)+nd7)=Kli7)'
and consequently g . R' = ^ (—, — j . It is easy to perceive by differentia
tion, that if we neglect the square of the perturbing force, the preceding
differential equation will become, by means of the two first equations (P)
/ x d y — y d x >^
V ~dl )
In the second member of this equation the coordinates may belong to
elliptic motion ; this gives ~rr constant and equal to V m a(l — e *),
a e being the excentricity of the orbit of /«. If we substitute in the ex«
pression of ^ 3 g for x and y, their values g cos. v and g sin. v, and for
^ ~" ^ , the quantity V /* a ( I — e '') ; finally, if we observe that
by No. (480)
m = n * a ^
we shall have
C a COS. vy n d t . g sin. v 4 2fd R + g (j — ) >
(^ — a sin. vy n d t . g cos. v I 2/ ^ ^ + i ( a) \
^g = m VI — e^
The equation (T) gives by integration and neglecting the square of
perturbing forces,
—2 r^j —  — = + — ffn dt.dRi jn at. pi ,— )
J a^ndt ^ m •^'^ ^ m*^ 'Vdg/ .,,v
dv= J (\)
V 1 — e*
x
\
(X)
Book I.] NEWTON'S PRINCIPIA. 109
This expression, when the perturbations of the radiusvector are known,
will easily give those of the motion of /x in longitude.
It remains for us to determine the perturbations of the motion in lati
tude. For that purpose let us resume the third of the equations (P) :
integrating this in the same manner as we have integrated the equation
(S), and making z = g 3 s, we shall have
a cos. vyn d t . g sin. v (—, — \ — a sin. v^n d t . g cos. vf ^ — j
3 s = — " ; (Z)
m V 1— e^ ^
5 s is the latitude of /* above the plane of its primitive oibit : if we wish
to refer tlie motion of <«. to a plane somewhat inclined to this orbit, by
calling s its latitude, when it is supposed not to quit the plane of the
orbit, s + 3 s will be very nearly the latitude of /* above the proposed
plane.
514. The formulas (X), (Y), (Z) have the advantage of presenting the
perturbations under a finite form. This is very useful in the Cometary
Theory, in which these perturbations can only be determined by quad
ratures. But the excentricity and inclination of the respective orbits of
the planets being small, permits a developement of their perturbations
into converging series of the sines and cosines of angles increasing pro
portionally to the time, and thence to make tables of them to serve for
any times whatever. Then, instead of the preceding expressions of 8 ^,
8 s, it is more commodious to make use of differential equations which
determine these variables. Ordering these equations according to the
powers and products of the excentricities and inclinations of the orbits,
we may always reduce the determination of the values of 5 g, and of 3 s
to the integration of equations of the form
equations whose integrals we have already given in No. 509. But we
can immediately reduce the preceding differential equations to this simple
form, by the following method.
Let us resume the equation (R) of the preceding No., and abridge it
by making
It thus becomes
no A COMMENTARY ON [Sect. XI.
In the case of elliptic motion, where Q = 0, g Ms by No. (488) a func
tion of e COS. (n t + « — '')» a e being the excentricity of the orbit, and
n t + e — « the n)ean anomaly of the planet [i. Let e cos. (n t  « — ■^)
= u, and suppose f * = p (u) ; we shall have
In the case of disturbed motion, we can still suppose g^ =z <{> (u), but
u will no longer be equal to e cos. (n t + « — «■). It will be given by
the preceding differential equation augmented by a term depending upon
the perturbing forces. To determine this term, we shall observe that if
we make u = 4/ (g*) we shall have
du d*. p* Xp^ Ap^
"¥ (?*) being the differential of vj/ (g*) divided by d.f ^ and 4^' (g*^) the
d^e^
differential of 4'' (§^) divided by d.f^ The equation (R') gives \ 
equal to a function of g plus a function depending upon the perturbing
force. If we multiply this equation by 2 ^ d f , and then integrate it, we
shall have  , f equal to a function of g plus a function depending upon
the perturbing force. Substituting these values of ' "^ and of ^^ —  in
the preceding expression of i — ~ + n * u, the function of f, which is in
dependent of the perturbing force will disappear of itself, because it is
identically nothing when that force is nothing. We shall therefore have
d^u . (\.^. p^ p^ d p^
the value of ^ —  + n^ u by substituting for — j —  , and —^ — , the parts
of their expressions which depend upon the perturbing force. But re
garding these parts only, the equation (R') and its integral give
i^20.
^^ = 8/Q,d,
Wherefore
^ + nMi = 2Q4' (g^)  8 r {s')/Q. s d {.
Again, from the equation u = <p (p ^), we derive d u = 2 g d f 4' (S ') f
this ^* = p (u) gives 2 g d g = d u. f' (u) and consequently
Book I.] NEWTON'S PRINCIPIA. Ill
Differentiating this last equation and substituting 9' (u) for — ^ — ? , we
shall have
_ 9" (u)
9' (u)
•^'"{n =:;z7;;v3.
<p" (u) being equal to — ' , , in the same way as f' (u) is equal to
■ ', ^ ■ . This being done : if we make
d u o '
u = e COS. (n t + £ — w) + 5 u,
the differential equation in u will become
dt^ p'(u)3«^^ f (u)
and if we neglect the square of the perturbing force, u may be supposed
equal to e cos. (n t + £ — »), in the terms depending upon Q.
The value of  found in No. (485) gives, including quantities of the
order e ^
g = al + e^ — u(l — e^) — u^u^
whence we derive
§2 =a«l+2e'— 2u(l— ie^) _u«— u^lrr p(u).
If we substitute this value of f (u) in the differential equation in b u,
and restore to Q its value 2 f d K ■{• ^ {—, — \ , and e cos. (n t + £ — «r)
for u, we shall have including quantities of the order e ^,
— a^{ ^ "*" i ^'~^cos. (nt + £—•!»•)— e^ COS. (2nt+ 2 e — 2^)1
— ?/ndt[sin. (nt+£^) U + ecos.(nt+£*)l 2/JR+g(^) }]{X0
When we shall have determined b u by means of this differential equa
112 A COMMENTARY ON [Sect. XL
tion, we shali have ^ f by differentiating the expression of j, relative to
the characteristic 3, which gives
f 3 9 1
3g=— a3u< 1 +7e*+2ecos. (nt+ g— w)+ e* cos.(2nt+2£— 2tir) l.
This value of 3 ^ will give that of 6 v by means of formula (Y) of the
preceding number.
It remains for us to determine d s ; but if we compare the formulas (X)
and (Z) of the preceding No. we perceive that 5 g changes itself into d s
by substituting (T — J for 2/d R + g (^i— ) in its expression. Whence
it follows that to get 3 s, it suffices to make this change in the differential
equation in d u, and then to substitute the value of 5 u given by this equa
tion, and which we shall designate by 8 u', in the expression of d g. Thus
we get
— /ndtsin.(nt + s^)U+ecos.(nt + .)}.(^);(ZO
^s=— a3u'l + ^e* +2ecos.(nt + « — »)+ — e*cos.(2nt+2e— 2«')
The system of equations (X'), (Y), (Z') will give, in a very simple
manner, the perturbed motion of /t in taking into account only the first
power of the perturbing force. The consideration of terms due to this
power being in the Theory of Planets very nearly sufficient to determine
their motions, we proceed to derive from them formulas for that purpose.
515. It is first necessary to develope the function R into a series. If
we disregard all other actions than that of .<* upon fi', we shall have by (513}
j^ __ /^'(xx^+ y/+ zzQ j«/
(x' * + y' ^ + 2' «# {(X' — x)« + (/ — y)^ + (z' — z)»i^ *
This function is wholly independent of the position of the plane of x,
y ; for the radical V {x' — x) * + (y' — y)'^+ (z' — z) \ expressing the
distance of /», /i', is independent of the position ; the function x ' + y '
+ z** + x' * + y' '^ + z" — 2 X x' — 2 y y' — 2 z zMs in like manner in
dependent of it. But. the squares x* + y * + z'^ and x'^ + y" + z'^
of the radiusvectors, do not depend upon the position ; and therefore the
(juantity x x' + y y' + z z' does not depend upon it, and consequently
Book I.] NEWTON'S PRINCIPIA. 1!3
R is independent of the position of the plane of x, y. Suppose in this
function
X = f COS. V ; y = f sin. v ;
x' = g' COS. v' ; y' = ^ sin. v' ;
we shall then have
P _ l^'\l i COS. (v' — v)+ z t!\ a/
{^ * + z") '" f '—2 % i COS. (v' — v) + g' '^ + (z'— z) ^\ ^ '
The orbits of the planets being almost circular and but little inclined
to one another, we may select the plane of x, y, so that z and z' may be
very small. In this case g and / are very little different from the semi
axismajors a, a' of the elliptic orbits, we will therefore suppose
g = a(l + uj; ' = a'(l + u/);
u^ and u/ being small quantities. The angles v^ v' differing but little
fiom the mean longitudes n t + s, n' t + ^'j we shall suppose
V = n t + « + V, ; v' = n' t f »' + v/ ;
v' and v/ being inconsiderable. Thus, reducing R into a series ordered
according to the powers and products of u^, v^, z, u/, v/, and 2', this series
will be very convergent. Let
— cos. (n' t — n t + «' _ g) _{a ' — 2 a a' cos. (n' t — n t f 1'— O+a'*}"^
= i A w + A (^^ COS. (n' t — n t 4 »' — + A ® cos. 2 (n' t — n t +«'— e)
+ A ® COS. 3 (u' t — n t + 6' — «) + &c. ;
We may give to this series the form ^ 2 A ^'^ cos. i (n' t — n t f t' — s),
the characteristic 2 of finite integrals, being relative to the number i, and
extending itself to all whole numbers from i = — oo to i = ao ; the value
i rr 0, being comprised in this infinite number of values. But then we
must observe that A ^~'^ = A ^^\ This form has the advantage of serving
to express after a very simple manner, not only the preceding series, but
also the product of this series, by the sine or the cosine of aiiy angle
f t + 0; for it is perceptible that this product is equal to
^2AW^^"' Ji(n't — nt+ t —i) +ft + t^l.
cos. ' ^ '
This property will furnish us with very commodious expressions for
the perturbations of the planets. Let in like manner
fa* — 2 a a' cos. (n t — n t ft' — t) + a'^]~^
= ^ 2 B ' cos. i (n t — n t + £ — t) ;
Bf"*') being equal to B ^''. This being done, we shall have by (483)
Vot. II. H
114
A COMMENTARY ON
[Sect. XI.
K = ^ . 2 A <o COS. i (n' t — n t + e' — t)
n! /d A ('\
■*" 2'"'^'*( j^)cos. i(n't — nt + s' — s)
, yl , ,/d A«\ . , , , ,
+ T "' ('dT')*^^^ 1 (n' t — n t + t^ —
— 9" ^^'' — ^'^ 2 . i A <') sin. i (n' t — n t + t' — «)
+ ^' . u/. 2.a«(^^A!l)cos. i (n' t — n t + s' — i
jti' /d'^A^'K
+  U/ u/ 2 a a' ( J — J— , ) COS. i (n' t — 11 t + g' — 1
a > a a d a /
A(.' /d* A ^'\
+ ^ «/^ 2 a' «(^^^)cos. i („' t  n t + .' ~ .
(if ' /d A Wx
— ^ (v/ — vj u, 2 . i a f ^ \ sin. i (n' t — n t +
,/dA('\ .
0
— ^ (v/ — V,) u/ 2. i a' ( ^^) sin. i (n' t  n t + .' —
— J (v/ — V,) 2 . 2 . i 2 A ''' COS. i (nM — n t + f' —
+ — /3 2^/4 COS. (n' t — n t + «' —
+ '^'^^'~^)' 2 B ^•) COS. i (n' t — n t + a' _
+ &c.
If we substitute in this expression of R, instead of u^, u/, v^, v/, z and z',
their values relative to elliptic motion, values which are functions of sines
and cosines of the angles n t + e, n' t + g' and of their multiples, R will
be expressed by an infinite series of cosines of the form ^' k cos. (i n' t
— i n t 4 A), i and i' being whole numbers.
It is evident that the action of a'''', /«.'", &c. upon fi will produce in R
terms analogous to those which result from the action of /*', and we shall
obtain them by changing in the preceding expression of R, all that relates
to /i, in the same quantities relative to /*'', /*'", &c.
Let us consider any term /j/ k cos. (i' n' t — i n t + A) of the expres
sion of R. If tlie orbits were circular, and in one plane we should
have i' = i. Therefore i' cannot surpass i or be exceeded by it, except
by means of the sines or cosines of the expression for u^, v^, z, u/, v/, z'
which combined with the sines and cosines of the angle n' t — n t + t' — s
Book I.] NEWTON'S PRINCIPIA. . 115
and of its multiples, produce the sines and cosines of angles in which i'
is different from i.
If we regard the excentricities and inclinations of the orbits as veiy
small quantities of the first order, it will result from the theorems of
(481) that in the expressions of u,, v^, z or ^ s, s being the tangent of the
latitude of /*, the coefficient of the sine or of the cosine of an anorle such
as f. (n t + s), is expressed by a series whose first term Ls of the order f ;
second term of the order f + 2; third term of the order f + 4 and so
on. The same takes place with regard to the coefficient of the sine or of
the cosine of the angle f (n' t + e') in tlie expressions of u/, v/, z'. Hence
it follows that i, and V being supposed positive and i' greater than i, the
coefficient k in the term m' k cos. (i' n' t — i n t + A) is of the ordar
i' — i, and that in the series which expresses it, the first term is of the
order i' — i the second of the order i' — i + 2 and so on ; so that the
series is very convergent. If i be greater than i', the terms of the series
will be successively of the orders i — i', i — 1^ + 2, &c.
Call w the longitude of the perihelion of the orbit of fi and 6 that of its
node, in like manner call »' the longitude of the perihelion of /«.', and ^
that of its node, these longitudes being reckoned upon a plane inclined
to that of the orbits. It results from the Theorems of (481), that in the
expressions of u^, v^, and z, the angle n t + « is always accompanied by
— w or by — 6; and that in the expressions of u/, v/, and z', the angle
n' t + e' is always accompanied by — v', or by — 6^ ; whence it follows
that the term /«.' k cos. (i' n' t — i n t + A) is of the form
At'kcos. (in't — int + V s — is — gw — g' w' — g" & — g"' <^),
g, g', g^', ^" being whole positive or negative numbers, and such that
we have
= i'  i — g — g' — g'' — g"'.
It results also from this that the value of R, and its different terms are
independent of the position of the straight line from which the longitudes
are measured. Moreover in the Theorems of (No. 481) the coefficient of
the sine and cosine of the angle «r, has always for a factor the excentricity e
of the orbit of /i ; the coefficient of the sine and of the cosine of the angle
2 ar, has for a factor the square e ^ of this excentricity, and so on. In like
manner, the coefficient of the sine and cosine of the angle 6, has for its
factor tan. ^ <p, (p being the inclination of the orbit of /a upon tlie fixed
plane. The coefficient of the sine, and of the cosine of the angle 2 6, has for
its factor tan.^ ^ p, and so on. Whence it results that the coefficient k has for
its factor, e «. e' «'. tan. «" (^ p) tan. «"' (i p') ; the numbers g, g', g", g'" being
H2
116 A COMMENTARY ON [Sect. XI.
taken positively in tlie exponents of this factor. If all these numbers are
positive, tliis factor will be of the order i' — i, by virtue of the equation
but if one of them such as g, is negative and equal to — g, this factor
will be of the order i' — i + 2 g. Preserving, therefore, amongst the
terms of R, only those which depending upon the angle i' n' t — i n t are of
the order i' — i, and rejecting all those which depending upon the same
angle, are of the order i' — i + 2, i' — i  4, &c. j the expression of
R will be composed of terms of the form
H e 8. e' s' tan. ^" {k<p) tan. s'". ( i ?^) cos. (i' n' t — i n t + V t'
_ i , _ g.. _ g'. ^' _ g/^ ^ _ g//^. ^'),
H being a coefficient independent of the excentrjcities, and inclinations .
of the orbits, and the numbers g, g', g'', ^" being all positive, and such
that their sum is equal to i' — i.
If we substitute in K, a (1 + u^), instead off, we shall have
/d Rx /d Rx
Kd7) = ndr)
If in this same function, we substitute instead of u', v' and z, their values
given by the theorems of (481), we shall have
/d Rn _ /d Rx
provided that we suppose e — w, and s — d constant in the differential of
R, taken relatively to e ; for then u^, v^ and z are constant in this differ
ential, and since we have v = n t + e + v^, it is evident that the preced
ing equation still holds. We shall, therefore, easily obtain the values
of gTi — V and of (. — V which enter into the differential equations of
the preceding numbers, when we shall have the value of R developed
into a series of angles increasing proportionally to the time t. The dif
ferential clRit will be in like manner easy to determine, observing to vary
in R the angle n t, and to suppose n' t constant ; for d U is the difference
of R, taken in supposing constant, the coordinates of f/^', which are func
tions of n' t.
516. The difficulty of the developement of R into a series, may be
reduced to that of forming the quantities M'^\ B ^% and their differences
taken relatively to a and to a'. For that purpose consider generally the
function
(a * — 2 a a' cos. tf + a' «) ~ '
Book L] NEWTON'S PRINCIPIA. 117
and develope it according to the cosine of the angle 6 and its multiples.
If we make — 7 = k, it will become
a
a' . i 1 — 2 a cos. ^ + » *
Let
(1 — 2 a cos. ^ + a «) "' = A b 0^ + b "' COS. & + b ® COS. 2 &
S 6 S
+ b ^3: COS. 3 ^ + &c.
s
b^% b^'', h^~\ &c. being functions of a and of s. If we take the logarith
mic diflferences of the two members of this equation, relative to the vari*
able df we shall have
. — b^'^sin. ^ — 2b(^)sin. 2^ — &c.
— 2 s a sin. 6
\ —2a COS. ^ + a 2 ^ b W 4. b (!) cos. <>+ b ^^'^ COS. 2 ^+ &c. '
S S 8
Multiplying this equation crosswise, and comparing similar cosines, we
find generally
(i_ 1) (1 4. «2)b"»^ — (i + s — 2)ab(i2)
b « = '^. ^ .2 ... (a)
(I —5). a ^ ^
We shall thus have b^% b'^^, &c. when b^*^^ and b^**' are known.
8 8
If we change s into s + 1, in the preceding expression of (1 — 2 a cos. 6
— s
+ a^) , we shall have
(1—2 a cos. d+a^) ~'~^ = ^ b (o^+b w cos. ^+b(2) cos.2 6+h^^^ cos.3<J4&c.
8+1 8+1 8+1 8+1
Multiplying the two members of this equation, by 1 — 2 a cos. tf + a%
and substituting for (1 — 2 a cos. ^ + a*) its value in series, we shall
have
^b») + b^i) COS. ^ + b(2) COS. 2 ^ + &c.
8 8 8
= (1 — 2 a cos. d\a^)\ b^o) + b^') cos. 6 f b^2) cos. 26 + Sccj
8 + 1 S+ I S+1
whence by comparing homogeneous terms, we derive
bW = (1 +a2)b») — ab»J5 — abC' + i).
8 8+1 S + 1 8 + 1
The formula (a) gives
i(l + a^)bW — (i + s)ab^'^)
b P+i) = .. , « + ' '' ^^ L±L ;
s+1 (1 — s).a
The preceding expression of b ^'^ will thus become
8 *
2s.ab^^) — s(l + a«)b«')
8 1 — s
H3
118 A COMMENTARY ON [Sect. XL
Changing i into i f I in this equation we sliall have
2s«bW — s(l + a2)b('+i)
\^ (i + I) — ___»±i 8 + 1
i — s+ 1
and if we substitute for b ('+^^ its preceding value, we shall have
8 + 1
s(i + s)a(I + a «)b (•')+ sf2(i — s)a2_i(l4.a«)2]bW
b P + 1) =z '^\ ^±i
I (l — s) (i — s + l)a
These two expressions of b ^'^ and b (' + '^ give
a 8
^l^t^.(l + a«)b« — 2.1^l±^ab^l+>)
substituting for b (' + ') its value derived from equation (u), we shall have
s
b CO — ? • ^ « . (c)
an expression which may be derived from the preceding by changing i
into — i, and observing that b ^'^ = b^~'^. We shall therefore have by
means of this formula, the values of b (% b ^^\ b ^^\ &c. when those of
i^i 84. 1 84.1
b(o), b^i), b(2), &c. areknown.
■ 8 •
Let X, for brevity, denote the function 1 — 2 a cos. ^ + a '. If we
differentiate relatively to a, the equation
X « = ^ b W + b (1) cos. ^ + b ® COS. 2 ^ + &c.
8 8 8
we shall have
dbW) dbW db(2)
— 2 s (a — COS. ^) X « 1 = i . — ? \ f— COS. 6 + — ,^— cos. 2 ^ + &c.
^ ' ^ da ' da da
But we have
— a + COS. S r= ^ ;
2 a
We shall, therefore, have
/, .^ .8 db^o) db('^
MJ_ZlfLJx«> — 5A_ = i_^+ _. cos.^ + &c.
a a '^ d a ' d a
whence generally we get
dbw sb«
_J>_ = ^(1— '') b 0) !. .
da a 8 + 1 '^
Substituting for b ^'^ its value given by the formula (b), we shall have
'^^^'_ i + (i + 2s)a' , 2(1^3+1) ,,^,
d«  a(l_a^) • , l_a« *, *
Book I.] NEWTON'S PRINCIPIA. 119
If we differentiate this equation, tve shall have
d « h P) d h <')
_i+(i+2s)a' "^ . f2(i+s)(l +a') i
d a* ~ a (1 —a') ' d
d b ^' + ')
8(is+ l) 1? 4(is + !)« .,.„
I_a2 • da (l_a2)2 "
Again differentiating, we shall get
d 3 b ('^ d * b (') d b ^>'
"" : _ i4(i+2.s) a" "^ r ■ gf (i + s)(i + «'') i >/r
da«  a(l_a2) • j^a i" ^  (1?— a^ a^ j da
j 4(i + s)«(3 + «') , 2i , 2(i — s+1) ^'^^'"^^
"•■\ (!—«')' "^a'/" 1 — a^ • da^
8(is+ l)a ^1'^''*"'' 4(i — s+l)(l+3«'') , ,
(l_a2)2 • da (l_a2)3 ^
Thus we perceive that in order to determine the values of b and oi
s
its successive d ifferences, it is sufficient to know those of b ^°^ and of b ^^\
8 8
We shall determine these two as follows :
If we call c the hyperbolic base, we can put the expression of X — * un
der this form
X" = (1 — ac*^'— 1)«(1 — a c — «V^— !)».
Developing the second member of this equation relatively to the powers of
c 9 V— 1, and c — ^ '^"^ it is evident the two exponentials c ^ ^ V— i, c — ' * V— i
will have the same coefficient which we denote by k. The sum of the
two terms k . c ^ * v — i and kc — J*v^— Ms2k cos. i 6. This will be the
value of b ^'^ cos. i d. We have, therefore, b ^'^ = 2 k. Again the ex
s s
pression of X— * is equal to the product of the two series
1 + sac* 1 + lil+lla^c^^Vi 4 &c.
i m l£
1 + sac«Vl + L(L+Jla2c2»V_I + Sac;
multiplying therefore these two together, we shall have when i =
k = l +S^a^ + (?(^±ii)'a'^ + &C.;
and in the case of i = 1,
wherefore
H 4
120 A COMMENTARY ON [Sect. XI.
That these series may be convergent, we must have « less than unity,
which can always be made so, unless a = a' ; « being = — 7 , we have only
to take the greater for the denominator.
In the theory of the motion of the bodies a, fi\ (il', &c. we have occasion
to 4cnow the values of b ^"^ and of b ^*^ when s = ^ and s = f . In these
( 8
two cases, these values have but little convergency unless a is a small
fraction.
The series converge with greater rapidity when s r= — , and we have
^, ^ r 2.4" 4*2.4.6 "4.62.4.6.8'' "4.6.8* 2:37:710 i " +^'^*
"" i
In the Theory of the planets and satellites, it will be sufficient to take
the sum of eleven or a dozen first terms, in neglecting the following
terms or more exactly in summing them as a geometric progression whose
common ratio is 1 — o *. When we shall have thus determined b ^"^ apd
b f"), we shall have b ^ in making i = 0, and s = — ^ in the formula (b),
\ \
and we shall find
(1 + a2)bW + 6ab<'>
KW) — zi zi.
2
If in the formula (c) we suppose i = 1 and s = —  we shall have
2ab(o^ + 3 (1 + a2)b">
b (1) = ^ ^ .
(1 — a^)«
t
By means of these values of b^^^ and of b^'^ we shall have by the pre
\ i
cedinf forms the values of b ^''> and of its partial differences whatever may
be the number i ; and thence we derive the values of b ^'^ and of its dif
f
ferences. The values of b ^^^ and of b ^') may be determined very simply,
I I
Book L] NEWTON'S PRINCIPIA. 121
by the following formulae
b (") b (')
b W = HI • b ('^ — 3 ~^
I (i^n^s ~ (ia^)^
2 2
Again to get the quantities A ^"\ A '^^, &c. and their diiferences, we
must observe that by the preceding No., the series
^ A (") + A (1) COS. 6 + A ^'^ cos. 2 () + &c.
results from the developeraent of the function
t^^ _ (a^ — 2 a a' cos. 9 + a'')~K
Si
into a series of cosines of the angle 6 and of its multiples. Making — ; = a,
this same function becomes
h h h
which gives generally
AW = _l.b<»);
2
when i is zero, or greater than 1, abstraction being made of the sign.
In the case of i = 1, we have
A« = ^  h^^K
a'* a I
We have next
db«)
/dA«N_ 1!^ ^ /d^x
I d a y ~ a' • d a Vl J'
But we have r— = ~; ; therefore
da a:
db«
/d_AWx _ __ J i_
Wa>/~a''^*da'
and in the case of i = 1, we have
dbfi)
V da ;~ a'« 1 da J
Finally, we have, in the same case of i = 1
d^b^'>
/ d^AW x _ _ J_ L,
\ d a^ / ~ 3'="' da' *
122 A COMMENTARY ON [Sect. XI.
/ d«A»\ _ _ 1 i
« V daW~ a'** da» '
&c.
To get the differences of A ''^ relative to a', we shall observe that A ^'^
being a homogeneous function in a and a', of the dimension — 1, we
have by the nature of such functions,
whence we get
zdM^x _ o/dA^x /d*A('\
f(\ * A W» /A A 0). ,(] 2 A (')x
^'^(d4r)=2A0) + 4a(ij^)+a^(^.);
, 3 /d ^ A Wx _ . p. „ /d A Wx _ , fd'A «v 3 /d^A «x .
&c.
We shall get B ^'^ and its differences, by observing that by the No. pre
ceding, the series
I B(") + BW cos. ^ + B(2) cos. 2 ^ + &c.
is the developement of the function
a' 3 (1 — 2 a COS. 6 + a^)"^
according to the cosine of the angle & and its multiples. But this function
thus developed is equal to
a's ci\^io) ^ b") COS. ^ + b(2) COS. 2 6 + &c.)
l"l i I i'
therefore we have generally
B(0 = ~b«;
Whence we derive
db('^ d«b(') 2
3 ; &c.
/ d B (') n _ J_ __ ; / d ' B W . _ 2
V da /~ a'** da V daW~ a'^'
da«
Moreover, B ^'' being a homogeneous function of a and of a', of the
dimension — 3 we have
Book I.] NEWTON'S PRINCIPIA. 123
whence it is easy to get the partial differences of B ^'^ taken relatively to
a' by means of those in a.
In the theory of the Perturbations of ^a', by the action of (i, the "values
of A ^') and of B W, are the same as above with the exception of A ^'^ which
a' 1
in this theory becomes 2 > ^ ^'^* Thus the estimate of the values of
a a ]
2
A ^'\ B ^'\ and their differences will serve also for the theories of the two
bodies /^ and fif.
517. After this digression upon the developement of R into series, let
us resume the differential equations (X'), (Y), (Z') of Nos. 513, 514; and
find by means of them, the values of 3 ^, 5 v, and b s true to quantities
of the order of the excentricities and inclinations of orbits.
If in the elliptic orbits, we suppose
^ = a(l + u,); /=a'(Hu/);
V = n t + g + v^ ; v' = n' t — «' + v/ ;
we shall have by No. (488)
u^ = — e cos. (n t + g — w) ; u/ = — e' cos. (n' t + s' — «r') ;
v, = 2 e sin. (n t + £ — w) ; \f = 2 e' sin. (n' t + g' — w') J
n t + g, n' t + g' being the mean longitudes of a*, /*' ; a, a' being the serai
axismajors of their orbits ; e, e' the ratios of the excentricity to the semi
axismajor; ^ and lastly «r, w' being the longitudes of their perihelions. All
these longitudes may be referred indifferently to the planes of the orbits,
or to a plane which is but very little inclined to the orbits ; since we ne
glect quantities of the order of the squares and products of the excen
tricities and inclinations. Substituting the preceding values in the ex
pression of R in No. 515, we shall have
R = 5 2 A ^'5 cos. i (n' t — n t 4 «' —
4M
d A ^'\ 1
e cos.U (n' t — nt+i' — g) + n t + g — v\
e' cos.[i (n' t — n t + g' — g) + n t H e — «^l;
the symbol 2 of finite integrals, extending to all the whole positive and
negative values of i, not omitting the value i = 0.
Hence we obtain
124 A COMMENTARY ON [Sect. XI
At' r /d^A('\ /dAW\ /rlA(i)% 1
¥{'&) + ^''(Tr) + 24^) + *A»'}..'cos.(nt+,W)
e COS. Ji(n'tnt+ s's)+ nt+ «w
(nn')n i v d a / ) J
+ n t + ^ — «'J;
the integral sign 2 extending, as in what follows, to all integer positive
and negative values of i, the value i = being alone excepted, because
we have brought from without this symbol, the terms in which i = : /i' g
is a constant added to the integraiy^? R. Making therefore
^ , 3/ d'A(0) ^ , ^ 2/dA(0)x^
2ii (n — n) — n t V da / n — n' J
i(n — n') — n(. \da/ J
nm 1 . ,/ d'A"'\ ,. ,, ,/dA('"v
taking then for unity the sum of the masses M + /"■» and observing that
(237) ^L+Jf = n ^ the equation (X') will become
„ i'.iu , ,, „ , , n'p,' ,/d A»v
_!i;^',a.(iA^) +^^a A <4cos. i (n'tn t+ ,'.)
2 iNda/n^n J ^
Book L] NEWTON'S PRINCIPIA. 125
+ n* At' C e COS. (n t + s — t?)
+ n * /»' D e' COS. (n t + £ — r,')
4 n V' s C « e COS. {i (n' t — n t + e' — «) + n t + « — t»l
+ n V 2 D « e' cos.{i (n' t — n t + £' — j) 1 n t + e — t^'l;
and integrating
ul ^ I V d a / n — n/ J • , i . * , / \
— ^ n « 2 . r— ^ 7.r 5  — cos. 1 (n' t — n t + f' —
2 1  (n — n')' — n
+ At' f^ e cos. (n t + e — ») + /«,' f/ e' sin. (n t + e — »')
— ^ C . n t . e sin. (n t + s — ^) — — D . n t. e' sin. (n t + i — n>')
"^ '''^' {i (n — nO — n p'^ITlT' ^ ^"^'^^ (n^ t — n t + ^' — + nt+«— «r]
+ ^'^ (i(nnOlrn' '^'''°'^'^'''^"""^'^'~'^'^"^"^'~"'^'
f^ and f/ being two arbitraries. The expression of 3 ^ in terms 3 u, found
in No. 514 will give
+ fn'..{ \da^ "° , }cos.i(n'tnt + .'0
s '^ 1 * (n n ) * — n * ^
— /*' f e cos. (n t + £ — zr) — yl {' ^ cos. (n t + « — '='')
+ ^ /i' C n t e sin. (n t + « — ar) + ^ a*' D « t e' sin. (n t + s — ■^)
+
/* *» 2 . / j i.(n_n')^ — n== U (nn')nj « — n ''J >
V. X e cos. Ji (n' t — nt+s' — £) + nt+j — ■a\ )
I
DW
— /.' . n '^ 2 . __,^__p_^e' cos. Ji(n' tn t+ a'— «)+n t+s— ^'j.
f and f ' being arbitrary constants independent of f ^, f/.
This value of 6 ^, substituted in the formula (Y) of No. 513 will give 8 v
or the perturbations of the planet in longitude. But we must observe that
n t expressing the mean motion of (i, the term proportional to the time,
ought to disappear from the expression of 6 v. This condition determines
the constant (g) and we find
1 /dA(o)x
g =  3 ''^ ("dir)
126 A COMMENTARY ON [Sect. XI.
We might have dispensed with introducing into the value of 3 ^ the
arbitraries f^ f/, for they may be considered as comprised in the elements
e and tr of elliptic motion. But then the expression of 8 v would include
terms depending upon the mean anomaly, and which would not have
been comprised in those which the elliptic motion gives : that is, it is more
commodious to make these terms in the expression of the longitude dis
appear in order to introduce them into the expression of the radiusvector »
we shall thus determine f, and f/ so as to fulfil this condition. Then if we
/d A^''\. /d A ^^\
substitute for af — . — j — jits value — A^'~^^ — af — r j, we shall
have
D = aA<..a>(^)i.(4;i'),
n — 1 (n — n') n — i (n — n') \ d a /
/d^ A^'^K
'd AW\ . , ,/d2Af°'
Moreover let
E 0> = _ «i^, a A <" + i'("n;).in+i( nn^)j^3n;
n — n' 1 * (n — n ) ^ — n *
1 n
(i_l)n i^Mn+i(nnO?_3n
G^*) =
« — IX i2(n — nO' — n'
r , /d A ^\ . 2 n . (.) 2 n ^ E t')
/d A(''\
(i _ ]) (2 i 1) n a A('») + (i — 1) n a« (—5^—)
2 Jn — i (n — n')]
2 n ^ D ») .
i^i— Jn— i (n — n')]''
DooK I.] NEWTON'S PRINCIPIA. 127
and we shall have
£ /d A«\ . 2n
a^e^V'daJ"^ 2^ * {^(n — nO*— n^ ^
COS. i (n' t — n t + e' — i)
— /i' f e COS. (n t + 1\«) — /f e' cos. (n t + ? — w')
+ l/C.ntesin. (nt + g — w) + ^/Dn te'sin. (n t + t — .r')
E (') c
jrp^ e COS. ii (n' t — n t + 1' — t) + n t+ e — ^\
+ n*/i'
« //^
n2{ni(nn')
, r ^ 2n3{a^(^^) + ^iLaA4 . .
_^' J n" . ^i,_^ t \da/'n — n^ J_ J sin. i
''2 ^ti(n — nO'^ '+ i(n — n').U%(n — nO' — n"r)
(n' t — n t + e" —
+ fjk' . C . n t . e COS. (n t + s — or) + <«,' D . n t . e' cos. (n t + s — J)
r FW . c I
r r. esin. ii(n't — n t + e' — e) + n t+f — =r?
n — 1 (n — n') ^ ■' ' ' * I
+ "^'M GO) M
I n — i (n — nO ^'^"'^^("'^~"^ + ''~'^ "^"^+'~"U
the integral sign 2 extending in these expressions to all the whole positive
and negative values of i, with the value i = alone excepted.
Here we may observe, that even in the case where the series represent
ed by
2. A ^') cos. 5 (n' t — n t + f' — e)
is but little convergent, these expressions of — and of b v, become con
3.
vergent by the divisors which they acquire. This remark is the more
important, because, did this not take place, it would have been impossible
to express analytically the mutual perturbations of the planets, of which
the ratios of their distances from the sun are nearly unity.
These expressions may take the following form, which will be useful to
us hereafter. Let
h = e sin. w ; h' = e' sin. ?/ ;
1 = e COS. w ; 1' = e' COS. w' ;
then we shall have
a 6 Vda/^2 \ i2(n — n')^— n* i ^ '
— /*'(hf+h'f')cos.{nt + — /*'(lf+l'n sin. (n t + »)
128 A COMMENTARY ON [Sect. XI.
+ ^ U C + 1' D] n t sin. (n t + — ^' {h C + h'D}n t cos. (n t + a)
2 ti(n — nO* i(n — n') U^ (n — nQ * — n^] J
sin. i (n' t — n t + i' — s)
+/{hC+li'D}.nt.sin.(nt+6)+/tqi.C+l'.D}nt.cos.(nt+£)
( n iln » sin.fiKtnt + s'~0 + nt+.} )
) hF^'^ + h'G^'^ t
(~ n— i(n — nO ^'^^'^"'^~"^+''~^^+"^+^U
Connecting these expressions of 3 ^ and 3 v with the values of g and v
relative to elliptic motion, we shall have the entire values of the radius
vector of Ao, and of its motion in longitude.
518. Now let us consider the motion of fi in latitude. For that pur
pose let us resume the formula (Z') of No. 614. If we neglect the pror
duct of the inclinations by the excentricities of the orbits it will become
. d^au' ^ 2 , , 1 /d Rn
the expression of R of No. 515 gives, in taking for the fixed plane that
of the primitive orbit of /i,
the value of i belonging to all whole positive and negative numbers in
eluding also i = 0. Let y be the tangent of the inclination of the orbit
of fji,', to the primitive orbit of fiy and n the longitude of the ascending
node of the first of these orbits upon the second ; we shall have very
nearly
z' = a' 7 sin. (n' t + «' — II) ;
which gives
(4^) = / . y. sin. (n' t+g' — n) — ^ . a' B « y sin.(n t+£— n)
^a'sB^'^'ysin. {i (n' t — n t + s'— O + n t + s— nj
the value here, as in what follows, extending to all whole positive and
negative numbers, i = being alone excepted. The diiferential equation
Book I.] NEWTON'S PRINCIPIA. 129
in 3 a' will become, therefore, when the value of (i — \ is multiplied by
n* a^, which is equal to unity,
,12/),,/ rt
= ""r + n'du' — fi'n\4«y sin. (u' t + ^' — n)
d t ^ a' ' ' ^
+ ^^ a a' B ") 7 sin. (n t + s — n)
+ ^^'aa'sB^'i'ysin. {i (n' t — nt+s' — + n t+s — n)] ;
whence by integrating and observing that by 514
8 s = — a 3 u',
fj/ n^ a ^ • , , , N
d s = 2 n • 72 7 sin. (n' t + «' — n)
n — n^ a^ ^ '
'J B <') . n t . 7 COS. (n t + £ — n)
/I'n^ a^a' B <' — ^>
' 2.^^— ^^— — ,.^7sin.{i(n'tnt+s'0+nt^^^^
2 n*_^n— i{n— nOF
To find the latitude of ^ above a fixed plane a little inclined to that of
its primitive orbit, by naming p the inclination of this orbit to the fixed
plane, and 6 the longitude of its ascending node upon the same plane ; it
will suffice to add to 5 s the quantity tan. p sin. (v — 6), or tan. tp sin. (n t
4 s — 6), neglecting the excentricity of the orbit. Call f/ and ^ what p
and 6 become relatively to /*'. If /j, were in motion upon the primitive
orbits of /ct', the tangent of its latitude would be tan. <p' sin. (n t + £ — 6');
this tangent would be tan. <p sin. (n t + e — ^)j if A^ continued to move in
its own primitive orbit. The difierence of these two tangents is very
nearly the tangent of the latitude of fi, above the plane of its primitive
orbit, supposing it moved upon the primitive orbit of /m' ; we have there
fore
tan. f sin. (n t+s — ^) — tan. f> sin. (n t+e — ^) = 7 sin. (n t+e — n).
Let
tan. <p sin. ^ = p ; tan. <p sin. ^ = p' ;
tan. p COS. ^ = q ; tan. ^ cos. ^ = q' ;
we shall have
7 sin. n = p' — p ; 7 COS. n = q' — q
and consequently if we denote by s the latitude of /x above the fixed plane,
we shall very nearly have
s = q sin. (n t + g) — p cos. (n t + s)
— ^ ^ (p' — p) B f') n t sin. (n t +
Vol. II I
130 A COMMENTARY ON [Sect. XI.
— '^ ^ ^ (q' r q) B t'> n t cos. (n t + e)
— n^TZF^ • h ^("1' ~ "l) """• (''' t + eO — (p' — p) COS. (n' t + 0
/ ct'n'. a'a\ Jn^—Jn— i(n— n')' '^ ^ ^t^ T5f
) ~^P'~.P^^^'7w2  cos.[i(n'tnt+8'3)+nt+€} (
519. Now let us recapitulate. Call (g) and (v) the parts of the radius
vector and longitude v upon the orbit, which depend upon the elliptic
motion, we shall have
g = (^) + a^; V = (v) + av.
The preceding value of s, will be the latitude of /a above the fixed plane.
But it will be more exact to employ, instead of its two first terms, which
are independent of /«.', the value of the latitude, which takes place in the
case where /i quits not the plane of its primitive orbit. These expressions
contain all the theory of the planets, when we neglect the squares and the
products of the excentricities and inclinations of the orbits, which is in
most cases allowable. They moreover possess the advantage of being
under a very simple form, and which shows the law of their different
terms.
Sometimes we shall have occasion to recur to terms depending on the
squares and products of the excentricities and inclinations, and even to
the superior powers and products. We can find these terms by the pre
ceding analysis, the consideration which renders them necessary will al
ways facilitate their determination. The approximations in which we
must notice them, would introduce new terms which would depend upon
new arguments. They would reproduce again the arguments, which the
preceding approximations afford, but with coefficients still smaller and
smaller, following that law which it is easy to perceive from the deve
lopement of R into a series, which was given in No. 515 ; an argument
i^hichf in the successive approximations^ is found for tliejtrst time among the
quantities of any order iiohatever r, and is reproduced only by quantities oj
the orders r+2, r+4', &c.
Hence it follows that the coefficients of the terms of the form
t . ' . (n t + s), which enter into the expressions of f, v, and s, are ap
cos. ^
proximated up to quantities of the third order, that is to say, that the
approximation in which we should have regard to the squares and pro
Book I.] NEWTON'S PRINCIPIA. 131
ducts of the excentricities and inclinations of the orbits would add nothing
to their values ; they have therefore all the exactness that can be desired.
This it is the more essential to observe, because the secular variations of
the orbits depend upon these same coefficients.
The several terms of the perturbations of g, v, s are comprised in the
form
sm
k . ■ fi (n' t — n t + e' — j) + r n t + r ??,
COS. ' ^ ' i»
r being a whole positive number or zero, and k being a function of the
excentricities and inclinations of the orbits of the order r, or of a superior
order. Hence we may judge of what order is a term depending upon a
given angle.
It is evident that the motion of the bodies /ji.'', ijJ", &c. make it neces
sary to add to the preceding values of f, v, and s, terms analogous to
those which result from the action of (jI ; and that neglecting the square of
the perturbing force, the sums of all these terms will give the whole va
lues of ^i V and s. This follows from the nature of the formulas (X'),
(Y), (Z'), which are linear relatively to quantities depending on the dis
turbing force.
Lastly, we shall have the perturbations of /i', produced by the action of
(I by changing in the preceding formulas, a, n, h, 1, £, zf, p, q, and /t' into
a', n', h', Yf s', ^', p\ q', and /i and reciprocally.
THE SECULAR INEQUALITIES OF THE CELESTIAL MOTIONS.
520. The perturbing forces of elliptical motion introduce into the expres
sions of gj J :7 9 and s of the preceding Nos. the time t free from the sym
bols sine and cosine^ or under the form of arcs of a circle, which by in
creasing indefinitely, must at length render the expressions defective. It
is therefore essential to make these arcs disappear, and to obtain the
functions which produce them by their developement into series. We
have already given, for this purpose, a general method, from which it re
sults that these arcs arise from the variations of elliptic motion, which are
then functions of the time. These variations taking place very slowly
have been denominated Secular Inequalities. Their theory is one of the
most interesting subjects of the system of the world. We now proceed to
expound it to the extent which its importance demands.
12
132 A COMMENTARY ON [Sect. XL
By what has preceded we have
pi — h sin. (n t + s) — 1 cos. (n t + «) — Sac.y
^^^\ + ^nC + y.Bl.nt. sin. (n t + I
[_— ~{h . C + h' . D} . n t . COS. (n t + e) + / S.J
d V
7 = n + 2 n h sin. (n t + «) + 2 n 1 cos. (n t + «) + &c.
— /*' U C + r Dl n * t sin. (n t + s)
+ fi [h C ^ h' B] n"" t COS. (n t + i) + /*' T ;
s = q sin. (n t + — P cos. (n t + «) + &c.
— ^ a 2 a' (p — p) B ^1). n t . sin. (n t + f)
— ^ a'^ a' (q' — q) B ^^\ n t. cos. (n t + s) + /i' %;
S, T, ;)(; being periodic functions of the time t. Consider first the expres
sion of 1 — , and compare it with the expression of y in 510. The arbi
trary n multiplying the arc t, under the periodic symbols, in the expres
d V
sion of ^— ; we ought then to make use of the following equations found
in No. 510,
= X' 4 ^. X" — Y;
= Y' + 6.Y" +X"~2 Z;
Let us see what these X, X', X'', Y, &c. become. By comparing the ex
d V
pression of t— with that of y cited above, we find
X = n + 2 n h sin. (n t + + 2 n 1 cos. {n t + t) + /m' T
Y z=fj/n^{hC + h'T>] cos. (nt+0— /n»{lC + FDl sin. (nt+0
If we neglect the product of the partial differences of the constants by
the perturbing masses, which is allowed, since these differences are of the
order of the masses, we shall have by No. 510,
X' = (i^) U + 2 h sin. (n t + s) + 2 1 cos. (n t + 01
f 2 n ( Y^) [h COS. (n t + g) — 1 sin. (n t + t)}
+ 2 n(^)sin. (n t + i) + 2 n(^)cos. (n t + ;
X'' = 2 n(^) {h COS. (n t j — 1 sin. (n t + *)]
Book I.] NEWTON'S PRINCIPIA. 133
The equation = X' + ^ X" — Y will thus become
= (^) U + 2 h sin. (n t + + 2 1 cos. (n t + i)}
+ 2n(^)sin. (n t + + 2 n(jJcos. (n t + e)
+ 2n {^ (^) + (^) }. Uicos.(nt+0lsin.(nt+01
— /i'n^hC+h'Dlcos. (nt+O+A^' n* U C+FD] sin.(n t+t).
Equating separately to zero, the coefficients of like sines and cosines, we
shall have
If we integrate these equations, and if in their integrals we change S
into t, we shall have by No. 510, the values of the arbitraries in functions
of t, and we shall be able to efface the circular .arcs from the expressions
d V
of i — and of g. But instead of this change, we can immediately change
6 into t in these differential equations. The first of the equations shows
us that n is constant, and since the arbitrary a of the expression for g de
pends upon it, by reason of n '^ = — 3 , a is likewise constant. The two
other equations do not suffice to determine h, 1, e. We shall have a new
d v
equation in observing that the expression of ^ — , gives, in integrating,
yn d t for the value of the mean longitude of /i. But we have supposed
this longitude equal to n t + £ ; we therefore have nt+E =:yndt, which
gives
! d t ^ d t '
and as we have :; — = 0, we have in like manner t— = 0. Thus the two
d t ' d t
arbitraries n and i are constants ; the arbitraries h, 1, will consequently be
determined by means of the differential equations,
^ = ^{\C + YT)]; (I)
ni = '2lhC + h'D}; (2)
13
134 A COMMENTARY ON [Sect. XI.
The consideration of the expression of r— having enabled us to deter
mine the values of n, a, h, 1, and s, we perceive a priori, that the differen
tial equations between the same quantities, which result from the expres
sion of f, ought to coincide with those preceding. This may easily be
shown a posteriori y by applying to this ejcpression the method of 610.
Now let us consider the expression of s. Comparmg it with that of y
citeif above, we shall have
X = q sin. (n t + s) — p cos. (n t + + z"' %
Y = ^ . a« a' B(i) (p — pO sin. (n t + «)
+ !^. a« a' Bf) (q — q') cos. (n t + 0,
n and «, by what precedes, being constants; we shall have by No. 510,
X' = {^) sin. (n t + _ (if) cos. (n t + s)
X" = 0.
The equation = X' + tf X'' — Y hence becomes
° = (d^) ''"• (" * + '^ "" dl ''*''• ("* + *)
— ^ a» a' B(i) (p — p') sin. (n t +
— ^ a« a' B^i) (q — q') cos. (n t + ;
whence we derive, by comparing the coefficients of the like sines and co
sines, and changing 6 into t, in order to obtain directly p and q in
functions of t,
^ = ^.a«a'Bn).(qqO; (3)
^^ = ^.a«a'B(MppO; (4)
When we shall have determined p and q by these equations, we shall
substitute them in the preceding expression of s, effacing the terms which
contain circular arcs, and we shall have
s = q sin. (n t + e) — p cos. (n t + g) + a^' %.
r\ n
521. The equation ^ = 0, found above, is one of great importance
in the theory of the system of the world, inasmuch as it shows that the
mean motions of the celestial bodies and the majoraxes of their orbits are
unalterable. But this equation is approximate to quantities of the order
Book L] NEWTON'S PRINCIPIA. 135
/*' h inclusively. If quantities of the order iil h *, and following orders,
d V
produce in t — , a term of the form 2 k t, k being a function of the ele
ments of the orbits of ^ and (jI\ there will thence result in the expression of
V, the term k t^, which by altering the longitude of /i, proportionally to
the time, must at length become extremely sensible. We shall then no
longer have
dt  "'
6ut instead of this equation we shall have by the preceding No.
^" = 2k;
d t '
It is therefore very important to know whether there are terms of the
form k . t ^ in the expression of v. We now demonstrate, that if
"we retain only thejirstpcmer of the perturbing masses, however Jar may pro
ceed the approximation, relatively to the powers of the excentricities and
inclinations of the orbits, the expression v mil not contain such terms.
For this object we will resume the formula (X) of No. 513,
acos.v/ndt^sin.v  2fdR+^\^ — j \ asin.v/hdt.gcos.v ■! 2/6?R+gfT— ) r
3 3= i — ^
^ m V l — Q
Let us consider that part of 5 g which contains the terms multiplied by t \
or for the greater generality, the terms which being multiplied by the sine
or cosine of an angle a t + /3, in which a is very small, have at the same
time a^ for a divisor. It is clear that in supposing a = 0, there will re
sult a term multiplied by t ^, so that the second case shall include the first.
The terms which have the divisor a \ can evidently only result from a
double integration ; they can only therefore be produced by that part of
h J which contains the double integral signyi Examine first the term
2 a COS. vyn d t (^ sin. \/ d R)
m V (1 —e^) •
If we fix the origin of the angle v at the perihelion, we have
a(le»)
1 + e cos. v '
and consequently
a(l— e^)— g
cos. V = — ^ ? ;
eg
whence we derive by differentiating,
p ^ d V . sm. V = — ^ ' .dp;
e
14
1S6 A COMMENTARY ON [Sect. XI.
but we have,
g* d V = d t V m a (1 — e'') =a*. n d t V 1 — c*;
we shall, therefore, have
a n d t ^ sin. v _ g d g
The term
2 a COS. vy n d t . {(; sin. v yrf Rj
m V 1 — e''
will therefore become
^f^/(f d g/rf R), or H^ Jp^rf R /g«. rf R^.
It is evident, this last function, no longer containing double integrals,
there cannot result from it any term having the divisor a \
Now let us consider the term
2 a sin. vy n d t ff cos. v/d R]
ra v^ 1 — e*
of the expression o?d g. Substituting for cos. v, its preceding value in ^,
this term becomes
2 asm.v/n d t. {g — &{l —e^)] .fdR
me V 1 — e*
We have
g = a {l+ie' + ex'h
^ being an infinite series of cosines of the angle n t + i, and of its multi
ples; we shall therefore have
JjlAl Jg _ a (1 —e')]/d R = a/n d t {i e + x']fd R.
Call ^' the integral y';i/ n d t ; we shall have
a/n d t. {Ie + X']fdn = ^ a e/n d t/d R + a%"/rf R—^/z'' • ^R
These two last terms not containing a double integral sign, there can
not thence result any term having a » for a divisor; reckoning only terms
of this kind, we shall have
___ 2 a sin, v/n d t f g cos , yfd R] _ 3 a'e sin, v/n dtfdR
m V 1 — e* ~ m V 1 — e»
^l±.^^fudt/dR;
n d t m"^ ^
(f)+(„dt)^Vndt./rfR;
and the radius j will become
dg \ 3 a
Book L] NEWTON'S PRINCIPIA. 137
—i^j being the expressions of ^ and of — —  , relative to the el
h'ptic motion. Thus, to estimate in the expression of the radiusvector,
that part of the perturbations, which is divided by a ^, it is sufficient to
3 a
augment by the quantity — . xyndt.yfiR, the mean longitude
n t + 'a of this expression relative to the elliptic motion.
Let us see how we ought to estimate this part of the perturbations in
the expression of the longitude v. The formula (Y) of No. 516 gives by
substituting — . —  .J'n d tfd R for 5 g and retaining only the terms
divided by a %
f 2gdY+dg' 1
5 V = l^IiililLJl L. ?^/n d t/^ R;
V 1 — e ^~ m
But we have by what precedes
1 ae.ndt.sin. V , i * ./i o
d e = ==^= ; p^dv = a^ndt v 1 — e;
^ V 1 — e^ ' ^
whence it is easy to obtain, by substituting for cos. v its preceding value
in^,
2gd«g + dg«
a^n^d t^ "^ _ d V .
V 1 — e* " d t' *
in estimating therefore only that part of the perturbations, which has the
divisor a', the longitude v will become
(v) andf — T^ being the parts of v and — p , relative to the elliptic mo
tion. Thus, in order to estimate that part of the perturbations in the ex
pression of the longitude of /u, we ought to follow the same rule which we
have given with regard to the same in the expression of the radiusvector,
that is to say, we must augment in the elliptic expression of tlie true
3 a
longitude, the mean longitude n t j e by the quantity — fn d tj'd R.
The constant part of the expression of ( — t— ^ developed into a series
of cosines of the angle n t + £ and of its multiples, being reduced (see
488) to unity, there thence results, in the expression of the longi
138 A COMMENTARY ON [Sect. XI.
tude, the term — /ndt/t/R. U d R contain a constant term
k ^' . n c] t, this term will produce in the expression of the longitude v,
3 a /i'
the following one, ^ . k n * 1 1 To ascertain the existence of such
terms in this expression, we must therefore find whether d R contains a
constant term.
When the orbits are but little excentric and little inclined to one ano
ther, we have seen, No. 518, that R can always be developed into an in
finite series of sines and cosines of angles increasing proportionally to the
time. We can represent them generally by the term
k fi' . COS. Ji' n' t f i n t I A},
i and V being whole positive or negative numbers or zero. The differen
tial of this term, taken solely relatively to the mean motion of fi, is 
— i k . /M,' . n d t . sin. {i' n' t f i n t H A^;
this cannot be constant unless we have = i' n' H i n, which supposes
the mean motions of the bodies /* and ft' to be parts of one another ; and
since that does not take place in the solar system, we ought thence to con
clude that the value of c? R does not contain constant terms, and that in
considering only the first power of the perturbing masses, the mean mo
tions of the heavenly bodies, are uniform, or which comes to the same thing,
T — = 0. The value of a being connected to n by means of the equation
n * = ^ , it thence results that if we neglect the periodical quantrties, the
a
majoraxes of the orbits are constant.
If the mean motions of the bodies /a and fi/, without being exactly com
mensurable, approach, however, very nearly to that condition, there will
exist in the theory of their motions, inequalities of a long period, and
which, by reason of the smallness of the divisor a,\ will become very sen
sible. We shall see hereafter this is the case with regard to Jupiter and
Saturn. The preceding analysis will give, in a very simple manner, that
part of the perturbations which depend upon this divisor. It hence re
sults that in this case it is sufficient to vary the mean longitude n t +■ «
3 a
oryn d t by the quantity — /n d t/d R; or, which is the same, to aug
3 a n
ment n in the integraiyn d t by the quantity /d R; but consider
Book L] NEWTON'S PRINCJPIA. 139
ing the orbit of /* as a variable ellipse, we have n * = — 3 ; the preceding
variation of n introduces, therefore, in the semiaxismajor a of the orbit,
. . 2aV^R
the variation ,
m
d V
If we carry the approximation of the value r— , to quantities of the
order of the squares of the perturbing masses, we shall find terms propor
tional to the time; but considering attentively the differential equations of
the motion of the bodies /<*, /«,', &c. we shall easily perceive that these terms
are at the same time of the order of the squares and products of the ex
centricities and inclinations of the orbits. Since, however, every thing
which affects the mean motion, may at length become very sensible, we
shall now notice these terms, and perceive that they produce the secular
equations observed in the motion of the moon.
522. Let us resume the equations (1) and (2) of No. 520, and suppose
(ti'.n.C f^;^ /.n.D
(0,1) =_!:__; 0,1
they will become
dh
^ = (o,i)]iMi';
dl
^^=(0,l)h + 0,lh\
The expression of (0, 1) and of 0, 1] may be very simply determined in
this way. Substituting, instead of C and D, their values determined in
No. 517, we shall have
(0,l) = ~^{a«(^) + xa3(d^)};
nm /^'n/ Am 2/dAWx ,(^1A!^\\
We have by No. 516,
db^o5 d^b^")
,/dAWx . , a/d'^A^oK 2 i j^ 3 t_.
^ (dT) + ^^'^ (dT^)=" dT^'^ d^'
db^») d^b^o)
and we shall easily obtain, by the same No. ^ and g^ in functions
of b^") and b^^^; and these quantities are given in linear functions of b^*^
hi a
140 ' A COMMENTARY ON [Sect. XL
and of b^'V this being done, we shall find
Sa'b^i)
^ ^ da ; + ^^ Vdi^J 2(1 — «')^'
wherefore
3/tt'.n.a«.b"?
(0,1)= ^
4 (1 _a^)2 •
Let
(a* — 2 a a' cos. 6 + a'^) «= (a, a') + (a, a')' cos. ^+(a, a')'' cos. 2 H&c.
we shall have by No. 516.
(a, a') = i a', b W ; (a, a')' = a', b (D, &c.
We shall, therefore, have
. _ 3/^^ na«a^ (a, aQ^
^"' ^^  4(a'2_a^)« •
Next we have, by 516,
db») d'b(')
2
Substituting for b ^^'> and its differences, their values in b ^°) and b ^'), we
h '
shall find the preceding function equal to
1 . ~i ~2 j
(1a^)^
therefore
Sa.^'n r(l + a«)bW+ I a.b(0)\
•i .^
IM =  2(i«r
or
_ 3 /i', a n(a'+ a^') (a> a')^ + a a^ (a, a')}
2 {a'^ — a^)
IM
We shall, therefore, thus obtain very simple expressions of (0, 1) and
of [O, 1[, and it is easy to perceive from the values in the series of b ^°) and
of b^^), given in tlie No. 516, that these expressions are positive, if n is
positive, and negative if n is negative.
Call (0, 2) and 10, 2, what (0, 1) and jO, Ij become, when we change a'
Rook I.]
NEWTON'S PRINCIPIA.
141
and /i' into a" and /»''. In like manner let (0, 3), and (0, 3) be what the
same quantities become, when we change a' and [if into a"' and iiJ" ; and
so on. Moreover let h", 1" ; h'"'', Y'\ &c. denote the values of h and 1
relative to the bodies yl', (jf", &c. Then, in virtue of the united actions of
the different bodies /«,', fi'\ /Lt'", &c. upon ix, we shall have
^ ={(0, 1) + (0, 2) + (0, 3) + &C.11  W}\.\ — M.r' &c. ;
dl
Y^ = —[(0, 1) + (0,2) + (0, 3) + &C.1 h + [OJll.h' + ^. h"+&c.
d h' d 1' d h'' d \"
It is evident that j— , ,  ; j— , ^ — ; &c. will be determined by
dtdtdtdt •'
expressions similar to those ofr — and of t— ; and they are easily obtam
ed by changing successively what is relative to /* into that which relates
to /a', (//\ &c. and reciprocally. Let therefore
(1,0),170]; (1,2), [ITU; &c.
be what
(0,1), JOH]; (0,2),j0g; &c.
become, when we change that which is relative to «, into what is relative
to At' and reciprocally. Let moreover
(2,0), g0[; (2,1), 1511; &c.
be what
(0,2), joTl!; (0, l),j^i; &c.
become, when we change what is relative to im into what is relative to fjf'
and reciprocally; and so on. The preceding differential equations re
ferred successively to the bodies ij^, /«,', {j/', &c. will give for determining
h, 1, h', r, V, 1", &c. the following system of equations,
^ = HO, 1) + (0, 2) + &C.1 1  IM 1'  Ml' &c.
dl
^^ = _J(0, 1)+ (0,2) + &c.}h+ [0,lh' + 0,2V^+&c.
dh'
^ = J(l, 0) + (1, 2) + &c.]l' — [1^. 1 — 1,2 F — &
&c.
dl'
~ = —^1, 0) + (1, 2) + &c.?h'+ V0. h + jl^.li" + &c.
dh''
Y^ = U2, 0) + (2, 1) + &c.] \" — \2^\. 1 — IM 1' — &c
dV
jj = —{{2, 0) + (2, 1) + &c.l.h''+ 2,0lhH2, lh'+&c.
&C
(A)
142 A COMMENTARY ON ' [Sect. XI.
The quantities (0, 1) and (1, 0), 0, 1[ and [1, 0 have remarkable rela
tions, which facilitate the operations, and will be useful hereafter. By
what precedes we have
m 1\  — 3^\n a».a^(a ,aO^
^"'^^~ 4.(a'« — a^)^ •
If in this expression of (0, 1) we change ij! into jOt, n into n', a into a'
and reciprocally, we shall have the expression of (1, 0), which will con
sequently be
___ 3^.na^^a (a/ a/ ^
biit we have (a, a')' = (a', a)', since both these quantities result from th
developement of the function (a * — 2 a a' cos ^ } a' *) ^ into a series or
dered according to the cosine of ^ and of its multiples. We shall, there
fore, have
(0, 1). fi. n' a' = (1, 0). (iJ. n a.
But, neglecting the masses ;», /«', &c. in comparison ,with M, we have
a' a'^
Therefore
(0, 1)/A ^/a = (1,0)/*' Va';
an equation from which we