i nmmm m ■ mam SINGULAR PROPERTIES OF THE ELLIPSOID, AND ASSOCIATED SURFACES OF THE N TH DEGREE. DEDICATED, BY PERMISSION, TO HIS ROYAL HIGHNESS PRINCE ALFRED. BY TIIE REV. 0. F. CHILDE, M.A., AUTHOR OF RAY SURFACES, RELATED CAUSTICS, &c. MATHEMATICAL PROFESSOR IN THE SOUTH AFRICAN COLLEGE. MEMBER OF THE BOARD OF PUBLIC EXAMINERS AT THE CAFE OF GOOD HOPE. ** * T H, DEPT, BOSTON COLLEGE &BIZARY l,aejL^} ii'D ^ I kiLLtLtt MASS. MACMILLAN and Co., Cambridge ; J. C. JUTA, Cape Town. MDCCCLXI. CAPE TOWN: W. F. MATHEW, STEAM FEINTING OFFICE, ST. GEOKGE STREET. cSS TO HIS ROYAL HIGHNESS PRINCE ALFRED, IN REMEMBRANCE OF HIS YISIT TO THE COLONY OF THE CAPE OF GOOD HOPE, IN THE YEAR 1860, AND OF HIS PERMISSION THEN GRACIOUSLY GIVEN, THIS VOLUME IS DEDICATED, BY HIS FAITHFUL, HUMBLE SERVANT, THE AUTHOR, . '■ ; ,.v * hOQ Ml »iX — ,0 .ha ioj/b <. . Ofii in - li <.<;■ jBJjiamqoljvob I'-' m iQ to aahsa tfilugm ■ -- u ;■ ■aaidyjji) fci INTRODUCTORY REMARKS. As the title of this volume indicates, its object is to develope peculiarities in the Ellipsoid ; and, farther, to establish analogous properties in the un- limited congeneric series of which this remarkable surface is a constituent. The more conspicuous Singularities which have been evolved are enume- rated in the Table of Contents ; among them it seems desirable to specify more pointedly, and as briefly as possible, those which follow. (1.) — The relation of the Circle-ordinate u to the co-ordinates. This relation is first noticed in (6), p. 2 ; and will be found to pervade the whole subsequent investigation. (2.) — The equation (13), p. 4; and the cognate relations developed throughout Chapter IV. (3.) — The series of equations exhibited in page 8. (4.) — The very peculiar relation of the intercept p m to the co-ordinates, investigated for the Ellipsoid in (4), p. 10 ; and, for the General Surface, in (4), p. 22. 5. — The summary of General Formulae com- mencing at p. 23. In the Table of Errata it is pointed out that the forms (5) (8) (9) are erroneous, and the requisite corrections have been there intro- duced. 6. — The remarkable system of equations representing successive planes in the ellipsoid, p. 30 ; and tangent-planes, p. 33. 7. — The developments of Chapter III. The geometrical peculiarities detected by means of these developments, pp. 51 and 56, deserve to be particularised. (8 ) — The singular series of expressions for the tangent perpendicular in consecutive surfaces. (9.) — Properties of the Circle-ordinates in pp. 97 and 102. (10.) — The axes of the h derived ellipsoid, p. 109. (11.) — Singular rela- tions involving the Circle-ordinates in consecutive surfaces, p. 113. (12.) — The singular relations combining consecutive co-ordinates and constants, in different surfaces, p. 125. (13.) — The relation uniting the Circle-ordinates in any three consecutive surfaces (XXXII), p, 130. (18 ) — The singular equation connecting successive Circle-ordinates in the general surface, p. 132. In Chapter VI, relating to the Ellipsoid, the area of a plane section, which is determined under the most general circumstances, has a singular symmetry of expression. It is possible that this result may be otherwise obtained in VI INTRODUCTORY REMARKS. some more compendious way, although no method more convenient has occurred to the author. The identification of the area with a second ortho- graphical projection of the parallel central section, given at page 140, illus- trates the clearness of interpretation which may attach to results of a purely analytical character, by reference to their geometrical equivalence. In applying the expression for a sectional area to the Volume of any portion of the solid limited by parallel planes, we are led to results of no ordinary simplicity regarding the volume or mass ; while the extension of this investigation in determining the attractive force, under the hypothesis that the attraction varies directly with the distance, may not be without its value, in the con- sideration of problems relating to this subject. With reference to this point, it should be remarked that, in the formula (2), page 148, the Density may be assumed to be any function of the distance, although, in the subsequent propositions, it has been taken as constant. That illustrations in greater detail, as well on this, as on some other topics, have not been given, must be attributed to the narrow limits within which the writer has been unavoidably restricted. If it is of acknowledged difficulty in the present day, to bring forward anything of novelty in mathematical researches, yet it may be hoped that such attempts are not often altogether futile, when entered upon in the spirit which is anxious for the investigation of Truth, and desires to add its contribution to the treasury of Science. To do this, in some degree, is the design of the investigations exhibited in the following pages. For whatever defect in its execution may be apparent, the reader’s indulgence is requested ; while, for the recognition of whatever truth has been elicited, the author is content to rest upon the candour of an impartial judgment. TABLE OF CONTENTS. CHAPTER I. General object of the investigation Determination of the circle-ordinate u Geometrical signification of u Elimination of co-ordinates Singular equation connecting the lines rpr t ..... Generalisation of the hypothesis Equations connecting the consecutive perpendiculars and radii Simplification of co-ordinates to the »»th point of contact . Connection of the tangent perpendiculars and radii with lines inter- cepted by a plane passing through the extremities of the principal axes . General inferences from the formulae PAGE. L 2 3 3 4 5 8 8 9 11 CHAPTER II. Extension of this analysis to the General Surface 2 = \ . .14 Determination of u, which has the same geometrical meaning in the General Surface as in the Ellipsoid 16 Determination of the general co-ordinates x m y m z m . . . 17 Determination of the general expressions for p m u m . . . 18 General formulae expressing co-ordinate relations . . . .19 Reduction of the general co-ordinates x m i/ m z m .... 20 Relation of the tangent-perpendiculars and radii to lines intercepted by a given plane 22 Recapitulation of the principal Fundamental Formulae ... 23 Deductions from the fundamental formulae* 24 Adaptation to the Ellipsoid, of formulae derived from the General Surface 26 * Note. — The formulae (5) (8) (9), at pages 24 and 25, which are inaccurate in the text, have been corrected in the table of errata. Viii CONTENTS. Consecutive planes containing the lines r m _i p m _ x r m in the General Surface ........... Consecutive planes containing these lines in the Ellipsoid . Inclination of consecutive planes in the Ellipsoid . The principal planes alone, in the Ellipsoid, can contain consecutive systems of lines Determination of the equations to consecutive Tangent-planes in the Ellipsoid, and in the General Surface Inclination of the consecutive Tangent-planes Principal Radii of Curvature, at the 02 th derived point in the Ellipsoid, expressed in terms of the co-ordinates to the initial point Adaptation of the general formulae to surfaces in which the index is negative CHAPTER III. Development of the General Eormulse Developments for the Ellipsoid Development for the surface 2 ( — J = 1 • „ >&)”= 1 ” ” ” 1 ' Geometrical singularity in this surface Consequent simplification of the developments Development for the surface 2 ^ = I . Geometrical singularity in this surface Consequent reduction of the developments Development for the surface 2 'j* = 1 . ” ” ” S (f) = 1 ‘ ” ” ” S &) 2= 1 ’ PAGE. 27 30 31 32 33 34 35 36 37 37 39 42 44 46 49 51 52 54 56 57 58 60 61 CONTENTS. IX PAGE. Law connecting the perpendiculars in consecutive surfaces ... 64- Modification of the General Formulae in relation to consecutive surfaces 66 CHAPTER IV. Equations of Relation in the Ellipsoid Relation between 6 consecutive lines . 33 33 8 ,j 33 33 33 10 33 33 General forms of the co-efficients General equation of Relation Development of the general co-efficients into series Recapitulation of the developed co-efficients General equation combining consecutive vectorial radii and perpen diculars . Developments of the general equation Relation between 6 consecutive lines 9 9 8 10 12 14 16 18 20 J) 99 99 99 99 99 99 Equations of condition between the general co-efficients Verification of the expansions 68 70 71 73 74 75 76 81 82 82 83 83 84 86 86 87 89 91 93 94 CHAPTER Y. Relations connecting a system of surfaces generated from the primitive under a given hypothesis 95 Application to consecutive Ellipsoids 95 Value of the Axes in the first derived Ellipsoid ..... 97 Property of the circle-ordinate u (5) 97 Generalisation of the Ellipsoids by successive derivation ... 98 Relation between the axes in consecutive Ellipsoids .... 99 Co-ordinates of contact of the mih Ellipsoid determined . . , 100 Co-ordinates expressed in terms of the axes of the mth surface . . 101 Singular relation connecting successive circle-ordinates . . . 102 X CONTENTS. Axes determined in the consecutive Ellipsoids Summation of the axes ; the derived surfaces being continued ad oo . The axes in consecutive Ellipsoids form a geometrical progression Evaluation of the co-ordinates, on the ^th derived surface . Sum of the volumes of consecutive Ellipsoids Singular analytical expression for w Relation between the two infinite series of volumes .... Generalisation of the hypothesis in page 95 The consecutive surfaces are similar, and their axes reciprocally propor- tional to those of the primitive Absolute determination of the axes in the ®th derived Ellipsoid . General connection between the consecutive axes . . . . Suggestion relative to more extended combinations (XY) . Extension of the preceding investigation to the surface of the %th degree, 2 =1 Absolute determination of the constants in the mth surface . Singular relations connecting successive circle-ordinates . . Verification of these relations Extension of the investigation in page 98, (IV), to the General Surface The related surfaces are similar Absolute values of the co-ordinates of intersection, and of the axes, in the ®th derived surface In each series of derived surfaces the axes are proportional, as well as the co-ordinates to consecutive points of intersection General expression of the co-ordinates for the »«th surface in terms of its axes Singular relation combining the consecutive co-ordinates Verification of the general results (XXIX) „ „ „ (XXX) „ , „ (XXXI) ..... General relation between any three consecutive circle-ordinates . Series for the summation of the circle-ordinates CHAPTER VI. Area of the section formed generally by a plane, cutting the Ellipsoid in any position Geometrical interpretation of the result, viz. — The area of any given section is equal to the second projection of a parallel central section ; projected , 1st, upon a given plane ; 2nd, upon its own plane fA-GrE. 103 104 105 106 106 106 107 107 108 109 109 110 110 113 113 114 115 119 119 123 124 125 125 128 129 130 132 133 140 CONTENTS. XI PAGE. Volume of any frustum or segment of the Ellipsoid .... 140 Problems. — Tbisection of the Ellipsoid 142 „ Quadbisection „ „ 144 „ Volume of a Cone with Ellipsoidal base .... 144 Volume of the Ellipsoid determined by this problem (Cor. 1) . . 145 Volume of a Cylinder, whose axis passes through the centre of an Ellipsoid, and which is bounded by the surface .... 146 Volume of any Cylinder circumscribing the Ellipsoid is constant . . 147 Mass of any segment or frustum of the Ellipsoid .... 147 Application to the Theory of Attractions 147 Attraction of a plane thin lamina 147 Attraction of any segment or frustum of the Ellipsoid . . . 148 Attraction of an Ellipsoid of uniform density 149 Attraction upon a particle placed on the surface 149 Comparison of Greatest , Mean , and Least, attractions .... 150 Attraction of of the Ellipsoid limited by principal planes . . 150 Resultant attraction of the octant 151 Line of action of the resultant attraction 152 TABLE OF ERRATA. PAGE. 8, (VII); 19, (IV); 21, (V). For v read v m . 10, line 5. For (VI) read (VII). 11, (IX). For v t read v /m . 24, (VII). In form (5), for 2 3 , in the numerator , read 2 3 ( n “ 1 ). 25, (VII). Instead of form (8), substitute , r* m ( n 1) x 2 3 m n(n— 1)— 3 a n — 3 (n— 1) X 111 (n-1) - 1 n— 3 25, (VII), In form (9), for 2 in the numerator , read 2 n_1 . 83, (XIII). For v read v m . 34, line 1. After page 25 read (corrected). 42, line 4. For a 9 read orK 43, line 1. For a 13 read x 12 . 65. In (8) (9) (10), invert — under 2. (t 67. For the Note, read , “ See Gregory’s General Theorems in the Calculus. Professor Ivelland on General Differentiation (Edinburgh Phil. Transactions), &c.” 73. After (3) insert j 73, line 9. For p read r. 103, line 19. In c 2 for b read v. 135, line 11. For x 2 y 2 read x 2 y 2 . 147, line 7. For fraction read function. SINGULAR PROPERTIES or THE ELLIPSOID. CHAPTER I. (I.) In the following pages it is proposed to investigate some singular relations which exist between lines connected with the Ellipsoid, and with the surface represented by the general equation, x n y n z a — 4. 4. — — 1 (i n ' b n ' c n The properties of the ellipsoid will be considered in the first instance, the examination being afterwards extended to the general surface. Let us suppose a tangent plane to be drawn at any point P on the ellipsoid (fig. 1), 0 being the centre ; let OP = r be the radius vector drawn to the point of contact from the centre ; Oj> — p the perpen- dicular from the centre upon the tangent plane ; OP y = r l the radius vector intercepted upon this perpendicular between the surface and its centre ; we may then shew that these three lines are connected by the equation, ry 2 r~ — (a 3 -f i 2 -f c 3 ) jfirf -f- (a 3 5 3 + a 3 c 3 Z» 3 c 3 ) r 3 — a 3 5 3 c 3 = 0 : in which a h c are the semiaxes of the surface. (II.) Let £ 7] £ be the general co-ordinates of the tangent-plane, x y z any point on the ellipsoid ; these two surfaces are defined by the equations, ;r 3 a 3 z 3 a 3 x£ + t/ yr) + c 3 zt = 1 . (1) B a 3 + b 3 + c 3 = 1 . (2) * 2 ELLIPSOID. The equations of the perpendicular from the centre upon the tangent plane are, a 2 z£ — c%£ = 0 "1 = 0 J b^zT] — By combining (2) and (8) we obtain the usual expression for p. 1 x 2 y 2 z 2 pi — 7T + a* 6 1 + (3) (4) Let x t y t z t be co-ordinates of the point in which p intersects the surface, then for the determination of these quantities there are the equations, £l + a 2 + yJL b 2 + crzx c 2 xz t = 0 ; b 2 zy t — c 2 yz t — 0 ; from which, after assuming x 2 y 2 z 2 1 a 6 O" c° we find as the values of the co-ordinates of intersection, a * ,2 _|_ y% + 6 6 + (5) x, = ±(^) 3 , », = ± (*)%. ., = ±(i)V Now, r 2 = + i = \7a + fr + r_ 64 c 4 / p- u 2 — pr t \ ( 6 ) from which it appears that the line represented by u is a mean propor- tional to p and r t . Hence we have, for the elimination of x y z, the four equations, x 6 w + x*_ a 4 ^ r_ b 2 64 + + = 1 p 2 (?) (8) ' 5E zww CO-ORDINATE RELATIONS. 3 a 6 "t" £6 + c 6 jp-rf a 2 -j- y 2 + z 2 = r 3 ( 9 ) ( 10 ) It should be here remarked that the auxiliary symbol u, employed in the reduction, has the following geometrical signification. Let the radius OP y be produced to meet the surface on the opposite side in the point Q ; P y OQ is then a diameter, and OQ = r r On Q p describe a semicircle in the plane OPP /} and draw the ordinate OL : then, OL 2 = OQ x Op = pr r Consequently, u is equal to the line OL ; which it is obvious might equally well be taken as the ordinate of any other circle, or a semichord of the sphere, described on Qp as diameter. (III.) In eliminating x y z from the four equations last written, it is convenient to adopt the notation, 1 1 1 _ ttm _ a 2m ’ P m — g2m > 7 m — c 2m » so that the suffixed symbols shall be subject to the law of indices ; e.g. ttm ttj, — ttm -j- n i then, ax 2 + Py 2 + yz 2 = 1 (7) a 3 # 2 + P*f + y a* 3 = P (8) a 3 a; 2 + P 3 f + y ^ — Q (9) writing, 1 pi = p. 1 - p“ r~ Q. By employing indeterminate multipliers, we shall find, X 2 — Py ■ (P + y) P + Q > a (a 1 1 y* = — ay + (a -j- y) P — P ( a — P) (P — y) Q 1 > (11) — afi (a + P) P + Q y (a - y) (/3 - y) 4 ELLIPSOID. Now, when x y z are eliminated from the equation (10), there will be found, r® = A — BP + CQ; (12) if we make, A — B a/3 ay y («— y) (P—y) P( a — P)iP— y) + a ( a —P) ( a ~y) a + /3 Py — « 2 + i 3 + c 3 . a + 7 + ^ ^ + ft ft. y ( a ~y) (P-y) Pfa-P) (p- y) a i. a ~P) ( a ~y) C y( a y) (P y) P(*—P)(P— y) + a (a— (3) (a— y) ,J 1 ' Hence the relation between rpr t becomes, finally, r-p-ry — (« 3 + £ 3 + c 2 ) ^ 3 r / 2 -+- {eft ft -j- « 2 c'~ + ft ft) rf — drift ft =0 (13) Cor. 1. If we suppose r — p — r t this equation resolves itself into, (: r 2 - a 2 ) (r 2 - 6 2 ) (r 2 - c 2 ) = 0 ; shewing that the assumed equality can exist only on the principal axes of the surface, as we know to be the case a priori : and if a = b — c, the surface being spherical, this equation is reduced to (r 2 - a 2 ) 3 = 0. If any one of the axes is supposed to be evanescent, r, disappears from the equation, which becomes that of the corresponding plane elliptic section. For example, let c = o, then, ( a 2 -f b 2 — r 2 ) p 2 = a 2 b 2 . Cor. 2. It is to be observed that the co-efficients and indices in (13) follow the law of combination which determines those of a complete cubic equation. Cor. 3. When any axis of the ellipsoid is supposed to become in- finite, that surface passes into an elliptic cylinder, and the equation (13) ought to exhibit the relation between p and r t in the plane ellipse. Writing c = oo , we find, 4 CO-ORDINATE RELATIONS. 5 (« 2 -f - 6 2 — p 2 ) r 2 — a 2 b 2 ; which will appear, by an independent elimination, to be the proper relation in the ellipse defined by, Cor. 4. A curious analogy exists between the foregoing form and the relation between p and r in the same ellipse, which is, (a 2 + 6 2 — r 2 ) p 2 = a 2 b 2 . By a comparison of the two equations it will be perceived that, if a point is taken on the ellipse at which the radius vector is equal to any tangent-perpendicular, the intercept on the first perpendicular will be always equal to the tangent-perpendicular corresponding to that point. (IY.) Let us now suppose a series of points on the surface, whose positions derive from one another consecutively, in the same manner as P, is determined by P ; that is to say, P m is a point in which the surface is cut by p m _ x the perpendicular upon the tangent plane of P m _ x : r m is the intercept upon p m -i, or the radius vector of P m : x m y m s m the rectangular co-ordinates of Pm : we have then the following series of equalities, 1 p% X 2 ~ a 4 + f b 4 + /v2 c 4 * II M 1 sd rvt 2 tV y ~ a 4 + q y; 6 4 + fi. c 4 ’ 1 Pm /v* 2 a 4 + }/m 6 4 i r • • • 2- 3 Again, considering the lines analogous to consecutive points, we have the series. U which correspond to the II Hi* x 3 + yl b ' ® + 0 2 c® - " ii Hlr xf + y? 66 + c® ' ii i • • « •I'm 2 «® + ym 2 b 6 + *m 2 C® ’ 6 ELLIPSOID. But, X , and, as the law of derivation is uniform throughout the series. hence will be obtained the general expressions, * m - (^z i) 2 (*2=l) 2 ... (±) Z x . ym = (**) 2 ( sj ±) 2 ... ft .) v = (^) s (iM 2 ... (-1) 2 Z. Coe. The ratios of the m th co-ordinates vary as the ratios of the first co-ordinates ; for, iPin_ _ n\ m X_ _ _ / £ V m £ _ |£i _ / £\ 2m y , ym ~ V*/ V’ Zm ~ Z ’ Z m _ 2’ or, ar m : y m : 0 m :: (3c) 2m a; : (ac) 2m y : (abf m z. (V.) We have next to calculate p? p 3 2 ... m 2 w a 2 ..., employing the formulae of the last article ; 1 X 2 y? Z 2 O pr — ^ ■ + $ + 1 x z • + * 2 7 ^ 2 = tf 4 £ a s b s c 8 5 similarly, — 0 Fa" = w 4 « 4 f a; 2 l? 2 + « 3 + Z 2 ] c 12 ) • • • and, generally, 1 • U* / x 2 y 2 * 2 3 *o II 3 1 1 ''m— * l a 4111+4 "T" ^4m+4 • g4m+4 \ CO-ORDINATE RELATIONS. 7 Again, and, generally, 1 u X“ = ^ + 1 4 J® 5 ? = u la 2 10 y_L + yl + fro 4 *.4 — %-i ^m-2 w- /-£_ + 1 0 4m+6 T + r r.10 }> ^4m+6 + r ,4m+(5 }• Now it has been shewn that v? — pr t ; and, by the uniformity of the law of derivation we shall have, consequently, — Pi ^“ 2 > w a 2 — ^2 ^3 ) ^3 2 — ^3 ^*4. > ••• ^mti- Employing these equivalents in the reduction of the preceding ex- pressions, and writing 2 to indicate the sum of three analogous quantities, we obtain, And, 1_ — Pm — i r m Pm—z 9"m — 1 ••• P % 8 ELLIPSOID. (VI.) After the preceding results have been arranged in order, we find that the following remarkable series of equations is derived from the co-ordinates of any point xy z in the ellipsoid ; according to the law previously stated : > tOm Wm ti’m 'm = •^m a b C in which it is to be noted that w 0 — to, &c. Hence we obtain, using the forms of (VI), Now, let p ra be the radius vector of that point iu which the plane is intersected by the m th perpendicular, so that p, is measured on p, and p 0 , or p, oa r; then Pm 2 = £m 2 + *?m 2 + Cm 2 , It has been shewn that, for the ellipsoid, (VI), page 8, 2 { ^ \ _ 1 . \a 4m+ V pm pm?- ! ... p 2 r m 2 r m 2 _ 1 ... r? ’ therefore, after lowering the index m in this expression by unity, substi- tuting in the equation (8), and extracting the square root, we find the formula, involving only the uneven powers of a b c. 1 Pm pm— i Pm_ a ■ ■ -P 1 a • • ( 4 ) f i PLANE OF INTERSECTION. 11 Cor. If this expression is squared, (_£_) + !j( 5 ) = V rt 4m+3 / ~ \ ft 2m+l £2m+l/ y-v 2 rf) 2 ^i2 a* 2 2 but from the second of the general formulae in (VI) after lowering m by unity, s , 1 \„4m+2/ . 2 v ( x v ^ JCm 2 -! ... J9 2 ^m 2 ?'m 2 _ x ... rf rj 1 — pm 2 a 2 a) 2 m2 2 a* 2 a* 2 Pm //m — i • •• p ' m 'm _ i ••• / y • ( 5 ) (IX.) The expressions (4) and (5) in (VIII), taken together with the general forms of (VI), contain the relations which it has been our object to demonstrate as connecting these singular lines in the ellipsoid. They will be afterwards reproduced, as particular cases of the formulae which belong to the general surface ; and, in the mean time, we shall close this chapter with some remarks which suggest themselves as corollaries to the results which have been already determined. (1.) If the axes are assumed to he equal, the surface is a sphere, and jPm_i ... p r m _ 1 ... r, = a 2m_1 ; the form (4) in (VIII) will then give, 2(*) _ 7 2m+l Pm — pm O 2m— 1 x + y + z ( 1 ) which can be readily verified. (2.) If v t is a line, such that, e 2m = Pm Pm- l ...^r m _ 1 ... r , , the general co-ordinates of (VIII) may be written 12 ELLIPSOID. but in (VII) we have deduced expressions, precisely analogous to these, for x m y m z m , in which V 2m = Pm- j P r n consequently. £m pm Xra Pm Zm Xm ( 2 ) ( 3 ) (3) By combining the general formulae, so as to determine the absolute values of the radii and perpendiculars involved in them, in terms of the first co-ordinates, we obtain. Hence we have, further, rm 2 pm 2 pm 2 Tm 2 REMARKS ON THE FORMULAE. 13 Moreover, since w m = pm rm^i, if the value of m in r. n is increased by unity, there will be found, (4.) In the equations of (VI), if all the lines are referred to a numerical unit, we have the following summation to m terms ; 1 + \ + -q— f / r p~ rf + jfipfrf + p^pfrfr^ 2 + = 2 l*. I a? z 2m — 1 + t b ..9m /V>2 + *-r + ~ + -} ■1 7/ 2 C 2m — 1 Z 2 d 2 — 1 « 2m ' b 2 — 1 b 2m ' c 2 — 1 e 2m ' If m is supposed to become infinite, 1 1 4- — + _ + _ p 2 p 2 rf p 2 p~ r~ + ... ad oo , (4) x i 2 — 1 + ir b 2 - L + c 3 - L ( 5 ) Again, the developement of (4) in (VIII) will give the series, to m terms. - + — + — 1 — P PiP PiP,pr, + Pz PzP t pr 2 r t + ... = + ± + ± + ± + ) la «3 ^ ^ a 7 + ••• j __ « 2m -l X a 2 — 1 a 2m_1 fl nd, if m — qq f b 2m —l y , c 2m — 1 2 7 0 -I 7 0~,' T b 2 —l b 2m “ 1 C 2 — 1 c 2m-l ’ ~ + P PiP + ax "c 2 — 1 + P*PiP r t by + ... ad oo , V 2 - 1 + cz c 2 — 1 ( 6 ) ( 7 ) 14 GENERAL SURFACE. CHAPTER II. (I.) We proceed to extend this investigation to the case of the general class of surfaces represented by the equation, + ?/ n + V = 1. ( 1 ) The tangent-plane at any point xy z is given by, x ,n— 1 £ + V ,n— 1 ■V + 7 n— 1 a‘ L b n c £ r] £ being the general co-ordinates of the plane. £ = 1 , ( 2 ) The perpendicular upon this plane from the origin is defined by the equations, a n 2 H-1 £ _ c n ^n-l £ — q . b n rj — c n y n ~ l £ = 0 ; (3) £ 7] £ being the general co-ordinates of the line. For the point in which the surface is intersected by the perpendicular we write the co-ordinates x l y t z t , and there are, for the determination of these quantities, the three conditions, x 1 + ^ + 1 = a n b n c n a u z ,lx x t — c n x n ~ l z t = 0. b n z n ^y t 7II 1 , c n y n 1 z, — 0. Assume u to be a line, such that „2n x n(n-l) a n ( n + !) + 2n + „2(n-l) + 2 2(b-1) _ /.2n * we shall then obtain for the intersection, v 2 z n ~ l y , _ v 2 y n 1 x = V 3 x n 1 a u (6) Now, let p be the perpendicular upon the tangent plane, r, the intercept on p between the origin and the surface ; i.e. the radius vector of the point of intersection : then, + y' 2 + s' 3 zH*- 1 ) P 2 = X‘ = * / ^ (n ~ 1} 4 . l u 2u ^ b 2 * + «*» J 16 GENERAL SURFACE. P and, Also, 1 = j» 2 l w 2 = pr t ; the line u is therefore a mean proportional between the perpendicular am its intercept, as in the ellipsoid. (II.) From what precedes we have the following simultaneous equations, involving r p r l ; ,.n — + y n + — — 1. aP c n 3.2(11— i) + 7/ 2 (n-l) + Z 2 (n— 1) _ 1 o 2u c 2n pp + + _ 1 fl n(n-j-l) ^n^n-J-1) ,pii sP + / + z 2 = F 2 . The elimination of x y z in the general case does not appear to be practicable ; in the case of the ellipsoid, where n — 2, it has already been effected, and, in any other individual instance, will lead to the corresponding relation f(rp r,) = 0 (III.) Let us now consider a series of points to be determined ac- cording to the same law which was before supposed to obtain in the ellip- soid, that is by the intersection of the surface with perpendiculars upon successive tangent planes ; and let x m y m z m be co-ordinates of the m th point of intersection, r m its radius vector, p m u m the corresponding symbols for p and u : there is, then, by generalising the formulae of (I), ( 5 ) ; Chapter II : x m — 2 n— 1 Um — 1 •'Tin — 1 aP 2 Wj ri- ll — 1 -1 ym-\ b n Zm 2 Wm_ n— 1 -1 Zm — 1 ]/m — CO-ORDINATE RELATIONS. 17 As the quantities to be determined are all identical in form, it will be sufficient to consider one of them only, as x m , and we find, 2 m — 2 m — 1 2 2(n — 1) 2(n — 1) 2(n— 1) 2(n— 1) u m tf(a-l). Um — 1 % — 2 Um — 3 ... U. Xm = — ' 2 m — 2 m — 1 a n a ii(n— ]) «n(n— -!)_ a n(n— 1) a n(n— 1) The number of factors in each term of this fraction is in , and the indices of the denominator form the geometric progression, n + n (n— 1) + n (n— l) 2 + ... = n (n— l) m — 1 n — 2 m — 2 m — 1 m 2 2(n — 1) 2 (a — 1) 2(n— 1) (a— 1) U X Um — 1 Um — 2 . .. u. Xm — m n fr - . 1 ! - , 1 n — 2 ( 1 ) a with similar expressions for y m Zm. For determining the consecutive perpendiculars there are the relations, P f xH*- 1 ) l ± v ( x^- 1 ) 1 1 a 2n J ’ p? L « 2n j ’ and, in general terms, Ain = ( 2 ) Similarly, for determining the lines u u t ... u m we have, in general, 2n v S x ^ n(n ~ l) i l S’ D ( 3 ) 18 GENERAL SURFACE, Now, if the values before ascertained are assigned to the co-ordinates, 1 jt?m 2 2 2 4(n— 1) 4(n— 1) Wm — 1 — 2 . . . m m+1 ■ 4(n— 1) 2(n — 1) U X m_l m 0 2n a 8n(n-l) -t> fl 2n(n-l) a 8n(n-l) , (4) 1 «m 2n 2 2n(n — 1) 2n(n — 1) u m — 1 U m — 2 m mi-1 • 2n(n — 1) n(n— - 1) u x 2 m .a n2 ( n-1 ) a n2 ( n_1 ) ... ^(n-l) a n(n+l) (5) In the expression (4) for p m it is to be remarked that the number of factors, both in the numerator and denominator, is (m + 1), and we have, for the index of a in the denominator, the geometric series, mil i n + in (n — 1) -f in (n— l) 2 + ... = in — — — . n — i The number of factors in each term of the expression (5) for u m is also (m- fl), and we find, for the index of a, the series, » 2 (»-l)-)- n z (n-l)~ + ... to m terms + «(« + 1) = — ^ . Hence, the general formulae become, Via 2 m mil 4(n — I ) 4(n — 1) 4(n— 1) 2(n— 1) = 2 u X mi- 1 2n l n a ( 6 ) Urn 2n = 2 2 m m 4 1 2n(n— l) 2n'n— 1) 2n(n — 1) n(n — 1) u m — 1 «m— 2 ... U X mil n(n — 1) — 2 - ■ (V a n \ I CO-ORDINATE RELATIONS. 19 It has been already shewn, page 16, that w 2 = pr t ; consequently, Um — Pm ^m + 1 ; « 3 m— 1 = p m — 1 r m : and, after reducing by means of these equivalent forms, we shall obtain the general expressions, x 2 (.2(n—l) m+1 mil ZnL ”- 1 )- 1 ■a 1 m 2 m* 2 2(n-J) 2(n-l) 2(n-l) 2(n-l) 2(n-l) PmPm-l ... p r m r m - 1 ... r ; X .n(n-l) lil + l m+1 n(n — 1) — 2 n— 2 L m m * n n(n-l) n(n-l) n n(n-l) n(n-l) Pm. Pm-l ... p r m+ i r m ... r t (IV.) We have now to shew that, for the general surface, m (n-1) - 1 Xm f i)2 "I n — 2 m = with similar expressions for y m zm : v being a line such that, 1 m m 2n( n - 1 >-_i n — 2 = 2 ■a #n(n— 1) in » n(n— 1)— 2 n — 2 (i) From page 17, (1), *£m — m — 1 m 2 2(n — 1) 2(n — 1) (n — 1) Wm — 1 % — 2 ••• 'll X m (n _l) ] a 20 GENERAL SURFACE. Again, from page 18, (7), 2n = 2 2 m mil 2n(n — 1) 2n(n — 1) 2n(n — 1) u(n — 1) Uxa — 1 Uxa. — 2 ... U X mTl n(n — 1) - 2 a Now, after extracting the n th root of this expression, the index m being lowered by unity, we obtain, m — 2 m — 1 2 2(n — 1) 2(n— 1) 2(n— 1) «n — 1 — 2 i • . W, u - \ tfn(n-l) 2 n n: m ^ 5 n (n — 1) — 2 n — 2 m o (n-1) - 1 n — 2 V 2 2(n — 1) 2(n — 1) — 1 W nl — 2 ... u m— 1 m ofr-D _ i n — 2 V. ( 2 ) Hence there will be obtained the following values of the co-ordinates of the nfi*- point on the surface, expressed in terms of those of the first assumed point, m ^ (n-1) - 1 X m = } ” -2 m (n-1) - 1 2/m — f v 3 ~) n — 2 \ w \ m y Zm — - I?) m (n-1) - 1 n — 2 m 2(n— 1). J (3) CO-ORDINATE RELATIONS. 21 Cor. If in = 0 these expressions will become, X 0 = X ; y 0 — y ; z 0 = z ; as they ought to do consistently with the previous assumptions; while the formula for v, (1), will be resolved into the equation of the surface, 2 (t)"= 1 (V.) It remains, further, to investigate the formula for the general surface, involving p m , which corresponds to (4) page 10, derived from the ellipsoid. The co-ordinates of the m th point of intersection with the plane 2 (j^) — 1 will be, as before, — — ; Vrn Wm _ y™ . w m ’ ICm 1C m = s (t> Now, employing the results of the preceding section (IV), x m — m > (»-D - 1 n — 2 m (n — 1) - 1 n — 2 m (n-l) x ; &c. ic m = iri „(n— 1) — 1 m ”n — "2 (n-l) V ' X m ,( n — 1) - 1 a + n — 2 22 GENERAL SURFACE. m ,(n-l) - 1 1 C m — V Cm — with similar expressions for rj m £ m . But, Pm a = £ m 3 + vJ + Cm 2 , Pm 2 = 2 m fr j) 1 l f ,*- ir ) \ m , r 1 m } I n (n — 1) — 2 | I I 2n (n_1) ~ 1 1 V n — 2 J v n — 2 J a a ( 1 ) ( 2 ) ( 3 ) In order to reduce this expression, we have from page 19, after lowering the suffix m by unity, 1 ___ m — 1 m — 1 2 2(n — 1) 2(n — 1) 2(n— 1) Pm — 1 ... P r m — 1 r, and the substitution of this value in the preceding equation will give, after extracting the square root, the formula, (n-J) x 1 m n(n — 1) — 2 a n — 2 ra — ] m — ] (n — 1) (n — 1) (n — 1) p ?'m — 1 • ■ • T t • ( 4 ) (n — 1) pm Pm — 1 Pm — 2 • * FORMULA. 23 (VI.) It will be convenient in this place to recapitulate the principal formulae of relation which have been investigated. We have, then, from the general surface, *{£} = *• Prom page 22, form (4), and from page 19, (n — 1) 2 x m n(n — 1) — 2 a n — 2 m (n — 1) m , ( n — i) — i n — 2 (n-1) X | n(n— 1 m - l )- 2 \ a u — 2 m — 1 in— 1' (n _i) (n _i) (n— 1) (n- -1) Pm Pm — 1 Pm — 2 ... p ?'m — 1 • • • f , 1 m — •1 m — 1‘ 2 2(n— 1) 2(n — 1) 2(n— 1) 2(n- -i) pm — 1 pm — 2 ... p rm — i ... r t 1 m- -1 m— r n n(n — 1) n(n — ]) n n(n— -1) p m — 1 pm . — 2 ... p ^ in • • • ( 1 ) By a combination of (I) (3) there is derived the relation, (n-1) , n l x m n(n — 1) — 2 U n — 2 (n-1) - 2 X in n(n — 1) — 2 a n — 2 n r m — n Pm m — 1 n n n(n — 1) n(n — 1) n n(n — 1) pm — 1 pm — 2 ... p r m r m — 1 .. m — 1 n(n— 1) 24 FORMULAE. (VII.) By combining the forms of (VI) we obtain the absolute values of r m p m p m in terms of the first co-ordinates. Increasing m by unity in (2), and eliminating by means of (3), 2n pm By (3) and (2), 2n fm By (1) and (2), 2 pm (n-1) x m n(n — 1) — 2 a n — 2 1114.I (n-1) \ 2 X \ a m.j.1 (n- 1) - 1 u — 2 m (n-1) 2 n X m ( n — 1) — 1 n — 2 (n-1) ) n 2 3 x m n(n — 1) — 2 \ a n — 2 m (n-1) X m (n-1) - 1 n — 2 (n-1) 2 3 x m n(n — 1) — 2 a n — 2 ( 5 ) (6) © ( 7 ) FORMULAE. 25 The combination of (5) (6) will give the expression, 2 2 s’m pm m (n- -D— l a n 2 m+l ( n — •1) X m+l n^ n -1) — 1 n ■ — 2 (8) By augmenting, in (5), the value of m bv unity, and using the relation — ?’m+l Pnu m (n-1) X Win m n(n — 1) — 2 a n — 2 mil (n-1) \ " X \ m+1 f n Cn— 1) — 2 (9) Lastly, by combining (C>) (7), s ( t. pm 1 n r m (n — 1 ) n(n— 1) — 2 (n-1) X m nfn — 1) — 2 a u ( 10 ) K u Pm (n-1) X m u(n — 1) — 2 n n L = Ln 2 ) a " (n-1) X I n(n-l) ■ . ^ a » — : (II) 26 ELLIPSOID. (VIII.) In adapting these general formulas to the particular case of the Ellipsoid, the indices are found to take the singular form and their special values must be ascertained in the usual manner. Then, m n(n — 1) — 2 n — 2 0 m m — 1 = (n — 1) **f~ mn (n — 1) = 2m i 1. m n (n-l I -l = 0 m in — 1 — = (n — 1) -+■ mn (u — 1) — 1 = 2m. 0 Hence the formulae will be for the ellipsoid, as shewn independently at pages 8 and 10, pm — 1 pm — 2 p ?’m — 1 ?'m — 2 ••• 1 — ] \ a 4m + 2 ) 2 2 Pm— l Pm — 2 2 2 2 p •''in 1 'in — I ( 3 ) 2 si \ fl am+l iSm+1 / ?’m Pm 9 , 9 9 09 pm pm — 1 pm — 2 ... p ?'ru ?'m — 1 The remaining formulae (5) — (10), for r m p m p m u m , will be found to agree with those which have been established in page 12. . \ . . . * si GENERAL SURFACE. 27 (IX.) In this section we propose to determine the equation of the plane containing the lines r p r t ; and, generally, the equation of the plane which contains the lines r m — 1 pm — 1 r m , in the surface of the w th degree. Let X Y Z be the co-ordinates of the plane in which are situated the lines r p r t ; x y z the co-ordinates of contact ; x t y t s t those of in- tersection : then, since the plane passes through the origin, its equation may be written, z = MX + NY. (1) Consequently, z II g + Ny. (2) * 1 — + Ny, • (3) But, from page 15, (•5), ?« 2 x n ~~ 1 r - — ri~ >/ ,v ' w 3 ^ n “ 1 a n 0 n / C n ~n- 1 _n — ] “ - - M + c n a n N-^\ b n (4) By combining (4) and (2) wc find, M o n (b n z n ~~ — c n yn-2^ ^ r n (a n y n ~ 2 — ^n-2) x N b n (a*z n ~* — C n A’ n ~ 2 ) 2 c n (« n y n ~" — /; n X n ~~) y ' The plane containing the lines r p r t is, therefore, represented by the equation, /7 n (6" r n ~ ~ — c n y n ~ z ) X X b n X + y 4- z 1 «# & c 9 ~ Px Pz Pi Pi P r 3 r i X ffii 4- y + z 1 PsPxPz Pi Pi P r X r 3 r i V i ' X ) l * V ) Pm Pm — i Pm— i ■ • ■ P ? m _ i 1 m-i 38 ELLIPSOID. ,r 3 a 3 + O r V + — c 2 = 1 X 2 + y* b* ,1 2 + — C 4 =r 1 f X 3 + ,2 4 - 1 b ' 8 c 6 0 o’ r r r X" + f i a 8 + a* pf f r f' X 3 + ;/ 2 2 3 4 - — ^ c 10 i 5 10 Pf P 2 ^*2 2 ? ’/ 2 ' a 2. + y* 2 + f— ^ c 12 1 « 12 P^pfp* r 2 2 r*' y / ^ 2 \ 1 V« 4 m ) j0 2 m-x i» 2 m_ 2 ... p* r 2 m _ 1 ... r y / « 2 \ 1 \a 4ni + 2 / />j2 ao2 7.2 / P m— i P m-2 •” P ’ 111 ••• '/ The expression (4), in page 26, will furnish the additional series, which may otherwise be derived by combining in pairs the terms of those which precede ; 2 2 2 2 { { { { + x z + F 1 1 _ v-p/ « 3 <5 3 a? c 3 6 3 c 3 J P? // 2 r 2 xy + .r z + yz 1 L - ^2 2 -p2 2 a 0 6 3 a 5 o 3 6 3 c 5 J Pa 3 J xy + a? z + yz 1 \ _ r 3 2 — p 3 2 al tP cP

V-P4~ a 9 6° ff® c 9 6 9 c 9 J 1 Pa 2 Pz X pf P? P* r A r *' r * r ? FOHMULiE DEVELOPED. 39 The formulae of page 12 will give the absolute values of the consecu live radii and perpendiculars, in the following order, the successive values of u will be found by combining the corresponding terms in the first and second columns ; e.g. (II.) Development for the surface, In this and the following developments for surfaces in which # is a positive integral number, in order to avoid repetition, we indicate each series by the number which marks its generating function in page 23. GENERAL SURFACE 40 (!•)- itl ill — m — m — m = 1, 2, 3 , 4, 5 , x~ X 1 ,10 ,22 X 16 ,46 X ,32 794 - + + + + + 1 PlP ’ 1_ P 2 Vi P 2 r ?‘ 1 P 3 P 2P? P 4 ? ' 2 2 r f 1 Pa Pz P*P?fir*r*r?' 1 P 5 Pa P z 2 P 2 4 P? P™ ^ a r z 4 r 2 3 ? 7 16 ' ( 2 -)- VI X !»-«- + p~ m — 2 _ 4 . 18 ' /V / /l ?» >■16 3 > -43 + p 2 2 l > 8 ?" 2 4 ? 7 8 ' »?, = a ? 32 4 ' + P-/ P2 4 P? P W r 3* r 2 S r } G ‘ HI — >>( 3 i r/186 r + — = Pa Pz i p2* P? G P U r A 4 r 3 Sf ’2 16 r ? 2 5 FORMULA DEVELOPED. 4 L ( 3 .)“ m = 1 , «ia + jfi rf m r 13 2 ’ 3» + js 3 p 6 r 2 3 rf in = r 24 Q _ i ’ „66 + P 2 6 Pi 6 P Vi r s 3 r a 6 r™' r 43 “ = 4 ’ ^ + of 3 n~i G of 12 of 24 o* 3 r 6 r 12 o* 24 r$ I J 2 r ' 4 ' 3 ' 2 ' i m — 5, ^282 + pf p 3 « P^ ppfPrfrf.rtf* r 2 24 r , 48 (4.) — The formula (4), in page 23, will give for this surface the series, r 3 _ n 3 ■Pi P 3 ^3 r 3 l a 10 + -} = ?' 2 3 — P 2 3 P 2 3 i^/ 3 />* 6 ^2 3 y / 6 ' ^ 3 3 P 3 3 J ,.16 i a 46 + ■" } 5 3 + ... * « 4G 7 »*— P/ & + -} -{© + — S PlPzP'l'P^P^ r A 6 r 3 6 r 2 1;3 r / 24 r^-pz l * ~ P 9 *Pt*P^Pr^r**r«rVr**rf G 42 GENERAL SURFACE. (III.) Development for the surface, ( 10 - m — 1 , rn = 2, m — 3 , m — 4 , m = 5 , + + a 1 7 X' ■27 753 + »81_ „161 + x- 243 7 485 + 1 P, P 1 PiP,&rf 1 PzPtPf P* r a*r*‘ 1 PxVzP'fpfP™ r^r^rW' 1 Ps P 4 P3 S P2 9 P^ P 8l ' r J r 3 9 rW r* r ( 20 - ?•?* — » = wt — 1, 2 , 3 , 5 , ar + ^32, ^54 ^104 #162 + + a ,320 + x‘ .480 fl 968 + 1 p 2 jfi r <> I P 2 2 pf JO 18 r 2 6 ?’ ( 18 1_ j9 3 2 J° 2 6 jp/ 8 jB 54 r 3 6 r 3 18 r, 54 ’ 1 i ? 4 3 i°3 6 i®2 18 i 9 / 6 ^ i^ 162 r 4 8 /■ 3 18 ^> 2 64 z/ 62 " FORMULAE DEVELOPED 43 (3.)~ ,13 * = 1» “go +-’ = pi r i ,.36 * = 2 ' 5» + pi p!2 r ^i r 12 r 108 * = 3 ’ ^ + ' p 2 4 p}^ p 36 r 3 4 r 2 13 r 3& ,.324 m = 4 > SR +— = jp 3 4 Pst^lPf^ i? 108 r 4 4 r 3 12 ;- 2 36 r ; 108 , 0:973 ~ 5 ’ ,1940 +- ~ P 4^ P 3 }^ P 2°^ P ^8 /* j. ^ 12^* 36 7 108 y 324 % * (4.) — The formula (4) will give, in this case, the series, I* 5 + . l- r*-p* U 5 1 P* p* r 4 l«w ^ ■•} -{&)*+ -] l _ 1 p^pfp 12 /’ 2 4 r, 12 (£+.. 1*53 T r-{(sr-'} _ r s 4 — P 3 4 P3 4 i ? 2 4 i ? / 13 / )3G r 3 4 ?, 2 13 ; '/ 36 {"— +■ -} -{(Sr)V - } _ V-P4 4 / P4 4 j p3 4 i ? 2 13 i ?36 i ?1 ° 8 ? '4 4 ?, 3 12 ^'2 36 ^ 10& ' («485 T HQ V "}= >- 5 4 -p/ \ • • • • • • 44 GENERAL SURFACE (IV.) Development for the surface, no- I fcV . ; -7T + *16 * =2, ^ + m — 3, m — 4, m = 5, x' 64 7 106 + g256 S36 + (.1024 A706 + ( 0 ) 1. P, P Pa P, P* PaPaP? P lG r^r™' Pa Pa p^p} % p^ r 3 4 r 2 16 PaPxPz^Pa ^i?® 4 ^ 256 7’ 4 4 f'g 16 7’ 2 64 ?' ; 2s6 ( 20 - vn, m — m m 4, 5, , x° m — 1» Tn + 10 2 > ^ + ,32 50 r 128 3, ^ + a 210 ^513 ,>850 + X ,2048 -.3410 + jr pf P 8 r f 1 Pa 2 Pf P S2 r 2 S r f 2 ' 1 P 3 *P 2 8 p™P™ r 3 8 7’ 2 32 7 - ; 128 1. V 2 P 3 8 P a 32 P}^ i? 512 ?' 4 ® 7'g 33 7’j 128 7', 519 FORMUL/E DEVELOPED. 45 (3.)- m = I, m — 2 , m = 3, m = 4, m p 20 530 + ... = p 80 5l30 p 320 5530 + • • • — + • • • — 5 , pl280 ,.2130 a; 5120 „8530 + ... p 5 r 5 ’ r . 5 m20 y 5 r 20’ P i P '2 ' / 1 _ p 2 ° P m V 3 5 r 2 ~ () r / 8 ° I m 5 , n 20 m 80 m320 r 5 r 20 80 r 320 P 3 Pi Pi P '4 '3 J 2 '/ + ... = 5 20 rj 80 n 320 ,,1280 r 5 r 20 r 80 r 320 r 1280' P 4 P 3 Pi Pi P 1 5 'a ' 3 'i ' i (4.)- st \ \a e+ " / - {© 5 M= a* 5 ___ 5 / i r/ P, 5 i? 5 r 6* f * 16 'I 1 5 ( ' 4 U+-J r - I W + ...} = J a; 64 /* 64 ' k 5 , } \^kT6 + -’- } H 1 («ioe> )+...} = f *256 '*256\ 5 , 7 \ o426 + ■” } -{< *■) -Pa‘ n 5 ^ 5 «j20 ~ 5 r 20 Pi Pi P 7 2 r / ?- 3 5 -P 3 5 „ 5 r , 5 m 20 m80 r 5 v 20 r 80‘ H3 pi Pi P 1 3 1 i ' i rj-p * 5 ft 5 rn 5 /v) 20 ^ 80 ^320 5 a» 20 80 ^320 Pa Pz Pi Pi P 7 A 7 3 7 2 7 U ’ 1024 )6 f/® 10 M \6 L ftl 706 ' • • • j | \^ 1706 / + ’”C“ 5. -Ps‘ n 5 m 5m 20m 80m 320ml280 r 5..2 0 r 8 0 r 3 20 r 1280’ Ps Px P 3 Pi Pi P ' s ' a, ’ 3 1 i '1 46 GENERAL SURFACE. (V.) Development of the formulae for the surface, ..10 > (!•)- 2^) = V« 10 / 1. m — 1, r 9 in + • = m — 2, £. + ... = o'Ol m — 3, x 72d + m = 4, 0.6561 + ... = fl 8201 T = 5, r 5 9049 4- — a 73811 ^ *’* P, P Pi PiP^ r f PsPiP?P sl r 2 9 r* l ‘ 1 __ _ PaPs P i 9 jo® 1 ^ 739 r 3 9 r 2 81 r 7 s ®’ 1 p5 Px 3 9 2 81 J» y 739 ^(65 6 l 7*^81 ^ 2 729 . ( 20 - 7n — 1 , »is ^20 + „9 ,.162 m — -200 ~*~ ;w — o, m = >» = 4, 5, j.1458 ^1820 pi p\% r y& pip\*pm r \z r ™r x 13122 ,16400 X 118098 a 147620 P*Pi*P 16 - pUaS r 3 18 rj 16 - r ; 1458 1 Pa 1 9 3 18 i? a 162 P , 1458 jo 13133 r 4 18 r 3 162 r 2 1458 / .•6561 ' .13122* / • • • • # FORMULAE DEVELOPED. 47 (3.)- " = 5ho+’ ,90 m — 2, m — 3, m — 4, m — 6 , >.810 ,1010 P 7290 ,9110“ X .65610 ,82010 #590490 jplO y* 10 pU) r ^ 10 ,.90 1 p^lO p90p8\0 ,. g 10 ,^90 ,. 810' V 3 10 V 2 90 /^/ 81 ° P‘~ M ^4 10 ?' 3 90 ?' 2 810 P 7290* a 738110 1 J0 4 10 JO 3 90 JO, 81° jt? 7290^,65610 10^^90),, 810 r 7290 r (4.)- {s-r-isr-i-ji 10_ n 10 r; p/ 0 ^1° r™ f®8! , ) 10 (7#81 ,10 («Toi + ... 1 "-{(£i"+ 4 r 2 l°-p 2 1 0 p 3 10^,10 JB 90 ra IQ r W- j^729 1 10_ f #729,10 1 l«9n + -} +••*}- r 3 10 P 3 10 Pz^ Pi l0 JS / 90 ^ 8 70 /* 3 10 ^90 r 810‘ f #6561 4 10 l55oi + — > ~ } -{C-m) + -} = r 10 n 10 ' 4 P4 P# 10 ^ 10 ^. 90 ^ 810^7290 r 4 10 r 3 90,. 810 f #59049 "I 10 1^73811 + 11° r ’/ #59049x10 I 1 “I \ fl 73811/ r 10 — n 10 P5 1 °i ? 4 10 i ? 3 90 - • • jp 65610 p 5 i°/ , 4 90 ... r 65610’ r 7290’ i 65610* 48 GENERAL SURFACE. (VI.) In each of the surfaces which have been considered, the absolute values of the consecutive radii and perpendiculars may be obtained from the forms in page 24. The developments of the general formulae are susceptible of indefinite extension. The series, however, cannot here be conveniently extended ; nor is this necessary for the object of the treatise, since the examples already adduced will suffice to illustrate the power of the general forms. This division of the subject may, therefore, be con- cluded by annexing one or two illustrations derived from surfaces in which the indices are fractional or negative. For surfaces of which the characteristic index is integral and negative, the formulae have been given at page 36 ; and general forms may be obtained expressing the corre- sponding relations when the indices are fractional. The expression of these relations does not, however, appear to be of much importance, since the adaptation of the original formula; to any particular instance, or to any series of surfaces of any order, can present no difficulty, while it may, perhaps, be practically more convenient. Before dismissing the surfaces with positive integral indices, it may be observed that the indices of a b c are necessarily integral in every case in which m is integral and positive ; i.e., in all cases which here fall under our consideration : for, m m if k = n — 1, \ 3 1 5 which is always integral. And, m n(n— 1) — 2 m m b- j (n- 1 ) + 2— -r which is consequently an integral number. In the same way it may be seen, at page 36, that the indices are integers if n is negative and integral ; and this will be self-evident, if we consider that in any case each of the indices in question represents the sum of a geometric series in which every term is separately an integer. As a negative check upon the numerical accuracy of the several ex- pansions, it is useful to notice that in every case the sum of the indices on either side of the development must present an identity ; e.g. in (3), page 47, 738110 — 590490 = 147620, which will be found to be the sum of the indices in the reciprocal equivalent. The general formulae are, of course, subject to the same restriction of homogeneity. FORMULAE DEVELOPED. 49 (VII.) Development for the surface, The general formulae, in this instance, are reduced to the following expressions : m 4 pm— 2 i S pm — 4* . . . 4 Fm— 1 1 8 F m — 3 pm pm— i i 4 is -1 pm — 3 pm- 2, 4 -5 ... fm- 16 -2 Fm — 4 . . . pm— 2 l 4 pm — 4 F m—l 1 4 Fm— 3 • . . 2 i i pm — 1 pm — 3 Pm — 5 1 2 ... Tm—2 8 Fm— 4 • • • i pm— 2 k pm — 4 1 4 r m — ] 18 Fm — 3 1 2 pm — 1 1 ¥ pm — 3 . . . 1 2 r m 1 ¥ Fm— 2 . . . (i-)- m — 1 , m = 2 , m = 3 , m — I , m = 5 , a'i xk + ... Xi a\ + oV x\ + art s «?s + aii xk _ J_ ~ PiP — Pill. Pi Pi — ft* r n i P 3 Pi P k r P ' _ V 2 g pi r 3 % r,i Pa P 3 P P r i* _ Ps i PP r A ir i i PsPaPIP 1 * r 3 i ( 1 ) ( 2 ) ( 3 ) H 50 GENERAL SURFACE, ( 20 - m = 1 , + ax 1 _ P m 2 , ^ + a 2 pr , m — — — r + Oi Xi _ P, P^P* rp i x $ , ™ — 4 , — 4 .. c > # P*P* r 3 rfi Pa Z PP r a 2 vi = 5, a\b Xw c + ■■■ ~ - P 3 PP r A PtP^pk r 3 * T P ( 30 - '° TV 2 P 2^ Pk r 5* V 3* r. : - FORMULA DEVELOPED. 5 t (4.)- \ ai i xi ( xk ta - + • f L 1 {a's'. vi j xis 1 aU + f 1 . + }- [ 0 3I Xai {£)* + -} + -} Ui ) 1 + •••} _ r k-p? Ppfi r t i _ ( r J—pJ)P* rp ! ^JL i * p 2 a p 2 ; 2 2 O' 3 2 P 3 2 ) .Pp ? 2 4 P 3 2 /; 2 I /Js r 3 i rp — O4 2 P4 2 ) P 2 4 ppsv^^ypz P 4 2 i?3 2 J?V ?> 4 2 ? V _i \ 2 . 1_ O '^—Ps^) pfc r x* ? v ,f dl.r 3 W P 5 2 Pi 2 P 2 8 _?* 32 ^ 5 ’ ^ 3 8 ^ ; 32 (VIII.) This surface presents the following geometrical peculiarity, derived from the development of (2). When — 2, it has been shewn that, (Hr) PZi. P? ’ consequently, Fj P/ 2 1 ; or, 7;; 2 p>\ so that, in this instance, p t is a mean proportional to p and r n and is therefore equal to the circle-ordinate u, (page 3). In figure (2), let OP = ;• ; OP, = r, ; OP 2 = r 2 ; OQ — p OQ, — p t ; OQ 2 — p 2 ; OQ, 2 - OQ . OP, ; and, since the law of derivation is invariable for all the consecutive points Q Q, Q 2 ... P P, P 2 ... it follows that, OQ 2 2 = OQ, . OP 2 , and, generally, OQ, m 2 = OQ m _, . OP m : 52 GENERAL SURFACE. (IX.) By employing the relation established in the preceding section, the developments admit of simplification, and may be expressed in the forms which follow. ( 10 - m — 1 , al xh + xh m = 2 , ^ + a% m = 3, aV x» + 1 Pi P ' 1 Pi' 1 Ps P? ' m 4 , oft m — 5, Xi ] l + Ps r 3 4 Pfi ( 2 .)- m — 1 , m = 2, ax xh ah + + m — 3, -3 — r + ... as x* A x~s . m = 4, -x + ••• at / 2 ' 1. 1 m — 5 , als ari's + ... = FORMULA DEVELOPED. (3.)- ru = 1, at xi + m = 2, % + at /V 3, a is A'is + »* = 4, — i + m = 5, -jr — - + #64 £ih r s*Ft ' r 4 i r 2 i M ^ iv’ 6 (4.)- 1 ai xh + - }*-{(?)* + “} {^i + -}‘ _ {fe) 1+ ■"} 4^1 + l «1S I—- \all xk }‘-{Q)‘ + -}= r ft~pp 1 pp P, r *i—p*l P? P ^ P‘2 * s ' 2 Pa- Pr> Pa* Pa PP i _ l > 4 2 — P 4 2 p 3 h pp p ^ Pa Pa* ' r 5*— Pi* P ^ Pa* Ps* P s PaV/V'e 54 GENERAL SURFACE. An examination of the equations will shew that, in the instance of this surface, owing to the singular relation p lx ? = p m -i ?'m, the develop- ment of (3) is included in that of (2) ; and that this connexion must exist will be seen by reference to the general formulae as adapted to the surface under consideration, page 49, in which, if m + 2 is substituted in (2), while m is written in the formula (3), both expressions will furnish coincident results. (X.) In illustration of the formulae for negative indices, as given in page 36, we may first select the surface, = 1 . ( 10 - m = 1 , m = 2 , m = 3, m — 4, vi — 5, m ~ 6 , Vi = 7, vi 1 + ... I zzz x~ Pi P x x a 3 + -. jo 3 rf P 2 Pi aP + •• 0 ^ V ^ r i _ 2 x 8 p 3 P 2 a 4 r* ad 0 + •• P 2 2 P 8 r 3 2 >’, 8 a 6 Px P 3 P? ?’ 2 4 ' a; 32 + •• — 0 ^ 0 ^ 7 * % 7* 8 r \ 3 F 1 ' 4 '2 P 5 Px p'f P™ r 3 4 ? ’/ 16 ' a 04 + •• 73 2 77 8 r .32 7« 2 7» 8 i« 32 r x rur '5 '3 ' / a 32 P 6 Ps P 3 4 .P/ 6 *\i 4 ’V®' ft 42 ^128 + •• •— /n 2 73 8 ,73 32 7« 2 7* 8 y 32 7 y 5 y 7 3 l '6 '4 '2 P 7 Ps P a. 4 P 2 16 jp 64 r fl 4 ?* 3 16 ?' 04 ‘ .C‘256 + .. n 2 „ 8 , w 32 ml 28 3 8 33 , 128 P 6 P X P 2 P 1 7 ' 5 ' 3 ' / a 36 P 8 P «' 4 P 2 16 P^ »Y* r 4 1 ^ = 3, « 6 + ... Cl TP < 1 .V 16 Pi 2 / rf = 4, + ... /Y) 4 /111 6 /)* 4 n* 16 — 2 r '3 ' / a w GO Cl GO CM CO - 5, a* 3 + ... _ p 3 4 p} ( ' r A 4 r 2 16 ^64 Pa P 8 P™ ^' 3 8 r ™' - 6, «128 + ... — Pa 4 Pi 10 P U ?, 5 4 r 3 10 r / 64 a 42 rr) 2 nr. 8 rn 32 8 y 32 1 J 5 1 J 3 Ft ' 4 7 2 786 7 14/ ' 33 + /T) 4 ry\ 1C rr) 04 n* 4 nr* 10 64 5 r 3 Fl ' 6 7 4 ' 2 Pa* Pa 8 Pi™ P m r s 8 r z™ r }™' ,.512 m = 8 , — + 5 a 1 7° m 4 n 16 m 64 m256 4 16 64 256 Pa Pa Pi P 'i '5 r 3 0 7) 2 m 8 m 32 /II 128 8 32 128 Pi Pa Pa Pi 'a 'a ? 3 56 GENERAL SURFACE* ( 3 .)- v> T 1 , .r 2 + ••• pr r — 2 , a 2 ,T 4 + ... = P, r 2 } » 2 rf 3 a* + ... P-i P 4 r a r 4 J J ff 2 V 2 2 4 a 6 + ••• P 3 P * * 16 P^P S ^’3 2 ?\ 8 ' a; 32 + ... Px P 2 4 y 6 r 5 ^ 3 4 ? '/ 16 a™ ^3 2 V X r 2 8 (5 « 22 + ... P 5 i^ 3 4 p} & r 6 r 4 4 r 2 16 X CA y ? 4 3 ^ 2 8 j ? 32 r 5 2 r 3 8 r 32 7 , + ... 7 ) 79 4 7 ) 16 7^64 a* 7 * 47 . 16 7 * 64 F& P X P 2 P ' 1 ' 5 7 3 7 / a 42 ^5 2 P 3^ P 82 ? ' 6 2 ?> 4 8 r 2 32 3, a 33 4" ... Pi Ptf P3™ P, M r s r 6 4 r x K r i *356 Pe 2 Px P 2 32 j » 128 >'? 2 ? ’s 8 r s 32 ? 7 (XI.) It appears from the last development that the circle-ordinate n, in this surface, is equal to r, the radius vector at the first point of con- tact. In consequence of this singularity we shall have, by attending to the principle of uniformity which governs the position of the consecutive points on the surface, fig. 2, « m = r m ; r 2 ' m Pm ?*m -1* 1 • in which it is to be understood that r and p are indicated by r 0 and p 0 - This peculiar relation will have the effect of reducing the preceding ex- pressions to others which involve only the lines r r t r 2 r 3 ... r m ; in the form which follows. FOBMULiE DEVELOPED 57 m m m ill Vi ill m m vi (XII.) Reduction of the developments for the surface, v = 1. ' x ’ ' X > (!•)- — I } 1 ;C 2 4“ • — = L. Pi l y.2 ' = 2, X* a? + ... — P 2 y*4e 7 ? = 3, a 2 + ... r 3 r ? ttfi P 3 r 2 3 r 8 x 18 + • . . r 3 ^ i' >.16 (fi Pi r 2 » 8 ’ ' 3 '/ - 5 a 10 + ... r 5 ~ 4 * 16 7 3 7 / .j.32 Ps 2 a* 8 a«32 '4 7 2 7 (2-) = 1, a~ X* + • •• — r? r i ■ = 2, x s + ... y % 2 r 8 a " r i 4 o a « + ... } % 2 7 3 , 8 x u ,.16- — 4 X& + ... V r 3 8 r 32 1 \ a 10 r 3 4 y 16 7 i a 22 + ... r 5 2 8 r 32 '3 ' 1 X Gl " y 4s 7 4 r ,w r Gi 53 UENERAL SURFACE. ( 30 - m m m = m - 1 , a >2 + ... = r 2 . = 2 , a 2 + ••• — r f_. r 4 — 3 * 8 + ... r 9 2 r 8 a 2 ? 4 — 4 + ... r 2 r 8 T ) a ? 16 r 3 4 ? 43 a -’ 32 + ... 2 r 8 r 32 7 4 7 2 7 / J , a 10 r 4 r 16 ’ ' 3 ' / (XIII.) Development for the surface, 2 ( 4 ) = i. v ar ' ( 10 - - 1 , a + ... 1 P, P - 2 , a ; 9 + ... = yO 3 r 3 Pi Pi — 3 a 13 + ... Pf r i % ,J 3 a: 2 ? Pz Pi P 9 r , 9 ' — 4 a ? 81 -f- ... jo 2 3 P V r 3 S V t 1 a 41 P A P 3 P ^ r -/ - 8 , ^6561 + ... p^pWp^p^rS a 3281 p s p 7 ; 7 s 9 Pa 81 p / 29 m = o, — i m — r 9 r 81 r 729' '6 7 4 . 7 2 FORMULAE DEVELOPED. ( 2 .)- B9 m 1 , V + x 6 r m m m = 2 , = 3, 4, x 18 Z 28 ^54 c 163 TIcT + + jt>® r ( ® /-/ a ’ V? rf jOg 2 ^ 18 r ; 18 6 „,54 6 r 54 /V f ' ?\ / lb 18 P 3 “ P ! r * r 13122 ** = 8 > “TH^TT + a' 6500 /? 6 6 p/ >l jOg 486 /? 4374 /' 7 r> r s 54 l ^ 486 ,.4374 P? 2 />,“ /^3 162 i 3 / 1458 >V S ;, 4 162 r a 14 ' 58 ' ( 3 .)- m — 1 , m — 2 , m — 3, »t = 4, *10 as x .54 *26 162 + + + + o 3 p ~ rf . v) 2 2 Z'/ 4 2 i >6 ^/6 ^ 2 3 ;; 18 r 3 2 r / 8 6 r 6 i ? 3 3 A 18 r/ 7 'g 18 jt?g G j » 54 r 3 fl r ; 54 «6562 ^13122 "* „ 2 „ 18 n 162 „ 1458 r 2 ,. 18 r 163 ... 1458 P 1 I J 5 7 3 3 Pi 7 8 '6 ' 4 '3 ^g 6 54 J 9 g 48G ^ 4374 r 7 6 r 5 54 r 3 486 ?' 4371 • • «•! • • • 60 GENERAL SURFACE (XIV.) Development for the surface^ (!•)- m — 1, «* * 4 + ... = 1 Pi P ’ m — 2, X 16 39 + ... = P* r * Pa Pi m — 3 , + ... Pi r 2 4 r 64 Pa P 2 i> 16 ^ 16 ’ m = 4 , a ,256 -f- ... — j» 2 4 j» 64 * 3 4 *® 4 Pi f 3 i»/ 16 r 2 16 ’ m — 5 , ffl 6l4 + ••• ^3 4 ^/ 64 >Y 64 *1034 Ps J»4 i^a 10 i? 258 r 3 18 r , 258 ( 2 .)- II *32 « 18 + ••• ' II II 03 «78 + ... GO . m. = 2, a™ X 4 * + •" n j 3 y 3 — J'j ' 2 Jjiz r 12 * = 3, ^92 a 114 + • • • j» 2 3 p 48 r 3 3 r 48 pYl r vi m — 4, a 462 a;768 + ... _ Pz’pf’ r A * r 8 48 jt? 2 13 JB 193 ?’ 3 13 r ; 193 m = K x im j» 4 3 jBj 48 ^ 768 r s 3 r 3 48 r/ 68 fl 1842 12 192 r 12 r 192 r 3 7'/ 7 4 ' 2 (XY.) Development for the surface, (!•)- »? = 1, J- as xl + ... ii >» = 2, x% cA + ... pi r). Pt Pi' w = 3 1 + ••• _ Pp **,8 at &•¥ P*P% P 9 * r i - i ' »* = 4, + - p 3 l p% r 3 l ;•¥ a?g PaPsP? r J ' = 5, all + ... _ A* ** J : PsPaP^P^ 1 " 62 GENERAL SURFACE, ( 2 .)- m = m — m — = 1, X a 3 + ... _ 1 f ’ - 2, x% oh + ... _ P s >' 3 O * Pi = 3, al + ... 3 a* 3 r 1 ' 2 x' 2 l P ^ p-i r,i ' = 1, x% 1 + ... p 2 3 p 3 l r 3 s r 3 l aS? p 3 °p?> )\J = 5, a% + ... _ p 3 p 3 l r 3 r 2 l P A *P , * p\'r M l r,V (3.)- VI — 1, ai xl + ... = pi r t i. m — 2, at xl + ... _ PP r ii 3 . 3 . pn rp m = 3, ala x\l + ... _ p 2 i r 3 i rjt pi rp m = 4 all + ... _ P 3 i P$ rj rj ^3 xU pp p% r 3 4 rj \l ' m = 5. xW + ... _ ppp 2 l p$r b i rj r,§4 all Pz* pfi r *p r 2® (XVI.) The instances in which the general formulae have been here developed will suffice to elucidate their application to particular surfaces. At the same time it is evident that it becomes necessary, at this point, to quit a very interesting, although laborious, field of investigation ; since, to work out in more extensive detail individual illustrations, would be incon- GENERAL REMARKS. 63 aistent with the purpose of this book. It may be worth while to call attention to the peculiar symmetry of expression characterising the equations of successive planes in the ellipsoid, as exhibited in page 30; and it is obvious that, in the expanded series, the developments may be rendered in terms of the consecutive values of u , as in the original ex- pressions of page 18, in place of those which have been given in terms of p m and r m . Further, it is impossible to overlook the singular relation borne by the symbol p m , in all the surfaces, to the other connected lines. In conclusion, it will be not uninteresting to examine the law by which the value of the perpendicular^ is governed in consecutive surfaces. It is deserving of remark, as indicative of singularity, although it might possibly have been anticipated, that the equation expressing the value of this quantity, as well as the equation of the surface itself, ought to be regarded only as a particular case selected out of an infinite number of connected, symbolised, geometrical relations. If the reflection seems to be unavoidable, that the co-ordinate relation which it has been customary to call, by long-established convention, “ the general equation” of a surface, or a curve, is more properly to be regarded in this light, as exhibiting one only of the aspects under wiiich its subject may be considered; and, in truth, as but a single term in an infinite series, each implying, individually, the essential properties of the figure, an additional reason appears to be presented for limiting, at present, our expectation of geometrical or analytical perfectibility. Considerations of this nature, no doubt, must tend to confirm the impression, which is forced upon our attention from many independent sources, that a far more perfect theory than we as yet possess of symbolised geometry, may be reserved for the research of the future.* It is, then, to be desired, that those who take interest in abstract speculations upon subjects hitherto but little, if at all, examined, should not shrink from diverging out of beaten tracks of thought; should en- deavour, indeed, to fix the impression of ideas which seem to be new; to accumulate, classify, and bring them into prominence : in short, to collect and combine those floating threads of mathematical truth which, singly considered, are barely to be traced; but, collectively, may lend a visible and well-defined support to the beautiful network of science. * See, on this subject, De Morgan, Double Algebra ; and Treatise on Related Caustics, page 12, Cor. 2, by the Author. 64 DEVELOPMENT OF PERPENDICULARS. (XVII.) Development of the perpendiculars in consecutive surfaces. It will be seen by a glance at the tabulated expressions which follow, that both the numerators and denominators in the value of the perpen- dicular are determined by the simple law of a geometric series in which 2 is the common ratio. Oo II 54 j e 1; II 99 (2.) »<£) = 1 ; 1 _ - ( 9 - (3.) ■G> = 1 j 1 P % 99 - (4.) 99 = 1 ; 1 _ pi 99 - (5.) Kl4) = 1 ; 1 pH s ($>)■ (6.) ■G) = 1 ; 1 _ pH * (9 (?•) s (£) = 1 ; 1 P 2 (80 k:4) = 1 ; 1 pH 2 (—\ Vale/ (9.) ■<£) = 1 ; 1 pH v (* U \ ^ V a 18/- (100 ■(5) = 1; I _ pH s (^\ V a 20/ I., ] DEVELOPMENT OF PERPENDICULARS. 65 The same law will be observed to regulate the values of the perpen- diculars in consecutive surfaces which are expressed under a negative index; as may be seen in the following series. (O 2 (v) “ 1 ; 11 2 (?)• (2.) <0 = 1 j i _ 9 F 2 (?)■ GO 2 (0 - 1 ; 1 _ f 2 (?> (*■) ' O = 1 ; 1 9 r 2 <;; )• (5.) oo = 1 ; i __ 2) 2 2 (O- (*•) • Consistently with the hypothesis respecting m, which in its first accep- tation has been necessarily assumed to be integral, n must be also integral, in the application of the last set of formulae ; if the surface is such that n is fractional they cannot be employed, in any sense in which the proper- ties of these expressions have been hitherto considered. It may, however, be not impossible that a consistent interpretation is assignable even to fractional values of m. At a first view this may appear paradoxical ; yet there does not seem to be any sufficient reason, e.g., why such an ex- GENERAL REMARKS. 67 pression as^i, or r%, in the preceding pages, should not, in an analytical sense, be as much capable of interpretation as And we cannot but a x% recollect that results have been obtained in the functional analysis which, before they had been established, were not less beyond anticipation.* If n — 2, and eonseqently m = 1, the surface being then the ellipsoid, the index of a , as it has been before remarked, will take the m m (m + 1) m (3 + log m) + m = 3 ; m m (ra + 1) m (1 f log m) + m — 1 — 2 ; although, perhaps, it may seem hardly necessary to reproduce this evaluation, since it has been already included in page 26, under the original form of the index. * See Gregory’s General Differentiation. &c. indefinite form — : then, 0 m (m tl) m — 2 m — 1 (m + 1) (m — ] ) m — 1 CHAPTER IY. (I.) The object of this chapter is to deduce, in some instances, equations of relation between consecutive radii and perpendiculars in the ellipsoid. It is clear that these relations may exist in infinite variety, and we have to select a few only of those which appear to present the most interesting features in point of form. Taking the second of the formulae given at page 26, - S*_l = 1 2 2 2 2 2 2 ' Pm— l pm— 2 ... p Tm — J Tm—2 ... r, Now, let it be assumed that ®m /?m 1 54m ; ym — 2 Pm — 1 I 2 2 2 2 pm — 1 ... p ?’m — 1 • • • T t 1 c 4m ’ then, if we take four consecutive equations, they may be written, a m X* + /3m y 3 + ym z 3 = P 3 m — 1 . (!) a m — 1 X~ + /3m — l y~ + II C* r-4 L P 3 m — 2- (2) a m — 2 X~ + /3m— 2 y 2. + ym— 2 Z z = P 3 m — 3. (3) am — 3 $ + /3m — 3 y 3 + ym— 3 S 3 = P 3 m — 4. (4) Prom any three of these, when x 3 y 3 z 3 have been determined in terms of the other quantities, their elimination from the remaining equation will lead to a relation between the radii and tangent-perpendiculars of the surface. I EQUATIONS OF RELATION. 09 As the conclusions which follow may be verified by any one who takes an interest in the subject, it does not seem necessary to express in detail the determination of the co-ordinates. The method which appears to be the most convenient is that of indeterminate factors, keeping in view that the constants a m /3 m y m are subject to the law of indices. After the requisite reductions have been made, there will then be, from the three first equations, *2 = rVi - 03 + y)P 3 m- 2 + p y PV-a a m _2 (a — /3) (a — y) Pjn_ L __— ( a + y) P 3 m -2 + ay P-m-3 Pm- 2 (a - P) (/ 3 - y) P 3 m- t - (<* + /?) PV, + a/3 P3 m _„ ym— a (a — y) (ft — y) (II.) Of the infinite number of equations derivable from the ellipsoid which may be taken for the elimination of these co-ordinates, we select, in the first instance, that which is next in order of sequence ; equation (4), page 68 : Consequently, a m— 3 T 3 m-] — (P + y) P 3 m- 2 + fty P 3 m- 3 _ am- 2 (a ~ ft) (a — y) /3m— 3 P 2 m-! — (a -t- y) I* "ill- 2 + ay P 2 m-3 , Pm - 2 (a — ft) (ft — y) ym— 3 l J2 m_ 1 — (a + ft) P 3 m _ 2 + °ft P 3 m_ 3 _ po ym— 3 ( a -y) 08 -y) Now, since the suffixed symbols of a /3 y follow, in this notation, the law of indices, a m_ 3 a \ Pm- 3 Pm— 2 ym -3 ym -2 • 111 — 2 a ft 7 70 ELLIPSOID. Let A, =: B, = C, = + * ( a ~ft) (“-?') ft (a-- ft) (ft-y) y (a—y) (ft-y)' ft + y a + y _j_ a 3-/3 a (a — P) (a — y) (3 (a— /3) (/?— y) fty _ ay + y (a-y) (/3 — y) a/3 a (a— (3) (a — y) /I (a — /3) (ft — y) y (a — y) (/3 — y) The equation which has been deduced may then be written, A^m— , - B / PV 2 + C / P3 m _ 3 = P 3 m_ 4 . It may be shewn, after the requisite reduction, that A = — = a 4 6 4 c 4 a fty Pq = 2 j-LJ = « 4 6 4 + « 4 6- 4 + 6 4 c 4 . C y =2 j — J = ^ + c 4 - After increasing by unity the value of m, which is entirely arbitrary, the resulting equation is finally, 0 0 0 Q Q Q Pm P ill— i p~ in— 2 ^ m ? ni_i ' m— 2 — (a 4 + b l -f- c 4 ) /v 3 m 1 ?' 3 m ?’ 3 m_ 1 + (a 4 6 4 + a 4 c 4 + i 4 c 4 ) y; m 3 r m 2 — a 4 6 4 c 4 = 0. (1 .) This equation expresses the relation between any six consecutive perpendiculars and radii from p t and r t upwards ; and, as regards its con- struction, bears a curious formal analogy to the equation (13) in terms of r p r n deduced in page 4. A verification of these formulae may be found when the surface becomes a sphere, in which case all the lines are equal to the radius ; in the present case we have, under these circumstances, (r 4 — a 4 ) 3 = 0 r = a. \ EQUATIONS OF RELATION. 7 L (III.) In place of the equation (4), page 68, we now select for the elimination of the co-ordinates the next in succession; viz. — + An -4 y % + ym-4 Z 2 = P 3 m_ s . We have, then, A, = G-m- d a m— 2 c, - + B, = a 2 (p—P) ( a ~ y) ftai a —P) P—y) y 2 ( a — y) (Z 3 — y)' JL (J_ ■ 1 a P y ' a « 4 Z 4 c 4 (a 4 -f Z 4 + c 4 ). fi+y _ a + y a 4-/3 + 77 + — ) • P y ' + a 2 ( a ~ P) ( a — y) £ 2 (“— /?) iP-y) y 2 0~y) (/?— y) /S ; v a y y y (a 4 + Z 4 ) (« 4 -f- c 4 ) (Z 4 + c 4 ) Py a 7 4* a/3 a 2 («— P) ( a y) /? 3 ( a -/3) (/5-y) y 2 («— y) (0— y) 7T ( a 2 @2 + a 2 y 2 +p2 y 2 +a 2 ( 3 y + a /3 + afiy 2 ) . a 2 P 2 y 2 a 8 + Z 8 + C 8 4- a 4 z 4 + fl 4 c 4 4 - Z 4 c 4 . « 4 (a 4 4- Z 4 ) 4- Z 4 (Z 4 4- c 4 ) 4- c 4 (c 4 + « 4 ). By substituting the values of the co-efficients in the equation, A 2 P 2 „ w 4- B 2 P 2 m _ 2 + C 2 PV 3 = PV_ 6 , we obtain, after increasing the suffix m by unity, P " m P^m— i P“ m_2 P^m— 3 ?' 3 m— 1 ?’~m_ 2 ^’“m- 3 — I ft 4 (ft 4 4- Z 4 ) 4 - Z 4 (Z 4 4- c 4 ) 4 - C 4 (c 4 4 - ft 4 ) ] y? 2 in p 2 m _ J r 2 m r 2 „,_ j 4 - (ft 4 + Z 4 ) (a 4 4-c 4 ) (Z 4 4-c 4 ) yi 2 m r 2 m — a 4 Z 4 c 4 (a 4 4- Z 4 4- c 4 ) = 0. (2.) 72 ELLIPSOID. This singular equation expresses a relation which obtains between any eight consecutive radii and perpendiculars, four of each, belonging to the ellipsoid. The peculiar symmetry of expression in this, as well as in the preceding equation (1), is deserving of notice ; but, in those of a higher order which will be given subsequently, it is not so much observable : although, still, in all the equations which may be derived from any of these eliminations, the mutual connexion between the lines here subjected to examinaton, no less than their connexion with combinations among the axes of the surface, cannot but be considered as very remarkable. On reducing the surface to a sphere we find, in this case, as a formula of verification, (r 4 — a 4 ) (r 4 + 3 a 4 ) = 0 r — a. (IV.) If now we take for the eliminating equation the next in order, ctni— 5 4 " fim— 5 y* T ym-5 — T m— g j there will be obtained, after the necessary reductions, t A 3 = a 4 b 4 c 4 | a 4 (a 4 + b 4 ) 4- l 4 {b 4 4- c 4 ) + c 4 (c 4 + a 4 ) j • B 3 — a 13 b 4 + a 13 c 4 4- b 1 " c 4 + a 8 b s 4- # 8 c 8 4- b 8 c 8 + 2 a 8 b 4 c 4 4- 2« 4 b 8 c 4 4- 2a 4 b 4 c 8 4- a 4 5 13 4- « 4 c 12 4- 5 4 c 13 . C 3 = (a 4 + 6 4 4- c 4 ) (a 8 4- 6 8 4- c 8 ) 4- a 4 & 4 c 4 . Eliminating these co-efficients from the equation, A 3 P 3 m_ 1 - B 3 P 3 m- 2 + C 3 P 3 ni— 3 = P 2 m_0, and increasing the value of m by unity, there will be found as the relation between any ten consecutive radii and perpendiculars, five of each, the resulting equation of condition, EQUATIONS OP RELATION. 73 n p*. m— i p"T 11-2 P " m -3 P^ m -i r "' va - ? '"ju -3 £ (a 4, + P + c 4 ) (a 8 f- IP + c 8 ) + (P P c 4 j p z m i? 2 m- x ? ,3 m / 3 m_ ! + | a 13 5 4 + a 13 c 4 i 43 c 4 -j- a 8 -j- « 8 4 c 4 + 2 a 4 /; 8 c 4 + 2a 4 6 4 c 8 + $ 4 i 43 + fl 4 C 43 -j- P C 43 1 p 2 m ? -3 m — a 4 Z» 4 c 4 |a 4 (V 4 + 5 4 ) + 5 4 (S 4 + c 4 ) + c 4 (c 4 + ff 4 ) — 0. (3.) On reducing tlie surface to a sphere we have the formula of verification, 7-20 _ 10 a 12 ; .8 + 15 a U r i _ 6 ft 20 _ 0 ; which is equivalent to the factorial expression, (r 4 — a 4 ) 3 (r 8 + 3a 4 p 4 + 6a 8 ) = 0 : r a. (V.) It is not necessary to adduce further, in this place, individual instances of elimination similar to those which have been here developed ; although it may be worth notice, that, as they are capable of endless extension, it is not unlikely that many very curious combinations might be elicited. Instead, then, of dwelling longer, at present, upon particular cases, we shall examine the general forms of the co-efficients. The general form of the eliminating equation is, a m— k + fiin—k y~ -j- I'm - k — Pfiu-k— i J and it is evident that, as the value of k becomes greater, the complexity of the final equation, as well as the number of lines which it involves, will be increased. L 74 ELLIPSOID. Let the corresponding co-efficients be written, Ak — 2 I B k -2 ; Ck— 2 3 a m— k 1 &e. : a m— 2 a k — 2 then, and we shall have the following expressions, 1 1 Ak_ 2 = + a k— 2 t.a-A) ( a -y) Ak - 2 ( a ~A A~y) 7 k _ 2 (a-y) (A-y) 13 Jk_ 2 A+y a + y + • + A Ck-2 = a k_2 (<*-A) O-y) Ak - 2 (a- A) (A-y) yk _ 2 ( a ~y) (A-y) fty ay , aft - + «k - 2 ( a -A) ( a -y) Ac - 2 ( a ~A) (A-y) yic_2 (a-y) (ft-/) consequently, ^ „ aic-2 A - 2 (a— a k-2 yk-2 ( a ~ y) + Ac— 2 yk- 2 (A— y) a k-2 ^k-2 yk-2 (a- A ( a y) ( ft-y ) p i a k _ 2 Ac— s( a 2 A 2 ) a k_ 2 yk— 2 ( a 2 72 ) ~t“ Ac— 2 7 k— 2 (A 2 7 2 ) a k -2 Ac -3 yk -2 ( a — A ( a — y) (A— y) C- _ “k-1 _A_ 1 (a— A — a k-i yk- i ( q — y)+ A -1 7 k-i (A— y) a k-2 Ak -> yk-2 ( a A ( a -y) (A— y) In all cases, the numerators of these fractions are divisible without remainder by the binomial factors in the denominators ; this is evident, because either of the assumptions, a — ft = Q a — y = 0 ; ft — y = 0 ; will cause both of the terms to vanish : and it would not be difficult to expand them into series, according to the method afterwards EQUATIONS OF RELATION. 75 adopted in regard to similar functions. There is, however, no necessity to exhibit these quantities under such a form, since, for adaptation to particular cases, the present arrangement is sufficiently convenient. (VI ) After introducing the values of the constants which have been now determined, into the general equation, viz. : Ak-a PV-i — Bk- P 3 i tl _2 + Ck _2 P 2 m -3 — P 2 in— k— i the general equation will result, 0 O O O l 1 m— i • • • P m_k ? 'in— i ••• ? m— k Ck — 2J9 2 in — 1 p"m . — 2 ?' 2 m — 1 ?' 2 m — 2 + Bk— 2 J5 2 m— 1 r 2 m— ] — Ak— 2 = 0. (VII.) The line r, the first vectorial radius, does not appear in any of the preceding equations, after (13), page 4 ; it may be introduced by selecting for the eliminating equation, 0.0,0 o a?- + y* + z = r ~- Then, from page 69, P-m 3 - Q8 -t- y) P~m— 2 + / 3 y P 3 m - 3 a-m_2 ( a — ft) ( a — y) P 2 m-I — (a + y) P 2 m— 2 + oy P 2 ni— 3 , 2 (“ - P) — y) P"m_ i — ( a + /?) P 2 m — 2 A a fi P'ni- 3 r 2 ym-2 ( a — y) 0 s — y) 76 ELLIPSOID. (VIII.) We proceed to develop into series the co-efficients involved in the preceding elimination; the first of which will be, y a m _2 /3 m— 2 ( a /3) a m -2 ym_ 2 ( a y) + ^m -2 ym— 2 (/3 y) «m _2 /?m- 2 ym -2 (a. — /3) (a — y) (/3 — y) Let the denominator be represented, at first, by L>, and by D / D 2 wdien the fraction has been cleared of the factors (/ 3 — y) and (a. — y) ; then. L m_2 u u—1 (/ 3 m— 2 ym— 2)“ a m— 2 (/ 3 m_ i — ym_ 1 ) + / 3 ;m— 2 ym— 2 (/ 3 ~yJ j — dm — 1 (/3m — 3 + /3 in — I y + ■ • • + /3ym — 4 + ym— 3 ) — am — 2 (/3m — 2 + /5m— 3 y + ... + /3ym — 3 + ym — 2 ) + /3m — 2 ym — 2 j" Since the expansion of — contains k terms, there will be here p — y (m — 2) + (m — 1) + 1 terms, = 2 (m —1); and, since this is an even number, they may be combined in (m — 1) terms of a binominal form ; then, after two reductions, ■Vm — 2 — yy- ^ (a — y) am — 2 (/3m — 3 + /3m — 4 y + ... + /3ym — 4 + ym — I 3). /3m. — 2 (am — 2 — ym— 2) /- • = -jy ^ a m — 2 (/3m— 3 + /3m— 4 y + ... + /3y m — 4 + ym— 3) — /3m- — 2 (am — 3 «m — 4 y + ... L aym — 4 + ym — 3) j- . 1 ■ EXPANSION OF GENERAL CO-EFFICIENTS. 77 The first of the two component series involves m — 2 terms, and the second an equal number ; in all, therefore, there are 2 (jn — 2) terms, which may be incorporated into m — 2 binomials, each of which includes the factor (a — /3) : after the expan=ions have been effected we shall find, ultimately. Am — 2 = —75 < ttm — 3 /3m — 3 + am — 4 /3m — 4 (a -)- /3)y am — 2 pm — 2 ym — 2 ( + am — 5 /3m — 5 (a2 -f- a/3 + /32) y2 + This series will always end with the first term involving a 0 /3 0 , a factor which is equal to unity. The number of terms included in this co-efficient is = 1 + 2 + 3+ .., to m — 2 terms. — — (m — 1) (m — 2). 2 For the determination of the second co-efficient we have, Hm — 2 — -jj- 1 am(/3m_ 2 ym— 2 ) a m— 2 (/3m ym) + /3m— 2 ym_ 2 (fi 2 T 3 ) j” • — j)- ^ am (/3m — 3 + /3m — 4 y + ... to m — 2 terms) — am— 2 (/3m— 1 — /3 m — 2 y + ... to w terms) + /3m— 2 ym— 2 (/3 + y) 73 ELLIPSOID. In this expansion there are, in all, (in — 2) + (m + 2) terms, = 2m, and vve remove the two last from the second line, combining them with the third line ; then, Bm— 3 = Xa m (/3m— 3 + /3m— 4 y + ... to mi— 2 terms) — am — 3 (/3m — 1 -j- /3m — 3 y + . .. to in — 2 terms) — (am— 3 — /3m— 2) /3ym— 2 — (am— 3 — /3m— 2) yin— 1 — — ^ am — 3 (a3 — /32) (/3m — 3 + /3m — 4 y + ••• ) — /3ym — 3 (am — 3 — /3m — 3) — ym- — 1 (am — 3 — /3m — 2) J In this arrangement there are m binomials, and in the next expansion we divide by (a— /3), calling D2 — am — 3 /3m — 2 ym — 3 (a — y) ; 33m — 2 — j— a m — 2 (a + /3) (/3 m — 3 + /3m — 4 y + . . . ) — (am— 3 + a m — 4 /3 + . . . ) /3ym — 2 — (a m — 3 + am — 4 /3 + ... ) ym- — 1 j- Ibis expression contains 4 (in — 2) single terms, which may be incor- porated into 2 (m— 2) binomials, each involving the factor (a— y). In order to this the two last serial terms must be inverted, and we shall have, Bm — 2 — jy a IU _ x /3m — 3 +a m _ 2 /3 m _ 2 + a m _ 1 /3 m _ 4 y + a m _ 2 /3 IU _ 3 y + am — 1 /3m — 5 y3 + a m — 2 /3m — 4 y2 + ... — (/3m — 3 + a/3m — 4 + ... ) /3ym — 2 (/3m — 3 + a/3ia — 4 + ... ) ym — 1 EXPANSION OP GEN Ell AL CO-EFFICIENTS. 79 — ym — l) /3 m — 3 -j- (am — 2 — ym — 2 ) a/3m — 4y + ... to vi — 2 terms. + (am— 2 — ym — 2) /5m— 2 + (am— 3 — ym — 3) a/3m — 3 y + ... to m — 2 terms. Bi ajn- /3m— 2 ym— 2 (a m — 2 + am — 3 y + ... ) /3m — 3 + (am — 3 -f- am — I y -j- . . . ) a/3m — 4 y + ... + (a m — 3 + a m — 4 y -j- ... ) /3m — 2 + (a m — 4 + a m — 5 y + . . . ) a/3m— 3 y + ••• In each of the two sets composing this co-efficient there are (m — 2) series, the terms of which form arithmetic progressions in regard to the suffixed symbols, with the common difference (— 1) ; hence, in 1st set, sum of terms = (ju— 1) + (m— 2) + ... 2)(tn+l) „ 2nd „ ,, = (m— 2) -f (m— 3) +... = — -(m— 2)(wa— 1) 3 consequently the whole number of terms in J3 m — 2 is = m(m— 2). 80 ELLIPSOID. The third co-efficient will be, — y- «m (/3m — 2 + /3m — 3 y + • • • ) — % — 1 (/3m — 1 + /3m — 2 y + • • • ) T /3m— 1 ym— 1. This expression contains Cm— 1) -f (?« + l) terms, = 2m, which may be incorporated into m binomials ; then we find, Cm — 2 — y- •j' am — 1 /3m — 2 (a — y) + a m — 1 /3 m— 3 y (a — y) + ... — /3 m — 1 (a m — 1 — ym — l). /3m — 1 (am — 2 *T am — 3 y + ... ). The last series may be resolved into (m— 1) binomials, each containing the factor (a— /3) ; hence there is, finally, am— 2 /3m — 2 ym— 2 1 am — 2 /3m- — 2 + am — 3 /3m — 3 (a-f-/3) y + am — 4 /3m — 4 (a 2 + a/3 -f- /32) y .3 4 - j£S8 . . 12 ®} APPLICATION OF GENERAL CO-EFFICIENTS. 81 (IX.) The suffixed symbol m may be increased by unity throughout the investigation ; and, before extending their application to particular cases, it is convenient to recapitulate the expansions which have been deduced for the several co-efficients : in doing this, since the quantities which enter into the series are similarly involved, the symbols /3y, in the development of B m — 1 , may be transposed, so that we shall have, Am — 1 — — n \ dm — 2 /3 m — 2 + «m — 3 (3m — 3 (a + /3) y am — 1 pm — I ym — 1 * -f- am — 4 (3m — 4 (a2 -f- 0.(3 -j- (3-2 ) y2 + ... Bm — 1 — ~ / (dm — 1 + dm— 2 (3 + ... ) ym — 2 dm— 1 /3m— 1 ym— 1 ( + (dm — 2 + dm — 3 (3 + ... ) a/3y m — 3 + (dm — 3 + dm — 4 (3 + . . . ) a2 (32 ym — 4 + ... “h (dm — 2 ■+■ dm — 3 (3 + ■•• ) ym — 1 + (dm — 3 + dm — 4 (3 + ... ) d/?ym — 2 + (dm — 4 + am — 5 (3 + . . . ) a2 (32 ym — 3 + ••• Cm — 1 = 75— ) a m — 1 / 3 m — 1 + dm — 2 (3m — 2 (a + / 3 ) y dm — 1 pm — 1 ym — 1 L + dm— 3 /3m — 3 (a2 + a(3 + (3%) y2 + ... M 82 ELLIPSOID. In these expressions, Am— 1 contains m [m — 1) 2 Bm— 1 (nfi — 1) Cm — 1 m (m + 1) 2 terms yy Con. It is evident that the co-efficient C m — 1 might have been inferred, without an independent development, by writing m + 1 for m in the numerator of Am— l. (X.) After m has been augmented by unity in (VIL), page 75 , and a proper arrangement given to the terms, there will be found, T 2 = Am — 1 P~m — Bm — 1 P 2 m- — 1 + Cm — 1 P~m — 2. which is equivalent to the general equation, j» 2 m j» 2 m— 1 ... iV P" r z m ?’ 2 m— 1 ... rf r 2 — Cm — 1 j0 2 m m — 1 ^ 2 m ?' 2 m — 1 + Bm — 1 p ~ m ?' 2 m — Am — 1 — 0. (XI.) Some developments of these formulae will here be given ; in each of them it is to be recollected that the series will end so soon as a term appears containing a 0 , ( 3 0 , or y 0 . (1.) — Let m — 2. A y = — i — = a 4 6 4 c 4 . a/ 3 y B = _!-/(« + /?) + 7) = \ + ±- + ±- = a 4 6 4 + « 4 1* 4 4 - 6 4 c*. a fly l J a/i ay jdy c, = + (. a +P)y\ = — + ~ + — = « 4 +6 4 +c 4 a p 7 L J a p 7 APPLICATION OF GENERAL CO-EFFICIENTS. 83 The consequent relation between the six lines r r, r 2 pp t p 2 is, then, p 2 2 -pf p 2 r 2 2 r 2 r 2 — (a 4 -f Z 4 + c 4 ) j» s 2 pf r 2 2 ?q 2 + (o 4 6 4 +o 4 c 4 + 3 4 c 4 ) y? 2 2 r 2 2 — a 4 Z 4 c 4 = 0. This equation has been already obtained by an independent elimination in (1), page 70, if we there write m = 2 ; considering r 0 = r. p 0 = p. (2.) — Let m — 3. A 2 = — / a/3 -j- ay -f /3y 1 = a 4 Z 4 c 4 (ft 4 + i 4 + c 4 ) . a 2 /3 2 y 2 L J I3 2 — — — — -j (a 2 -fa/3 + /3 2 )y + (a-f /3) a/3 + (a + /3) y 2 +a/3y 1 . a 2/5 2 y 2 l J = (a 4 + 6 4 ) (a 4 -f- c 4 ) (Z 4 -f c 4 ). C 2 — — -y { a 2 /? 2 + a/3(a + /3)y + (a 2 + a/3 + /3 2 )y 2 \ . a 2 p 2 y 2 *- J = a 4 (a 4 + Z 4 ) + Z< 4 (Z) 4 + c 4 ) -f- c 4 (c 4 -j- a 4 ). Hence, as in (2), page 71, P 2 3 j » 3 2 j »/ 3 ^ 3 r 2 3 r s 2 r 2 r 2 — |« 4 (a 4 -t-5 4 J + Z 4 (& 4 + c 4 ) + c 4 (c 4 -f-a 4 )} j» 2 3 jo 3 2 r 2 s ? 2 2 + (« 4 + 5 4 ) (a 4 + c 4 ) (7/ 4 + c 4 ) y? 2 3 r 2 3 — a 4 Z 4 c 4 (a 4 + Z 4 + c 4 ) — 0. (3.) — In the developments which follow, the co-efficients will be ex- hibited, without any further reduction into factors, under the symmetrical forms which they naturally assume. 84 ELLIPSOID. Let m — 4. A 3 = ft 4 b 4 c 4 £ (ft 8 + ft 4 b 4 + b 8 ) + (ft 4 + b 4 ) c 4 + c 8 1 • B 3 = (a 13 + a 8 b 4 + a 4 b 8 + £ 13 ) c 4 + ( a 8 + a 4 b 4 + b 8 ) c 8 + (ft 4 + b 4 ) c 13 4- (ft 8 + a 4 b 4 + b 8 ) a 4 b 4 + (a 4 + b 4 ) a 4 b 4 c 4 + a 4 b 4 c 8 . C 3 = (a 13 + a 8 b 4 + a 4 b 8 + b™) + (a 8 + a 4 b 4 + b 8 ) [c 4 + (a 4 + b 4 ) c 8 + c 43 . Hence, tlie relation between ten consecutive lines is found to be, /Y& 4^2 nrQ f) 2 4)2 4*2 4*2 4*2 a * 2 a »2 __ / 4r 3 / 2 Ft Jr ' 4 7 3 7 2 7 i 7 { (ft 43 + « 8 b 4 + a 4 5 8 + £ 13 ) 4- (a 8 + uA b 4 + b 8 ) c 4 + (ft 4 + b 4 ) c 8 + c 43 } p 2 x p~ 3 r 2 A r 2 3 + | (ft 13 + ft 8 b 4 4- a 4 b 8 + 5 13 ) c 4 4- (ft 8 4- a 4 b 4 4- b 8 ) c 8 + (ft 4 + b 4 ) c 43 4- (ft 8 + a 4 b 4 + b 8 ) ft 4 b 4 + (ft 4 + b 4 ) a 4 b 4 c 4 4- « 4 b 4 c 8 | p 2 4 r 2 x — a 4 b 4 c 4 1 (a 8 4- ft 4 b 4 4 b 8 ) 4- (a 4 4- b 4 ) c 4 4- c 8 1 = 0 . APPLICATION OF GENERAL CO-EFFICIENTS. 85 In verification of this equation, the reduction to the sphere will give, 1 — 10-1-15 — 6 = 0, identically. (4.) — Let m = 5. ft 4 Z 4 c 4 | (a 12 4- ft 8 Z 4 4- a 4 Z 8 + Z 12 ) + (a 8 + « 4 Z 4 + Z 8 ) o2 a -2 A — Q ^ 5 / 6/5 ' 6 ' 5 ^ J, 5 r 6 ’ 6 5 — u * A 5 = «4 54 c 4 4 a 13 £4 -j- a 8 5 8 + eft 5 13 + 5 16- ) + (a 12 + a 8 54 + aft 2 8 + 5 12 ) ft + (a 8 -f «4 54 + 5 8 ) ft + (a* + 54) C 12 + B 5 = (« 20 + a 16 54 + fl ia 58 4. «8 312 + a 4. 516 + 520) c 4 + fate + a 18 34 + ft + ft 512 4. 416) c 8 4- (a 13 + a 8 54 + «4 58 4- 5 13 ) 6*12 4- (a 8 + «4 ^4 4. 58) C I6 + {aft + 54) C 20. 4. ( a i6 + fl 12 54 + a 8 5 8 4- « 4 6 12 + 6 16 ) «4 54 4. ( a 12 4. ft 54 4. «4 58 4- 512) a 4 ft ft + {aft + «4 ft 4- 58) ft ft ft 4- {aft + 54 ) «4 ft c 12 4- «4 54 C I6 APPLICATION OP GENERAL CO-EFFICIENTS. 87 C s = («20 + 0 18 ^4 + «12 £8 + a 8 512 + a 4 £16 + £20) + (a 16 + «!2 &4i + «8 £8 + a 4 gl2 4. fllfi) c 4 + (« 12 + a 8 -f- cfc 5 8 4- ^2) c 8 + (a 8 + «4 §4 4 §8) c 12 -f- («4 4 ^4) C 1G + C 20. In A, there are — ni (m— 1) terms 5 2 B 5 „ „ (» 8 - 1) C 5 „ „ — m(m-t-l) <0 = 15. = 35. = 21. As a verification, when the surface is reduced to a sphere, the equation becomes, 1 — 21 + 35 — 15 = 0; identically. (6.) — Let m — 7. The equation between the first 16 radii and perpendiculars is then, 4)2 2 ,» i 2 lf St , yj 2 / r \ 2 ^2 n «2 a .2 n 2 ^*2 ^*2 ^.2 /p 2 **2 P T P $ P 5 P A P 3 P 2 Pi P / 'l / 6 / 5 / A / 3 ? 2 7 i 1 f) a.2 ,ii2 a. 2 *»2 _L /n2 ^2 __ A — Q ^6 P 7 P 6 '7 '6 » ^6 P 7 ' 7 6 — 88 ELLIPSOID. In which there will be found, for the values of the co-efficients, the following expressions ; A 6 = a 4 A c 4 £ ( a 20 + a 16 A + « 13 A + a 8 Z» 13 -f- a 4 6 16 + Z 20 ) + a 16 + a 43 Z 4 + a 8 A + a 4 6 12 + A 6 ) c 4 + (a 13 + a 8 Z 4 + a 4 Z 8 + 6 43 ) c 8 + (a 8 + a 4 A + & 8 ) c 43 + (a 4 -f £ 4 ) c 16 + c*»] • B 6 = (a 24 + a 20 Z 4 + a 16 A + a 12 Z 42 + a 8 Z 46 + a 4 6 20 + 6 24 ) c 4 + (a 20 + a 16 Z» 4 + a 43 Z» 8 + a 8 & 42 + a 4 6 46 + & 20 ) c 8 + (a 16 + a 42 A -f- a 8 -f a 4 Z 43 + 5 16 ) c 42 + (a 13 + a 8 a 4 + a 4 A + £ 42 ) c 46 + (a 8 + a 4 Z 4 + 6 8 ) C 20 + (a 4 + b 4 ) c 24 + (a 2 0 + a 16 IA + a 12 b 8 + a 8 Z 42 + a 4 b 18 + ^0) ft + (a 16 + a 12 A + a 8 Z 8 + a 4 5 42 + 5 46 ) a 4 Z 4 c 4 4 (a 42 4 a 8 Z 4 4 a 4 A 4 & 42 ) a 4 A c 8 4 (a 8 4 a 4 A 4 i 8 ) a 4 6 4 c 42 + (a 4 4 5 4 ) a 4 A c 16 4 a 4 A c 20 . APPLICATION OF GENERAL CO-EFFICIENTS, 8S C G = (g 24 + a~° ft + g 16 b 8 + a 12 ft 2 + g 8 6 16 + g 4 6 20 + 5 24 ) + 0 20 + g 16 6 4 + g 12 6 8 + g 8 6 12 + ft ft 6 + 6 20 ) c 4 + (g 16 + G 12 6 4 + G 8 5 8 + G 4 i5 12 + £ 16 ) C 8 -f (G 12 + G 8 ft + G 4 b 8 + £ 12 ) C 12 + (g 8 -j- G 4 -j- b 8 ) c 16 -j" (ft- -j- ftft ft 50 + ,,24 In A 6 there are here, — m (m— 1) terms = 21. B, C, (m 2 - 1 ) „ 4 - » (m+1) tv 48. 28. The reduction to the sphere will give the identical equation, 1 - 28 + 48 - 21 0 . (6.) — Let m — 8. The equation between the first 18 radii and perpendiculars will be, ai2 nyt rrp* rr& /n2 nj 2 ^2 ^ 2 /n2 ^*2 ^«2 a»2 *2 4*2 a»2 4*2 4*2 *»2 z 7 8 1 J i r 6 r 5 r x r 3 r 2 rj r / 8 / 7 / 6 / 6 / 4 / 3'2 / / 7 C 7 /- 8 ^ 2 7 r 2 8 ?” 7 + B 7 p\ r\ - A v = 0. 90 ELLIPSOID. A 7 == « 4 b 4 e 4 £ (« 34 4« 20 d4« 16 & 8 4« 13 d 3 4« 8 d 6 4« 4 ^ 30 + ^ 34 ) 4 (« 30 + « 16 b 4 + « 43 5 8 4« 8 5 13 + « 4 5 16 4^ 20 ) c 4 + O 16 + « 13 6 4 + ft 8 b 8 + a 4 6 13 + d 6 ) c 8 4- (a 13 4 # 8 5 4 +« 4 5 8 4- d 2 ) c 13 4- (a 8 4 « 4 6 4 4 b 8 ) c 16 4- O 4 4 d) c 30 4 c 34 J- B 7 = (a 38 -4 a 24 b 4 4 « 30 £ 8 4 « 16 5 13 4 « 13 d 6 4 « 8 ^ 20 4 « 4 6 34 4 5 28 ) c 4 4 - (« 34 4 a 30 d 4 «, 16 J 8 4 « 12 fl 12 4 « 8 b 16 4 « 4 0 20 4 b ~ 4 ) c 8 4- (a 20 4 fl ic ^ 4 a i2 58 4 ft s 512 + a 4 gie + ^0) c i2 4 (« 16 4 d 3 5 4 4 a 8 6 8 4« 4 d 3 4d 6 ) c 16 4 (« 13 4 « 8 $ 4 4« 4 b 8 4 d 3 ) c 20 4 (a 8 4 a* 4 £ 8 ) c 34 4 O 4 4 5 4 ) c 38 . 4 (a 24 4 a 2 0 54 4- a 16 6 8 4 d 2 4 o 8 d 6 4 « 4 ^0 4 d 4 ) « 4 b 4 4 (a 30 4 a 16 6 4 4 a 1 2 £ 8 4 a 8 d 2 4« 4 d 6 4 d°)« 4 h 4 c 4 4 (« 16 4d 2 5 4 4« 8 & 8 4& 4 d 2 4d 6 ) a 4 £ 4 c 8 4 (« 13 4 a 8 5 4 4 a 4 b 8 4 3 : 12 ) « 4 d d 2 + (a 8 4 a 4 &> 4 «®) « 4 6 4 d« 4 (a 4 4 5 4 ) a 4 5 4 C 20 4 a 4 d c 24 . APPLICATION OF GENERAL CO-EFFICIENTS. 01 c , = (ft 28 + a 24 6 4 + a2 0 b 8 x fl 16 glS + fl 18 6™ + „8 5 20 + a 4 624 + 528) + (« 24 + « 20 6 4 + « 16 6 8 -fa 1 2 6 42 + « 8 6 16 + ffi 4 620 _j_ 624 ) c 4 + («20 + «16 64 + «12 6 8 + a 8 6 42 + 646 + 620 ) c 8 + (« 46 + a 42 6 4 4- a 8 6 8 + fl 4 612 + 6 46 ) 6‘i 2 4- (« 13 + a 8 6 4 + a 4 6 8 + 612) C 16 + (a 8 + a 4 6 4 -f 6 s ) c 20 + (« 4 + 6 4 ) c 24 + c 28 la A, there are — m[m — L) terms = 28. 2 b 7 («/ 2 - 1 ) = 03. C, m(in + 1) 36. The reduction to the sphere will, therefore, give the identical equation, 1 - 36 + 63 - 28 = 0. (7.) — Let m = 9. The equation between the first 20 radii and perpendiculars will be, 7)2 7)2 7)2 7)2 7)2 7)2 7)2 7)2 7 ) 2 7)2 )»2 7*2 7*2 7*2 7*2 t *2 7*2 7.2 7 * 2 7*2 7' 9 V 8 V 1 V 6 r 5 V 4 r 3 P 2 l J l Jf ! 9 ' 8 7 7 ' 6 ' 5 r A ' 3 7 2 7 i 7 Q 7)2 7)2 7*2 7*2 u 8 / 9 / 8 7 9 ' + B 8 / 9 r 2 0. A c = « 4 6 4 c 4 1 (« 28 + a 24 6 4 + « 20 6 8 + a 16 6 12 + a 42 6 46 + « 8 6 20 + « 4 6 24 + 6 28 ) c 4 + (« 24 + a 20 6 4 + « 16 6 8 + a 12 6 42 + a 8 6 46 + a 4 6 20 + 6 24 ) c 8 + (a 20 + a 16 6 4 + a 42 6 s + ft 8 6 42 + a 4 6“ + 6 20 ) c 12 + (« 16 + a 42 6 4 + a 8 6 8 + a 4 6 12 + 6 46 ) c 16 + (24 + «20 6 4 4- a 46 6 8 + fllS «12 + a 8 b 16 + a 4 620 + 6 24 ) « 4 6 4 c 4 + («20 + fl 43 6 4 + fll 2 58 _|_ a 8 J] 2 + 516 + 520) a 4 54 c 8 4 - (a 16 + « 12 b 4 +a 8 b 8 + a 4 b 42 + 6 43 ) a 4 6 4 c 42 4 - ( a 12 4 - «s 54 + 58 + 512 ) fl 4 c 4 c ie 4 - O 8 + a 4 b 4 4 - 6 8 ) « 4 6 4 c 30 4- (a 4 + 6 4 ) a 4 b 4 c 24 4- a 4 b 4 c 28 C s = O 32 4 - a 28 b 4 + a 24 b 8 + « 20 6 42 + a 13 6 43 + a 43 6 20 + a 8 6 24 4 - « 4 b 28 4- 6 32 ) 4 - O 28 4 - a 24 6 4 + « 20 5 8 + a 16 612 4- « 12 6 43 + a 8 b 20 4- a 4 6 24 4 - 6 28 ) c 4 4- (a 24 4- tf 20 6 4 4- a 16 b 8 4- « 42 612 4. a s 516 _j_ a 4 520 4_ 524) c 8 4_ («20 + flie 54 + fl i2 58 + a s 512 + a 4 5 4G 4 - 6 20 ) c 42 4 - (« 46 4 - « 12 6 4 4- a 8 6 8 4-« 4 6 42 + 6 43 ) c 46 4 - (a 42 4- a 8 6 4 4- a 4 b 8 4 - 6 42 ) c 20 4 - (a 8 4 - o 4 6 4 4 - 6 8 ) c 24 4 - (a 4 4- 6 4 ) c 28 4 - c 32 . CONNECTION OF GENERAL CO-EFFICIENTS. 5)3 (XII.) In concluding this part of the subject, it is to be remarked that the co-efficients A m —1, B m — 1 , C m — ], which have been employed in these expansions, are subject to three separate conditions; which may be incorporated into a single equation. On examination of (1), (XI), page 82, we observe the relation, e 4 B, = (C, - c 4 ) c 8 + A ; . A, — c 4 B ; + c 8 C ; = 4 y* + (~y consequently, eliminating by (4), (I), 1 r* 3 y 3 a 3 af l a? b 2 + A} = c A J .*. v? — aa. — bb. — cc 1; (5) From the foregoing equations we obtain, 0/2 a, — 7 u« 6 ‘ = T - (0 It lias been shewn, in page 2, that «^ 3 = pr t ; so that the connection between the axes of the co-related ellipsoids, is given also by the equations, El . b a ’ 1 J>r, . r - P r , . T 5 ' 7 ’ (?) expressions determining in magnitude the axes of the derived surface which passes through the point P y , and has the plane in contact with that point at right angles to r, the vectorial radius of the primitive. Con. 1. From the equation (5) of this section it appears that the line u is a mean proportional to each symmetrically expressed pair of the six semi-axes ; while, from (6) it is evident that the greatest and least axes in the two surfaces interchange their directions ; the least axis of the derived surface coinciding in direction with the greatest axis of the primitive, and conversely. The mean axis retains the same direction in both. Cor. 2. From (7) the following proportions are obtained ; *i_ _ p_ m A — iL. £l = JL. r, a ‘ r, b ' r, c’ o 98 RELATED ELLIPSOIDS. (TIL) Again, let an ellipsoid be drawn through the point Q, fig. (2), in which the tangent-plane of the first ellipsoid, in contact with it at P, is met by a perpendicular from the centre. Let the tangent-plane in contact with the new surface at Q cut OP, or r, at right angles ; then, if p t y p are the co-ordinates of the point Q, there will be obtained by writing n 2 in the general co-ordinates of equation (6), page 15, x — r X. mV- r , X c~x iV_ 0 c-y b°~z' Let p p p be the semi-axes of the ellipsoid, generated under this hypothesis ; then, as in the preceding case, x r P r P y_ z r ,y . 70 > P ? X consequently, ,c~ c~x V 0 <‘~U . hr b^z a p = 6 b = c p. (IV.) Let it now be supposed that ellipsoids are consecutively derived in a series from one another, in the same manner as the surface drawn through P ( was generated from the primitive ; according to this arrange- ment, the tangent-plane of the second derivative will meet y at right angles, that of the third r, and so on : so that r will be perpendicular upon the tangent planes, of the 1st, 3rd, 5th ... derived surfaces, while p is perpendicular not only to the tangent plane of the original ellipsoid, but to those also of the 2nd, 4th, 6th ... derivatives. Each surface is, therefore, considered as a primary with respect to that which is next to it in sequence ; or, rather, to all its subordinate surfaces. Let a m b m c m be the semi-axes of the nfi 1 derived surface, y m ~m the co-ordinates of contact with its tangent-plane. * RELATED ELLIPSOIDS. 99 From what precedes, the following primary relations will subsist between the axes of consecutive surfaces. consequently, a a t = 5 i, — cc r II h , h 2 — C l C 2 ■ H CC Cl © b 2 b 3 — C 2 c 3 ^m— i II O* 2 ' — a a m = b bm C C m . 0) For the co-ordinates of the several points of contact in the consecutive surfaces, taking into account only the positive sign, there will be, In these expressions it is necessary to be observed that the symbols ■r m ij m and u m are used in a sense differing from, although analogous to, that which has been assigned to them in the developments which have preceded. Hitherto they have been applied in reference to consecutive points on the same surface, but they now indicate the position of similar points on consecutive surfaces. There will then be the relation, 1 r 2 1 •* m yjC) M m O. m «/2 y m . * m m * js in b m ( 2 ) 100 RELATED ELLIPSOIDS. Now, u , u 2 ... are the circle-ordinates in consecutive surfaces, corresponding in their character with u , in the original surface. We have then, as stated above, — (“m-i ) 2 r v _ \ 2 „ __ / !( m-i — V } A m_i* — \j j ym- \ • % — I j / z m-i a m—\ i ^ra— i Consequently, the co-ordinates of the point of contact on the surface are given by the expressions, tW th — V m — 11m— i 11m— 2 • • ■ d in— i Urn— 2 11m— i Urn— 2 ^m_l ^m— 2 • ■ • b Urn— i 11m— 2 ••• u Cm— i o to c y- In these expressions it should be noticed that m may assume ah integral values, from unity ; and that u 0 , tf 0 , &c., indicate u and a, as in former cases. Each of the terms in the fractions included in x m y m z m contains m factors. For the reduction of these co-ordinates, we find, page 97, ii a 7 n b ‘ = T ; a * = K = T t * C n u; dm — 11“ m_ j i * * * u 2m 1 II rH 1 ej O* w 2m 1 C2m— -1 = u 2m a w 2(m - 1 ) 6 v 2(m— !) In the expressions for the axes of the alternate surfaces, commencing with the given ellipsoid, any integral value may be assigned to m ; and, in those which belong to the alternate surfaces beginning from the first derivative, all integral values may be taken for m, from one to infinity. Let 2 (a 0 ) and 2 () li 2( C/ ) 2 (c 0 ) =: A : C _ a t t? 3 v°~ — u 1 — - ; consequently, a the same forms being necessarily true for each series of axial lines. (VIII.) From the values assigned to the several systems of axes in (Vll), combined with the equations (3), page 101, it results that : * Note. — Assuming that — is a proper fraction, we have, v “t") 2 (J) > 1 > and, consequently, after reduction, a d (^4_ c l)2 yi z % q_ (V 1 — C 4 ) 3 iC 3 2 2 + c 6 (a 4 — 5 4 ) 3 A’ 3 if > 0 : which it is clear will be generally true, p 106 RELATED ELLIPSOIDS. 1st. If m is any even number, as 0 , 2, 4, •Till / u Y 1 / u \ m / u (tJ x ' Vm ~ w y ' Zm ~ w *’ 2nd. If m is any uneven number, as 1, 3, 5, Xrx (tHt)*- ym = (— ) U w j 2 ^, z ^ u ® ^ 2 „ \ 6 (IX.) Let the sum of the volumes of the two infinite series of ellip- soids, which have been here considered, each series having reference to alternate surfaces, be represented under the symbol 2 ; then 2 (V 0 ) = — \abc + 2 c 2 + O - (V,) = y [ a > h i c i -f* CL o b 3 C 3 *4* After these series have been represented in terms of the values of the consecutive axes, as given in pages 103 and 104, there will be found, 2 (Vo) 4 7t (the 3 { 1 + 12 ) 13 • j 4 7 r abc » 13 3 Y 2 '(V,) = 4 7T ifi 3 abc { 1 + + -} 4 TT V 6 3 abc v ti — u 6 Hence may be derived, involving the value of it, the singular expres- sion, £. (u e +v«)K = A v 0 AAA. ^ 4 2i (V 0 ) _3_ 4 V 0 2a (V,) j J 1 n 6 | 3 Y 2 5 «.l s a* 2* (V 0 ) 1 Tv6~ + 2V^i2~2Vtd8 + ... Or, TT RELATED ELLIPSOIDS. 107 When V 0 is replaced by its proper value, the symbol 7r will disappear, leaving, as a relation between the two infinite series, 2k (V,) _ v?_ (; u 6 + V e )i (V 0 ) a b c (X.) In (I), page 95, figure 2, the first derivative has been assumed to pass through the point P y , subject to the condition that its tangent plane at that point is intersected at right angles by the first vectorial radius. Let it be supposed that this method of derivation is generalised, so that ellipsoids are drawn to pass through each of the consecutive points P ; P 2 ... P m under the hypothesis that the tangent plane of the surface passing through P m is pierced at right angles by the radius r m _ 1 . The semiaxes of the first derived surface being a t b t c n it has been already shewn, in page 97, that, and, by adapting the investigation to the second surface, it will be found that, a 2 b n = Similarly, (XI.) In the general case we have, for the equation of the derived surface, 7/ 2 y~2 m . y m . * m t ~a*~ ~ i ' u m u m c m But, from page 8 (corrected), 108 RELATED ELLIPSOIDS. Hence we may find, d — b bxa — C Cm ; and, consequently, Urn _ Um-x 7 , u to - 1 I'm — — • U' m— i There is, then, the following series of relations, aa t — bb t — cc r aa 2 — bb 2 = cc 2 . ddm — bb m — CCj n» (XII.) The equations of the preceding section imply the conditions, a t _ b j _ c, d 2 ^2 ^2 dm~ i b m __ j Cm— i dm b m C m The surfaces, therefore, which are generated under this hypothesis, are all similar to each other. Further, we have, a, __ b c b t _ c b t a ’ c t a ‘ c, b shewing that the axes of the first derivative, and, consequently, of all those which follow, are reciprocally proportional to the axes of the primary. V BELATED ELLIPSOIDS. 109 (XIII.) From the preceding considerations it may be shewn that, By these expressions the values of the axes, in each of the con- secutive surfaces, are fully determined, as functions of the initial point of contact assumed upon the primary ellipsoid. (XIV.) Beverting to the expressions in page 107, we observe the relations, aa t — w 2 = pr t . n-l If the foregoing examination is extended to the second derived surface, it will be found that, uf xf 3 «n-l ufyf 3 . 4“-i 5 C 2 U 2 ~n-2 ,.u-l hut this is not necessary, since it is evident, from the nature of the case, that we may at once adopt the generalisation, r/H -1 O n — Q W~m_ i y m— i Cm — « 2 m— ! ^“V., (XVIII.) By writing m — 1 for m in the expressions deduced in pag., 20, the symbol v being corrected as in the errata, it will be found that, m— n-2 l m—1 (n-1) - 1 m—1 (n-1) (n-2) X a in = m— 1 2 2 (n— 1) — 1 , Um— l H'm— 1J a .(n— : 1) (n— 2). Ill 1 fl n(n— 1) — 1 with similar expressions for b m c m given by the appropriate substitutions. X RELATED SURFACES. 113 These values of the axes in the m th surface may be expressed more symmetrically under the following form. a m — ft bm — b m — 1 m — 1 ® m _, \ 2 ( n - 1 ) / x yc*- 1 ). (v) m — 1 (&)’ (f) y m_ i m — 1 m— 1 Cm = CzfX W* (t) n(n-i). In these formulae we have to notice, from page 20 (corrected), that, m — 1 (n-I)-l 1 ) u — 2 ^m _2 *0n— 3 U . (XIX.) Some singular relations involving the circle-ordinates u, u n ... may be here appropriately noticed. Since a m — - /)->n 2 a in— i 1 a n_1 a m ^.2 a m-i = U' ^ m— i m — i It is evident that siiniler relations are true for the remaining semi-axes But, Xm—l ym — 1 — 1 is a point upon the primary surface ; .-. 2 ( X -J^) = 1. V a n J From this it will follow that, ?< 2 m _ "m 2 I Um ,,3 _x — * m-1 r -J- y m_x + Cm o a m— i ■ 111— 2 Cm_ , o . i 2 i c m-x 2 ^ m- 2 + -— 7 } J m-2 + * m -2- a 0 C 114 RELATED SURFACES. U 2 _ — X? + — y, b + £2 c * /y A z t ’ u 2 _ — X 2 + r + z 2 . c Again, from page 16, (III), — '« 2 rn — 1 x n 1 rn — ] ; with similar expressions for y m z m . - • #m — 1 A’m «"m— 1 £ n m— 1 &c. Consequently, A'm — 1 A’ m + ym — 1 l/ui + Zm — 1 z m Hence we have, M 2 m— 1 = #m — 1 Xm. + ym— 1 ym + Zm— l Zm . z< 2 m — 2 = .Tm — 2 Xm — 1 + ym — 2 ym — 1 + Zm — 1 Zm . « 2 = X x, + yy, + Z z r In verification of these expressions for the circle-ordinates, since xyz and x l y l z l are points taken upon the lines r r n which inter- sect at 0 ; let i be the inclination of these lines, then, r r t cos. i — x x t + y y t + z z t — vr == p r r COS l p_ r r ■ ft RELATED SURFACES. 115 (XX.) In order to complete this part of the subject, the investigation commencing at (IV), page 98, as applied to the ellipsoid, should now be extended so as to include the general surface of the w th degree. In the figure (3) P P / ; P y P 2 ; ... are portions of surfaces drawn after the same law as the ellipsoids in the section to which reference has been made ; i.e. the primary surface P P ; has 0 Q, or p, for the perpendicular upon its plane of contact at any given point P : again, 0 Q /} or r, is perpendicular to the plane which touches the first derivative P ; P 2 at the point P r Regarding the consecutive derivation of these surfaces from each other, according to this hypothesis, as unlimited, we have to ascertain their mutual relations. This will involve the determination of their axes, and of the co-ordinates to those points in which the consecutive surfaces are pierced by the lines p and r. Now, the point P 3 in the surface P ; P 2 is derived from P, in precisely the same way in which P, is derived from P in the given surfaces P P ; ; and the same law of derivation applies to each of the following surfaces. Thus, if u t u 2 ti 3 ... are lines in the consecutive surfaces of a character analogous to that of w in the primary, the co-ordinates of each successive point of intersection being expressed according to this principle, there will ensue the relations, ir a" *n-l / A — h- 0,n-l. y n > X, = — " 2 x n— 1 W q — = U m ~i .-pn-l m— 1 m-i * 116 RELATED SURFACES. By the reduction of these fundamental expressions there will be, in the general case, x m — 2 2 2(n—l) 2(n — 1) % — 1 Mm — 2 Mm — 3 - n n(n — 1) n(n — 1) Cm — 1 Mm — 2 Mm — 3 ■ • • m- 2(n — 1) m u (n-1) m — 1 X n(n— 1) ( 1 ) The co-ordinates y m z m are included under the same form by the inter change of a and x with b and y , or c and z. (XXI.) The consideration which is now presented to us consists in the determination of the absolute values of x m y m z m , and of the axes M m b m c m which belong to the consecutive surfaces. In page 111, it has been already shewn that, iu the first derivative P P _£l a , c n— 1 #n-2 1. a n-l 2 n~2 c , c n~l yi-2 1. £n-l z n~2 Again, the second derivative P 2 P 3 is related to the first, P, P 2 , in the same connection which this latter surface bears to the primary. We have then, representing the tangent plane of the second derived surface at P 2 , The equations of the perpeudicular O Q, will be, £ = z > V = &t. z , consequently, x , - r n * n— 1 U 2 2 y, _ cll 2 y 2 n 1 r/ n n-1 u 2 ^2 z , 6% z 9 n_1 K A / 1 / RELATED SURFACES. 117 This relation would necessarily arise from the law of mutual interdepend- ence which has been assumed to unite the consecutive surfaces ; and, by generalizing the application of this law, there will be found the following- series of equalities. On the line 0 Q, i! £ 1 *-* y, c n y n-\ z , a a 2 11 ' 1 ' z , b 11 s 11_1 » x, n n v n— 1 ^2 ^ 2 y. n 7/ 11 X c 2 // 2 z , « 2 n n_1 ' z , x i n n— 1 a y 1 z i «4 n V* ' z t h n ~ n— 1 U A °A Again, on the line OP, X c, n xf 1 y _ c n v n-1 1 >7 1 z « ; n z b n gn-V X n 11 rv\ 11 1 O3 xXj 3 y _ c 3 n y 3 n ~ l z «3 n 23 n_1 ' z b n ~ n— 1 3 z 3 X C 5 n * 5 n-l y _ c s n .y 5 n_1 z «5 n 2 S n-1 ' z b n ~ n— 1 5 Z B • • • ... ... The tangent planes of the alternate surfaces are, evidently, all parallel. (XXII.) We have, in the next place, to consider the relations existing between the consecutive points of intersection. Since the line 0 P passes through each of the points ( xyz ), ( x 2 y 2 z 2 ), (x d y 4 z d ), ... there will be the equations, X z x 2 = x -± Z A = • • • y_ = y_2_ y± z z % z 4 ■ • • 118 BELATED SURFACES. Again, since the line 0 Q passes through the points (x t y t z), (x z y 3 z 3 ), Os V 5 Z 5 ) ••• it is plain that, x t X 3 £s. _ *7 *3 Z 5 II _ y_3_ — y_s_ _ z , Z Z *5 (XXIII.) From the expressions given in (XXI) we have, */ = (M n Oil) 11-1 0~ 2 ' = (M n (JLY 1-1 'a 2 ' ' ^ ' consequently, ’ ^ 2 C a 2 a similarly, C -^~ = 6 2 b Again, we have, and, by a similar reduction, there will be found, * J BELATED SUBFACES. 119 When the relations which have been established are regarded in their application to the series of surfaces generated according to the law which has been assumed, the following conditions, as might have been anticipated, will be found to co-exist ; viz. : It appears, therefore, that all the derived surfaces in which 0 Q, or meets at right angles the tangent planes, are similar to each other, and to the primary ; while those in which O P, or r, is perpendicular to the tan- gent planes are similar also to each other, and to the first derivative. The second set of surfaces are not, however, similar to the primitive, since. (t r ar except only in the case of the ellipsoid, as was previously noticed in page 111. (XXIV.) In the preceding section, the ratios of the axes have been established ; and it has been shewn that they are constant in alternating surfaces. It remains to ascertain the absolute values of these lines, as well as of the co-ordinates to the points in which the consecutive surfaces are intersected by the lines r or p ; the second investigation will be found to be implicitly involved in the former. 120 RELATED SURFACES, From page 116, (l), we have, x 3 uf tt 3 ( n 1 ) a , n a n ( n_1 ) #( n *) 2 But, a , = u~ x n -z rtH-l and, after the introduction of these equivalents, there will be found, -cv.. RELATED SURFACES. 121 consequently, C 2 = ' it ' b. ei II «2 = (i)V x u ' similar investigation, it will appear that, «. = (±y* V !,/ /• a s = 0)V ^ u/ «| ^ II CO r = (-)"«, v U ! '■ = (*)*'< c 3 — 0)* x 2 = —L x ; cc y 2 = •y; z n — u o — ~ # • * ; %C 4 — - t(/ 3 a , y 3 = — = — 3 3 l ’ = fk cto *2 i ^4 — 4 ^2 i ~ _ u 4 - . *4 — — »2 » which may be immediately reduced to the form. a. x, = — x ; a y< t y > X e\ a x-. y 2 b Z o = X , = — a 3 a x-, y 3 = "3 y> ° 3 * ; fl!m >Tm = — .T. a y m y- — Cm / ' VERIFICATION OF FORMULAE. 125 (XXVIII.) By combining the expressions deduced in the preceding page the following remarkable singularity may be demonstrated. Since, & l ^2 — Ct ^ Ct 2 £)• ' a t « 2 ' x a consequently, = 1. a 2 In the same way, it will appear that, 2 / A ’/ x i x 3 __ ]_ . 2 ( X ‘ Xq A ‘ 3 X x \ #y ^3 / vb ®2 ®3 ^4 / and, generally, *#3 *^3 • ■ • Cl 2 ® 3 ... y \2- |m This relation may be otherwise shown in the following manner, ar - ■■ ■ * \ n _ . a ) = (- X X ’ • • • — — p = (fi. Xq 1 • • • y V a m ‘ > a a a i t #2 dm' py ^2 — ..?m \ _|_ p/^2 ,.••• ym\- + ~ — ]„ / 'Oy » 2 ... t 'C, C 2 ... C m • (XXIX.) As a certain degree of intricacy attaches to many of the pre- vious investigations, it may be desirable to give, in conclusion, some verifications of the results which have been deduced for the general surface, by reference to the particular case of the ellipsoid; since the corresponding expressions applying to that surface have been obtained independently. 126 RELATED SURFACES, • “l/l In order to ascertain the ratio — we find from (3), page 17, the u fundamental equation, 1 _ s f ayd 11 - 1 ) ] ^m 2n 1 « m n (n+l) ) consequently, But it has been shown, in page 111, (XVII), that, x / = v? x n 1 V? x n 2 and from these relations it will be found that, Now, when the general surface is reduced to the ellipsoid, by the assumption n = 2, this expression takes the form, (— } = 2 ( a' 2 x 2 ). ' u / But, in page 102, it has been assumed that, e 4 = 2 {cfi x 3 ) ; consequently, — l — — u v In the general surface of the n th degree it has been shewn, in page 120, that, \ VERIFICATION OF FORMULAE. 127 In the ellipsoid, therefore, these formulae will be reduced to the expressions. These relations will be verified by referring to the results recorded in (VII), page 103 ; and in (VIII), page 105. Again, if we wish to verify the expressions which have been given for x 3 , there will be found, after the requisite reductions, l 2n ' u/ Consequently, if n = 2, / ( «n(n— 1) 1 =!>] -3 j T = ©• = #)*■*• U 2 u, u; , = (a) 2 (a) ' u ' ' u ' But, from page 126, — — u v consequently, and. a , = v V 7 (— j */I 0 7 ffl a result which agrees perfectly with that which was demonstrated in the case of the ellipsoid, in page 1 04. These verifications, by reference to particular instances which may be multiplied to any extent, appear to suffice in confirmation of the formulae which belong to the general surface of the n th degree. 128 RELATED SURFACES. (XXX.) It will now be shewn that the formula for « m 2n , as given in page 122, or 25 (corrected), identifies itself with the fundamental ex- pression (3), in page 17. Let m = 0, then 2 n-1 (— ^ = 1; and, after inverting the terms of the formula in page 122, we shall obtain, 2 { «n(n— 1) | ffl n(n + 1) J ’ which agrees with (3) as given at page 17. Again, if m — 1, the index of a in the denominator of the general form in page 122, will be n(n— 1) — 2 0 — + 1 ; n — 2 consequently, the expression reduces itself to, vn— 1 f 1 Un(ntl)/ « y 2n = y. x n(n—l) 2 Xn + l) U 2n M 2n(n— 1) - M Sn(n-l) ^ a?n(n— 2' 1) 2 fl n (n hi)' 2 ' a.n(n— 1) 2 a n ( n h l) But the formula at page 17 becomes, when m = 1, 1 u 2n v / x^-Y) ) _ X «a(ntl) ) ’ in which, x, v? a? n — l The substitution for x t of its equivalent value, will yield an expression identical with that which has just been deduced from the form given in (XXY). I * VERIFICATION OF FORMULAE. 129 (XXXI.) In employing the general formula of (XXV), page 122, m may be assumed to have any positive integral values, as in the last section, under the restriction that the lines are confined to a single surface, — e. g., the primary; or, otherwise, that they belong to consecutive surfaces passing through points determined upon the primary by the law of tan- gential intersection described in (III), page 16. When, however, this formula is applied to consecutive surfaces which are generated according to the hypothesis in (XX), page 115, we have to write m = 0 in the indices, at the same time replacing x y z by y l z t ..., Xm ym 2m- Then, for the surface, m (n-i) vn-l m n (n — 1) — 2 a n vu-1 J n — 2 n = i. mil (a-1) x m + 1 n(n — 1) — 2 a n — 2 ( A’ ra n (a-1) 1 v f gm 11 ^- 1 ) ) %n 2n ~ l «m n ( I1+1 ) / ’ Now, in the formula (8), page 1 7, the value of u m has been deduced for consecutive lines in the same surface ; and, in order that it may be applicable to lines in the consecutive derivatives which are now under con- sideration, a m is to be substituted for a : the expression is then, consis- tently, coincident with (1). Cor. It is evident that the results obtained from the value of u m in page 122, ought to be identical with those which issue from the general form (8), in page 28. In illustration of this, let m = 2 in that formula; the result will be, s 130 RELATED SURFACES. / s \ (n-1) 2 \ £ -t n(n— 1) — 2 ) a n — 2 ' n 1 (P, r 2 ) J1 iP ^ / ) n ( 11 ~ 1 ) 1 « ® n ?^2n( n -l) { 2 ' #n(n— 1) 2 «n(n*l) which agrees with the expression obtained in page 128 by writing m — 1. (XXXII.) If any three consecutive surfaces are considered in the series generated according to the hypothesis in (XX), page 115 ; and if n m _ i u m n m + j are the circle-ordinates connected with the three surfaces respectively; these lines are always subject to the relation, — Um — 1 tot 1. From (1), (XXXI), 1 w 2n m + 1 But, from page 121, (VC*- 1 ) X ( 8 ?i +1) ) «mtl consequently, « 4n n(n ]) m+i Bl — 1 m— 1 ^n(n + 1) 4n u ^ n (n f ] ) ' mtT m m— 1 RELATION OF CIRCLE-ORDINATES. 131 Hence, we obtain, U~m % — 1 Win + 1 . From this proposition, then, we obtain the singular property, that the circle-ordinate in any of these derived surfaces is a mean proportional to those which belong to the two surfaces immediately preceding and following in the series ; so that, when consecutive values are assigned to m, w 2 = u w 2 w 2 a = u t w 3 w 2 3 = Un U A Cor. 1. — When the surfaces are ellipsoids, we find, by employing the formulae of (VII) (VIII), pages 103 — 106, the curious series of relations following : w 2 = u t v. w 3 = w 2 t> 3 . n 4, = u 3 « 3 . n m = Um .— 1 The elimination of v from these equations will reproduce the expressions which have been determined, connecting the successive values of w m . 132 ■RELATED SURFACES. Cor. 2. — Since x m — — page 124, the value of u m may be ex- a pressed in terms of the axes of its proper surface. Thus, we find, Cor. 3. — Hence, in the general surface, + (irni) + (t fx-d This includes the relation in page 102, which has been established in- dependently in reference to the ellipsoid. # 9 . CHAPTER VI. (I.) As a question connected with the main subject of this volume* it is now proposed to determine the area of a section of the ellipsoid, formed by a plane cutting the surface in any manner whatever. In the particular case when the plane is drawn through the centre, the expression for a sectional area is well known, and will be found in most works which treat of Solid Geometry ; but the author is not aware that any solution has hitherto been given to this problem in its general sense. Since a determination of the area will lead readily to that of the volume of any segment or frustum, this investigation, independently of such elegance as may attach to the expression deduced, appears to be not altogether devoid of a practical interest : while the value of the results seems to be still further enhanced, in regarding the facility with which, as will be seen, they may be made to bear upon the solution of many ques- tions relating to the Attraction of the Solid. Let A be the area of any section, which can be shewn to be, generally, elliptic. p a perpendicular drawn from the centre to the secant plane. p t a perpendicular upon the tangent plane which is parallel to, and least remote from, the secant plane. The expression representing the area of a section will then be found to be. A 7 rabc. jpf (,P? - f) ■ in which ah c are the semiaxes of the surface. 134 ELLIPSOID. (II.) Let the equation of a plane catting the ellipsoid in any manner be, z = mx -f- ny + e . By eliminating z, between this equation and that of the surface, viz., 2 {^y) — L we obtain the equation of a vertical elliptic cylinder which includes the section ; or, which amounts to the same thing, an equa- tion to the orthographical projection representing the sectional curve upon the plane of xy. There is, then, x 2 a 2 V_ b 2 (mx + ny + e) z c 2 1 : from which will be obtained, as the equation of the elliptic projection, ( n 2 Jr -f- c 2 ) a 2 y 2 + (m 2 a 2 -J- c 2 ) 3 2 x* + 2 m n a 2 & 3 xy -f- 2m cc b~ e x -f- 2 n a 2 b~ e y + a 3 6 3 (e 2 — c 2 ) = 0. (1) (III.) Let h and k be co-ordinates to the centre of this projection (1) ; x t y t those of the curve measured from its centre : so that, x = x t + h. y — y t + k. By introducing the new co-ordinates in (1), the constant term being denoted by (h k), there will be found, (k 2 b 2 + c 2 ) (v- yf -j- (m 2 a 2 -f- c 2 ) b 2 xf + 2m n a 2 5 2 x t y t + (hk) = 0 ; (2) the equations of condition being given, in the usual manner, by assuming the co-efficients of x t and y t respectively = 0 : then, (» 2 b 2 -f- c 2 ) a 2 k + m n a 2 £ 2 h -f- a 3 J 3 e = 0. (3) (w? a 2 + c 2 ) 3 2 h -f m n a 3 5 2 k + m a~ W e = 0. (4) SECTIONAL AREA. 135 From (3) (4) the co-ordinates of the centre are ascertained in the terms following ; k ?» 2 a 2 + n 2 b 2 + c 2 m 2 a 2 -j- n 2 b 2 -f c 2 n b 2 e There is, further, the constant term, {h F) = a? £ 3 c 2 (l — nd a 3 + » 2 £ 2 + c : (9) ,)• (V.) For determining the dimensions of the principal axes, we have from (7), A 2 = | (id 5 3 -f c 2 )« 3 cos 3 0 -f (?» 2 a 2 + c 2 ) i 3 sin 3 (9 — m n a? Id sin 20 j 2A 2 = (ra 2 « 2 + c 2 ) Id + (yd S 2 + c 2 ) a 2 — £ (m 3 cd + c 2 ) S 3 — {yd Id -f- c 2 ) « 3 | cos 20 — 2m n cd Id sin 20. But, from (8), it will appear that, | (m 2 ad + c 2 ) £ 2 — {yd £ 2 + c 2 ) « 2 j cos 2 0 + 2m n ad £ 2 sin 2 0 — 2m n ad & 2 cosec 2 0 ; consequently, A 2 = -i- 1 (m 2 a 3 + c 2 ) 6 3 + (» 2 td + c 3 ) a 2 — 2m n ad Z» 2 cosec 2 0 | . Similarly, B 2 = -i- 1 {yyd cd + c 2 ) i 2 + {id 5 3 + c 2 ) e 2 -f n ad 5 2 cosec 2 0 | . If « 2 5 2 are the semi-axes of the projection, it will be represented by. V + x , ( 10 ) 2 1 ; - .. SECTIONAL AREA. 137 and, by comparing this equation with (2), we shall have, finally, | (m 2 a 2 + c 2 ) S 2 + (w 2 5 2 + c 2 ) a 2 + 2»& w a 2 & 2 cosec 2 0 j a ' (»/)* = |(m 2 a 2 + c 2 ) 5 2 -f- {rfi 5 2 + c 2 ) a 2 — 2m n a 2 & 2 cosec 2 0 j-’* (YI.) Now, if A! is the area of the projected ellipse, A' = 7r a a b 2 : in which expression, ^2 ^2 2/ 1 1 (m 2 a 2 + c 2 ) S 2 + (a 2 5 2 + c 2 ) a 2 j- ~ — 4?» 2 rfi a 4 ^ 4, cosec 2 2 $ j' • Let D be written for the denominator of this fraction; then, after eliminating 6 by means of the equation (8), we shall obtain, D = 2 a b c (m 2 a 2 + w 2 £ 2 -f- c 2 )A When the value of / has been introduced from page 136, the area of projection will be ascertained under the following expression. A' = 7 r ale m 2 a 2 -f- » 2 £ 2 + c 2 ) a m 2 a 2 -f- 'ffi + C 2 :)• CD (YII.) In order to simplify the equation (11), let x t y t z t be co- ordinates of that point in which the ellipsoid is in contact with a tangent plane drawn parallel to the secant, the equations of the two planes will be, x x , i V V, i * z ! a 2 6 2 c 2 — m x -j~ n y + e 138 ELLIPSOID. and by combining these expressions, under the condition of parallelism, there will ensue, for contact, "6° of xy, *1 = 4. Vi - + x t = 4 * 1 p / 3 1 — i ib 1 is the inclina sec i — p t sec i = Iso, p = c 2 (w 2 a 2 4- w 2 £ 2 + c 2 )i n& (;» 2 a 2 4 w 2 4- c 2 ) s mc& (m 2 » 2 + re 2 $ 2 4- c 2 )i ot 2 + w 2 + 1 \i ,m 2 « 2 + « 2 & 2 + c 2 )* ( 12 ) e cos * ; ( w 2 « 2 + m 2 £ 2 + c 2 )2 _ i£i P and, Hence, 2 - y; 2 ) 4-- (13) (YIII.) Since A' is the orthographical projection of the given elliptic section A, there is, A = A' sec i — A' • P , TT (lb C , o c)\ A = — - — (jo 2 — jp 2 ). /v 3 (14) ■ . SECTIONAL AREA. 139 Cor. 1. — If p — 0, the secant plane will pass through the centre ; let this section be called A y : the general expression then takes the well- known form, 7r a be A Cor. 2. — Consequently, A (VIII.) The relation deduced in Cor. 2 of the preceding section suggests an elegant geometrical interpretation of the formula (14), which expresses generally a sectional area of the ellipsoid. In figure 4, let 0 P, a vectorial radius to the point of contact P, cut the secant plane in k. Suppose O n to be drawn from the centre of the surface, perpen- dicular to the tangent-plane at P ; and cutting the sectional plane in the point m. Let 0 n be produced to meet in the point n t the parallel tangent plane, on the opposite side of the surface ; aud, upon n n : as diameter suppose a circle to be described, in the plane 0 P n, which contains the vectorial radius OP together with the tangent-perpendicular On. Produce k m, in the plane OP n, to meet this circle in Q. Join O Q , and assume the angle QO» = 0. From this construction we obtain, p — p i cos 0 ; 1 1 1 A 140 ELLIPSOID. Now, let A 2 be the orthographic projection of A y , the central section, upon a plane which contains the line 0 Q, at the same time that it cuts at right angles the primary plane QOP. The inclination of A, to this new plane is the angle Q O L , = A 2 consequently, A A / cos (-£■ ~ #) : A 2 cos (I- - 0 ). Finally, let A 3 be the area of A 2 , when re-projected upon the original central plane section A y ; i.e., the plane L L l in the figure 4 : then. A a 2 cos (JL - e) A.. (IX.) The considerations adduced in the preceding article lead to the singular property of the ellipsoid which is included in the following pro- position. Theorem. — The area of any section formed by a plane cutting the ellipsoid is equal to the second projection of the parallel central section. The projection being made, orthographically ; — 1st, upon a plane deter- mined in position from that of the original section : 2nd, by the re- projection of the first projected area upon the plane of the primary central section. (X.) From the formula which has been given in (14), for the area of a section of the ellipsoid, may be derived a general expression for the volume of any portion of the surface, limited by parallel planes. If Y represents the volume contained in any segment of an ellipsoid, of which the base is A, there will be ultimately the differential equation. d V = A dp . 7r abc Pi {P? ~ P 2 )dp. d Y V VOLUME OF FRUSTUM AND SEGMENT. 141 The integration of this expression will give the volume of any portion of the solid which is bounded by parallel planes ; viz. : V + c = (3 rft - /)• 0) 6 Pi Cor. 1. — When this integral has been corrected between the limits — p t and + p n there will be found, for the volume of the whole solid, the usual expression, y 4 7 rabc Cor. 2. — In the formula (1), V may be regarded as a function of 6, from the relation in (VI II) p — p t cos 6 ; then, V + C = (3 cos 6 - cos 3 6) : the limits of 6 being 7r and 0, for the whole solid. Cor. 3.— To determine the volume of any segment or frustum of the ellipsoid. Let p 2 be the distance of any plane section from the centre. The formula (1) will then give, C = (SpfPz — P s 2 ) 3p; 3 V = { 3^ 2 - (/ + pp 2 + A) | (p ~ Vz) = ( 2 ) expressing the volume of any frustum , of which the limiting planes are at distances p 2 and p from the centre. If, in (2), p — Pp we find for the segment , V 7 r abc zp? ( Pi ~ P*? (?Pi + Pa)- ( 3 ) p 2 being the perpendicular drawn to the base of the segmei. 142 ELLIPSOID. Cor. 4. — In (1) or (3) suppose p or p 2 to be perpendicular to the tangent plane of a similar concentric ellipsoid ; then - or — is constant. Pi Pi Hence is evident the well-known property that the volume is constant intercepted by a plane touching the ellipsoid, and cutting an external similar surface. (XI.) In this section we adduce examples in illustration of the formulas which have been demonstrated, in reference to the volume of an ellipsoid. Problem 1. — It is required to trisect the ellipsoid. If jd and p 2 are perpendiculars drawn from the centre upon parallel planes which limit the lateral dimensions of any frustum of the solid, p t being a perpendicular upon the tangent plane which is parallel to the former ; we have, from (X), (2) ; V/ = Tp' { 3j& ' 2 “ + pp * + p*^}(p~p*)- C 1 ) In the case under consideration the terminal segments are equal, and each equal to the frustum, which will be symmetrically related to the centre. In order to express the last condition analytically, we have to write Pi = — p ; then, 2 it abc ~3 JF (3 pi _ j,8) P ' From (X), (3), the volume of a segment, in relation to which p 2 is a perpendicular drawn from the centre to the limiting plane, is, v 2 = O/ - p 2 ) 3 (2p / +p»)i 6 Pi and this will be contiguous to the former section if we make p% = p : then, v, = (p, - P ) 2 (*/-, + i ») 3 pf ) ■ - ' PROBLEMS. 143 Equating V y and V 2 we find, 3js 3 — 9 pf p + 2pf = 0. (2) The solution of this cubic will give the value of p required. Since p = p, cos 6 , let cos 6 = v ; •\ 3 v z — 9 v + 2 = 0. On applying Sturm’s Theorem we find that there are, analytically, three real roots, two of which are excluded, as being beyond the possible limits of cos 6 ; and, by employing the trigonometrical method of solution, there will be obtained, as the only admissible value, v = 0.2261 = cos 6. p — 0.2261 p r This determines the bounding planes of the two segments, the thickness of the frustum being, consequently, 2 p — 0.4522 p r Cor. 1. — It is to be remarked as singular that the value of p is always the same fraction of p l , in whatever position the parallel trisecting planes are taken ; in this respect, therefore, the property of the sphere is retained in the general surface. Cor. 2. — If the ellipsoid is reduced to a sphere, p t is constant for all positions, and = a the radius. Let x be an abscissa measured from the centre ; then, from the usual formula Y = tt dx, there is, 2 7 - a 3 1 - '»' 3 V - - t)- 7 r (. a — (2 a + x). V, = *»•(«** - £). 144 ELLIPSOID. After equating the foregoing expressions, we shall find, 3# 3 — 9ft 2 # + 2ft 3 = 0 ; an equation which coincides with (2), (XI), obtained from the ellipsoid, and affords a complete verification of the formulae which have been esta- blished. Problem 2. — It is required to quadrisect an ellipsoid, the plane of contact between the two frusta being drawn in any direction through the centre. Let V, V 2 be the volumes of the two central blocks, V 3 V 4 those of the segments ; then, for determining the former, by writing p 2 — 0 in (X), (2), Cor. 3, we have, v, = J^f(3 = v a : o p t and, for the segments, V 3 = = V„: d Pi consequently, for the quadrisection, (3 pf - f)p = (p, - ft (2ft + p). Hence, p z — 3 pf p + pf — 0. or, cos 3 6 — 3 cos 6 1 = 0. The equation for 6 or p has three real roots, two of which only are admissible, and the solution will give, for the position of the dividing planes, p = 0.3473 iv Problem 3. — It is required to determine the volume of any Cone, of which the vertex coincides with the centre of a given ellipsoid, and which is bounded by that surface. > / ■■■'• ■ - % PROBLEMS. 145 Let V be the volume of the solid which it is the object of the ques- tion to determine ; then, V = V, + v # . Let a plane elliptic section, anywhere situated, be the base of the Cone, V, its volume. If A is the area of the elliptic base, it has been shewn in (VII), (14), that, . 7 T d 6 C y q c)\ A = -rr 0>,' - !>-)■ il Lrom the known expression for the volume of a cone, which has a plane base, v, = | A „ V ' = f^T W-&P- Again, if V 2 is the volume of the segment, contained between the elliptic base and the surface, we find from (X), (3), V. ~T P? @Pi + P)' Hence, after the reductions requisite, we find, y 2 7r abc p t — p 3 1\ Cor. 1. — If p — 0, V = ^ p ^c ? as ^ s p ou u p e= If P — — P,, V = 3 4 t r abc so that this problem affords a determination for the volume of an ellipsoid. Cor. 2. — Let V' be the volume of the semi-ellipsoid, then, V : V' :: p, — p : p, which cannot but be regarded as a singular and beautiful relation. / 146 ELLIPSOID. Problem 4. — It is required to determine the volume of the solid, which is cut from a given ellipsoid by a Cylinder, of which the axis passes through the centre in any direction. Let each of the two equal and opposite plane elliptic sections be repre- sented by A ; then, if Y ; is the volume included between these limiting sections and the lateral surface of the cylinder, Yj = 2 Ap. Let V 2 be the volume of the segments, intercepted between the two terminal elliptic planes and the surface ; y 2 = - 2 .f - a \°- (p, - P? (2 p t + p). If Y is the volume which it is required to determine, including the elliptic cylinder, together with the two equal ellipsoidal segments, V = V, + V 2 ; After the reductions requisite, it will be found that, Y 4 it ale j)f — 3 pf Con. 1. — If Y^ is the volume of the whole ellipsoid, the last formula will give the singular relation, Y : V :: - f : v f. Cou. 2. — A being the area of the elliptic base, V 4 3 — P 3 A. If p ~ 0, in this last expression, the ellipsoid is enveloped by the cylinder, then = -§■ (2A A). V 4 * •i / J I « * MASS OP A SEGMENT OR FRUSTUM. 147 consequently, as in the sphere, the ellipsoid is equal in volume to — ■ of O any circumscribing cylinder. Cor. 3. — Fiona this it appears, that the volume of a cylinder , in any 3 position , circumscribing the ellipsoid , is constant , and equal to — the volume contained by the circumscribed surface. (XII.) To determine the mass of any segment, when the density of each parallel section varies as a given fraction of its distance from the centre of the ellipsoid. If p and d M are the density and mass of an infinitessimal section A, then, from (X), page 140, d M — p dV ; ••• M + C = fW-f)pdp. (1) d Pi J Cor. — Let p, be the co-efficient of density, and suppose the density of each parallel section to vary as p n ; then we have. M + C * ahc l x ( n + 3) p , 2 — (« + !) nH vf (n + 1) (n + 3) (XIII.) The formulas which have been established in this chapter suffice to determine many questions of interest relating to the Attraction of an Ellipsoid, upon a particle situated externally or internally; several of which will be now considered. (XIY.) It is required to ascertain the attraction of a plane lamina, of very small thickness and uniform density, upon a particle anywhere situ- ated ; the attractive force being directly proportional to the mass of the attracting elementary molecules, and to their distance from the particle attracted. Let t and p be the thickness and density of the lamina. Let any point 0, figure 5, be taken as origin ; m the particle attracted; m Q, a perpendicular upon the plane. Take ^A as any elementary area at a point P ; did the attraction at d A on m in the direction m Q. 148 ELLIPSOID. The attraction at d A in the direction mV is, mass x »iP = ptdk x mV; and, after this is resolved at right angles to the plane, writing m Q = k. dV = pi d A x to Q = k pt d A ; Integrating between given limits, we have, therefore, F = k pt x A. If M is the mass of A, M == pt A ; F = 30. (1) Hence it appears that the attraction towards the lamina is constant, wherever the particle attracted may be situated in a plane parallel to the lamina. The attraction is, further, the same as if the particle were placed in the vertical passing through any point O, figure 5, and attracted by the whole mass of the lamina, supposed to be concentrated in that point. (XV.) It is required to determine the attraction of any segment of an Ellipsoid, upon a particle anywhere situated ; when the attractive force varies directly as the mass of each plane section parallel to the base, and directly as the distance of that section from the point of attraction. Let A be the elliptic area of any plane section ; p the perpendicular drawn to it from the centre ; p, the perpendicular upon a tangent plane parallel to the section. Then, A = ^ (p? - p*). Let Ji be the perpendicular drawn from the centre upon a plane which includes the particle attracted and is parallel to the section A ; k a per- pendicular from the particle upon the section : so that, k = h — P- consequently, dV - 7rfi! f c pf (]i-p) (pp-p 2 ) p dp ; F + r. - -* ah( > fo-p) P dp- (2) J ) i ■ \ * ATTRACTION. 149 (XVI.) An ellipsoid, of uniform density, attracts a particle anywhere situated ; the attractive force varying with the distance of the attracted particle from each parallel section : the position of the plane sections in regard to direction being entirely arbitrary. It is required to ascertain the attraction of a segment or frustum of the solid. Since the density is constant, we find from (2), (XV) ; F + C = { 4 h p (3 pf-f) - 3 f ®pf-f) } . (1) Cor. — W hen the integral (1) is taken between the limits — p t and + p t , it will be found that, F _ 4 7r pabc_ h ' (2) It appears, therefore, that the attraction is the same, according to this law, as though the whole solid were condensed into a particle of equal mass at its centre. (XVII.) The variable section of the surface may be supposed to move parallel to a tangent plane which includes the attracted particle ; and this can take place in an infinite variety of ways. It is required to ascertain the attraction, under these circumstances. (1.) — Let the limiting section be central; then, from (1), (XVI), F = - ZfVpf-J?)}. (1) If we consider the attraction of half the solid, the second limit of p is p t ; and, by the hypothesis, we have, also, h — p t : t-, 5 7 rpabc (2.) — From (XIV), page 147, it appears that the supposition here made is equivalent to that of placing the particle attracted upon the surface ; and we have for a particle so situated, when, 150 ELLIPSOID. Pi - a ; Greatest attraction = SirpaPbc 12 on the major axis. Pi = Mean „ = 57 Tpalfic 12 ,, mean ,, Pi = c ; Least „ = 57 rpabc 2 12 j) least • • • G : M : L :: a : b : c. (XVIII.) In the preceding sections the attraction of the semi-ellipsoid which is nearest to the particle has been considered ; it is required to ascertain the attraction of the remoter half of the solid. In order to this the integral (1), (XVI), must be corrected between the limits — p,, 0. Hence, if F 2 is the attraction of the remoter half solid, and F, that of the nearer half, L = (8i - 8J>,). F a = (8* + 3 p,). Cor. 1. — If the particle is placed anywhere on the surface, li = p, ; then, F, O 7T p a b i 12 P, F, 11 7T p ahc 12 Pi F, F 2 :: 5 11 . Cor. 2. — From the general expressions for F, and F 2 it appears that, in any half of the ellipsoid, the attraction is neutralised in a plane at the 3 distance h = — -p t from the centre; p t being the perpendicular upon a tangent plane drawn parallel to the base of the semi-ellipsoid. (XIX.) It is required to ascertain the attraction of the eighth part of an ellipsoid, bounded by the principal planes, on a particle anywhere situ- ated upon one of the principal axes. ■ . - * ■ . ATTRACTION OF AN OCTANT. 151 4 /V The mass of a section parallel to one of the limiting planes is = (p 2 — p 2 ) ; in which p t has either of the values a, b, c : Trpabc , 2 = ^ “ -P) ^ ~ /) d P- The integration from p — 0 to p — p, will give, F = vpa^c (8J _ By placing the particle, alternately, at each of the three vertices, and denoting by F a F b F c the corresponding attractive forces, we find, F a 5 rr p a 2 bc 5 7 r p ab 2 c 48 48 ts 5 7r p abc 3 c 48 ' (XX.) It is required to ascertain the resultant attraction of an ellip- soid octant upon a particle anywhere situated. Let F a Fb F c be the attractions towards the limiting planes. Let a (3 y be the co-ordinates of the particle, then, by (XIX), F a = ir P alc (8a - 3a). F b = 7r f )ahc . (8 [3 - 3 b). 48 48 T? 7T p (l b C / q q \ Fc = (87 - sc). Hence, if F represents the resultant attractive force, the three attrac- tions being imagined to operate simultaneously, F = | (8a- 3a) 2 + (S/5—35) 2 + (8y-3c) 2 j * Cor. 1. — If a/3y are all negative, F = £(8a + 3a) 2 + (8/3 + Sbf + (8y + 3c) 2 J * Hence it appears that, for constant values of a (3 y, the attractive force upon the particle is greatest when it is placed in this position ; and, generally, it has greater intensity when the particle is placed towards either of the plane faces of the solid. 152 ATTRACTION OF AN OCTANT. Cor. 2. — If a/ 3y are respectively = 0, the attracted particle U placed at the angle of the octant; in this case the resultant attractive force is, * = (<* S + i- + <$■ Cor. 8. — The attraction is zero when, 3 a 3 , 3 a — — a ; fi = — b ; y = — c; but it may be shewn that these co-ordinates indicate the centre of gravity of the octant : when, therefore, a particle is placed at that point, the attraction upon it is neutralised. Cor. 4. — Let a y fi i y t be co-ordinates of the particle, measured from the centre of gravity ; then, P = T Pf c („! i + JS“ + yff. = T (^1 W + ^ + V, 8 )t = mass of the solid x distance of the attracted 'particle from the centre of gravity . The solid, therefore, attracts with the same force as though it were condensed into a molecule at the point of neutral attraction. Cor. 5.— Let 8 be 1 th the distance of the attracted particle from the point of neutral attraction ; 9 if/ the inclinations of this line to the principal axes : then, 8 = 2i(8a — 3a) 2 , and we find, cos 6 = 8a — 3 a l cos 8/3—35 8 ‘ COS if/ = 8y — 3 c 8 These expressions determine the line of action of the resultant attraction. W. F. Mathew, Printer, 59, St, Gecrge Street, Cape Town. - - ■ \ V Date Due css Boston College Library Chestnut Hill 67, Mass. Books may be kept for two weeks unless a shorter time is specified. Two cents a day is charged for each 2-week book kept overtime; 25 cents a day for each over- night book. If you cannot find what you want, inquire at the delivery desk for assisstance.