# Full text of "A commentary on Newton's Principia. With a supplementary volume. Designed for the use of students at the universities"

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Digitized by tine Internet Arciiive in 2007 witii funding from IVIicrosoft Corporation littp://www.arcliive.org/details/commentaryonnewtOOwrigiala 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 Glasgff-jo 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 self-eulogy, 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 tres-remar- 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 au-dessus 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 mis-spent 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 introductoi-y to the succeeding part of the work. It comprehends the substance of the metliod of Exhaustions of the Ancients, and also of the Modej-n 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 self-evident, 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'^4-cx™ + &c.), A X, that is the Ratio of the Differentials ofa + bx'^4-cx'" + &c., and X. A+Bx"+Cx°' + &c. 15. Required the Ratio of the Differentials ^ a4-bx' + 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''-»4-mCx'"-'+&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 (a4-bx'+ 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)(y-2) 273 (a — 1) + &c., it is easily seen that the coefficient of y in the expansion is , (a-Jip (a-l)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. .-. A4-2Px + 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 + -2-x'' + 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 (e-1)* (e-l)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 Latus-Rectum = 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 non-continued 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 curvatiu-e 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 -l-L'-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 radius-vector 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 \ MT-s 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 = SP-SN = , + 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 T--T=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. (n-l)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+I5-3 &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(a-h) = 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(a-h)=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(a-h) = 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 tangent-at 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 latus-rectum is p, to find a similar one, whose latus-rectum 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 semi-parabolas, and therefore parabolas, aj-e 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 semi-diameters are a, b, if x be measured along the axis, when x = a, y = b. Hence b , y = — . X ^ a and x' or the semi-axis major being assumed any whatever, this value of y' will give the semi-axis 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 i-equired parabola) a' c = -,. (S — |3') V(/3 + /3')^ + a^ whence (a') is known, or the latus-rectum 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— TT-oj 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 rjr-rp- 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 2-2 :?-?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^ = —, . T-wo — -.. = — 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 curve-surfaced 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 curve-surfaced 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 curve-surfaced 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 curve-surfaced 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 curve-surfaced, 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 curved-surface, 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 tangent-plane at that point, whose diameter shall be the limit of the diameters of all the spheres that can be made to touch the tangent-plane or curved-surface — 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 curved-surface 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 sharp-sighted 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 non-accordance 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*)tx5-L 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 axis-major 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 , ^, p-HTf + <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"4-e~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+^ /j-q:^ = ^+ 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. (a-bx) + 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. Z-e^ 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' 4-4ac 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)4-Q-(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 + Q|8+ &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 = 1-v 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 dxa-|-b/» dx a + b /• d x * J a^x — x^ ~ J ~x~ "*" ^2 J a — x 2""-' a -f- x = alx-^-±^l(a-x)_^-+-^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 = —^ _ ^^ (4-a' + 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 — «)n-i^ (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 „n-l^ 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 = C-S^^j,-^^ + l(K-4) = C + i|E|? + l(K-4). 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 ^-(X-I) 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 = (x-r. (x-i.) 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. 2p-f l.w = — I yi'n-i 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 y-r- — ^ , 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 ■ g-Z— 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. C-l+l) (2p^+a) ^ _1_ n n sin. 8 and -P = sin. '"-'•>• (^P- + '>X-l^- ■ 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 i-esult 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 + Q-x) _ 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 ofthefm-m 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')«J-i + ^*'' P + Qx F + Q^x . . "^ X* + a X + b ■•■ x^ + a X + b' "*■ ^ R + Sx R^ + S-x . ,. 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^---- . u--P•••-^duXF(^?^^^^^^=^^u-P•••,u-P •,&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 + bx-cx')= ^"""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 ^m-i 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 + q-idu 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, /Xa-dx = 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--(n-l)/(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 (X-x) d X &c. = &c. we have /»Xdx_ Xx X^x ^^ f X^")'dx ^(Ix)"- (n-l)(lxr-> „_i.n— 2.(lx)--^ *''' ^ (n-1).. .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 — Iz-j • «. 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. ^4-n . (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 — -fr-o 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 , _^ (a-b)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 + b-a)~fx = X.(b-a) + ^^^.i^^^ + &c. Again, make X = a then A, A', &c. become constants and we obtain f(b)-f(a) = A(b-a) + ^. (b-a)« + ^3 (b-a)» 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 m-f-n . 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 radius-Dector § and traced-angle 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^ = g-y 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 o-R- ~ 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 i-adius 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 semi-axis 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 gi-eat 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'-l-dz^) 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)4-f-'(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 qf-s. 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 = u-j + 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 dv-i-vXdx = . (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 e--f^^\ 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« + 6x--6) 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)(^+i-E)o-v' •••(") 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 f-v — 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 xJ-1 (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 = semi-base 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 fi-om 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«n-i ^ 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 = ?-P-iJ 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^ = j-Tz — + 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 g-p .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^df-gT^''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 z-A 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^p-ys- But in this case S P = P V. 1 32^r* 1 .-. F « o-rTS , 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 p-pi = 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* ~ a-b* and differentiating, and dividing by — 2, there results dp _. i p^ d f a-b" 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/U-g 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^2-t-"-+ ^.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 axis-major\ 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 axes-minores. 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 axis-major 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(l-e2r where a is the semi-axis 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(l-e^)- 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 TTTT-P or -gL'^SP* gL '"SP2 a being the area described by the radius-vector 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 r-j \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' V-fr 2gA and p2V2fr ^ (f^-^O =/d?ff '•x(p-2— .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 = z-in 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-:: A-i' : 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 semi-axes 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 / q-t-*) 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^4-ax *'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 « Tj-pg . 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 ^QR-QR.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 p-t 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 =r-t' ^ ^^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 1-92. 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, gi-av. = 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-* = —1-69— = ^^^'^^^ ^^"^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.R-g.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 traced-angle, 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 "^V-dt^-dt^ e X ^^^P (125) or _ P X V ^ Vg' — p' "" P g = PVx^(pi-l) . . . . (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 =2a-g 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^ = 2PVx-|x 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 ...^" = i|l^'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 — sz-T-X L.atus-Kectum. * 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 axis-major be given in length, viz. 2 a. Then die per- pendicular S T ( = p), and the radius-vector 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 axis-major 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) Ai-tan.^ g d ^ Also from equations (a) we easily get ^4^ = - gsin.(._«) (1) COS. (6-a)= ^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 axis-major; 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(l-e') 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- axis-major 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 (l--e^)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(l-e^) ^ 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 fi-om 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 latus-rectum. 2 \/2 But ASP = AOP--SOP=|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)-f4-Vdf V(g-r) c 2V J|^^/(^_r)_|(^_r)^} c -3^' V(^— r)x(g+2r) But which gives c= PV= V2g^r (229) .•.t=--^X(g+2r)(g-r)^ . . . (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 semi-latus- 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. - d-GH _ 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) = ^(AQ-SR). 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 (FG--sin.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::r-eX^'-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 axis-major 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 — - = r-T=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 -j-rr = o-fp* •'• 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 ■j-rr- = -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. (-r-p) » 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 = -T-j (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.H-P ABFD V« V« ,. - ••• D-FG-E = 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 g-j 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 velocity-curve 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 Time-Curves. (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 V-curve is a parabola. (3) Let F a DS .-. D F a D C^ and V-cui*ve is a parabola the axis parallel to A B. (4) Let F a Yj J ••• D F a ^r^ , .*. V-curve 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 ^^ , .*. T-curve is an hyperbola between asymptotes. (6) If a body fall from oo distance, and F a jjj , V a -^ , .♦. the ordinate of the time-curve D, .*. T-curve is a straight line. (7) If a body fall from oo , and F a ^ , V a -— , .'. the ordinate of T-curve x \^ D C, .*. T-curve is a parabola. (8) If a body fall from cc, and F a ^-3, V a ^, .-. the ordinate of T-curve a D C*, .*. T-curve 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 . p-p- 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^ = ^-f-rCa""^'— 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 = —j-q • 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'(-i--L)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 velocity-curve 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, whence-the 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 P-C 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(j-L-)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)'^ = V-2- 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.M--sin.*^)) = 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=r-i=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" + ' «a-4^--{'-(Tr'}"' when t = 0, X = a, . n f.l a, 1.3 a . o \ •••-a-4>-(-{'+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 = V-Fi. 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 /-T-o-Et-f: 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= ^^^Qbfd-z') • • • • (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.VD2-P2 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. ^ V2-2ff /(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^ ^ ^(Df-e^-DP + P^) 4 V D t=: ^^g'''/^{(p-£)"lo-^)»} and making f g" = " 2 D t = VD /_ (^ + -2)^" V2 =VsT.x Jc-.'{(p-2:).-«^}+|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''{(p-t)^-"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) ^ 2-a* ••^ ^^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 Trajectcn-ywhen F= j — g/.X (D-2— g-2) — D3 ^ g2 .'. if V and P belong to an apse we have ,.o g** D2— P2 V 2 = g-2 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 PV-y ^« - 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 n-3 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 = -T-r 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(rg-P2) 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'+ m-l ) J. |-L/ -+rP /— ^ .., r ^(m-l.f +r^-mP)-V(r-mP^) --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 /t-J5- / , . ''^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 Pn-3 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 ^/(x2-a2) £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 = x-Ldi^(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 z-kCp 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^(2a-e)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-(l-e-) "-G'^"' and again differentiating, &c. we have d^u _ F^ G^ — F" dT^ + " ~ G»a(l— e*) ■*" G*^ ^ "' But if 2 R = latus-rectum 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^ x-L . .... (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 cx-4-2 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 : -r-j . 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 : : ip-g : -^3 r- 11- .xr . F« ^ RG' — RF .*. F m ellipse at V : m n at p : : «r3 • 1 '- T-3 ■ 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 X-f 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* g-b^+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 latus-rectum 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 latus-rectum. 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 radius-vector 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 T-r? 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 — 3T2x4-3Tx2 — 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» - °^ A-3 1— 4c „ F« „ .-. X'a Al^^"^a Ag1-3 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 above-mentioned 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 (p3-n 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 Fa-1 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 = 2-J^ (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 great-circle, 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 e-r^ 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 knff-jon 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(h-x) 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 = Y-R^. dt ds Hence dyd*x — dxd^y ^j ttj .tjj —^ -V--2 ^ = 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(h-x) .••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 radius-vector, 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' dt2-dg**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^ + P-a-F--^-2Q (5) which gives, putting for d t its value • >/Uc-2Q)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 measm-ed 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-^)(7-z)- 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 tei-ms 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(k-z)-h»(l-i)} Book 1.] NEWTON'S PRINCIPIA. 285 ^ _ 8g(k-z)-h'(-l-(-^) " d&^ ~ h2(l + p-i) It is requisite to express the right-hand 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(k-z)-h'(-i-l) 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{k-z)_h-(l-l) 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 radius-vector (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 \{C-2f¥'d^')e-m 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^' + ^ =C-2/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 ^^; = C-2/(Fd^ + Zdz). Also d_(j= — d s = _ gMr - _ hj dt* ' ~ dt^ " §•' Hence ^'=C-2/(Fdg+Zdz) 889. To find the tension of a pendulum moving in a spherical surface. C-2/(Fdg + Zdz) = 2g(k-z) 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(H-p'^+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 (pdz4-dx)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(h-x)d^y d s d s d x h_ dx __^/i^ \ ^^y dy ^^ g •2'V7h — x)- ^ ^^~''^^'l^~Ts'2V (h^l^ The right-hand side is obviously the differential of V(h-K)||-, hence, integrating |. V{h-x) = V(h-x).^ + 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 f-j , &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 r-pxdx /»px°dx h^ h^ h-2- must be of the form that is hence or if 2n + l C X 2 -in + l '* h-i~ 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 fm-ce 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» . C-2g/Fdg ^Fp •• r s also r = — -,— ^ dp Book I.] NEWTON'S PRINCIPIA. 309 .•.C_2g/Fdf = -S||i 2dp_ — 2Fdg •'• p -C-2g/Fdg and integrating P^ = CqC-2g/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 C-2g/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^^ = 3-o'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 '••^^<i-0-a-s = » <«) 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.(yH-XyO 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'k-l)a'k— ^' = (a'k-l)(-^'+- + ^') 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 — yn-i 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 _. Pnyn-1 _ 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 ^n-1 a„ /(in a„ i^" and this will be integrable, if the right-hand side of the equation be redu- cible to this form — k (yi + X y2 + X' y3 + &c.). That is, if _ Pi ^-Pz /i-i 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'(n-S) 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 for-m, 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 (a-s)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 = (a-x)^.-^ + 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 = A|l-k(a-x)+|-](a-x)^-^^.(a-x) + &c.} ..(2) Book I.] NEWTON'S PRINCIPIA. S26 Here A is B C, the value of y when x = a. When x = 0, y = ; .•.l-ka + -^^ __ + &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 semi-circumference 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) x-a+ 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 + a-v .-.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 semi-ellipse 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 = x4-2V(2ry— y2)| /3 = -y i da _ 2 r — y dx ~ y d5__dv__ /2r — y dx ~ dx~ V y "da" >\2r — y~ 'S2r-f)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 t-v-\ .'. - — : — 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::VS-fP: 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 « p-rrTj , .*. 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 ^ \ = lm-3lm( ) 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. 44-5. Cor. 5. Hence the body P, caet. par., will recede farther fi-om 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 —^ 5-3- , 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' p-j- , .*. 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 ab-A' 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 {i-i^ } "■ d^ I « d _ Ma' r . , . — 3-3 = 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 j-j — sin. t nearly, since T-j- 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 = |^3XST-ST = ^^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 radius-vector 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 = -^ + —^ g-jj- (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 shoi-ter 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 -t-j 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 -r-j increases. .*. the quantity decreases, I — c A increases 1 and -T-r 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 -r-j . 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 « j-3 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 : p-Fpa , 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 ^7-1^7-3 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 = -K-p- + 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 radius-vector 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. 3-77 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. Twenty-four 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*4-2ax-f-x» __ a*4-2 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 = 2z-MCH = 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^7-3X7^. 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 . . p-g-j . ;;-^ .-. 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 -r-r ^ 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 LS--g 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.(LB-LA)- 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 ba-dx 528. Cor. 2. Letthe force a -j^ — r , then T K = — - , .-.attraction = PA. FK.Ff= PAy-5^^ 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^n-2 (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 m|n 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 non-resisting 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 non-resisting 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 non-resisting 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 non-resisting 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 non-resisting 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 non-resisting medium = A C D. In the same way, let Ay be velocity lost in a non-resisting 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 non-resisting 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 non-resisting 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 T-T 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 "" "^ "S-R- 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 — ^ axis-major = ^ axis-minor = 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==-f-2AB 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 non-resisting 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. l-j-2av-f-v' .". by Demoivre's first formula, f. or time = when f. -; ;r — ; = — ^ X cir. arc. rad. = g and 1 + 2a Y-4- 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*-}-2avi-a'' = 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 .■.AbNK-DTV= ^^'-^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 non-resisting 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 non-resisting 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-=SP-iVQ-X^_&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 Ty-jj 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 non-resisting 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 ::AS2-f-BS*+ &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 :: (l-l) 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 non-resisting 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 non-resisting 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(x-fgrdz, 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 i-esistance 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 = CB--OB = 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 ^ = -y-s 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 pei-pendicular 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*— 2rab-dy'+ 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 J-i 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 (a-aO*+ (/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 radius-vector, 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-- dius-vector 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 (y-z)^: (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)^ - {y-yj (/3_y)' + (/3-/3')^ (y-z)' = a2 IV INTRODUCTION. are essentially different from the last. Consequently by (6 vol. 1.) (a — a')^(7--z)* = (y_/)=!(a_x)« 0—^') ' (y-z) 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 y-y"=p(x-x-) whence by substitution, . (X — x')' + aMx-xO* + ^[(x — xO*= 1 (X -x")^ + a'» (X - X-)'- + ^ (x-x'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 y-v'= ^(x-x') y-/'=^(x-x'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' + (y--y')' + (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 pn-o- 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'(/'a-7a'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)y-t- 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)4-cos.)y'y)cos.(z'y)4-cos.(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(x-x)« + (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')(y--y") ^ ^>"(x'W) (y-y" ) _^ g^^ ^ /*V(V-x") (y -y") _^ ^V"(x"-xj)(y-y"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 tQ-eliminate 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 = x-r^ + 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 xdx-j-ydy = 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 mdx4-ndy = 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 — pdx-xdp= ^^/^^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 ai-e 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_I-dx) =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) +'''(d-y) = 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*,2ay-f-x*' 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, xdy-fydz= 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 zdz-fydy = 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 arbitrai-y function of one or more variables^ tofnd the equation of partial dijfh-ences 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'4-2ax=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 pat-ts, 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«-f-x') = |^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.5-p^^ 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 = xpy4--N|/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= {y-x (ra — x)y'] + Jy + (n-x)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 semi-axis. 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 + '3-3d*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+(M-d N+d 2 P-...)3 y+ (/^d i^...)3z}=0 C (n-dp + d2q...)3x-f(N-dP+d2Q_...)3y + (»-d*...)dz (C)...^ +(p-dq + d==r...)d3x+(P-dQ + ...)d3y + (^-d;^...)d3z (.+(q-dr...) 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 a7-e 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 the-curve 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 het-Aeen the given radii-vectors A M, A K is the least possible^ We have, (vol. I, p. 000)> >f ^ be the radius-vector, 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 (d-f).*"' + 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-±X-L) = 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 4-btan.-'/; 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(x-c) + ^\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+'^^r-c'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(X-h) »-»-kA' h « (X-h '^) ...=k m h X'"-i-jA'+m(m-l)}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— yx-i = 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, y-2 • • • 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 = (y--i)% . 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.(z-h)A^ , z(z -h)(z-2h)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) Q-Zi &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(Al--Bu) = 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' ^' xm-2 , 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»+ -l-xMi — 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" = -TT-i &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.— ^-.-^j-b = 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'' + dT-a +«" 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 ^(x-l)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 P-l 1.2.3...'(p-l) 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_i-4. 1.2 _ 2 2 ■*"3'*'6+10"' x(x+l) ~1 x+1 + i. + i4.i+ 1-2.3 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 P-1 p-2 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 summing-term. 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 summing-teiia 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 "d-p- = ''"^(^'.«) ^i^ + "" f (?^'^) V— + ^^- • ^^ i\,2 fl, 3 IT^ " -/*'f (?1.2)^^=^ + /*'"f(f2.3)^^— ^^ + &C. "^ gl,2 i2,3 " "^ ?1, 3 g-2, 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 semi-axis 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 semi-axis 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 semi-axis would be a, and periodic time q-i/ <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 semi-axis 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^, =/(x-xO + /x"(x-x'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.)(x-x) ^' = (/* + /*' + &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*(y-y)_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 (t-tI) 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 ^^y-l 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, obsei-ving 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)(dy--d y)-y-y)(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. M-dg + 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 diffh-ent 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" + ^/O-r (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 radius-vector 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 semi-axis-major which we shall designate by a ; e is the ratio of the excentricity to the semi-axis-major ; 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-(^-^)V-i 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(n-i) _ a (n-i) 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 radius-vector § 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(-d-r)+<^)=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 M-y^+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 »_ydz--z d y dt '""- dt ''^- dTt n-f _L ^ J°^ dys+dzn y dy.dx zdz . d x • 17 ar^ — / + dt* + dt^ (V) ^ O-f X v/™ dx'+ dz^ -( xdx.dy zdz. dy n-f'a.^/"^ dx' + d y') , xdx.dz , ydy.dz 0_t +z|- dT^~f + dt^ + dt« . m 2m . dx« + dy*+dz* .«=T--T"*'"~ d-t^ 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 radius-vectors 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 semi-axis major of the orbit, or the mean distance of ^t from M ; and VO-i^a) 30 A COMMENTARY ON [Sect. XI. is the ratio of the excentricity to the semi-axis 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 V-da". 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^J-V 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'j-v 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 J-vI/' (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 + 5-5 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"'-i-t: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 t--5. 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"V-i+ 1' 1 + V (l—e*) Book I.] NEWTON'S PRINCIPIA. 45 we shall have *^ _ c V X i_xc"V-i ' and therefore — log.(l-^c — "V— ») — log.(l~Xc"V--i) v_u-f- \^'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 radius-vector 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 /=VV-2l^-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 semi-axis 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 radius-vector, 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(l-e)^=.^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 radius-vector 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 axis-major of the orbit, the chord of the elliptic arc, the sum of the ex- treme radius-vectors, 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(l-e^) ^ 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.«/3|cos.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 semi-axis 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 semi-axis 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 axis-minor 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 = 2a-f-f, P = 2a — c whence / 1 c — a z' = — 1, z = a and the formula gives ^ a«Tf -iCj-a ; 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 axis-major. 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« = 33|8; &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 • (d7-0 + «"=• 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'/3-t-(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 thirty-five 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)' (d-sO' (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 ( .--) , (g-pj) 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 radius-vector 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 radius-vector. 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-«){-L-jl^^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 (f-j from the first and suppose ^,_ (d-t) (dT')-^rt) (a-t^)+<d-t) (?t) "'"•^+(dT) '""•'"°-^'' ^ j-^ sin. ^ COS. 6 cos. (A — ■ ) + ( i .) sin. (A — a) we shall have -^•{p-i[^} (') 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 ( j-r2) 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 ^ , (-1-73) and (^t-t-s) ^ 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(l--E^. 1-E' ^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 radius-vector 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 semi-axis 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 semi-axis 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 radius-vector § 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 pei-ihelion. 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~\ {-\— ^-(R-l)cos.(A-a)} \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. 4-2.(^^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 (t-t) 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 radius-vec- 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 semi-parameter of the conic section of which a is the semi axis-major, 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 ^-Y-T^ is equal to - . ^ ^ ; in substituting for g'^its value ^+2RrX ^d Rn , /d A> cos. (A — a) + R% and for (-j-r) 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 radius-vector 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 = -x|l.tan.^+2-3;+ 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 radius-vectors 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 radius-vectors 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 semi-circumference, the side opposite to the angle jS' — |S will be the angle at the sun comprised between the radius-vectors 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 radius-vectors 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 half-day, 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 ^^* ^^® radius-vectors. 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 (l--e)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 radius-vector ^. 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 semi-axis 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} + Fdt|i^-t-F}+&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 -r--4 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 - ii-y + 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 y-j , -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;^. (^ + 4-3^.(-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 "^O-Y+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 — ^ , (-r-T-) > &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)^ Cy-r ^ , &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; 52-v}/ = h''-'^ -^ = 0; changing therefore successively in the equation (x) the function f into -v}/, 3 -v]/, 6 * -vj/, &c. we shall have rd-vI/> 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'^ 0-113 + ^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 semi-axis 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^ J-fe - 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 radius-vector 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- dius-vector, 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.dR-i 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 radius-vector 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 d-u 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 = a|l + 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 — ||/ndt|sin.(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 radius-vectors, 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- axis-majors 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) = n-dr)- 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 gT-i — 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(i-2) 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(i-s+ l) 1? 4(i-s + !)« .,.„ 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(i-s+ 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^^V-i 4- &c. i m l£ 1 + sac-«V-l + L(L+Jla2c-2»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 "4.6-2.4.6.8'' "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 ~ -(i-a^)^- 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'(H-u/); 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- axis-majors of their orbits ; e, e' the ratios of the excentricity to the semi- axis-major; ^ 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- «' — 4-M 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't-nt+ s'-s)+ nt+ «-w| (n-n')-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't-n 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(n--nO-lr-n' '^'''°'-^'^'''^"""^'^'~'^'^"^"^'~"'^' 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't-nt + .'-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 (n-n')-nj « — n ''J > V. X e cos. Ji (n' t — nt+s' — £) + nt+j — ■a\ ) I DW — /.' . n '^ 2 . _-_,^__-p-_^e' cos. Ji(n' t-n 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