Analpticae. 
 
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 THE UNIVERSITY - 
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 MATHENATIOS 
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 4 f —leditattones = Analpticar. 
 
 : . 
 
 
 
 By 
 WILLIAM SPOTTISWOODE, 
 
 OF BALLIOL COLLEGE, OXFORD. 
 
 LONDON: ee | eu 
 1847. : Leg Ae ae 
 

 

 
 TO THOSE 
 
 ee WHO ) 
 LOVE TO WANDER ON THE SHORE 
 
 TILL THE DAY 
 WHEN THEIR EYES SHALL BE OPENED 
 
 AND THEY SHALL SEE CLEARLY 
 “THE WORKS OF GOD | 
 
 IN THE UNFATHOMED OCEAN OF TRUTH, ; 
 
 These Papers 
 ARE INSCRIBED. 
 
 ek ; 7 
 SEND te a Se ’ > ae 
 

 
PREFACE. 
 
 Tue following Papers have been written at various periods, as 
 the subjects presented themselves to notice from time to time. 
 If leisure had been afforded, an attempt would have been made 
 to draw some of them up into a distinct treatise; but it was 
 thought that even in their present form they might interest 
 some of those who take pleasure in the pursuit of Mathematical 
 Science. Some of the papers are entirely original; some are 
 partly taken from foreign Memoires, and these chiefly from the 
 writings of M. Cauchy. 
 
 In the solution of the problems presented to notice analysis 
 has been almost invariably employed ; the comprehensiveness 
 and uniformity of that method being sufficient apology for 
 the exclusion of geometrical proofs. In the form of the 
 equations symmetry has been preserved, wherever the circum- 
 stances of the case would permit ; for although such expressions 
 are sometimes longer than the corresponding unsymmetrical 
 ones, yet being more readily committed to memory, more 
 expressive of their meaning, and to one familiar with them 
 more easily applicable, they have been thought worthy of 
 
 attention. 
 
Vi 
 
 The following theorems, on account of their frequent 
 occurrence in the following pages, are transcribed from 
 Gregory’s Solid Geometry. 
 
 Teas ee ae 
 
 i Favib-dh bia 
 each of these ratios is equal to 
 Ca ea &c.)? 
 (0? + b2 + b2 + &e.)' 
 na+ na, + na, + &e. 
 NO 10 Oe ee 
 
 
 
 
 
 and to 
 
 n, 2, N,, &c. being any quantities whatever. 
 For assuming each of the ratios equal to 7, we have 
 a=rb, a,=rb,, a,=rb,, &c. 
 Squaring and adding, | 
 Gt, bapsh &C. 920 th ber Oat OCC 
 
 whence, extracting the root and dividing, 
 
 CSO, als te) ey 
 CREE ee prey b yao: 
 Again, na=rnb, na,=rnb, n,a,=7n,b,, &e. 
 
 By addition, 
 
 (na + na, + n,a, + &c.) = r(nb + n,b, + 0,6, + &c.) 
 
 Wh na+ na, + na, + &c. LL eth 
 
 ee nb +n,b, + nb, + &e. ~~ b= b= 
 II. If we wish to determine the variables from three simultaneous 
 
 equations of the form 
 
 AE OY Nee 10 ee, es ee (1), 
 LUTE YC — Cte eae ie (2), 
 O06 Me Za, sae te. (3), 
 
 instead of eliminating first z and then y, in order to determine x, we 
 may eliminate both at one operation by the following rule: Multiply 
 
Vil 
 
 (1) by dc, —¢,6,; (2) by cb, — be,; (3) by bc, — 6c, and add: it will 
 be found that the coefficients of y and z are identically equal to zero, 
 and we have 
 
 _ d(b,c, — ¢,b,) + d,(cb, — bc,) + d,(bc,— b,c) 
 Tinae bc,— €,b,) + a,(cb,— be,) + a,(be, — b,c)’ 
 
 with similar expressions for the other variables. If d=0,d,=0, d,=0, the 
 
 
 
 equations contain only two independent variables, since we may divide 
 all by any one of them; and the condition that the equations should 
 coexist is 
 
 a(b,¢,—c,b,) +a,(cb,— bc,) +a,(bc,— b,c) =0. 
 We shall frequently refer to this process under the name of cross 
 multiplication ; and the student is recommended to make himself 
 
 familiar with the forms of the multipliers, as a ready use of the process 
 will be of great service to him. 
 
 III. The sum of three squares of the form 
 (on bayer (em az) + (bz cy)" 
 may be put i a shape which is very convenient, especially in geometrical 
 
 investigations. or if we add and substract from the preceding expres- 
 sion the three squares | 
 ine by’, Gee 
 the expression may be transformed into 
 aera. tty t 2-) (art bY ez )*, 
 ‘ ‘ « 9 9 9 OV + ) e ae Z 1 
 or (a + b?+ ca (Ger + Ge + Zz") / l = ve ae Gz) Toa? ON I 
 cdr ot Ca) (ay ey? eae) 
 
 Now, if a, 6, ¢ be taken as proportional to the direction-cosines of 
 
 some one line, and 2, y, z of another, the expression 
 
 merge eg ea ee 
 (@+B+e(e2@tyt+2) 
 is equal to the cosine of the angle between the lines: let this be 6; then 
 the sum of the squares is equal to 
 
 Came Ge a (tates Yat z*) (sin $)’. 
 
Vill 
 IV. If we obtain as the result of any process that a function of wv is 
 equal to a function of y in which y is involved in a manner similar to that 
 in which « is involved in the other, then, as there is nothing to distinguish 
 one co-ordinate from another when they are symmetrically involved, we 
 may say that each of these functions is equal to a similar function of z, 
 and this is the consequence of the general symmetry of our expressions. 
 Thus, if we have two equations 
 la + my + nz =0, 
 Vx + m’y + n'z=0, 
 and eliminate z between them, we find 
 (In' — Un) « + (mn' — m'n) y=0, 
 ay 
 
 or = ns 
 mn —-mn ln—-In’ 
 
 
 
 here the two sides of the equation are symmetrical, one with respect to 
 
 x and the other to y. We may therefore say that each is equal to 
 
 ior this being the corresponding symmetrical expression with 
 
 respect to z. 
 
 Balliol College, Jan. 1, 1847. 
 
MEDITATIONES ANALYTIC. 
 
 Symmetrical Investigations of Formule relative to Plane Triangles. 
 
 In any plane triangle we have the relations 
 
 
 
 
 
 
 
 
 
 
 
 
 
 sin A = sin B = sin C (1) 
 a b c 
 __ sin Ccos B+sin Bcos C_ sin Acos C+sin C cos A _ sin Bcos A +sin A cos B (2) 
 ~ ecosB+bcosC acosC +eccosA bcos A + acos B i 
 sin (B + C) i sin (C + A) mn sin (A + B) (3) 
 ~ ecosB + bcos C acosC +ccosA beosA + acosB 
 at sin A Pe sin B at sin C (4) 
 ~ ecosB+b6cosC acosC +ccosA bcosA +acosB 
 whence 
 Sree vec Cree cod Chae cos | A. 28 oa A) S cos B (5) 
 a a b b of c 
 whence 
 
 (Ee a ee cos B _ cosC 
 
 a c b 
 
 
 
 
 
 
 
 
 
 
 
 
 
 b a c 
 
 eae 
 ( DR: i 
 
 c b a 
 
 
 
 
 
 
 
 | 
 Raa erect cor 8 t (6) 
 | 
 
 whence, multiplying by 
 
 be, V.caser ab, 
 
2 
 
 respectively, there result 
 b? — ¢ 
 
 a 
 
 
 
 cos A = bcos Saad 
 
 
 
 ae ie 
 ; cos B= ¢eos © — a cos A 
 
 
 
 
 
 
 
 
 
 2_ 72 
 # PeosC = acon A — beosB | 
 c 
 whence 
 (b?— c*) cos *A + (c?— a?) cos 7B + (a*— 8°) cos 7C = 0 
 and 
 2 2 2 Eee 2_ #2 
 da cos A + © — cos B + 2% x cos C = 0 
 and 
 cos A cos B cos C 
 
 a(b? + c’— a’) i b(c? + a*— b*) Ma c(a* + 6? — c?) 
 
 also, since 
 sin AcosA _ sin Bcos B _ sin C cosC 
 
 a.cos A). | 4h eos Bde be cose” 
 “(osu 2AY asin 2Ber ein 2 
 
 QacosA 2bcosB 2c cosC 
 
 consequently 
 
 Bud 
 a 
 
 a? — b? 
 c 
 
 ' c—a?.. 7 
 sin 2A + ~ sin 2B + sin 2C = 0 
 
 
 
 
 
 
 
 b? 
 
 ae+e—a) (e+a—b) eat e—e) 
 also by geometry each of the ratios (10) is 
 
 
 
 
 
 
 
 il 
 ~~ Yabe 
 hence 
 cosA , cosB , cosC_ a+ +e? 
 a b 1 cies Qabe 
 ») (F,2,.2 2 272) (4 BA at 
 a cos A 408 cob B Ac cos Ce cere Cae a) a Cah ace! 
 Qabe 
 be , ca a) eye Oe 
 =i eg) ca atin aT 
 
 and similarly other formule might be obtained, 
 
 (10) 
 
 (11) 
 
 (12) 
 
 (13) 
 
 (14) 
 
 (15) 
 
 (16) 
 
Again, writing 
 b?74 2&—a?= 2 C+ &— b= p? a+ B—c?= y? 
 we find by means of (1) and (10) 
 tan A = p? tan B = v* tan C 
 also, since in every plane triangle 
 tan A + tan B + tan C = tan A tan B tanC 
 vatan Age. tat) Db 
 
 —<<_ ——— ee EL 
 
 1 l 1 1 1 1 
 ” Fe F ee 
 _ tan Atan BtanC _ 2? tanA yp? tan By” tan C 
 ThA LAE PRT he exe at 
 yang pov + Ur +p 
 
 ne be Vv 
 r® tan 3A a Hw tan °B if ¥ tan °C 
 
 
 
 a wv? + Pr? + r2u? ai wv? + vr? +? oo pe? + Pr? + 2? 
 “. tan A = p* tan B = v* tan C =o (wv? + vd? +72n7) 
 also from (10) 
 Chst Ate COa bee a COstGu aL 
 
 
 
 an bp cP abe 
 sn A sinB _ snC _ W(p?v? +0? +22?) 
 Bape c es oats cy Fre Qabe 
 and if we put 
 b+.¢ 
 a ee 
 2 
 
 we shall find by actual multiplication that 
 pev? + vr2+ A2u?= 16S(S — a) (S—3d) (S—c) 
 and the above will agree with the usual formule. 
 
 
 
 (18) 
 (19) 
 
 (20) 
 
 (21) 
 
 (25) 
 
On some Theorems relative to Sections of Surfaces of the Second 
 Order. 
 
 
 
 I.—oON THE SECTIONS OF THE CONE.* 
 
 Tue plane sections of a surface are given by combining the equation 
 to the surface 
 
 F (2yz) = 0 (1) 
 with the equation to the plane, which will be 
 e+ py + vz=0 (2) 
 or 
 Ma—a) + wly—B) + (2-7) = 0 (3) 
 
 if it passes through a point « 6 y. 
 
 Instead however of finding the nature of the section by a trans- 
 formation of coordinates, a process necessarily long and tedious, we will 
 make use of the distance of a given point from the surface, as that 
 quantity is independent of the coordinate axes. 
 
 Let the equations to a line passing through the point « 6 y be 
 
 ges 3 oie ashes. (4) 
 Then if (4) les in (3) we have the condition 
 M+ pm + un =0 (5) 
 
 Combining (4) and (5) with (1) we shall find the equation to the 
 plane section in terms of all the values of r in it. 
 
 Before proceeding, we will first determine how curves of the second 
 order, in plano, are distinguished. ‘The general equation of the second 
 order between two variables is 
 
 Ax’ + 2Bry + Cy?+ 2Dzr + 2Ey + F=0 (6) 
 and this represents an 
 
 ellipse <0) 
 
 parabola | scorn as B? — AC is | 0 
 
 hyperbola > 0 
 
 
 
 
 
 * The first part of this section will be found in Gregory’s Solid Geometry ; it has been 
 inserted in order to make the latter part intelligible. 
 
5 
 
 and if we substitute for 7 and y in terms of r from the equations 
 
 z—a_ yf _ 
 age Tata (7) 
 
 
 
 we find 
 (AP + 2Blm + Cm?)r? +..... = 0 (8) 
 and the discriminating condition is equivalent to. saying that the curve 
 is an ellipse, parabola, or hyperbola, according as the coefficient of 7”, 
 equated to zero, leads to impossible, equal, or possible values of the 
 ratio /: mor m: /. 
 We shall find this condition equally applicable in three dimensions. 
 The equation to the cone is 
 Pz? + Py? — P’? =0 (9) 
 whence 
 Ce a te ee ere. a ole at (10) 
 and the discriminating condition depends upon the nature of the 
 expression 
 
 Pi? + P’m? — P”n? = 0 (11) 
 combined with 
 Al + pm + m = 0 (12) 
 eliminating 7 between these we have 
 (Po? — P/r™2)2 + (Pe? — P"u2)m? — 2P”rplm = 0 (13) 
 
 and the discriminating condition becomes 
 P2442 — (Py? — P’n2) (Pv? — P’p?) 
 
 or 
 
 ~— PP? + P’P/22 + PP2y2 
 or 
 
 (PP? + P’Pu? — PP'r*) (14) 
 and the nature of the conic section will depend on the sign of the 
 quantity under the bracket. 
 
 Suppose the cone to be a right one on a circular base 
 a be (15) 
 and the condition becomes 
 y{PP”(r2 + w2) — Prt 
 
 and since 
 Ve weal (16) 
 we have 
 Py {P”(1 — v’) — Pr} 
 or 
 
 P24 Pi (Pop By} 
 
*, we have an 
 
 ellipse > p” 
 parabola {sont as v “| Ss Pap” 
 hyperbola 
 Now, since the equation to the cone is 
 Pe ap ye ee 0 
 we have 
 pp” 
 ‘Be 
 where COB = semivertical angle of the cone. 
 And » = cosine of 7 between a normal to the plane and the axis of 
 x =sin OED, where OED = angle between the axis of the cone, and 
 the cutting plane. 
 
 Hence we have an 
 
 tan COB = 
 
 ellipse > : 
 parabola facsoncng as sin ‘own = ne 
 hyperbola < see (COB 
 > 
 1e.as Z on =| Z COB 
 “ 
 
 
 
 II.._oN THE PRINCIPAL AXES OF SECTIONS OF SURFACES OF THE SECOND 
 ORDER.* 
 
 Ler the surface be a central one, then its equation will be 
 
 Brel yi ke eee (1) 
 
 and the equations to a line in the plane of section will be 
 Pete Gd Ne ee 5 
 l ae ea (2) 
 
 and the condition that it lies m the plane, the direction cosines of a 
 normal to which are 
 
 Xs My DY, 
 will be 
 M+ pm + vn = 0 )) 
 from (2) we find 
 emE+dr y=n+mr GG 4 nr (A) 
 
 
 
 * The following investigations were partly communicated to me by the Rev. B. Price, 
 of Pembroke College. 
 
~y 
 
 ‘, substituting in (1) 
 PE + Py? + P’S? + 2(PE + P’nm + eat 
 + (PP + P’m? + P’n?)r? = H 
 then in order to find the principal axes of the section, we must make r 
 pass through the centre, or have the two values of r equal and of 
 opposite signs; and then find 7? a maximum or minimum. Hence we 
 
 (9) 
 
 have 
 PlE + P’mn + P’nt = 0 (6) 
 
 and 
 Rai Eee big ech Sy) (7) 
 Pl + Pl? + Pn 
 Now, in order that 7 may be a maximum or minimum, it will be 
 sufficient that the differential coefficient of the denominator of this 
 fraction be put equal to zero; /, m, n, being subject to the condition 
 P4+m+n=1 (8) 
 Our object now is to eliminate /m mn from the equations (3), (7), 
 and (8). Hence differentiating as above mentioned, we have 
 
 
 
 
 
 Pidl + P’mdm + P”ndn = 0 
 Adl + udm + vdrn = 0 (9) 
 ldl + mdm + ndn = 0 
 whence, introducing two indeterminate multipliers +, +, we have 
 Pl+orx.—7l1=0 
 Pin ten — rm = 0 (10) 
 P’"n + ov — m=0 
 multiplying by / m n respectively, and adding, we find 
 PP + Pm? + PP’? = 7 
 or by (5) 
 1 Flies GPE opm + Pe’) (11) 
 or writing 
 Da Pear agra ty Co == (12) 
 
 hence (10) become 
 
whence 
 
 on om ov 
 
 —l= 3 = 9 — ’ 
 ve Te 
 
 
 
 
 
 
 
 whence, multiplying by « » » respectively, and adding, we find by 
 means of (3) 
 r2 pe yp 
 + TE =0 
 pro zs pi Q p” Q (15) 
 
 r* 7? 
 
 
 
 
 
 
 
 from this equation we may easily show that the two greatest and least 
 values of r are at right angles to each other, for let 7’ 7’ be two roots 
 of this equation, then writing down the equation for each of these values 
 
 of r, and subtracting, we find, 
 
 QQ C03) Pry Py] m= 
 CHEE CDE F-DEDis 
 
 or by (14), if 7 is different from 
 1 2 
 lL +mm,+nn = 0 (17) 
 i.e. the two lines are perpendicular. 
 Also clearing (15) of fractions we have 
 
 (7-8) (-8)w+ (m8) (Pye (P-9) (PB) roo as 
 
 whence 
 (P°P’A? + P’Py? + PP’r’)r? \ th 
 —2Q{(P + PY) (PO P)pA ep (Pie Eye er ea 
 whence, if a 6 be the principal semiaxes of the section, we have by the 
 theory of equations 
 
 
 
 
 
 ath? = Q? 
 PepPAte ee Pai PPy4 
 or 
 _ H— (Pe 4 Pat 4+ P'C 
 w= CprpTye + P”PiE + PP8)* 2) 
 and 
 2 i ia (P’ + P”)a? + (PZ a P)u? “ (Pe als P)y? (21) 
 
 
 
 PP? + P’Pp? + PP? 
 
In the case of the cone 
 ne 0 (22) 
 and one of the coefficients as P’’ is negative; hence we find real finite 
 values for a and 6 only when 
 VPS is Se PEPS + 2 P7P 
 
 i.e. only in the case of the ellipse (by a former paper). In the case of 
 the parabola they become infinite, by the same paper. In that of the 
 hyperbola they or one of them becomes imaginary. 
 
 Also the area of an elliptic section 
 
 = mab 
 Pé? + Pn? — P’C? 
 = cranes meme a pe y 1g (23) 
 /PD// jue eT eee 8S Se 
 ire pent i ae ees Eee 
 
 
 
 JII.—Ir a sPHERE BE INSCRIBED IN A CONE, AND A SECTION OF THE CONE 
 _ BE MADE BY A PLANE TOUCHING THE SPHERE, THE POINT OF CONTACT OF 
 THE PLANE AND SPHERE WILL BE A FOCUS OF THE SECTION. 
 
 Ler the equation to the sphere be 
 
 Ded sied Yorks 2am (1) 
 then the curve of contact is given by the equation 
 ax + by+cz=7 (2) 
 
 where abc are the coordinates of the vertex of the cone. Let.é 1% be 
 the current coordinates of the cone, then the equation to the generator 
 may be written 
 
 ae a Ee ee (3) 
 
 subtracting (1) from (2) we have 
 
 x(a—x) + y(b—y) + 2(e—z) = 0 (4) 
 whence, by means of (3) 
 a(E—2x) + y(n—y) + 2($—z) = 0 (5) 
 or 
 Extay+ @=7r* (6) 
 whence (3) 
 wee Oo eee Te oT (7) 
 
 
 
 @+eP+e—r £at+nbt+&e—r 
 C 
 
10 
 
 whence 
 (fa+ n+ &e—ryP= (24+ C—r\(a? +0? 4+ —7°) (8) 
 which is the equation to the cone enveloping the sphere (1). 
 Now if we take the plane of the section for that of € y 
 
 f=-—r (9) 
 hence the equation to the conic section is 
 {fa + nb + r(c—r)}? = (& +n") (a? +B? +c? —7*) (10) 
 i.e. the radius vector 
 p=V(E +7) (11) 
 
 is a rational function of the coordinates; i.e. the origin is the focus. 
 Similarly the point where the plane of section is touched by another 
 sphere on its other side will be the other focus. 
 The equation to the enveloping cone may be put in a remarkable 
 form as follows ; multiplying out we find 
 E’a® + °b? + bc? — Qr*(Ea + nb + &) 
 + 2(nfbe + SEca + Enab) 
 
 — ea + b? am Ce 7) + n? (a? +6? + c?— 7?) + C(a?+b2+ ee 7 (12) 
 — 72(a2 +B? + c2) + rt 
 or 
 (P+ C—P)E + (2 +a? =r)? + (P+ BR — 7°)? \ ds) 
 2(be&m + caf E+ abEn) + 27r?(aE +bn +cf) — r*(a? + a c) = 10 
 or 
 
 {en — bh)? (ab—e8)" 4 Means 2 rye yt bt ee (14) 
 
 r r r 
 
 
 
 a convenient form for finding the distance of any point on the cone from 
 the vertex. 
 
pd 
 
 On the Reduction of the general Equation of the Second Order. 
 
 ].—ON SURFACES OF THE SECOND ORDER. 
 
 Tuer general equation of the second order between three variables is 
 Ax? + A’y? + A”2’ + 2Byz + 2B’zr + 2B’zy + Cx + Cy + C’2+E=0 (1) 
 but since this in its present form contains so many terms, it will be worth 
 while to investigate the possibility of simplifying the expression by a 
 change of coordinates. 
 Let the direction of the coordinate axes be changed by the formule 
 emaktant+al’t y=bE+ ont zackientec’t (2) 
 if then the system be still supposed rectangular, the nine direction 
 cosines will be subject to the six conditions 
 ata + a= 1 P+ 674+57=1 C4+ceeA4+ce"7=1 (3) 
 be + De’ + Oc’ = 0 ca + ca’ + c’a"=0 ab + a’l’ + a”’b’=0 (4) 
 whence also the inverse system 
 @+R + 2=1 ateh2¢c=1 a? +2 4ce7=1 (5) 
 aa” + Ub" + cc” = 0 a”a + bYb + c’c =0 aa’ + bb’ + c’=0 (6) 
 may be easily deduced. i 
 On the introduction of the new variables £4 ¢ by means of (2), the 
 equation (1) takes the form 
 AE + Bn? + AC? + QB nt + QHB'CE + 298"En + OE + On + O"$+E=0 (7) 
 where 
 A = Aa? + A’b? + A”c? + 2Bbc+2B’ca+2B"ab, A’=...., @’=.... (8) 
 95 = Aa’a”’ + A’b/'b" + A’c'c’ + B(b’c” +.0"c’) + Bea” + c/a’) + BY’ (a’'b" +00’) 
 98’ = Aa’a + A'S") + Alec + B(b"c+be") + Bi(c’a+ca”) + B’(a"b +ab") (9) 
 45” = Aaa’ + A’bb! + A’cc’ + Bibe’ + b’c) + B(ca’ + ca) + B’(ab’ + ab) 
 @=Ca+Cb4+C%c, G/ =Ca’ + C+ C’e’, EC” =Ca” + Cb” + Cc” (10) 
 Since however the nine quantities 
 Ay Dy Cs a bata Be Ci 
 are subject to only the six relations (3) and (4), or (5) and (6), it is 
 allowable to assume three more conditions respecting them ; let those be 
 B=0 W=0 B”=0 (11) 
 Cc 2 
 
12 
 
 When these relations are satisfied the coordinate axes will be callec 
 principal axes. 
 
 It now remains to be proved that when the equations (11) are satis- 
 fied the quantities @’@’Q” are real, i.e. that the conditions (11) are 
 possible ; in order to do this, let the last two of (11) be written thus 
 
 (Aa + B’ce+ B’b)a’ + (A+ B’a+ Be)d’ + (A”ce+ Bb+4+ B’a)c’=0 \ (12) 
 (Aa+ B’e+Bb)a” + (Ab + B’a+Be)b” + (ANe + Bb +Bia)e” =0 
 whence, by symmetrical eliminations, 
 Aa+ Be+B% _ Ab + Bla + Be _ A’e + Bb + Ba 
 b’c” — bc’ ca” — cla’ ab” — ay’ 
 
 But by similar eliminations from the last two of (4) or of (6) there 
 
 result 
 
 (13) 
 
 oe See ee (14) 
 hence (13) may be written 
 Aa + Be + B”) rm A’b + B”’a + Be a A”c + Bb + Bla a (15) 
 a b c 
 with similar expressions for @’@’’. Now (15) may also be written 
 (A—A)a + B’b + Ble = 0 
 B’a + (A’—A)b + Be = 0 (16) 
 Bia + Bh + (A”—Aye = 0 
 
 whence, by cross multiplication, 
 (A—AV\A—A'(A—A”)—B(A— A) — B(A—A)—B’(A—A”)—2BB’B’=0 (17) 
 The same equation would evidently have been deduced if instead of 
 (15) we had taken the analogous system involving a’b’c’Q’, or that 
 involving a’b"c"’@"". It may thus be shown that the three roots of (17) 
 are all real; the equation (17) and the two others, which involve @’, @”’ 
 respectively, being all cubics, must each have one real root at least. 
 If however any two of these roots were equal, there would result 
 
 (A—Aja’ + BY’ + Bl’ =0 \ ag) 
 (A—Q)a” 4 B’b” 4 Be’ = 0 
 whence by means of (14) there may be deduced the system 
 
 and similarly, by the assistance of (4) or (6), the analogous systems. 
 
 mo A’—-G B B’ B A”—-QG 
 Rs i 7 a’ a p” oe Toma, (20) 
 
 
 
13 
 
 whence by means of (6) 
 
 B’(A—@) + B”(A/—@) + BB = 0 
 and similarly for the other quantities; whence may be deduced the 
 system 
 
 
 
 
 
 
 
 
 
 acai, 
 B’B 4 Ae 
 sg t Av + A — 28 = 0 t (21) 
 at A+ A’ - 2@=0 | 
 or, as they are usually written, 
 iD, wt a Ae DO i 7 ey DD'y J ’ ie 
 AS sa’ SPRAY TATA +A" 28 (22) 
 
 Consequently, unless the conditions (22) are satisfied, no two of the 
 roots are equal to one another; but whatever is a root of one of the 
 cubics is a root of the other two; hence the equation (17) has in general 
 three real unequal roots. 
 
 If none of the quantities AQ'A” vanish, it will evidently always be 
 possible to destroy the coefficients of the first powers of the variables 
 by means of removing the origin, without again changing the direction 
 of the coordinate axes. The equation (1) will then be finally reduced 
 to the form 
 
 Pé + Qr? + RS? = H (23) 
 
 It may be observed that in this form of the equation no change is 
 effected by altering the sign of any one or more of the variables £1 ¢; 
 from this property the present position of the origin is called the centre 
 of the surface. If however any one of the quantities @ @’ @’ vanish, the 
 last transformation is evidently impracticable ; the condition that this 
 may be the case will be derived from equating @ to zero in (17); there 
 results 
 
 AA’A” — AB? — A/B? — A”B” + 2BB/B” = 0 (24) 
 this is, therefore, the condition that the surface may be one without a 
 centre. 
 
 It was shown above that (22) were the conditions that (17) should 
 have two equal roots; when this is the case it is evident that the values 
 of a’'b'c! would be equal to those of abc respectively ; but as this would 
 not satisfy (6), the values must be imaginary, and their ratios equal to 
 the ratios of the values of a 6 c. 
 
14 
 
 Hence the result will be of the form 
 
 but since by (4) the axis a’b’c must be perpendicular to the plane of the 
 axes of abe and a’b’c’, the relations (25) represent that, if the axis 
 a’b’c’ were drawn in a plane perpendicular to its proper plane, it would 
 coincide with that of abc, which is the usual interpretation of the 
 symbol “—. It may also be remarked that the first two of (6) will 
 then be reduced to one equation only, and consequently the plane of 
 the axis a’’b"c’’ being determined, the direction of it in that plane will 
 be indeterminate; i.e. it may lie in any direction in that plane. The 
 surface will in this case be one of revolution, i.e. it may be generated 
 by the revolution of a plane curve round the axis whose direction is 
 determinate. 
 
 The conditions that (1) may represent a surface of revolution may 
 also be deduced in the following manner. 
 
 Adding and subtracting the three quantities 
 
 B’/B” 2 B’B 4? BB’ 
 B iB’ ay "Dy * 
 
 
 
 (1) takes the form 
 (a-* Bet + (aS )rt (4-42 
 
 (26) 
 copeyate +) + Cr+ Cy +C'z+E=0 
 
 Now, when the surface is cut by a plane perpendicular to the axis of 
 revolution a circle is formed, and a sphere may always be drawn through 
 this curve; hence, when the coordinates of the surface satisfy the 
 equation to the plane they must also satisfy the equation to a sphere. 
 The conditions that this may take place are evidently 
 
 ¥Y 
 a+o+p=0 (27) 
 and 
 BBY fe aR De 
 A‘— ore te Ce =A B’” (28) 
 
 hence (13) is the equation to a plane perpendicular to the axis of 
 revolution, and (14) are the general conditions for a surface of reyo- 
 lution. 
 
er 
 
 It remains for us to determine the position of the principal axes 
 relatively to the surface ; for this purpose let 
 
 Pere Yi Se Ley (29) 
 
 be the equations to a principal axis; the substitution of these in the 
 numerators of (15) gives 
 sae Se AYR Mca a9) 
 
 a 
 
 = Bia=2) +Aty=8) + Bem 
 
 
 
 (30) 
 
 
 
 = B(x-a) + Bly—-8) + es | 
 
 c 
 which may also be written thus 
 
 Ax + B”y+ B’z+C = B’x + A’y + Bz + C’ " B’x + By+ A”z + C” (31) 
 a b c 
 
 if «, B, y be so chosen that 
 Aa + B’B + By +C = | 
 
 
 
 B’a + A’B+ By + C= 0 
 Ba + BB+ A”y + C’=0 
 
 From these equations it is easily seen that «, 8, y are the coordinates 
 of the centre of the surface when referred to the original axes. It is 
 also observable that in the values of «, 8, y, deduced from these equations, 
 the common denominator is 
 
 AA‘A” — AB? — A/B? — A”B’? + 2BB/B/ 
 
 and consequently, if the centre of the surface be removed to an infinite 
 distance, this quantity must vanish, which in fact agrees with the con- 
 dition (24). Hence the principal axes pass through the centre of the 
 surface. Moreover, the equation (1) may be written 
 (Ax+B’y+B/z+C)x + (B’2+ A’y + Bz+C')y+(Ble+By+A’z+C”)+E=0 (33) 
 and the surface may consequently be considered as the envelope of a 
 system of a planes, the direction cosines of whose normals are subject 
 to the conditions (31). Hence when 
 
 ae ae yoy ma (34) 
 i.e. when a principal axis meets the surface, it will be a normal. 
 
 Hence there are generally in surfaces of the second order three 
 principal axes passing through the centre of the surface at right angles 
 to one another, and coinciding with the normals at the points in which 
 they meet the surface. 
 
 (32) 
 
16 
 
 I].—on PLANE CURVES OF THE SECOND ORDER. 
 
 Tue above method has the advantage of being immediately applicable 
 to the equation to plane curves of the second order. 
 In this case (1) becomes 
 Az? + A'y* + 2Bay + Cr + Cy + k=O (i) 
 the transformation of the coordinates is to be effected by means of the 
 two equations 
 x=aé+a'n y = bE +04 (2) 
 and the condition that the new axes shall be rectangular will be | 
 a@a+a?=1 P+bh?=1 
 au“+ ab! =0 } 8) 
 which, as in the case of three dimensions, may be transformed into the 
 inverse system 
 a+h=1]1 foe ta) 
 teats + bb’ cs } (4) 
 the new coefficients will be given by the equations 
 @= Aa + A’? + 2Bab 
 A’ = Aa? + Ab? + 2Ba’d’ 
 93 = Aaa’ + A’bs’ + Blab! +0'b) (9) 
 C=Ca+ Cb W=Cr + CH 
 In order then to destroy the coefficient of the product of the variables 
 we must, as before, have 
 13 = 0 (6) 
 which by means of (4) and (5) will give the relations 
 (Aa + Bb)a’ + (A’b + Ba)b’) = 0 
 aa’ + bb’ =90 I (7) 
 the result of which is 
 Aa + Bb _ A’b + Ba 
 a b 
 which may also be written in the form 
 (A—A)a + Bb= ot 
 Ba + (A’/-A) =0 
 the elimination of a and 6 from which produces the quadratic 
 (A—&)(A’—A) — B? = 0 (10) 
 
 at (8) 
 
 (9) 
 
17 
 
 The condition that the curve may not have a centre will be deducible 
 from (8) in the same way as in the case of surfaces, by equating to 
 zero the coefficient of the square of one of the new variables ; this will 
 give 
 
 Aa + Bb=0 A’‘b + Ba = 0 (11) 
 the elimination of a and 6 from which produces the relation 
 AA’ — B?=0 (12) 
 
 If the two roots of (10) were equal there would result the relations 
 A@A—-A_B EA a 
 
 
 
 (15) 
 
 b a b a’ 
 which by means of (4) gives 
 B(@—A) + B(A—A’) = 0 (14) 
 or combining this with (10) 
 Bio Bie BAS (15) 
 
 This condition may be also deduced in the same way as in the case 
 of surfaces. For, if the curve represented by (1) is a circle (which is 
 the case corresponding to surfaces of revolution), on writing (1) in the 
 form 
 
 (A—B)z? + (A’—B)y? + B(a+y)? + Cr + Cy +E=0 (16) 
 the conditions in question evidently coimcide with (15). 
 In order to determine the point through which the principal axes are 
 drawn, and the angle at which they cut the curve, let 
 x—a _ 
 sali baie a7 
 be the equation to any principal axis; the combination of this formula 
 with (8) gives 
 Arma) + BY A) _ AYA) + Bisa) Aas) 
 P 
 which may be written in the form 
 Ax + By+C _ Bx+Ay+C 
 b 
 
 a 
 
 
 
 
 
 
 
 (19) 
 if « and B be chosen so as to satisfy the equations 
 
 Aa+ BB+C=0 Ba + A’‘B+ C=0 (20) 
 i.e. if « and £ be the coordinates of the centre of the curve. 
 
 Again, writing (1) in the form 
 
 (Az+ By+C)z + (Br+A’y+C’)jy + E=0 , (21) 
 
 and the curve in question may be considered as the envelope of straight 
 D 
 
18 
 
 lines drawn perpendicular to a straight line, whose equation is (18) ; 
 consequently whenever 
 
 Kar yay (22) 
 i.e. when xy is a point on the curve, or when a principal axis meets 
 the curve, it is also perpendicular to it. 
 
 Hence in plane curves of the second order there are in general two 
 principal axes passing through the centre of the curve, at right angles 
 to one another, and coinciding with the normals at the points where 
 they meet the curve. 
 
 It will be worth while here to enter more particularly into the deter- 
 mination of the quantities a 6, a'b’, «6. For this purpose, multiplying 
 the equations (9) by 6,a respectively and subtracting, the quantity 4 
 will be eliminated, and there will result 
 
 
 
 B(?—a2) + (A—A’)ab = 0 (23) 
 or writing 
 a= sind (24) 
 and consequently instead of the first equation of (4) 
 sin 70 + cos*0 = | (25) 
 (23) may be written 
 . B(cos ?6—sin 70) + (A —A’) sin @ cos @ = 0 (26) 
 whence 
 2B 
 tan 20 = Aran (27) 
 
 if the conditions (15) are satisfied the angle § becomes indeterminate, 
 i.e. any rectangular axes are principal axes. 
 
 Again, from (20) the coordinates of the centre of the curve are easily 
 deduced ; they are 
 pling Wore i Ae g—BC— Ac 
 
 AA’ — B? A A’— B? 
 
 and the centre will be removed to an infinite distance if the condition 
 (1d) is satisfied. 
 
 From equation (10) it is evident that the nature of the roots, and 
 consequently the form of the curve, depends upon the quantity 
 
 AA’ — B? 
 
 and the curve will be called an ellipse, a parabola, or an hyperbola, 
 according as the quantity is greater, equal, or less than zero. 
 
 (28) 
 
Fo 
 
 II].—iInVESTIGATION OF THE VARIOUS KINDS OF SURFACES REPRESENTED BY 
 THE GENERAL EQUATION OF THE SECOND DEGREE. 
 
 It was shown in § I. that, whenever all the roots of the cubic, formed 
 for finding the coefficients of the squares of the variables, were finite, 
 the coefficients of the products of the variables might always be 
 destroyed ; and, moreover, that unless the condition (24) were satisfied, 
 the coefficients of the first powers of the variables might also be made 
 to vanish by a suitable choice of the origin of coordinates. When 
 all these transformations have been effected, the form of the equation 
 to the surface will be the following, 
 
 Px? + Qy? + R2=H (1) 
 or, writing for convenience, 
 Hl ae oo 
 P= 2 3 R= * (2) 
 ea ai (3) 
 
 As however some of the quantities P, Q, R may in certain cases 
 become negative, the most general form of (3) will be 
 
 eS ey (4) 
 b= *o- 
 
 This equation evidently comprises eight others; viz. 
 One, when of a’b’c’ all are positive. 
 
 Three, - - two are positive, and one negative. 
 SETIEC sm f= - - negative, and one positive. 
 One, - - all are negative. 
 
 These it will be worth while briefly to consider. 
 (1) Let a?b’c’ be all positive; the equation under consideration will 
 then be 
 
 x2 y? a 
 a a a ee (5) 
 When 7 = 0; z= 0 there results 2 = + a 
 Se UU y=eb (6) 
 z=0,7=0 =k C 
 
 consequently the surface cuts each of the three coordinate axes in two 
 points, whose distances from the origin are determined by equations (6). 
 The lengths a, b, c are called the principal axes of the surface. 
 D 2 
 
20 
 
 Again, writing (5) in ae forms 
 
 2 
 
 oe eo 
 
 l (7) 
 0) CD | 
 Gre) Hee) Mal 
 
 it is evident that, if the surface be cut by planes parallel to the three 
 coordinate planes respectively, the sections so formed will be all ellipses, 
 having their centres in the axis perpendicular to which the cutting plane 
 is drawn; it is also observable that the principal axes of the ellipses 
 are decreased as the cutting plane recedes from the centre, by quantities 
 varying directly as the square of the distance of that plane from the 
 centre. It is also evident from (7) that, when 
 
 c= bha y=0 2) 
 af SE 0 21) =o} (8) 
 Zi eee Lreats) vi 
 
 and consequently that the surface cuts the coordinate axes only in the 
 points determined by (6) or (8). 
 
 Also, 
 
 if.2)> ea, i> ee, or itz $+ ¢ 
 the curves (7) become imaginary; and consequently the surface in 
 question does not extend beyond the limits 
 rata aim aD ae (9) 
 
 this surface is called the ellipsord. 
 
 (2) Let one of the quantities a’, b’,c? be negative; the equation 
 under ate will then be one of the ae 
 
 2 22 2 2 2 
 = MEE Ge | PR eae Nee ite 
 n+ b Bi Coa i c. as Cie 
 Taking de first of these equations 
 
 When y = 0, z = 0, there results 2? + a? = a 
 
 ak (10) 
 
 wget Sls 780 
 me 0,5, 7 = 0, a are 
 
 (11) 
 
 hence the surface in question never meets the axis of x, but cuts each 
 of the axes of y and z in two points determined by the equations (11). 
 
21 
 
 Again, writing the first cae of the oe (10) in the forms 
 a) ie 2) my 
 0-1) C2) ie 
 
 a eel 
 HC) *O=5) 
 
 it is evident that, if the surface be cut by planes parallel to the plane 
 of yz, there will be formed a series of ellipses (as in the case of the 
 ellipsoid) whose magnitude is extensible ad infinitum, the smallest being 
 formed in the plane of yz. The sections made by planes parallel to the 
 plane of zx are hyperbolas, the magnitude of whose principal axes 
 decrease as the cutting plane recedes from the origin to a distance = +), 
 when they vanish, and again increase ad infinitum, as the cutting plane 
 recedes still farther from the origin. It is also observable that the 
 position of the curve traced on the cutting plane is turned through an 
 angle of 90°, as the distance of’ that plane from the origin passes through 
 either of the values + 4. Precisely similar results will be obtained by 
 considering the sections of the surface made by planes parallel to the 
 plane of zy. This surface is called the hyperboloid of one sheet. Since 
 the forms of the surfaces represented by the two remaining equations 
 of the system (10) are similar to that of the one considered, their 
 position only being altered, it will be unnecessary to examine them any 
 further. 
 
 
 
 
 
 (3) If two of the quantities a’, b’, c? be negative, the equation under 
 consideration will be one of the system 
 
 Gini Ss =, ees ee Se eo 
 ee eek ea ok eee a te is + a 1 (18) 
 
 Taking the first of these eee 
 
 acest A), ot a Oe y+h=0 
 z=0, y= 90, 2+c¢ceé=0 
 
 When y = 0, z= 0, there results x =+ a 
 fe 
 
 hence the surface in question cuts the axis of x in two points at a 
 distance of = +a from the origin; but never meets the axes of y or z. 
 
22 
 
 Again, writing the first ies of the a (13) in the forms 
 WH 
 "G) — 1) e( —- EGaD 
 MN! 
 e CD D 
 
 a 
 
 Gn" 70+) | 
 
 it is evident that, if the surface be cut by planes parallel to the plane 
 of yz, there will be formed a series of ellipses as long as the distance of 
 the cutting plane from the origin is greater than +a; the ellipse will 
 vanish when z= +a, and will become imaginary when w is less than +a. 
 The sections made by planes parallel to the other coordinate planes will 
 be hyperbolas, the magnitudes of whose principal axes are extensible ad 
 infinitum. ‘This surface is called the hyperboloid of two sheets. In this 
 case (as in that of the hyperboloid of one sheet) it will be unnecessary 
 to examine the remaining equations of the system. 
 
 (4) Ifall the quantities a’, 6’,c’ are negative, the locus of the equa- 
 tion will evidently be entirely imaginary. 
 
 If the quantity H, in (1), vanishes, the equation to the surface will 
 take the form 
 
 to 
 
 z 
 c 
 
 apigiee Ta (16) 
 
 SS. 
 to 
 
 (1) If all the quantities a’, 6°, c’ are positive, the locus of the equation 
 will be entirely imaginary. 
 
 (2) Ifone of the quantities a’, 6’, c’ be negative, and the remaining 
 two negative, the equation under consideration will be one of the 
 system 
 
 LE ee, = 0 Eh en ee a 
 Dane a’ TO wc 
 
 In all of which the three variables may vanish together, i.e. the 
 
 equations 
 
 eee y=0 Zo (18) 
 are consistent with (17). Hence the surface passes through its own 
 centre. 
 
23 
 
 Again, writing the first equation of the system (17) in the forms 
 2 2 2 a 2 
 ie os ae or = Weds ae it (19) 
 a a & b° c a 
 it is evident that if the surface be cut by planes parallel to the plane 
 of yz, there will be formed a series of ellipses, the magnitudes of whose 
 principal axes, varying directly as the square of the distance of the 
 cutting plane from the origin, may be increased ad infinitum. If the 
 surface be cut by planes parallel to the remaining coordinate planes, 
 there will be produced a series of hyperbolas, subject to the same 
 variations as the ellipses on the planes parallel to the plane of yz. 
 If the cutting plane be in the plane of yz, then 
 z 
 
 2 
 r=0and += =0 (20) 
 
 
 
 
 
 
 
 which is satisfied only by 
 =0 z=0 (21) 
 consequently the trace of the surface on the plane of yz is a point. 
 If the cutting plane be in the plane of zx, then 
 
 2 
 
 eared ne 0 (22) 
 Gay mec 
 which represents two straight lines mutually intersecting. 
 If the cutting plane be in the et of xy, then 
 zee) and % -¥=0 (23) 
 which again represents two straight lines mutually intersecting. 
 This surface is called the cone. 
 If one of the coefficients of the squares of the variables (as that of Q) 
 vanishes, the equation may be reduced to the form 
 Pz? + Qy? + Rz = 0 (24) 
 which may be written, as in aa cases, 
 
 Uhre 2 s 
 1 ea a (25) 
 This formula includes the two following ones, 
 ze a z 
 oe pe ae (26) 
 
 The sections of this class of surfaces made by planes parallel to the 
 plane of xy are evidently, in the first case ellipses, in the second hyper- 
 bolas, which vanish in the plane of xy, and the magnitudes of whose 
 
24 
 
 principal axes, varying directly as the distance of the cutting plane 
 from the origin, may be increased ad infinitum. The sections made by 
 planes parallel to the two remaining coordinate planes are parabolas, 
 the magnitude of whose parameters, varying directly as the square of 
 the distance of the cutting plane from the origin, may be increased 
 ad infinitum. 
 
 It is observable that in the second case the direction of the parabolas 
 in the planes parallel to yz and xz are opposed, while in the first case 
 they are the same. ‘The, first of these surfaces is called the elliptic 
 paraboloid, the second the hyperbolic paraboloid. | 
 
 The remaining forms which the equation to the surface can take are 
 the following, 
 
 6s 
 0 
 19 
 
 to 
 
 = eee eee (27) 
 Ee ae ee) ere oO ae a (28) 
 Lb ec aides ota re eer 
 fact on =) (29) 
 Seo OM een (30) 
 
 all of which are comprehended under the general term of cylindrical 
 surfaces. 'These, it is clear, being independent of one variable, will 
 retain the same form, whatever value be given to that variable; they 
 may consequently be considered as generated by the motion of a straight 
 line drawn parallel to the axis of that variable whose coefficients vanish 
 in the equation to the surface. 
 
25 
 
 On the Partial Differential Equations of certain Classes of 
 
 Surfaces. 
 
 ].—DEVELOPABLE SURFACES. 
 
 Ler the equation to the developable surface be 
 
 tex 0 (1) 
 and the equation to the plane of which it is the envelope 
 an + by + cz = (2) 
 
 where a, b,c are variable parameters. 
 Differentiating (1) and (2) we have 
 Udz + Vdy + Wdz = a 
 adx + bdy + cdz = 0 
 whence by means of an indeterminate multiplier 
 
 Again, on differentiating the first of (3), there results 
 Ud?x + Vd*y + Wd?z + dxdU + dydV + dzdW = 0 (5) 
 or by means of (4) 
 r(ad?x + bd?y + cd?z) + dxdU + dydV + dzdW = 0 (6) 
 but on differentiating the second of (3), there results 
 ad?x + bd*y + cd*z = 0 (7) 
 = Elence 
 dUdx + dVdy + dWdz = 0 (8) 
 
 combining which with the first of (3), by means of an mdeterminate 
 
 multiplier, we find 
 dU _ dV _dW_ (0) 
 
 or 
 BV = w'dx + vdy + u'dz (10) 
 WwW =v'de + w'dy + wdz 
 E 
 
 wU = udx + w'dy + | 
 
26 
 
 Eliminating p, dx, dy, dz, from these equations with the assistance of 
 the first of (3), we arrive at the usual result, viz. 
 U? (vw—u?) + V?(wu—v”?) + W?(uv—w’”) \ 
 
 11 
 + 2VW(v'w’ —uu’) + 2WU(w'u' — vv’) + 2UV(w'v’ —ww’) = 0 ee 
 
 
 
 II.— TUBULAR SURFACES. 
 
 Ler the equation to the tubular surface be 
 
 L=0 (1) 
 and the equation to the sphere of which it is the envelope 
 (x—a)*? + (y—b)? + (z-—c)? = (2) 
 
 where a,6,c are variable parameters. Differentiating (1) and (2) we 
 
 have 
 Udz + Vdy + Wdz = 0 
 
 (x—a)dx + (y—b)dy + (z—c)dz = 0 
 Whence by means of an indeterminate multiplier 
 
 (3) 
 
 
 
 r—a —b z—c r 
 F Udi io, Na eee 4) 
 where 
 Pe 14 V2 y= (5) 
 hence 
 U 
 
 Pdz = rdU — =—dP 
 ae = rdU P 
 
 V 
 Pdy = rdV — —dP 
 y =r P t (6) 
 
 Pde = rdW — 1 dP 
 whence eliminating 7, dz, dy, dz, we arrive at the usual result, viz. 
 P4+rP{u(V? + W?) + 0(W? + U2) + (U2 + V2)—2u’VW— 20’ WU—2w'UV} 
 + r°{U?(vw—w?) + V?(wu—v?) + W?2(uv—w’) (7) 
 + 2VW(v'w’— uu’) + 2WU(w'u’ — vv’) + 2UV(u'o’— ww’)! = 0 
 It may here be remarked, that on r becoming infinite this condition 
 coincides with that given in § I. for Developable Surfaces. 
 
 
 
27 
 
 On some Theorems relating to the Curvature of Surfaces. 
 
 J.—ON LINES OF CURVATURE. 
 
 Der. A line of curvature on any surface is a locus of a series of its 
 consecutive points, such that normals at each point shall meet the 
 normal at the consecutive one. 
 
 Let the equation to the surface be 
 
 L=f(%, 42) = 0 (1) 
 then, writing for convenience, 
 dL dL, dL 
 U= _— =o = 2) 
 dx ‘4 dy nm dx (2) 
 
 the quantities U, V, W will be proportional to the direction-cosines of a 
 normal at any point (2, y, 2). ‘The equations to this normal are 
 
 Cs A ral IE Nove Sl 
 
 Unit 1a any Wao at | (8) 
 
 where r is the distance from the point (z, y, 2) on the surface to a 
 
 pot (1, ¢) on the normal and distinct from (2, y,z). Also, the 
 
 equations to a normal at any consecutive point on the surface («+ dz, 
 y+dy, 2+ dz) and meeting the former normal in the point (£, 1, ¢) are 
 
 Mere Ctl Cee a ts dite ae 2. eG 
 
 U+dG V+d¥V W+dW 
 
 whence, multiplying up and subtracting (3) from (4), there result 
 
 
 
 =r-+dr (4) 
 
 dy = Vdr + (r+dr)dV 
 
 dz = Wdr + (r+dr)dW 
 whence, eliminating 7 and dr by cross-multiplication, there results 
 (VaW—WaV)dx + (WdU—UdW)dy + (UdV—VdU dz = 0 (6) 
 which, being independent of £, », %, will hold good for any point on the 
 
 surface. Hence (6) is the differential equation of a surface, by the 
 E 2 
 
 dx = Udr + wrsana | 
 (5) 
 
28 
 
 intersection of which with (1) the lines of curvature are formed. It is 
 observable that since 
 dU AW ede ?L ’L d aL 7 
 
 di i 
 
 a ay), et -—_ d= d. ee es 
 DOT Tet aid Ge ae + yds ede 
 
 dV dV dV aL aL Ll 
 
 V= —_ aE —- =) ae (E sree " 
 
 dV = a gpd as aa ed cae (7) 
 re dW dW a? ?L aL 
 
 1W — ah / el eee], Se cee dL: eae eee | 
 
 : dx is dy Ye dz dxdz *Y ayde Has dz?" 
 
 the equation (6) will be of the second order in the differentials dz, dy, 
 dz; and consequently will in general give two values for the ratios 
 dx: dy:dz; in other words, will represent two distinct lines traced upon 
 the surface (1). 
 Again, suppose the equation (6) to have been integrated, and that 
 the result of the integration is 
 BS (8) 
 the differential of which is 
 dc + Wdy + dz = 0 (9) 
 where @, G, G1 are quantities analogous to U,V, W. Combining this 
 with (6) there results the following system, 
 VdW — WdV _ WdU — UdW _ UdV — VdU 
 
 
 
 (10) 
 
 at oD cet 
 hence, using the factors U, V, W (as in Theorem I. Pref.) there follows 
 UG + VP + We =o (11) 
 
 whence the following theorem may be enunciated ; ¢f two surfaces cut one 
 another in their lines of curvature, they cut one another at right angles. 
 Again, let three surfaces whose equations are 
 L= Z=0 Pi . (12) 
 cut one another at right angles; the conditions that this may be the 
 case are 
 
 QuU+0V+Maw=0 
 UU+VV+WW=0 (13) 
 U@ + V0 + WU =0 
 where U, V, W are quantities analogous to U, V, W, or &, ©, Ga. 
 But on differentiating (12) we have 
 Udx + Vdy + Wdz =0 
 Clie + Fay + uo} (14) 
 Udx + Vdy + Wdz =0 
 
 
 
29 
 
 from the first and second of which there result 
 dx i dy s dz (15) 
 VEi—-DPW WRA-GIU UY-Gv 
 and using the factors 
 VdW—WdV, WdU—UdW, UdV—VdU 
 
 the numerator of the result will be 
 
 (VdW — WdV )dx + (WdU—UdW)dy + (UdV—VdU)dz (16) 
 and the denominator 
 (U? + V?+ W?)(@idV + Pav + Wadw) (17) 
 Hence, writing for convenience 
 U?+ V?+ W? = P? (18) 
 
 each of the ratios (15) is equal to 
 
 (VaW— Wav) os + (WaU— aw) + (UdV—VaU) e 
 . (19) 
 GAdU + DV dvV + GW 
 Again, using the factors 
 
 Dd-TdY, 3 =9§dA- Aa, = Ad - Pda 
 each of the ratios (15) would be equal to 
 
 (aw —- way) = + Marta”, + May —yay = 
 pe 3p AP" P (20) 
 Ud@i + VdV + Wddet 
 But differentiating the first of (13) we find 
 Ud&t + VdV + Wd@@i + CAdU + DadV + GAdW =0 (21) 
 
 Hence, writing 
 VdW — WdV = P*dA WdU — UdW = P?dB Udi Vall = P'dC- 
 VdtA—TGAdV=P-da Gud W— Ud GA = 3p2d 38 adv —DdG=Pd€ | (22) 
 VdW — WdV = P°dA WdU — UdW= P?°dB UdV — VdU= P*dC 
 and forming quantities similar to (19) and (20) for U, V, W, we should 
 have the system 
 (d&+dA)dx + (d%3+dB)dy + (d€+dC)dz = 0 
 (d4+dA)dx + (dB+dB)dy + (dC+dC)dz = 0 | 
 (dA+dA)dzx + (dB+d3B)dy + (dC+d€)dz = 0 
 
 (23) 
 
 or 
 dAdx + dBdy + dCdz =0 } 
 
 dAdx + diBdy + d&dz = ey (24) 
 
 dAdz + dBdy + dCdz = 0 
 which are the equations to the lines of curvature on the surfaces L, £, L 
 respectively ; hence if three surfaces cut one another at right angles 
 
30 
 
 the lines of intersection of any one of the surfaces with the other two 
 are its lines of curvature ; which is Dupin’s Theorem. 
 
 From this it also follows that the two lines of curvature on any surface 
 are perpendicular to one another. 
 
 There is one other remarkable property of lines of curvature which it 
 will be worth while to investigate before proceeding further. 
 
 Writing for convenience 
 
 H =u + yu (Uw —Vo'— Woo! | 
 
 V if / / : 
 K =o + gy (Vo —Ww'— Uw) t (25) 
 WwW , LENT. 
 L=w + tv (We Uw — Vv’) | 
 there results 
 si fs H? + Km? + Ln? (26) 3 
 Pp 
 where , is the radius of curvature of the surface, and 
 _ de _ dy _. dz Og 
 is ds Can ds im ds Sh 
 also from the equation to the surface 
 Ul + Vm + Wn = 0 (28) 
 
 Now in order to find the variations of p corresponding to the variations 
 of 1, m,n, we have from (26) and (28) 
 
 Hidl + Kmdm + Lndn = sa(= ) (29) 
 Zea We, : 
 
 and 
 Udl + Vdm + Wdn = 0 (30) 
 
 but 
 P+mtn=l (81) 
 
 hence 
 ldl + mdm + ndn = 0 (32) 
 But on a line of curvature the relation 
 Hl(Wm— Vn) + Km(Un— Wi) + La(Vi—Um) = 0 (33) 
 is satisfied, which gives by cross-multiplication from (29), (30), and (32), 
 P 
 
 d{—~)=0 34 
 () (34) 
 
 Hence on a line of curvature the radius of curvature is a maximum 
 or minimum : or, the lines of curvature pass through the principal sections 
 of the surface. 
 
31 
 
 IJ.—vmBILIct. 
 
 Der. An umbilicus is a point on a surface round which (for all 
 points situated indefinitely near the point in question) the curvature is 
 the same in every direction. The directions of the lines of curvature 
 will consequently become indeterminate at an umbilicus. 
 
 In order to find the conditions for the existence of such a point, 
 we have 
 VdW — WdV = (Vo!— Ww’)dzx + (Vu' —Wv)dy + (Vw— Ww')dz (1) 
 also 
 Udx + Vdy + Wdz =0 (2) 
 therefore eliminating dx 
 VaW — WdV =— WKdy + VLdz (3) 
 Modifying the other terms of (6) of § I. in the same way, and arranging 
 the result, the transformed equation becomes 
 U(K—L)dydz + V(L—H)dzdz + W(H—K)dzrdy = 0 (4) 
 hence if 
 HoK=S 1 (5) 
 the equation to the lines of curvature is satisfied independently of 
 dx, dy, dz. But since when either U, V, or W vanish these con- 
 ditions become indeterminate, it will be necessary to make some 
 transformations, in order to put them into a determinate form. 
 Suppose then that U=0. Multiplying (5) throughout by U V W, 
 and substituting for H, K, L, from (25) of $I., we see that when U 
 vanishes 
 
 Vo’ — Ww’ = 0 (6) 
 also writing for convenience 
 Vol — Ww! = O (7) 
 (5) becomes 
 Base (O= Un) aioe C0 = Ut} (8) 
 UW UV 
 r Wv 
 Whence, using the factors VW there results 
 WwW Vv U 
 —v0 + —w — 2’ 
 po Naan Os, a 
 Veer V. 
 Van Ww 
 
32 
 
 Hence we obtain the following systems, any one of which are the 
 
 conditions for the existence of an umbilicus. 
 eh Vigo — 2VWw + Ws 
 
 
 
 
 
 
 
 6-10 Vo’! — Ww’ = 0 Ve We (11) 
 me Wu — 2 WUv’ + Uw 
 2 / 2 
 We='0) .9 Ua oo i oe set (13) 
 
 II].—on THE LINES OF CURVATURE ON AN ELLIPSOID. 
 
 Varrous methods have been proposed for investigating the nature 
 of the lines of curvature on an ellipsoid ; but as all have involved certain 
 transformations more or less laborious in order to integrate the equation, 
 it appeared that these tedious operations might be advantageously | 
 avoided by the following course. ‘The equimomental surface was first 
 brought into connexion with the problem by Mr. Cayley, in the Cambridge 
 and Dublin Mathematical Journal, May 1846. 
 
 Let the equation to the ame oo 
 
 aot y “4 =1 (1) 
 then the equation to the ee of curvature is 
 Q at 2 a, Y 2 a, = 
 -— ia (C7 — ee on is == 9 
 (b ee a Se A ) 0 (2) 
 Now if «, 6, y be so chosen that 
 eo—@=l —-BP=ac—y=h (3) 
 the equation (2) may be replaced by 
 oo nee 9 eae: 2 Ohh he pee 
 (VE + (Fat) £4 (a8) Z=0 (4) 
 
 But the equation to the ellipsoid whose lines of curvature are giyen 
 by (4) is 
 
 or substituting from (3) 
 xv y z 
 = - . = 6 
 Beh Fon eee, (6) 
 
 Hence the lines of curvature on the ellipsoid (1) coincide with those 
 on the surface (6), but (4) are the conditions that (6) shall be confocal 
 
 
 
33 
 
 with (1); consequently, the lines of curvature on a central surface of 
 the second order are formed by the intersection of that surface with 
 confocal surfaces of the same order. 
 
 Again, if instead of (3) the following system had been employed 
 
 &—a®=P—PP=e-y? =k (7) 
 we should have had instead of (6) 
 wae Tas Pt cg ATA eS aed 1 (8) 
 
 atk +k o+k 
 
 Now subtracting (6) from (1), (8) from (1), and (8) from (6), there 
 result 
 
 x 2 2 a 2 2 
 Her HET Teer? ASEH RED exw = ° 
 
 mo) F: y? as, Co gee 
 (2+h)(@+h) (PAP +A | (2+h)(C+A) 
 
 Hence the lines of curvature on a central surface of the second order 
 are formed by the intersection of that surface with cones of the same 
 order. 
 
 Again, from (9) there result by symmetrical elimination 
 
 (9) 
 
 
 
 
 
 
 
 
 
 ot y? 2 
 a a be Ke Ce 
 FELINE — CHAFEE — (EDN SINC HA) 
 E 2 (10) 
 — (h+h)at(P? —c) +b —a?) +.c#(a? — 07) ] + a'(b?—c*) + 8c? — a?) + (a? 0") 
 = 1 | 
 ; a‘ (b? —c*) rm b*(c? —a?) a c#(a?— 6?) 
 whence 
 ‘ rPah+tkt w+ P+ (11) 
 and (8) may consequently be written thus, 
 2 ¥ 2 cs 
 Ea Cy | + et te ep eet t (12) 
 
 Hence the lines of curvature on a central surface of the second order 
 are formed by the intersection of that surface with an equimomental 
 surface (which in certain cases takes the form of the wave surface). 
 
 The perpendicularity of the lines of curvature on an ellipsoid follows 
 immediately from (9); those equations being in fact the conditions that 
 the system of surfaces shall be orthogonal. 
 
34 
 
 On certain Formule for the Transformation of Coordinates. 
 
 
 
 Mr. Caytey has, in a paper in the Cambridge Mathematical Journal, 
 quoted some very beautiful formule for determining the position of two 
 sets of rectangular axes with respect to each other, employing rational 
 functions of three quantities only. The geometrical method, however, 
 by which he has deduced them is somewhat complex; it is hoped, 
 therefore, that the following analytical investigations will be found as 
 elegant and more simple. 
 
 Let the nine direction-cosines be 
 
 iT} mmm”, nn'n”’ ; 
 then since the two systems are rectangular, they are connected by the 
 following six formulze of relation, 
 P+ rtt+nt=1 P28 me ne 1 PS gl? 4 on? — (1) 
 Ul’ + m'm’ + nn" = 0 ll + mm + n/n = 0 U' + mm'+nnr’'=0 (2) 
 whence we easily deduce the inverse systems 
 2s (7? 4+ 77=1 m +m? + m’?= 1 FU ik YE * cok a (3) 
 mn + mn’ + m’n’= 0 nl + nl +n''l’= 0 lm + I'm’ +1'm’” = 0 (4) 
 
 Now subtracting (3) from (1) 
 
 m? + n& — 1? —0? = 1? 4 n? —m? — mM? = 1? + mi? — n? — n®= 0 
 whence 
 n? — mM? =I"? —n? = m — I? (5) 
 
 Let us now assume 
 r 5 YD La 
 7m’ Pon m—?l (8) 
 combining which with (6) we arrive at 
 (n’ +m) = (Ul +n) = (m+) 
 or 
 
 pv vr Ape 
 nan! VA gn en ee 
 
35 
 
 Now we have introduced three quantities, but have made only two 
 assumptions concerning them; we are therefore at liberty to make 
 another ; let this be 
 
 (6) = (7) (8) 
 
 Then we easily deduce 
 
 
 
 
 
 
 
 put rN_ w+ em _AMtVY_ py—-rA_ WAM _dA— (9) 
 n’ 1’ m m’”’ an a * va I’ 
 whence 
 eee ge ee 1—P a 4(1—P) 
 Que Ooty (ev + Ota)? I++ e+) (14 
 wee ® at 1 — m? a 4(1—m’) 10 
 (YAP (Apt)? (uyv—r)?+(ud+v)? (L074? +07)? — (14+ ?— YP —r?)? { ) 
 n° n’? 1— nn” 4(1—n’’) 
 
 
 
 
 
 
 
 
 
 
 
 (A—pe (ptAP (A—p)P?+ (wa (LFA +e +P (1+ — pF)? 
 
 Now it is obvious that we may from inspection find a solution which 
 will satisfy these equations, but it will be more satisfactory to solve 
 them directly ; we shall arrive at the same result. 
 
 We will write (10) for convenience thus 
 
 i i Hh an ma a) Sig NITi] 
 
 l 
 o 7 a’ B oe B” y yy’ 
 The values of 
 
 
 
 aa’, BRB", yy 
 are obvious; our object is to determine 
 
 Now since the two systems are rectangular we have the conditions 
 
 UIT+ m’m + nn =0 (12) 
 
 UN + mm”! + nn’ = 0 
 W + mm’ + nn’ =0| 
 
 whence by (11) 
 
 
 
 0.0+8'T+y¥u=—aa" 
 cio Or += — 88 (13) 
 a’o + Br + Ov =— yy 
 whence by cross multiplication 
 o oe T = Uv 
 aa’ Bry — Bn (BP +7) BY BY a’ yal" (y? +a?) ona" B"— a’ Ba’? +B’) (14) 
 1 
 
 ~ al@"y+ a” By 
 F 2 
 
36 
 
 But 
 a’ Bi'y + al Bry! = (pv +X) (vN+ BAW +Y) + i Re eek (15) 
 — 2(r2u2v? + w?v? + v?A? + r2u?) 
 and 
 aa’ Bry — By (BP + 9°) = (pe —v*) (NV? — pw?) — (wv? —V*) (L +?) (Ww? + a (16) 
 = (Np? + wy ele m?)(1 +A? — ps? — v”) 
 
 whence forming symmetrical quantities for + and v, we have 
 
 7 FAM ms p= ee Seed (17) 
 
 Lib Am ee ae ope iti ee Dee 
 whence by (10) 
 
 K=14+NV4p'+r (18) 
 whence we immediately arrive at the system given by Mr. Cayley 
 Kd=1+N—p’?—v Km = 2AAp+r) Kn = 2(vA—p) 
 KU = 2(Au—yv) cm =14+—rv?—r? Kn’ = 2(vu+r) f (19) 
 Kl’ =2(rv+ p) Km’ =2(uv—D) qn’ =14+7?—)d?2— pw? 
 
37 
 
 On the Principle of Virtual Velocities. 
 
 By the Rev. B. Price of Pembroke College. 
 
 
 
 Tue object of the following paper being to deduce the equation of 
 virtual velocities from more abstract principles than is attempted in the 
 Mécanique Analytique, it may be allowable to make some few remarks 
 on what Lagrange has there given as the proof of the principle, but 
 which appears to the writer of the present paper to be nothing more 
 than an illustration. 
 
 Lagrange having acknowledged that the principles of mechanics 
 adopted by the older mathematicians are nothing more than particular 
 expressions of the equation of virtual velocities under different points of 
 view, conceives that the equation depends on a principle different to 
 that of the lever and the composition of forces, viz. on that of the 
 pully ; and by means of this constructing and measuring a system of 
 forces, which act on a body or system of connected particles, proceeds 
 to give what has been usually termed by English writers, Lagrange’s 
 Proof of the Principle of Virtual Velocities. 
 
 It seems, however, to the writer of the present paper, that no advan- 
 tage is gained by the introduction of the pully. ‘The pully is not nearer 
 to the first principles of mechanics than the lever, the inclined plane, or 
 the parallelogram of forces; whereby Lagrange, in this proof, is not 
 nearer to mechanical axioms than was Archimedes or Varignon. Re- 
 actions of various kinds, tensions, rigidity of cord, friction, and so on, 
 are forces introduced in the system of pullies and blocks of which no 
 account is taken; and no principles or axioms are previously laid down 
 which authorize us to neglect these. Lagrange seems to make the 
 principle, which is nothing short of every problem in mechanics, to 
 depend on a single problem, viz. that of the pully and the block. He 
 deduces the general type of every mechanical equation from an equation 
 which he, in a manner, proves to be true in the case of forces acting 
 through a particular combination of the pully. It seems to the writer 
 of the present paper, that if Lagrange intended his problem to be a 
 
38 
 
 proof of the principle, he is guilty of the fallacy of concluding the 
 general equation from the particular instance. 
 
 In the method pursued in the following pages the three laws of 
 motion are assumed as axioms, and upon them the reasoning is founded. 
 It is conceived, however, that they are not empirical laws, but general 
 truths, adapted to the principles of mechanics, which we may thus 
 state :— 
 
 I. “ There is no effect without a cause ;” which may be thus adapted, 
 ‘a particle or body at rest remains at rest, and a particle or body in 
 ‘¢ motion continues to move uniformly in a straight line, unless acted 
 ‘¢ on by some force external to itself.” 
 
 II. “‘ Every cause produces its own effect;” and we may thus state 
 the adapted mechanical law, “ a force acting on a body under the 
 “ action of other forces produces its own effect equally as if the other 
 “ forces did not act.” | 
 
 III. “* Action is accompanied by an equal and opposite reaction ;” that 
 is, ‘* whenever a mechanical force acts there is a reaction simultaneous, 
 ‘‘ equal, and opposite.” In what manner action is to be estimated must 
 be derived from experience ; and on this subject more will be said below. 
 It may be also worth while to notice that the methods of measuring 
 velocity and accelerating force by means of differentials (not differential 
 coefficients) are considered to be known; that D’Alembert’s principle 
 is assumed as axiomatic; and that wherever the differential calculus is 
 introduced it is considered to be a calculus of differentials or infinitesi- 
 mals. But before proceeding it 1s necessary distinctly to understand 
 what is meant by virtual velocity, in what manner virtual velocities 
 are to be estimated, with what signs to be affected, and then formally 
 to enunciate the principle. 
 
 99 
 
 I. Meaning of the expression “ Virtual Velocity.” 
 
 By virtual velocity of a point or of a particle is meant, that velocity 
 with which the point or the particle would begin to move supposing it 
 to be disturbed from its position; or it is that velocity with which, at 
 the first instant of its motion, any point or particle moves, supposing 
 such a motion to be possible. Hence, when a number of forces act on 
 a body or a system of points connected in any manner so as to produce 
 equilibrium, were a slight motion consistently with the geometrical 
 relations of the points to be given to the system, the small spaces 
 
39 
 
 described by the point of application of each force relatively to the 
 direction of the force in a very short time @ is a correct measure of the 
 
 , os 
 virtual velocity of the point, that is of the force, inasmuch as _ is the 
 
 velocity of the point ; and if @¢ be the same for all the points, the virtual 
 velocity must vary as és; and relatively to this direction of the force, 
 because the force can produce motion or pressure only in its own direc- 
 tion, and therefore whatever small motion the point of application of a 
 force may have in any other direction, that is due to some other force or 
 forces ; as, for instance, suppose P, Q, R to be three forces acting on a 
 particle A, and to be so related in intensity and direction as to keep it 
 at rest ; conceive the point A to be moved over a very small space to A’, 
 the ahaohike distance over which A has moved is 
 AA’; but the virtual velocities of the forces are 
 proportional to the several projections of AA’ 
 on the several directions of the forces: viz. the 
 virtual velocity of P is measured by Ap, the 
 virtual velocity of Q by Aq, and of R by Ar; 
 the displacement being so very small that we 
 may consider the directions of' the forces still to be parallel to their 
 former directions in the undisturbed system; and the virtual velocity is 
 to be considered positive when the projection of the absolute space over 
 which the point of application of the force is moved is towards the 
 point whence the force acts, as is the case in P and R, and negative 
 _when it falls in the direction produced backwards, as in ase . 
 
 
 
 * As the parallelogram of forces is with great facility deduced from the diagram given 
 in the text, it may be worth while to anticipate the equation of virtual velocities for the 
 purpose of exhibiting the several steps. 
 
 Let? AAP = @ Lu ess, RAQ—a, PA =), QAP =y, 
 the equation of virtual velocities is in this case P. Ap —Q.Ag+R.Ar=0 
 and substituting for Ap, Ag, Ar their values, we have 
 
 P. cs cos 6 + Q. 6s cos (06+y) +R. és cos (B—6) = 0 
 dividing this by cs, expanding the cosines of the multiple arcs, &c. we have 
 P+Q cos y + R cos 6 + (R sin B—Q sin y) tan 6= 0 
 whence as @ is indeterminate 
 P+Q cos y=—R cos (1) 
 Q sin y= R sin B (2) 
 Squaring (1) oe (2), and adding R2= P24 Q?+ 2PQ cos y, the parallelogram of forces; 
 
 and from (2) ——— Q pall from the symmetry, the triangle of forces. 
 sina sin inp sin y 
 
40 
 
 II. Enunciation of the principle of Virtual Velocities. 
 
 The principal of virtual velocities is this,—when any number of forces 
 are so combined in both intensity and direction as to produce equilibrium, 
 the sum of the products of each force and its virtual velocity is equal to 
 zero; or if any number of forces acting on a body or system of particles 
 are in equilibrium, the forces are inversely proportionate to their virtual 
 velocities ; whence if P be the general type of a force, and ¢p the 
 general type of the small distance relatively to its own direction over 
 which the point of application of P moves, owing to a slight displacement 
 
 of the system, then 
 > Rop = 0 
 
 and we may thus formally state the theorem : 
 
 If any system of points or of bodies acted on by any number of forces 
 is in equilibrium, and if a small motion is given to the system con- 
 sistently with its geometrical conditions, by means of which the points of 
 application of the forces move through small spaces, these spaces rela- 
 tively to the directions of the forces are proportional to the virtual 
 velocities, and the algebraical sum of the product of each force and its 
 virtual velocity is equal to zero; the velocities being considered positive 
 when the spaces described are towards the points whence the forces act, 
 and negative when in direction of the force produced backwards. 
 
 III. Explanation of the Principle. 
 
 This principle involves two considerations, 
 
 Ist. In what manner are forces producing pressure or motion to be 
 
 estimated ? How are their effects to be measured ? 
 2nd. What do we mean by equilibrium ; that is, in what particular 
 forms and in what proportion, as well as to intensity as to 
 direction, are forces to be combined to produce equilibrium ? 
 As to the first, daily experience shows us that the effect of a force 
 depends on two circumstances, the mass of the moving body which 
 contains and is acted on by the force, and the velocity with which it 
 moves; the effect we perceive to be greater the greater the moving 
 mass, and the greater the velocity ; whence, in mathematical language, 
 we say the effect is a function of the mass and the velocity ; but what 
 function? What we may call the “ proof experiment” of Attwood shows 
 that pressure producing motion or moving force is to be measured by 
 the product of the mass moved and the accelerating force; that is, 
 
+t 
 
 in order to estimate the statical effect of a moving mass, we must 
 multiply the abstract accelerating force by the concrete mass ; or, as we 
 may state the case under another point of view, in order to impress 
 on a given mass a certain velocity in a certain direction (7. e. in 
 order to produce a certain effect) a force must be applied, the intensity 
 of which must be proportional to the product of the mass to be moved 
 and the velocity with which it is to move, and the direction of course 
 the same as that in which the body is to move; this product of the mass 
 and the velocity is called the quantity of motion. Or again, supposing 
 a body to contain P particles, each equal to m, m being the unit 
 particle, and suppose the body to move in a given direction with a 
 certain velocity measured by p, the distance over which it passes in 
 a short time, then the effect of each particle is equal to mp, and there- 
 fore the effect of the body being the aggregate of all the effects of the 
 several constituent particles is equal to Pp; for although it may appear 
 that some of the effective forces, that is, some of the motion, is lost, 
 owing to the mutual connexion of the particles, as for instance the 
 molecular forces, yet by the principle of D’Alembert, as long as the body 
 is rigid, and there is no motion of the particles relatively to one another, 
 these forces neutralize each other, and the body notwithstanding produces 
 a force of precisely the same intensity as the sum of the forces due to 
 each particle m, were each to move independent of and separate from 
 the others ; whence then it appears that the correct measure of the effect 
 of a force arising from a moving body is the product of the mass moved 
 and the velocity with which it moves ; and this being so, 
 
 The second consideration comes in; in what manner must forces 
 measured as to their effects in the way we have just explained be 
 combined so that a body under the action of them may be at rest? 
 Forces are, I say, in equilibrium, when the absolute sum of' their effects 
 measured as above is less than in any other state, when the system is 
 slightly deranged :—for conceive a system of bodies or of points which 
 is in equilibrium to be slightly disturbed by the introduction of a new 
 force external to the system, the equilibrium of the former system is 
 thereby destroyed, but the new system will arrange itself in a position 
 of equilibrium, and the resultant of all the forces in the former system 
 will be such in intensity and direction as exactly to counterbalance the 
 force which has been introduced; thus the sum of all the forces of the 
 
 G 
 
42 
 
 deranged system is greater than the sum under the former balanced 
 system by twice the resultant, that is by twice the force which we have 
 introduced in deranging the system ; whence it appears that if the body 
 or the system be at rest, the sum of the forces is less than if it be 
 disturbed, and therefore the sum of their effects measured as above is a 
 minimum for a position of equilibrium. 
 
 Let therefore P,, P2, P3, . . . P, be m forces acting on a system, 
 and let us conceive (as we always may do) that these forces act from 
 or towards certain points in the lines of their directions, which points 
 we will, after Lagrange, call centres of force, and let p,, ps, Psy» + + Dn 
 be the distances between these centres and the points of application of 
 the forces; it is plain then that the several forces P,, P., P;, . . . P, 
 remain the same in the deranged as in the undisturbed system, so 
 that the only quantities which vary owing to the displacement are 
 
 Fis Dern Dove Pavepscn 
 t= SPp: 
 
 then in a state of equilibrium H is a minimum, and therefore 
 oll =0 
 soe Lop =O (1) 
 that is, the sum of the products of all the forces and the virtual 
 velocities is equal to zero. 
 
 IV. Other circumstances of equilibrium. 
 
 Are there not however other ways in which the forces may be com- 
 bined so that =. Pap = 0, and yet these not be such states of equilibrium 
 as the one we have supposed? ‘The theory of infinitesimals, as applied 
 to the discussion of curves, shows that 811 = 0, when [1 is a constant, as 
 well as when II is a maximum or a minimum-—a maximum or a minimum 
 I say, because in either case we have such a position of equilibrium as 
 has been supposed in the last section; the absolute sum of the effects 
 of the forces having been taken, without reference to change of sign ; 
 for (to adopt the ordinary language of mathematicians) if a body or a 
 system of particles is in equilibrium, whether stable or unstable, an 
 additional force is required to keep it in a position slightly deranged 
 from its position of equilibrium, and therefore the absolute sum of the 
 effects of the forces is, as before stated, greater in the displaced than in 
 the undisturbed system by twice the force which is required to keep 
 
43 
 
 the system in its displaced position; but suppose II to be a constant, the 
 criterion of equilibrium is satisfied, and the body or the system is in 
 equilibrium in other positions of slight displacement; for although by 
 the first law of motion a force is required to change the position in 
 which the body rests, yet as II is constant, not admitting of increase or 
 decrease owing to any small displacement, we have no force to call into 
 action to be equal to and counterbalance the external force we have 
 applied to move the system, and therefore the system or body would 
 continue changing its position until some other force (as friction, 
 resistance of the air, &c.) acts to neutralize the applied force, and 
 thus to produce rest; whence it is manifest that as soon as such 
 a force does act the system or body is brought to rest and there 
 remains; and as this counteracting force may be brought into action 
 at any instant of the body’s motion, it follows that there are an infinite 
 number of positions in which the body or system can rest under the 
 action of the impressed forces. Such a state is what is usually called 
 “ Neutral Equilibrium,” but what may well be named “ a position of 
 stationary action.” 
 
 V. Method of applying the formula. 
 
 The difficulty in applying the equation of virtual velocities to the 
 solution of statical and dynamical problems consists in the calculation of 
 the virtual velocities %p,,%p,, ... ép, for the several points of application 
 of the forces. The simplest method, however, is as follows, — to 
 consider two different positions of the system which are consistent with 
 the geometrical relation of the parts, one of equilibrium, the other 
 slightly deranged, and to express the virtual velocities of the pomts of 
 application of the forces in terms of the coordinates of the points of 
 application and of arbitrary quantities which have been introduced in 
 deranging the system, and as these latter are indeterminate, equating to 
 zero their several coefficients, we shall have a sufficient number of 
 equation to determine the position and conditions of equilibrium. 
 
 Let then P,, P,, P;,... P,, be the forces acting on the system, 7, y; 2, 
 Lo Yo 2 - » +» Ln Yn%, be the coordinates to their points of application, 
 A, 6, C,, A, b, Cy, . +. A, 5,C,, be the coordinates to their centres, defined 
 as in Art. III., and let their direction-cosines be cos «, cos 6,, COS ys 
 COS a2, COS Fx, COS Y2) «++ COS %y COS B,, COS yn, and let the distances of their 
 
 G 2 
 
44 
 
 centres from their points of application be p,, p., ps... P,» so that we 
 have the following system of equations, 
 
 Pr = (%-HYP + (%-hY + (4-4) 
 
 Px = (%.— 4)” + (Yo— 5g)? + (2-62)? (2) 
 
 Pa = (®_= Gn) + (Ya—5n) + (Zn on)” 
 it is manifest that in any small derangement of the system, such as we 
 have conceived, we may suppose the centres of force to remain fixed, and 
 the points of application to move; but before we proceed further we 
 must make some remarks of great importance. 
 
 VI. Analysis of motion into that of translation and that of rotation. 
 
 Suppose a body or a system of points of invariable form to change 
 its position in space, and consider two successive positions of the body 
 independently of the forces acting and of the time consumed in passing 
 from one position to the other, it is manifest that there is an infinite 
 number of different ways in which the change may have taken place. 
 Which, then, of all these is the most simple? Which is the best adapted 
 to the methods we have adopted of determining position ? 
 
 Whatever displacement a system may have undergone, we may con- 
 ceive the motion to have taken place as follows: Ist, All the points 
 may have moved over equal and parallel lines in the same direction ; 
 2dly, Considering some one point in the body or system to be fixed, and 
 to be a centre round which the body turns, though this point may be 
 continually changing, yet we may give such a revolution to the body or 
 the system, that the several parts of it shall after these two motions hold 
 relatively to space that position which the body has after its real motion ; 
 and 3dly, There may be a motion of the particles of the body or system 
 relative to one another, such as to cause a change of the mutual position, 
 influences, and distances on each other, as is the case with gases, elastic 
 fluids, machines in which one part slides on another; this however not 
 being the case with rigid bodies, to which alone it is the object of the 
 present paper to apply the principle of virtual velocities, and as the 
 position of a body in space, with reference to fixed axes and planes, is 
 not affected by this third kind of motion, we shall at present confine 
 ourselves to the first two, into which, though they be of distinct character, 
 all motion, even the most general, may be resolved. 
 
45 
 
 And as the principle of virtual velocities is true for the most general 
 motion conceivable, the velocities are to be estimated in their utmost 
 generality, and therefore the system must receive a motion both of trans- 
 lation and rotation, though not simultaneously but at successive times, 
 as the principle of superposition of small motions, which is equivalent to 
 that of infinitesimals, authorizes ; and =.P%p is to be calculated for both 
 these motions ; we proceed, therefore, 1st, to give a small motion of trans- 
 lation to the whole system, along each of the three rectangular coordinate 
 axes, whereby all the points of application of the forces will be moved over 
 equal small spaces parallel to the axes; and 2dly, we will turn the body 
 or system successively round the three axes through small angles, the 
 effect of which two motions will be that the points of application of the 
 forces will be moved in the most general way possible, and we shall 
 derive from the results all the conditions of equilibrium. 
 
 VII. Application of the principle to motion of translation. 
 
 Recurring to formule (2) Art. V., and taking the variations, we 
 have i 
 Tie a) it 91 mel 
 dp, = oa da, + Pe oy; tape oz) 
 and substituting the values of the direction-cosines, we have 
 dp, = cos a, O27, + cos By dy, + cos y, dz, 
 and similarly 
 
 Sp. = Cos a2 Sx, + cos By Sy + COs ¥ dz. ! 
 1 (3) 
 
 SPn = COS An O52, + C08 By OY + COS Yn SZq 
 where @p,, 9p. . . . 8p, are the distances relative to the direction of the 
 action of the forces over which the points of application of the forces are 
 moved. 
 
 Which may also thus be shown — 
 
 Let 8s be the absolute displacement of the point (7, y, 2,) owing to 
 the motion of translation, then the direction-cosines of this line és are 
 = Z, a and if ¢ = the angle between the line és and P’s direction, 
 
 dp = 6s cos ¢ 
 
 = be{ cos @ + by J cos B + 5 cos 7} 
 os 8s 
 = dx cosa + dy cosB + 8z cos ¥ 
 
46 
 
 Applying these values in the equation (1) of virtual velocities, and 
 remembering that all parts of the system move over equal and parallel 
 distances in the same directions, and therefore 
 
 O2p ice 08) O23 Sie t of Ot 
 Sy) = Deiat eee ae = *| (4) 
 62; = 62, = 023 = De ch phigh OS a 
 
 the equation >. Pip = 0 becomes 
 >. P {cosa dx + cos B dy + cosy dz} = 0 
 or, as we may write the equation on account of conditions (4), 
 =>.P cosa dr -+ =.P cos B dy + =.P cosy dz = 0 (5) 
 ax, dy, 'z being outside of the sign of summation, and putting 
 00s aut 2 COS po oY ae 00S 7 
 Xdx + Yoy + Zdz = 0 (6) 
 If then the system or body is entirely free to move in space, we have 
 no other relation given between z, y, z, and their variations, besides (5) 
 or (6), and therefore we have 
 Sai Pecos 10 =. P.cos BR = 0 ee Pacisays— (7) 
 that is, the sum of the parts of the forces resolved severally along the 
 three axes of coordinates must separately be equal to zero. 
 For the cases where the system is not free, but confined to move on 
 a given curve, and for other deductions from (5), reference may once 
 for all be made to the Mécanique Analytique. 
 If the forces P, P, . . . P, act in parallel directions, we have 
 
 Cy Se Og es Se i Be oe Se tL, 
 PB} =B =h3=.-++-+ = Bn 
 V1 =— Yo = Ya = 2 2 ee ee = Yn 
 and conditions (7) are reduced to the single one, 
 Pe tet Ps tetbs beter ar as ca eens le oP ee eo ee (8) 
 
 VIII. Application of the principle to motion of rotation. 
 
 The theory of rotatory motion, as laid down by Poinsot, authorizes us 
 to conclude that every motion or tendency to motion round any axis 
 may be resolved into three motions round three rectangular axes, passing 
 through some point in the axis about which the body turns or has a 
 tendency to turn; and hence it follows, that if we turn a system through 
 small angles round each of three rectangular axes, the resultant of these 
 several motions is equivalent to the most general motion of rotation that 
 
aN 
 
 the system can receive round any axis passing through the origin of 
 coordinates. 
 Let 2 Y 2) %2 Yo 2% +++ Xn Yn % be the coordinates to the points of 
 application of the forces P,, P., P;,... P, 3 and let 
 2,7 + yy + 2," = n° 
 2 2 2 ay 
 oh bier ” 
 zt ie Yr? 4 a =_ 72a 
 and let the projections of these several radii vectores be on the plane of 
 yz Tie Tor Tae © + «© Tne 
 Me eli Oh} aT ay 
 LY Vig Tox T3z 26 0 8 Try 
 Let us first turn the system through a small angle #4 round the axis 
 of x, 6 being the angle between 7, and the axis of y, 
 
 y = 7, cos 8 2 =a ty sO 
 sy = — r, sin 080 dz = 7, cos 060 
 oe oy = — 260 éz = yo0 
 
 taking the variations of y, z, 6, which are the types of yy... Yn 
 Pera eetcing Ui Us et Uae 
 
 Similarly let the system be turned through a small angle % round the 
 
 axis of y, ¢ being the angle between 7, and the axis of z, and we have 
 
 dz = — xdh dz = 25h 
 and let the system be turned through a small angle #) round the axis 
 of z, / being the angle between 7 and the axis of x, and we have 
 
 dx = — yd dy = xd 
 ZY, 2,9, and ¥ being, as before, types of the several quantities which 
 correspond to the several forces. 
 
 And by the principle of superposition of small motions, being autho- 
 rized to neglect these quantities which are variations of variations, 
 inasmuch as they become infinitesimals of a higher order, and therefore 
 inappreciable in an expression which involves infinitesimals of a lower 
 order, the total variation of these several points of application of the 
 forces is equal to the sum of the variations due to the several rotations 
 round the several axes of coordinates; hence we have the following 
 typical values of the variations 
 
 dx = 26h — yor 
 dy = xd — a0 
 dz = ¥59 — «dh 
 
 (10) 
 
48 
 
 and since the expression (3) gives 
 dp = cos abx + cos Bdy + cos ydz 
 substituting the values (10) we have 
 dp = (y cos y—z cos 8)60 + (z cos a—=z cos ¥)6p + (x cos B—y cos a)oy 
 
 whence substituting again in (1), remembering that 2%, , @) are the 
 same for all the forces, and therefore may be written outside the signs 
 of summation, we have finally 
 =. P(ycosy—zcos8)d0+ >. P(zcosa—x cosy)op + =>. P(x cos8B—ycosa)iy=0 (11) 
 and if the system of the body be entirely free to revolve round any of 
 these axes, that is, if we have no other relation between 4, ¢, and J than 
 equation (11), then 4, ¢, } being independent of each other, putting 
 
 =.P(y cos y—z cos 8) = L 
 
 =. P(z cos a—2 cos y) = mt 
 
 =.P(x cos B—y cos a) = N 
 we have L=0, M=0, N =0 as the conditions of equilibrium of rotation ; 
 that is, since L, M, N are the sum of the moments of the forces tending 
 to turn the system round the three rectangular axes of coordinates then 
 of x, y, z respectively, we conclude that if there is equilibrium the sum 
 of these moments must severally and separately be equal to zero. 
 
 Thus then having deduced from the equation of virtual velocities 
 the equations (6) and (11), the six equations of equilibrium which are 
 ordinarily given in treatises on mechanics follow immediately in the 
 manner we have indicated. 
 
 (12) 
 
 Addition to the foregoing Paper. 
 
 Suppose the forces which we have been considering in the last paper 
 to act in parallel direction, then we have 
 
 Oy == 0 10 ee 
 Roem eey (1) 
 YY as ae a ae 
 and the equations (7) are reduced to the single one, 
 Pret P. + Pee ces ee etn 10 
 
 and equations (12) become 
 cosy >. Py —cosB =.Pz =0 
 cas. Peco. Pe= 0} (2) 
 cos8 =.Pr—cosas.Py=0 
 
49 
 
 whence we have 
 =.Pr_ =. Py_ =. Pz 
 cos a cos 8 cos y 
 
 Now conceive the parallel forces and their pomts of application to be 
 
 so arranged that 
 
 Sree Pee aa asa (4) 
 then «, 6, y become indeterminate, and there is no pressure of rotation, 
 no tendency to make the system turn round any axis, whatever be the 
 direction of the forces relative to the coordinate axes. 
 
 Whence it appears that if a system of parallel forces subject to the 
 conditions expressed in equations (1) and (4) act on a body, the system 
 is in equilibrium, whatever be the direction in which the forces act; and 
 this leads to an easy method of determining the centre of parallel 
 forces. 
 
 Conceive a number of forces of which P is the type to be acting on a 
 body, «yz being the types of the coordinates of their points of appli- 
 cation, and conceive the forces to be so arranged that we have one of 
 them P, the coordinates of whose point of application are «yz, to be such 
 in intensity and position that the conditions (1) and (4) are fulfilled, 
 viz. 
 
 =” {(5. Pzr)?+(S.Py)?+(=. Pz)*} (3) 
 
 
 
 
 
 
 
 
 
 P=P,4+P.4+ a wore batik 
 
 Pre = Piz, + Po t+... + Pit, = 3S. Pz 
 
 Py = Pi Paysite 20s 2) + Pap = S.Py 
 
 Deted Picola Peta bide pd Tub ney > 2S Pz 
 then we have 
 
 oe Fon: Ps Popa Py tne ob 
 
 este! me Pe at asp 
 
 Whence it is easy to determine the coordinates to the centre of gravity 
 of a body, for inasmuch as it is that point at which if the whole weight 
 of the body be collected the moment of rotation about a given axis is 
 the same as the moment of the body constituted as it is, we have a case 
 precisely similar to the one supposed above, and therefore if wy z are the 
 coordinates to the centre of gravity, we have 
 
 ~ fxdm ~ _ fydm ~ _fedm 
 7 = fdm I= fdm a= fam 
 
 
 
 
 
 
 
50 
 
 On infinitesimal Analysis. 
 
 
 
 Tue calculus of infinitesimals appears to be a branch of analysis to 
 which much attention has not been paid by English writers. The 
 method of limits has probably been supposed more tangible and more 
 satisfactory to minds previously unacquainted with considerations of this 
 nature ; and indeed it is not without difficulty that the mind first forms 
 a conception of a system of quantities differing infinitely from one 
 another in magnitude, and yet all distinct from an absolute zero, and 
 learns that many mathematical expressions are formed only approxi- 
 mately by means of neglecting terms of certain orders of infinitesimals. 
 Now although this is the case, there still exists a difficulty of a not 
 entirely different nature in the doctrine of limits, so that the latter 
 method is perhaps not so eminently superior to that of: infinitesimals as 
 to demand the complete exclusion of the latter. In fact its frequent 
 employment in applied mathematics, as in the theory of curves of double 
 curvature, of small oscillations, including the undulatory theory of 
 light, and other cases, appears to call for some explanation of its funda- 
 mental principles. Moreover, the two methods obviously leading to the 
 same results must in fact come to the same thing; and it may not be 
 inappropriate here to mention the point at which the coincidence takes 
 place. The limit of any continually increasing or decreasing quantity 
 or ratio is that quantity or ratio to which it always tends, but to which 
 although it may approach nearer than by any finite quantity or ratio, it 
 never becomes actually equal. The difficulty then consists in passing 
 from any given value of the quantity or ratio under consideration to 
 the limit, for in doing this (however near we may approximate to the 
 limit) a saltus must finally be made; because the essence of a limit 
 consists in this, that by no continuous approximation is it possible ever 
 to arrive at it. Now it is obvious that nothing short of the limit itself 
 will give accurate expressions in the result, and, consequently, it is 
 necessary either to reach the absolute limit or to remain satisfied with 
 
ot 
 
 approximations. ‘The former course is pointed out by the method of 
 limits, the latter by that of infinitesimals. In order to compare the two 
 methods it will be well to consider a simple example. In both cases a 
 curve is considered to be approximately a polygon, the degree of ap- 
 proximation increasing with the number of the sides ; and in the method 
 of limits it is said that the two figures coincide accurately when the 
 number of sides becomes infinite, or, in other words, that in the limit 
 the magnitudes of the sides absolutely vanish ; while, in the infinitesimal 
 theory, it is supposed that the magnitudes of the sides always have an 
 existence, and that they themselves are indefinitely divisible. Now, if 
 it be required to consider the deflexion of the curve from the tangent, 
 that is, the angle between any two consecutive tangents, it will, in the 
 former theory, be necessary generally to consider the second tangent as 
 drawn through some point at a small distance from the first point, and 
 afterwards again to have recourse to the limit, by making the distance 
 between the two points actually vanish. Hence, geometrical representa- 
 tions can be formed of an infinitesimal no less than of a finite quantity, 
 because they differ from one another only in magnitude ; but since in 
 the limit all magnitude is lost, no complete geometrical representation 
 can be formed: and, in fact, (returning to the example noticed above, ) 
 as long as any appreciable angle exists, in general between the two 
 tangents (the two points of contact actually coinciding) the curve 
 becomes a polygon, and we find ourselves employing the imfinitesimal 
 method. It therefore appears that the method of limits is only a 
 particular form of that of infinitesimals, and that, from the geometrical 
 indeterminateness arising from the notion of the limit, the infinitesimal 
 quantities are not actually destroyed or eliminated, but only lost 
 sight of. 
 
 Perhaps no better authority for the definition of infinitesimals and 
 their use can be quoted than Carnot, who makes the following remarks : 
 “ Jappelle quantité infiniment petite, toute quantité qui est considerée 
 “comme continuellement décroissante, tellement qu'elle puisse étre 
 ‘¢ rendue aussi petite qu’on le veut, sans qu’on soit obligé pour cela, de 
 “faire varier celles dont on cherche la relation. Lorsqu’on veut 
 “ trouver la relation de certaines quantités proposées, les unes con- 
 “* stantes, les autres variables, on considére le syst¢me général comme 
 “ parvenu a un état déterminé que lon regarde comme fixe: puis on 
 
 H 2 
 
52 
 
 “compare ce systéme fixe avec dautres états du méme systéme, 
 “ lesquels sont considérés comme se rapprochant continuellement du 
 ‘“ premier, jusqu’a en différer aussi peu qu’on le veut. Ces autres états 
 ‘du systéme ne sont donc a proprement parler eux-mémes, que des 
 ‘“‘ systémes auxiliaires, que lon fait imtervenir pour faciliter la com- 
 ‘“‘ paraison entre les parties du premier. Les différences des quantités 
 ‘“ qui se correspondent entre tous ces systémes peuvent donc étre sup- 
 ‘“‘ posées aussi petites qu’on le veut, sans rien changer aux quantités 
 ‘“‘ qui composent le premier, et qui sont celles dont on cherche la 
 “relation. Ces différences sont done de la nature des quantités que 
 ‘““ nous appelons infiniment petites : puisqu’elles sont considérées comme 
 ‘ continuellement décroissantes, et comme pouvant devenir aussi petites 
 ‘“‘ quon le veut, sans que pour cela, on soit obligé de rien changer a 
 ‘‘ Ja valeur de celles dont on cherche la relation.” “ L’analyse infinité.. 
 ‘“ simale nest autre chose que l’art demployer auxiliairement les 
 ‘“‘ quantités infinitésimales, pour découvrir les relations qui existent 
 ‘entre des quantités proposées.”* In addition to which let it be 
 observed that the ratio of an infinitesimal of the (7+1)th order to one 
 of the mth is an infinitesimal of the first order, and the ratio of any 
 two infinitesimals of the same order is finite. 
 
 
 
 A quantity or function is said to vary continuously from one value 
 to another when it is capable of receiving any value between the given 
 ones as its limits. In this it is to be observed, that any values (which 
 lie between the limits) chosen arbitrarily and in any order of succession 
 may be given to the quantity or function without violating the Jaws to 
 which it is subject ; and consequently the difference between any two 
 consecutive values may itself have any value not greater than the dif- 
 ference between the two limits. This difference may, however, obviously 
 be made as small as is desired by taking the two consecutive values 
 sufficiently near to one another ; and by taking the same value twice in 
 succession, it may be made absolutely zero. Hence a series of these 
 differences, considered as a system of quantities, are as arbitrary as the 
 values themselves, and their differences may consequently have any 
 value not greater than the greatest of the first differences. These new 
 
 
 
 * Carnot, Reflexions sur la Metaphysique du Calcul Infinitésimal, Chap. I. 
 
53 
 
 differences (or, as they will be called, second differences) may also 
 obviously be made as small as the difference between the first differences 
 whose values approach nearest to one another in value ; and as small as 
 is desired, and even zero, by a proper choice of the first differences, if 
 the latter are independent, i.e. subject to no conditions. This process, 
 being evidently subject to no other limitations than those above given, 
 may be continued as long as is desired; and a difference of any order, of 
 the mth for example, is consequently as arbitrary in magnitude as the 
 first assumed values, provided always that no conditions exist which can 
 affect the differences of any order inferior to the ath. 
 
 Let these differences be represented in the following manner ; suppose 
 that wis any quantity or function, and wu wu’ two values of wv, then their 
 difference 
 
 uo —u = du; 
 similarly if du, (dw)’ be two values of dw, then 
 (du)’ — du = ddu; 
 and similarly the successive differences of uw may be thus represented 
 du, ddu, COCA as fee 
 
 Now it is clear that the superior limit of the magnitude of any 
 difference of the nth order depends upon the magnitude of the greatest 
 difference of the (n—1)th order, this again on that of the (n—2)th, and 
 so on until that of the (n—n)th, i.e. until the given values themselves ; 
 but the inferior limit of the magnitude of any difference of the nth 
 order depends upon the distribution of the magnitudes of the differences 
 of the (n—1)th, these again on those of the (n—2)th, and so on until 
 the (n—n)th, i.e. finally upon the distribution of the original values 
 chosen between the given limits. Consequently when the given limits 
 differ infinitely from one another the differences of any given order, as 
 for instance the mth, may be made infinitely great, finite, or infinitely 
 small; while in every case they may be made infinitely small. These 
 remarks will be sufficient to explain the general theory of the various 
 orders of infinitesimals. 
 
 It was seen above that by choosing the primary arbitrary values of 
 the given quantity or function sufficiently near one another, the first 
 differences may be made as small as are required ; in other words, they 
 may be made infinitely small in comparison with the values themselves ; 
 in this case they will be called differentials. Again, by repeating the 
 
54 
 
 process the second differentials may, in accordance with the remarks 
 made above, be made infinitely small in comparison with the first 
 differentials ; and so on without limit. 
 
 It must however be remembered that these differentials, although 
 infinitesimals of any order, however great, are still actually different from 
 zero; because it is only in the case where two consecutive values of 
 the (n—1)th differences have been taken actually equal to one another 
 that the nth differences vanish ; which is a case distinct from the one 
 indicated above. ‘These remarks are evidently applicable whatever be 
 the order of infinitude of the primary quantity or difference under con- 
 sideration. Hence, of any quantity, whether infinite, finite, or infinite- 
 simal, there may be an infinite number of differentials, each differing from 
 one another by any required order of infinitude. Now these differences 
 or differentials of the orders 1, 2,3,.. . . exist, as has been seen, when 
 the function under consideration varies continuously from one given 
 value to another between its given limits. Similarly those of the orders 
 2, 3, 4,.... exist when the first differential varies continuously between 
 its proper limits; and so on for others. But on the other hand, these 
 differentials evidently may exist also in the case when the function varies 
 no longer continuously but per saltus, never receiving any other values 
 than those contained in a given system. For the differences of these 
 fixed values will be the differences of the first order, and the differences 
 of these first differences (if the first differences be unequal, as must be 
 supposed in the general case, ) those of the second order.’ The differences 
 will however no longer be arbitrary, as in the case of continuous varia- 
 tions, but will be determinate as soon as the saltus of the primary func- 
 tions are given. But in order to make a differential of any given order, 
 as the mth for instance, vanish, it will be obviously necessary to perform 
 upon the differentials of the (—1)th order an operation which is only a 
 particular case of the general one above considered, and differing, not in 
 nature but in tensity (if it may be so termed), from that which was 
 performed upon those of the (n—2)th. Since however this would intro- 
 duce much complexity, especially as im the various problems which 
 present themselves it would be convenient to employ a larger or smaller 
 number of orders of differentials according to circumstances (which 
 would destroy that uniformity of method which it is always desirable to 
 combine with an unity of principle), the method usually adopted, and 
 
65 
 
 the one which appears most desirable for general use, is the following ; 
 that the differentials of any given orders, as of m,n,p, ... . should 
 coincide in magnitude with infinitesimals of the same orders respectively ; 
 hence, the precisely same operation which converts a differential of the 
 mth order into one of the (n+1)th will convert one of the (n+1)th into 
 one of the (n+2)th; and so generally r operations of the same nature 
 will convert a differential of the mth order into one of the (7+7)th.* 
 
 * An example of this is the theorem of Taylor for the expansion of f(#+dz) in a series 
 whose well known form is, 
 
 fete) =f(x)$ fey +P" oes. 
 
 the number of terms being infinite. The usual expression for the remainder after n terms is 
 dx” 
 ——— f(a + Oe: 
 eae nt Ea 
 
 where @ is some quantity between 0 and 1. If however 
 
 f (2) 
 represent that the (z-+1)th differential of f(x) is to be made to vanish, the series in 
 question might be also written, 
 
 2 NAB. pit gy a de We, 
 nae Ce ee et to tea de) 
 which however is not so convenient as the former, since a less clear idea of the remainder 
 after m terms is given by the new operation ‘f than by the quantity 0. 
 Another instance of various symbols of operation is one proposed by M. E. Lamarle in 
 Liouville’s Journal de Mathématiques, Juillet 1846 ; it is as follows ; 
 
 « Lorsqu’on écrit 
 
 S(a@+h) — f(x) = hf'(x+6h), (1) 
 on se borne, en général, a faire observer que 6 désigne une quantité comprise entre 0 et 1. 
 On sait cependant que les premiéres notions de l’analyse algébrique permettent de fixer 
 d’une manitre extrémement simple et tout a fait précise le sens de l’équation (1). 
 Ajoutons qu’en l’écrivant sous la forme que nous venons d’indiquer, on est forcé d’en 
 restreindre l’application aux cas ou la fonction et sa dérivée demeurent continues dans 
 Yintervalle que l’on considére. 
 « Cette remarque suffira sans doute pour justifier la préférence que nous accordons a 
 
 la formule suivante: 
 S(a+h) —f(®) = h§$asr’s (2); (2) 
 
 ou la caractérisque J# se trouve définie par l’équation de condition 
 
 SA’ (x) = lim lz (2+4)4p(et 2) +(e ae)t +f (e+ a 
 
 a ” ”~ a ” 
 ” “ " a * 
 
 n 
 ns 
 
 a 
 a 
 
 a 
 “ 
 
 
 
 {aoe 
 
 “ et qui peut s’énoncer en ces termes: ‘ Dans tout intervalle ow la fonction demeure con- 
 “ tinue, son accroissement a pour mesure l’accroissement de la variable, multiplié par la 
 “< valeur moyenne de la fonction dérivée.’ ” 
 
56 
 
 There are however certain particular cases in which this process, viz. 
 of making the differentials of certain orders vanish, is used with great 
 advantage ; as for instance, if there are (m—1) equations involving 
 n variables and their differentials of various orders, any one of the 
 differentials of any of the variables may always be determined in terms 
 of the remaining differentials of the (»—1) variables, the differentials of 
 the mth variable, and the variables themselves. Consequently one of 
 the differentials of the mth variable is arbitrary, i.e. subject to no restric- 
 tion; it may therefore receive any value at pleasure ; the value usually 
 assumed is zero; and the differential equated to zero is usually the 
 second. The variable subjected to this arbitrary condition is called 
 the independent variable; and it is easily seen that any one of the 
 n variables may be so treated. It is also observable that unless second 
 differentials are involved there is no clue by which we can discover the 
 independent variable. It may further be observed that the second 
 differential has in preference to all others been subjected to the con- 
 dition above stated, because it is of the lowest order which we are at 
 liberty so to treat. This will easily be seen when we consider that from 
 the manner in which these differentials have been formed it appears 
 that the giving the differentials of any order (the mth for instance) 
 any arbitrary value (such as zero) is a condition which affects not only 
 the differentials of the mth and all higher orders, but also those of the 
 (m—1)th order, inasmuch as the values of the differentials of the latter 
 order to be chosen for the formation of those of the mth are no longer 
 arbitrary, but restricted to equidistant values. Hence the first dif- 
 ferentials could not be made to vanish without subjecting the values of 
 the variables themselves to limitations which the conditions of the 
 problem would not justify. It is also clear that if the second dif- 
 ferentials vanish, the third, fourth, . . . . and all others do so also. This 
 is the whole theory of the independent variable. 
 
 In actual practice it will of course be impossible to retain an infinite 
 number of orders of differentials, as we should in that case have an 
 infinite number of quantities to contemplate and take account of at the 
 same time, and our faculties are such that this is beyond their power ; 
 hence we are obliged to have recourse to a method of approximation, 
 and the one naturally adopted is to neglect those small quantities which 
 do not affect the truth of the results. This may at first sight appear 
 
57 
 
 an unsatisfactory method of operation, and one which overturns all 
 the preconceived notions of the absolute truth and correctness of 
 mathematical results ; but it is nevertheless the method actually adopted 
 (or at least equivalent to it), although often brought to notice under 
 different aspects. But although this is the case, it is still equally true 
 that the error committed by this method can never become appreciable, 
 for the neglect of infinitesimals of any order, as those of the mth for 
 instance, can for each term so discarded produce no errors of an order 
 lower than the nth; but no sum of a finite number of infinitesimals of 
 the nth order can produce a quantity of the (n—1)th; similarly no sum 
 of a finite number of infinitesimals of the (7 + 1)th order can produce 
 a quantity of the mth; and so on throughout all orders. Suppose now, 
 for example, that above the (x + r)th order none but finite sums exist, 
 or in other words that by neglecting differentials of this order no 
 quantities of lower orders are discarded, (7m and 7 may, of course, have 
 any value from — «© to+ oo). Now the only case where an error of an 
 order lower than the (7 + 7)th can be committed by neglecting terms 
 of this and higher orders is that in which there are an infinite number 
 of terms of this order; but as in no expression which can occur in 
 practice is it possible to write down these at full length, the terms in 
 question must themselves (if they exist at all) appear under some other 
 form (such as a sum or product, for example), which in fact will present 
 them as a quantity of the (7 +7)th or some lower order; but as terms 
 of the latter order have by hypothesis been retained, no error of an 
 order lower than the (7 +7)th will have been committed; and as terms 
 of this order have been considered as inappreciable, the possible errors 
 will themselves also be inappreciable. On the other hand, if after having 
 retained no quantities of orders higher than the (n +7)th it be thought 
 desirable to introduce those of the (rn +7+ 1)th, a similar train of 
 reasoning will be applicable, and no theoretical difficulty will arise. It 
 has now been shown that the usual method of differentials 1s a conve- 
 nient one, that if it be adopted terms of certain orders must be 
 neglected, but that in neglecting these terms no error of any order 
 lower than the lowest neglected order can be committed ; the extent of 
 the accuracy of the investigations is consequently always determinate, 
 and the method when carefully pursued is free from danger. 
 
 When the orders of differentials follow those of the infinitesimals, as 
 
 indicated above, they will be denoted by the symbol d, so that the 
 I 
 
58 
 
 successive differentials of any quantity or function wu will be thus 
 
 denoted 
 la eid, te Se 
 
 or as they are more usually written, 
 
 du, Pu, Bus... s. (1) 
 Since we are best acquainted with quantities which are called finite, 
 and less so with those which are indefinitely greater or indefinitely less 
 than these, it has been thought advantageous to commence the notation 
 of the powers of the symbol d from this as their zero point. Now since 
 the powers of any symbol (as in the case of d) may be increased in a 
 negative as well as a positive direction, without limit, it will be necessary 
 to determine the meaning of quantities when affected with the symbol 
 of operation d raised to any power. 
 
 It must here be recollected that we have not proved, nor do we 
 assume, that the law of the operation indicated by the symbol ¢ follows 
 in general the index or any other law; but since it is known 4 priori 
 that any mathematical symbol (be it one of quantity, quality, or any- 
 thing else) is liable to be subjected to any algebraical operations, the 
 same must be the case with the symbol now under discussion, namely d ; 
 the question therefore now is, what operation does the symbol dz repre- 
 sent, (1) when 2 is a positive integer, (2) when it is a negative integer, 
 (3) when it is fractional. Now the first case is easily determined, for from 
 the manner in which the differentials have been formed, i. e. from their 
 very nature, we see that the symbol d does follow the index law when 
 the index is a positive integer; and this also because we have not given 
 the term differential or differential coefficient any meaning which will 
 require to be afterwards further elucidated (as, for instance, differential 
 coefficient has been defined to be the coefficient of the first power of h 
 in the expansion of f(x+h) in a series of ascending powers of h), 
 but have shown the actual connexion with the operations of differen- 
 tiation with the fundamental operations of all algebraical processes. 
 For this purpose it is easily seen, that in order to pass from the dif- 
 ferential of any quantity of the mth order to that of the same quantity 
 of the (n+1)th order, it is sufficient merely to write dd*u, i.e. d**'u, for 
 du, i.e. multiply the differential of the mth order by the symbol d. 
 Hence, in the method now adopted, the raising of the index of the 
 symbol d by unity turns an infinitesimal of the mth into one of the 
 {n+1)th. Similarly, m order to pass from the differential of any 
 
59 
 
 quantity of the nth order to one of the (n—1)th, it is sufficient merely 
 to divide the differential of the nth order by the symbol d. Hence, in 
 the method now adopted, the decreasing of the index of the symbol d by 
 unity turns an infinitesimal of the mth order into one of the (m—1)th. 
 In order however to explain more fully the nature of the operation 
 
 mdicated by the symbol d7’ or s, it will be necessary to revert to the 
 
 original formation of the quantity du. It was above seen that the 
 various values which can be given to du are formed by taking the 
 differences of the arbitrary values of the given function which lie 
 between those limits within which the function remains continuous. 
 This being the operation indicated by d, it is now required to determine 
 the nature of the inverse operation indicated by d~’. But since by the 
 operation d the quantity under consideration, w, was divided into a 
 number of indefinitely small portions, whose general type was dw, and 
 the sum of which was the difference of the values assumed by w at the 
 given limits, the required operation will be to deduce from any given 
 differential, as du; for instance, the quantity w,; this will obviously be 
 effected, if to the inferior limit of w (w,) there be added the sum of so 
 many of the differentials whose form is du as will be equal to the dif- 
 ference between wu, and u,;; or, if more convenient, by subtracting from 
 the superior limit w, the sum of so many of the differentials as will be 
 equal to the difference between wu, and uw, This result may also be thus 
 analytically represented : 
 
 Suppose uw to be any function of x, so that 
 u = f (2) 
 f'(2) = f(e+ de) — f(x) = f(z) 
 f'(@) =f(e + 42) — f(z) 
 
 f(atdz) =f(x+2dzr) — f(r+dz) 
 fi (a2+2dz) = f(x+8dz) — f(x +2dz) 
 
 also let 
 
 * But 
 
 f(eti—ldz) = f(x+idr) — f(x+i— dz) 
 Consequently 
 S'(z) +f'(2+dz) +... + f'(«+i—1dz) =f (z-+ tdr) —= f(x) 
 hence | ines 
 f(z +idz) = f(z) + f(z) +f (e+dz) +. . ++ f(eti—1dz) 
 
 12 
 
60. 
 
 Now if from the superior limit of f(x) (f(#,) suppose) there be sub- 
 tracted the inferior limit (f(2,) suppose) the result viz. f(7,) —f(.«,) is 
 called the definite integral of f’(.7); suppose moreover that 
 
 Doe ae ees Ae 
 then 
 
 ST (an) — f (20) =f (to) + f'(%o + dz) #.. 2 +f" (zt. + n= 1dz,) (2) 
 
 that is, the definite integral of any function is equal to the sum of the 
 values assumed by the differentials of that function between the given 
 limits, inclusive of the inferior and exclusive of the superior limit. Such 
 then is the nature of the operation indicated by the symbol d~’; and it 
 is also clear that the same may be repeated any number of times without 
 limitation; hence it will not be necessary to consider further the nature 
 of the operations indicated by the symbols whose general type is d~”, 
 where m is any integer. A similar train of reasoning would show that 
 the symbol d may be raised to any positive or negative integral power, 
 and any root of such power be taken, and that the result will always have 
 an intelligible meaning; thus, for instance, if d be raised to the pth 
 power the result willbe d@’, and if the gth root of this result be taken 
 
 the final result will be dt, and if 
 
 13 
 {| 
 3 
 
 we should have 
 
 dy =d’ 
 that is, the result will be the same as if we had arrived at d’ by a direct 
 process. 
 
 The meaning of the operations indicated by the symbols d* d~”, where 
 nm is any integer, having been determined, the following question also 
 suggests itself; what interpretation is to be given to a symbol of the 
 
 form d=? In the first place it may be remarked that a quantity affected 
 with the symbol d raised to any power (the mth for instance) is of any 
 order of infinitesimals one degree higher than the same affected with the 
 symbol raised to the (n—1)th power. Now this it must be remembered 
 was a particular method which it was thought desirable to employ. 
 Suppose, however, that the symbol d denote the operation of differen- 
 tiation when there is no limitation respecting the connexion of the 
 
61 
 
 orders of differentials and infinitesimals, then, as before, the results of 
 successive operations of this nature on any function wu will give the 
 series 
 
 isn Gate vOGdzs ct. ass 
 or as they may also be written 
 
 Cue Ce United chek ts (8) 
 in which any conceivable relation may exist between the symbols 
 d and d; but among all of these the one with which we are now con- 
 cerned is that in which 7 operations of the nature d are equivalent to m 
 operations of the nature d, i. e. 
 
 da = 0" 
 or as it may be written symbolically 
 di=d™ or d = d*4, 0rd" = d (4) 
 
 Now it is observable from equation (4) that by performing the 
 operation d 7 times any given quantity w is changed from a finite 
 quantity to an infinitesimal of the mth order, or that by performing 
 the operation d once wis changed from a finite quantity to an infini- 
 
 tesimal of the — order. This is what was required. Now, although 
 
 we have arrived directly at infinitesimals of fractional order, yet it 
 must not on that account be expected that instances of such quantities 
 can be given; for on account of our inability to determine the exact 
 limits of finite and infinite (whether infinitely large or infinitely small) 
 quantities, we cannot determine whether any given quantity lies at 
 
 “th or 1th part of the distance from one to the other. But the same 
 m 
 
 reasoning which shows the possibility (mdglichkeit) of the various 
 orders of infinitesimals involves also the necessity (nothwendigkeit) of 
 all orders, and consequently of fractional as well as integral. Thus 
 analysis, being the representation of real laws, gives as some of its 
 results these partial infinitesimals, as they may be called; which, since 
 they lie between two given orders of infinitesimals, must have a real 
 existence, although there is an apparent indeterminateness about them. 
 But this indeterminateness is itself a result of the absolute magnitudes 
 of the various orders of infinitesimals which have been chosen as near 
 together as was possible without destroying the general distinctness of the 
 various orders. _ From the actual existence of these partial infinitesimals 
 
62 
 
 it follows that they may be used in any analytical expression with no 
 jess propriety than the total infinitesimals; and although the former 
 are not in such general use as the latter, yet there are some cases in 
 which they may be employed with great advantage. 
 
 [The following is an example; in vanishing fractions (i.e. when for 
 a given value of x, as 
 
 the fraction 
 £(a) 
 S(@) 
 
 takes the form © ) certain cases present themselves in which, however 
 
 many times the numerator and denominator may be differentiated, the 
 value of the fraction still appears indeterminate, so that 
 
 
 
 
 
 df (zx) d’f (x) 
 Vl) ae ee uO 
 
 df(x) — 0 d(x) a, 0’ 4. e+e fe he oe 
 dx dx? 
 
 The readiest way to evaluate these is to find the general forms of 
 d"f (x) d"f(z) 
 ieee a ies 
 and then to give n some fractional value, the result will then be found 
 determinate. ] 
 
 From the manner in which differentials and. integrals have been 
 formed, viz. by subtraction and addition, it is obvious that the order in 
 which these are performed when any function is affected with symbols 
 of both operations, is indifferent, so that 
 
 anpru = frau (5) 
 
 Hitherto functions of only one variable have been considered ; but as 
 there are some points in which functions of several variables differ from 
 those of one variable, it will be worth while briefly to notice them. 
 Consider then the function of any number of variables 
 
 FT (is Ys Estee eee 
 the total differential of this is obviously 
 f(at+dz, yt dy, z+dz,..-.) —fl%y, 2%». -) 
 
But as in certain cases it may 
 Une. «: - 
 
 63 
 
 happen that only one of the quantities 
 
 . may be considered variable while the rest remain constant, 
 
 the partial differentials so formed will be represented by the symbols 
 
 JS (Fy Ue eek one) 
 
 (a+dz, yt+dy, z».. +) —f(x, ytdy, z -.+ +) 
 
 eee L(rs Ue Ze oh on) 
 
 doh wih (ts Ya Zs « + +) 
 - -) —f(e+dz,y, z .. ») 
 -) + f(x,y, 2 « » +) 
 
 dj dy des 
 the suffix denoting the quantity which is considered variable: 
 Now, 
 d,f (a, Y;% » » +) = f(z+ dz, y, 2; . 
 also 
 df(t, I, % -- =f 
 — fir ar, Y Ze 
 Again, 
 dy f (as Ys 2 > © +) =f ytdyy2s - 
 Ady f(x, Ys 2 + + +) = f(xtdz, ytdy, Zz, - 
 — f(a y+ dys 2%» + 
 hence 
 
 dd. f (23 Y; 250% + 
 
 oP df (X45 2 « ++) 
 
 Similarly it may be proved that 
 
 Addy « «+ fl%s YZ 
 or symbolically 
 
 (EE LEAT EL 
 
 Oaee rat, a Las Ue Suh ote- s) 
 
 (6) 
 
 (7) 
 
 where P represents the continued product of all the symbols taken in 
 
 any order whatever. 
 written 
 f(x + dz, y+ dy, z + dz 
 = f(z + dz, y + dy, z+ dz 
 + f(t, y+ dy, z+dz...) 
 + f(a, y,2+ dz...) 
 
 a ea ae a ee ee ee ee eee 
 
 But 
 S(2+dz,ytdy,z+dz. 
 
 =d[ f(x, y+ dy, z+ dz. 
 
 consequently 
 
 f(a + dz, y+dy,z+dz. 
 —f(x« + dz, yz+dz. 
 = d,| fle + dr, y,z + dz. 
 
 Again, the total differential of f may be thus 
 
 oe) —f (2 Ys % - re} 
 ~~ f(t, y+ dy,z+dz...) 
 — f(t, y,2+ dz...) 
 — f (Xs Ys 2% + + +) 
 
 Se &5 @. 6) Of 6. 6. 6 
 
 — f(%s Ys Z + + +) 
 
 ..)—f(, y+ dy, z+dz...) 
 
 } 
 
 (8) 
 
 — 
 
 ~)—f(e y+ dy z+dz. mat 
 -)tf(ay2z+dz...) 
 - )—-alf(z,y 2+ dz...)] 
 
 = dd, f(t, f, 2 + dz. -) 
 
 which is an infinitesimal of the 
 
 second order; consequently repeating 
 
64 
 
 the process for the other variables, it would be at length found that the 
 expression 
 f(z + dz, y+dy,z+dz...)—f(z,y + dy, z+ dz...) 
 = df fl2,.y + dy, 2i4,dz ."..)] 
 differs from | 
 fe t+ dn Qe) — fly, 2e ee) = alfa ee 
 only by infinitesimals of orders higher than the first, and similar results 
 would be all deducible for the other terms of (8); consequently, as far 
 as infinitesimals of the first order 
 f(at+dtytdyz+dz...)—f(%y2z.. -) 
 = f(x + dz, ey, Ets) — J (te tecr ie) 
 + Rep y dy, zit ai as paz ae e) 
 
 or writing 
 uF (x, 9, So0 s ) 
 du=du+tdu+t+dut+... 
 or multiplying and dividing the several terms by dz, dy, dz, . 
 respectively, there results the usual formula 
 
 du du du 
 mt beth, ath Fa ae a te eate Che 
 Du ae z+ dy ly + a, + 
 ‘ du , 
 where D represents the total differential and dg: the ratios of the 
 
 partial differentials of w (considering as variable only those quantities 
 whose differentials appear in their respective denominators), and the 
 differentials of x,y, 2. . ., or as they are usually called the partial 
 differential coefficients of wu with respect to 7, y, 2... 
 
65 
 
 Examples of the Application of the Infinitesimal Calculus. 
 
 Tue following are some miscellaneous examples of the application of 
 the above methods to actual practice. 
 
 i 
 
 To find dy in the following cases, calculating the expressions as far 
 as infinitesimals of the first order only. 
 
 yrHarer, 
 *, dy = (a+a+dz) — (a+z) = dz. 
 y=a—?, 
 “. dy = (a—u—dz) — (a—z) = —dz. 
 Yaa ae, 
 “. dy = a(x+dz) — av = adz. 
 a 
 ie ae 
 Me Tepe es a adx 
 ORS TE dat ge a(a + da) 
 adx dz adx 
 =— —_(]—— oe re Ve mh ce 
 x ( mi ) ao 
 eee 
 “. dy = («+dz)*— x* 
 = axt—"dz + GE a eae ia = ON. 
 Time Os 3 
 o's Gif = atte gt = af(a*—1) 
 2 
 = a*(log .adz Sees +...) = log.a.a‘dz. 
 
 ae diy = TE a G% = a (alt 1) 
 
 2 
 = e(de+ + OY dr, 
 
 K 
 
66 
 
 y = sin 2, 
 .. dy = sin (x + dr) — sing 
 = sin x (cos dxr—1)+ cosa sindr 
 
 = —tsinz sin*dx +...+cosz sindz 
 = cosx sin dx = cos zdz, 
 since cos dx = /1—sin*dx = 1—4sin®”dr +. . ,, 
 and sin dx = dz, approximately. 
 y = cos@, 
 
 .*. dy = cos (w1+dz) — cos x 
 = cos x(cos dx—1)—sin z sin dx 
 = — sing sindr = —sin zdz. 
 y = tana, 
 .'. dy = tan (v+dz) —tanz 
 _ tana + tandz 
 ~ I—tana tan dz 
 1 + tan *z 
 1—tanz tan dx 
 
 — tan z 
 
 = tan dx = (1 + tan*z) dz. 
 
 S 
 Il 
 Ns 
 
 _a+dx x _ zdx—adz _ zdx—xdz 
 
 2+ den) eit 2(z+ dz) io eae 
 
 
 
 I].—rTAYLor’s THEOREM. 
 
 Ler 
 Ua (ey, ce ae) 
 be any function of the variables xyz... .; and let 
 Ly Ys Sys 
 receive increments 
 AL, dYpleys = 5 
 respectively, and let 
 wu’ = K(a+dz, y+dy, z+dz....) 
 If then we consider the increments 
 dx, dy, dz, . 
 as infinitesimals of the first order, and 
 Du 
 
67 
 
 a corresponding quantity relative to the function wu, calculated as far as 
 the first order only, we shall have as a first approximation 
 
 uo —u= Du (1) 
 
 or 
 u=u+ Du (A) 
 But since the function w’ may involve the products of the differentials 
 dG dy dz 4+. - , the quantity Dw will not represent accurately the 
 
 difference wu’ — uw, but will require certain terms of higher orders to be 
 added to it. Hence as a second approximation we shall have, in a 
 similar manner 
 wu —u— Du=D(Dz) = Du 
 or 
 u =u+ Du + Du (2) 
 
 Now neither of the values of w’ given by (1) or (2) is a correct one, 
 and that given by (2) may be either too great or too small (according 
 to the signs of the succeeding terms) ; hence we must take as the most 
 probable approximation the average of the values given by the several 
 approximations: Hence adding (1) and (2) and dividing by 2 (the 
 number of approximations), we have for the next value of w’ 
 
 Dit Dx 
 
 iad ehagairse @) 
 
 Treating these in a similar way we shall have next 
 Du _ Du _p Duy Lu 
 
 eae OW ANT O/,  12 
 or 
 ; Dien e tu one 
 = | — 
 u ut Tae aio a 1) (3) 
 
 and, as before, adding together (1) (2) (3), and dividing by 3, we find 
 Du , D?u D®u 
 
 
 
 OT Me ae See ere (©) 
 Similarly the next value would be 
 A? Dai D*x D®u D*u 
 ere) MO sl 5.5.4 Oo 
 until we should have as the mth approximation 
 ae Die Du D*u D"u 
 I Ue otis a oS * 1.8.90. sn Gu 
 
 Kk) 2 
 
68 
 
 Moreover, carrying on the approximations in a similar manner ad 
 infinitum, the remaining terms of the series giving the accurate value of 
 w’ in terms of w and its differentials ae be 
 
 : De ape Be eee te \ 
 n 
 
 Lee Dis): ky OD a (n+ 2) (n+3) 
 which quantity is evidently 
 1 n+ 
 AOKI OST 
 and 
 1 n+1,,/ 
 SON Pama (TORE 
 1G. 
 1 n+1 
 1 D3 cs oe Uhal) s Oke 
 and 
 
 “¢ D*t!F (2+ dz, y+dy,z+dz....) 
 
 1.2.8 :.4006 nda) 
 Hence if 6, 6’, 6” ... . be some quantities between O and 1, the 
 
 remainder will be accurately represented by 
 ] 
 
 
 
 a See Sy 6 6’ De shee. 
 [2.8 A ee eee 
 and the correct value of w’ in terms of w and its differentials will be 
 u’ = F(r+dz, y+dy, z+dz... .) 
 eal So af, 2 one at + DE (sy, 2 ea) DF (2,9, 2 Set) Ova 
 oe th | aes 
 ] 
 
 typ t te ee ee ac 
 
 os D+F (2+ dz, y+ 6’dy, 2+6’dz .. .) 
 
 leet ee (nd) 
 Where by the usual rules of differentiation 
 du du du 
 oe Pedy shoe 
 Di = ie gag 
 
 4 PU 72 ch 
 
 
 
 
 
 
 
 
 
 ome i ey ody 
 Diu = — de" + 7 
 
 
 
 du du du 
 OU dydz +  dede +. 
 Wyte | a. ideds eal -) 
 
 ee 
 
69 
 
 tee 
 
 From the above formule we may deduce an extended form of 
 Maclaurin’s theorem ; putting 
 r=0,y=0,2=0.... 
 Chie ae Oo A at Ug 2 mat aL oe a 
 
 and 
 d d d 
 Pint Eada sich TOO 
 we have 
 ] 1 t 
 Fay z...)=FO)+ VF) + 7, VF) +... 
 1 1 
 —— "F(0 ee eee eI MAIR (Oa by: OS, 
 De sree a Meno. 8s TORE F(Gx, Wy, 82. « .) 
 
 
 
 L[V.—LAGRANGE’S THEOREM. 
 
 To expand F(y) in a series of ascending powers of «, where 
 
 pas tery) (1) 
 (1.) As a first approximation, let 
 p= (2) 
 this gives 
 eo (3) 
 and consequently 
 F(y) = F(z) (4) 
 
 (II.) In order to make a second approximation, let the value of y 
 given by (3) be substituted in (1); this gives 
 
 y =2 + af (2) (5) 
 and consequently 
 F(y) = Fiz + of(z)} (6) 
 or by Taylor's theorem 
 Fly) = P(z) + of (2) F(z) (7) 
 
 which gives the value of F(y), as far as terms of the first order. 
 (III.) In order to make a third approximation, let the value of y 
 given by (8) be substituted in (1); this gives 
 
 yo2t f{z+ af(z)} } (8) 
 =2+4+ 2f(z) + vf(z)f') 
 
70 
 
 expanding by Taylor’s theorem as far as terms of the second order. 
 Consequently, instead of (7) we shall have | 
 
 Fly) = Fe + af(z) + of (2)f"(2)5 | 
 
 = Fl) + of Fe) + fe) @P +7 5l/l@)PF'2) (9) 
 expanding by Taylor’s theorem as far as terms of the second order. 
 This may also be written thus, 
 
 F(y) = F(2) + f P(e) + DALE}. (10) 
 
 (IV.) In order to make another approximation, let the value of y 
 given by (8) be substituted in (1); this gives 
 
 y=zt af{z t+ af(z) - aS (2)f (z)} ; 
 = z+ xf(z) + 2f(z) f(z) + ef (z)(f(z))? + 3 (flz)) 7h" (2) (11) 
 
 == + ole) + EY + *D.AMDFE) | 
 
 expanding by Taylor’s theorem as far as terms of the third order. 
 Consequently instead of (10) we shall have 
 
 = Fle + af(z) + 2%fl2f'(e) + a Lf) (2)} | 
 = F(z) + af (z) F(z) + xf(z ue (z) F(z) +, D,[( (fer (z)]E"(z) 
 
 + a ))?E"(2) + PEPE) +1 MDPOE 2) 
 = Fe) + ofl)F(2) + 55 Serta 2 _paxriaF@) | 
 
 and soon. Suppose that by continuing the approximations there had 
 
 been found 
 3? e L(f(2) )SF’(z)] 
 (13) 
 
 (12) 
 
 
 
 
 
 
 
 Fy) = + 5f(2)F( (z) + aa D, AS (2))F')1+5 
 
 Vee) of cas: "17 ( f(z) a 
 
 
 
 then for the next approximation we should have the general form of the 
 
 series (5), (8), or (11); viz. 
 sDLMNF (2)] +. + -- cl 
 (14) 
 
 
 
 y=2zt af(z) + x f(z) f(z) + 
 ct ep Bite 2 oa 
 
71 
 and this being substituted in (1) will give 
 Fly) = Fe) + 24 fe) +S feyf'@ + 2 DSO FE! 
 teed DPE (OIL PE) 
 + Sf re) + FrOso +4 DLL FO) 
 Derr rah Fe 
 
 
 
 feet 5S DI rorr ny F(z) 
 
 D.LFENFED 
 DF) £0) ay Fer#N(2) 
 
 
 
 ye ea ) +2 fle)f 
 
 Feee + 
 
 
 
 | 
 EP yal | (15) 
 +p aa SO +FfOS') + DAGON! | 
 
 xz” 
 Le Dackans 
 or writing for convenience 
 
 
 
 J (2) =u 
 du aw dyn! 
 F(y)=F = gene, Poe a Og OT \ / 
 WF) tay ttrat M93 hot as. itety fon 
 x du? cane drynt} : 
 eae rs eae | Ry 
 i ls ope ocs weve eet. 2.5,. aah} (?) 
 Ae A 16) 
 2 du? Pus Cur 
 ——— —— fe ae n(n) 
 +rescal tT Se SP were ta eH maaGEll yi 
 grt us dyn) n+l 
 pameeeee ee ro iy = ar Fe (z 
 +123. ean" tigate ‘F seat (2) 
 But writing for convenience 
 du* due 
 geen gS — O 17 
 a oe baa aa Ne 
 
 it will be found by ordinary processes that 
 yt 1 _— uy” +I 
 du™*+! = (n+ 1)u"du 
 _n+1 
 
 
 
 x coefficient of x in the expansion of 0* 
 
 d?u"+1 = (n + 1){nu"—1(du)? + u"d?u} 
 
72 
 
 — 
 
 - x coefficient of 2? in the expansion of 07-! 
 
 Ss 
 
 
 
 s 
 — 
 
 
 
 1.2 
 deur *) = (n + 1) {n(n — 1)u*-?(du)® + Snu”—"dud?u + u*dut 
 
 n+1 , : : 
 + x coefficient of xin the expansion of 1"? 
 
 
 
 
 
 —na—2 
 1.2.3 
 . at 1 F , 
 so that the coefficient et ae in the (n+ 1)th line of 
 (15) will be 
 FO) (z)(f(z))"*" 
 Spd Be Aitg 3) Ab in the nth 
 F(z) D,(F(2))?* 
 oe eds Net Woks aoa in the 3d 
 n= Dp (De f(2))"*" 
 vSels ie Ne he tote ds aed ce te eae ee in the 2d 
 F’(z)D n 7 (2) Pt 1 
 Hep. oie ts ae ee. in the Ist 
 BY(2)DA(f(2))"*? 
 i i 
 in the expres- 
 
 consequently a whole coefficient of 1. S33 anda (aa) 
 
 sion for F(y) will be 
 
 FDNSOY + T PED (fey 4-4 
 apnea aoe 
 
 n. ee ee z)Dy"( f(z)" + wb 
 
 which by Leibnitz’s theorem 
 = DA {(/@OY"""F'(} (19) 
 
 so that the series (13) is generally true whatever be the value of 7 
 
713 
 
 V.—VANISHING FRACTIONS. 
 
 Ir wu be a function of x of the form 
 f(z 
 f(x) 
 where f(x) and f(x) are any functions of w, which remain continuous 
 between the limits 
 
 (1) 
 
 a 
 
 ea) 
 
 zx=aandr=a+e 
 eee 2) 
 exclusive of the first of each pair of values, it sometimes happens that 
 for certain values of the variable x, such as 
 aes (3) 
 the functions f(x), f(7) sensibly vanish; so that, if U be the value of w 
 when (3) is satisfied, | 
 f(a) _ 0 
 U= ete 4 
 fla) ~ 0 . 
 and the value of U is consequently indeterminate ; the question then 
 arises how the value of U is to be found. Now, according to the present 
 theory of infinitesimals, the values of f(a), f(a) will not in general be 
 absolutely zero, but merely infinitely small, so that the solution of the 
 problem depends upon the determination of the orders to which the 
 functions in question respectively belong. Ifthe order of f(a) be higher 
 than that of f(a) the value of U will evidently be infinitely small; but 
 if the order of f(a) be higher than that of f(a) the value of U will be 
 infinitely great ; in fact, if f(a) be of the order m, and f(a) of the order 
 n, U will be of the order 
 
 
 
 m—n 
 and this will give rise to three cases, 
 
 (1) Ifm > n, (m—n) is positive, and U will be infinitely small, 
 
 (2) Ifm < n, (m—n) is negative, and U will be infinitely great, 
 
 (3) Ifm=n, (m—n) = 0, and U will be finite. 
 
 Now, since the functions f(x), f(#) remain continuous between the 
 limits determined by the equations (2), an approximation may be made 
 to the values of f(a), f(a), by assigning to x some value a little greater 
 or less than a contained between the above-mentioned limits. If then 
 
 L 
 
74 
 
 - be an infinitesimal of the first order at least, the value of U will be 
 given nearly by one of the equations. 
 _ f(a+e) _ f(a—e) , 
 © = Aare)" ” = Rla=e) o 
 Now, by Taylor’s theorem, 
 
 
 
 
 
 (ate) = f(a) +-0(a) + a eer eve | 
 (6) 
 fare) =fla) +f @+ pal O+---- 
 and | ; 
 f(a — s) =) f(a) — ££" (a) 4 22 f(a) =v) 
 Lae m 
 fa-2) =f) ~ Ff) + poh" ---- 
 
 Hence if these values be substituted for f(a+<), f(a+e), f(a—s), 
 f(a—<) respectively in (5), we shall have (remembering that by the 
 hypothesis 
 
 f(a) = 0 J (2) = 9) (8) 
 
 f’ (a) +f" (a) oe an 
 [center ee srenenenneermen (9) 
 F(a) +f" (a+... 
 
 or 
 
 f’ (a) — 5” (a) sel? 
 Uo (10) 
 f(a) ~2f"(a) +... 
 from both of which expressions there would result, if infinitesimals be 
 neglected in comparison with finite ones, 
 f’(a) 
 = i i 
 Fe) wi 
 If the first derived functions themselves vanished, and in consequence 
 of the conditions 
 
 
 
 fMa)=0 f(a) =0 (12) 
 the expression (11) became indeterminate, it would be necessary to 
 take in another term in the expressions (6) or (7), im which case 
 (9) or (10) combined with (8) and (12) would give 
 
 Lea 
 
 U =v 
 KO ve 
 
75 
 
 And generally, if the derived functions up to the (n — 1)th vanished 
 together with the functions themselves, the value of U would be given 
 by the equation 
 
 a £07) ( a) 
 Pua) 
 It may here be remarked that the conditions (8) and (12) and 
 generally the conditions 
 
 fP@=0 fM(a)=0 (15) 
 
 represent (as is known by the theory of equations) that the equations 
 (8) have each (2 +1) equal roots, a factor corresponding to each of 
 which is expelled by each differentiation; so that the degree of the 
 factor (which in the present case will be an infinitesimal one) is con- 
 tinually reduced by unity, until when it is integral, it at length becomes 
 actually 0; in which case, if there be no other infinitesimal factors, f"*”(a), 
 
 f“*(a) have finite values, and the value of the fraction is determinate. 
 
 Sometimes, however, it happens that the index of the degree of the 
 infinitesimal factor is fractional, so that, although it is continually 
 decreased by unity, it never vanishes, but passes from positive to nega- 
 tive ; so that the fraction always appears under one of the indeterminate 
 forms 
 
 
 
 (14) 
 
 Swe (16) 
 
 and its value consequently cannot be determined by the above process. 
 Suppose then that the order of the infinitesimal factor is fractional and 
 
 = '" and suppose that 
 n 
 
 mMaprtl 
 n Pp 3 2 a!) 
 where p is a whole number. Then by p ordinary differentiations 
 
 the order of the factor will be reduced from — to 4, and it is then 
 
 required to reduce an infinitesimal of the order / to a finite number, or 
 n 
 
 in other words to decrease the order of the infinitesimal by the ith part of 
 
 unity; this must be done, as was seen in a former paper (on Infinitesimal 
 
 Analysis), by choosing « of an order =“. But as it was seen in the 
 n 
 
 L 2 
 
76 
 
 same paper that this can scarcely be called practicable, the general 
 
 formule for 
 
 d, fx d’ f(x) 
 Sid and mS (18) 
 
 
 
 must be found, and afterwards 7 must be put equal to z. This, according 
 
 to the principles of infinitesimal analysis, will produce the same result. 
 
 VI. 
 
 Der. <A plane is a sphere whose radius is infinitely great. It is 
 proposed to find the analytical expression for a plane according to the 
 above definition. Let the equation to the sphere be 
 
 (z—a)* + (y—5)" + (ze) = (1) 
 or 
 ety + 2% —Qar+bytez) + 0+P4+C=P°r (2). 
 
 Now in this expression it is obvious that a, 6, c, the co-ordinates of 
 the centre of the sphere, are of the same order as r. Hence dividing 
 by r? 
 
 2 2 2 2 24 22 
 am el a 2 (ar + by +c2) + ai aes =] (3) 
 Ts rT 72 
 which, if we neglect infinitesimals of the second order, becomes 
 az + by+cz= sla +e +e—2) (4) 
 or, writing for convenience 
 e@+Ph+e=k (5) 
 rs, +oz= Ay (pee ri) (6) 
 kin oh eer . 
 
 Now the second side of this equation may also be written thus 
 
 on 
 
 and 7 and & are approximately equal to one another, because the origin 
 is supposed to be ata finite distance from the surface ; also (k—r) is 
 the shortest of the two perpendiculars from the origin upon the surface, 
 and if we represent it by p (6) may be thus written 
 
 a 
 
 b c 
 pa OY tae ae (8) 
 
 k 
 
we 
 
 m which also 2 ib are the direction cosines of the line joining the 
 
 origin and the centre of the sphere, and consequently also of the perpen- 
 dicular (p) on the plane. This equation may also be put under another 
 form, for if «, 6, y be the co-ordinates of the point in which p meets 
 the surface 
 
 p=la+mB + ny (9) 
 where 
 etme. fet (10) 
 
 and consequently (8) becomes 
 
 Kx—a) + my—B) + n(z-y) = 0 (11) 
 If the origin be taken on the surface we have 
 =. 01810. = 0 (12) 
 and consequently 
 Ix + my + nz = 0 (13) 
 
 the equation to a plane passing through the origin. 
 
 There is one other form of the equation worthy of notice, which is as 
 follows ; dividing (8) by p throughout we have 
 
 i y a oe} 
 
 an oe (14 
 a b c 
 or writing for convenience 
 k k k 
 f=?, g=%, ha? (15) 
 te Yom = 
 oe ef ee |b (16) 
 ee. See 
 
 in which f, g, A are evidently the projections of the perpendicular p on 
 the three co-ordinate axes respectively, or the intercepts of those axes 
 between the origin and the plane. 
 
78 
 
 VIL. 
 
 Der. A point is a sphere whose radius is infinitely small. In this 
 case we may neglect 7, and consequently 7”; hence (1) becomes 
 (x—a)? + (y—b)? + (z—c)? = 0 (17) 
 vhich is satisfied only by 
 Ci ay == Ome, (18) 
 These are, therefore, the analytical expressions for a point. If the origin 
 be taken at the point, we have 
 
 a=0,3=0,c=0 (19) 
 and consequently 
 z=0, y=0, z=0 (20) 
 VIII. 
 
 Der. Two straight lines are said to be parallel when their point of 
 intersection is at an infinite distance. 
 
 Among the various ways in which the analytical expression for this 
 definition may be found the two following appear worthy of notice. 
 
 Suppose first that the two lines be not nearly parallel to any of the 
 three co-ordinate axes. 
 
 Let the equations to the two straight lines be 
 
 
 
 i m n 
 in which expressions 2, y, z are the co-ordinates of the point of inter- 
 section ; combining (1) and (2) we have 
 
 
 
 
 
 e— all yt ban ele, (3) 
 t—adl y—Um z—ecn 
 
 which may also be written thus 
 
 
 
 
 
 
 
 i--—, j UP aaa 
 1 a sy eee et 
 ae y “a 
 
79 
 
 Ifthen the point of intersection be removed to an infinite distance, 
 the co-ordinates x, y, z will become infinitely great, and the ratios 
 
 Gi ee Ged Che 
 
 Tec ee ye NZ 
 infinitely small; they may consequently be neglected in the expressions 
 (4), and the conditions of parallelism become 
 
 or 
 
 as usual. 
 
 If the two lines be so placed that one of the co-ordinates of the pomt 
 of intersection does not become infinite, the above method fails; and 
 although by a transformation of co-ordinates the difficulty might be 
 obviated, yet it will be simpler to treat the question thus : the equations 
 to the two lines may be written in the following form, 
 
 t=ac+lr y=b+mr Z=c+nr (7) 
 
 asa +r = Db! + mr’ z=c4+ nr (8) 
 
 in which expressions, 7, 7’ are the distances of the point of intersection 
 
 from the points of a,b,c and a’, b’, c’ respectively. Subtracting (8) 
 from (7) we have 
 
 6b — b' + mr — m’'r’ = 0 
 e—c’ + nr—nr =0 
 
 a—-a+fr—Tr =0 
 ; 
 
 or, neglecting a, 6, c, a’, 6’, c’, in comparison with 7 and 7’, we find 
 
 as before. 
 
80 
 
 IX.—To FIND AN EXPRESSION FOR THE RADIUS OF ABSOLUTE CURVATURE 
 IN CURVES OF DOUBLE CURVATURE. 
 
 Der. The circle of absolute curvature is a circle which passes through 
 three consecutive points of the curve. 
 
 If the co-ordinates of the first point on the curve through which the 
 circle of curvature is to pass are 
 
 Ly Ys Zz (1) 
 those of the consecutive point will be 
 a+de ytdy z+d (2) 
 
 and those of the third 
 
 a+ dx + d(a+dr) =x + Qdx + d*x 
 14 ty + dy sty)=y +20 + ey} (3) 
 z+ dz + d(z+dz) = z + 2dz + d*z 
 Also let the equation to the sphere of which the circle is a section be 
 (z—a) + YB) + (2—9)° = (4) 
 then since it also passes through the consecutive point, the equation 
 (4) must be satisfied by the co-ordinates (2), in other words the following 
 equation must also hold good, 
 
 (w+du—a) + (y+dy—By + (z+dz—y) = p* (5) 
 and similarly also the following one 
 (2+ 2dx + d?x—a)* + (y+ 2dy+d*y—B) + (2+2dz+d*z—yy = p® (6) 
 Expanding (4) and (5), and subtracting one from the other, there 
 results, and writing for convenience 
 ds? = da? + dy? + dz’ (7) 
 there results 
 
 (2—a) + (yh) + (2-9) Ga-F (8) 
 
 or, neglecting infinitesimal quantities, 
 
 (ea) + y—h)M + (e-n) F = 0 (9) 
 
81 
 
 Also, expanding (4) and (6), and combining (8) with the result, 
 we have 
 
 
 
 a ete Z 
 : 10 
 ty tied wads es wt mee (dx)? + (dy)? + (2z)"} 
 or neglecting infinitesimals of the ae and gee orders, 
 
 (z—a) = e ~s) 42 b+ (ei =v l (11) 
 
 Again, since the circle is a eens curve its op) ankles must satisfy the 
 equation to a plane as well as the equations (4), (5), (6); let this be 
 
 U(z—a) + m(y—B) + r(z—y) = 0 (12) 
 
 then since this plane passes through the points (2) and (3) also the 
 
 equation (11) must be satisfied by the co-ordinates of those points ; 
 
 hence, besides (11) the two following must also hold good, 
 
 l(a+dzx—a) +m(y+dy—f) + n(z+dz—y) = 0 (13) 
 
 Ux+2dx+da—a) + m(yt+2dy+d°y—B) + n(z+2dz+d’z—y) =0 (14) 
 
 The combination of which last three equations produces the two followmg 
 
 Idx + mdy + ndz = 0 (15) 
 Id?x + md’y + nd*z = 0 (16) 
 
 eliminating /, m, , in turn from these, we find 
 Ee ‘ (17) 
 
 dyd’z — dzd’y ~ ded: — dad’z  dad’y — dyf’x 
 eliminating /, m, n, between which and (11), we have 
 (a — a) (dyd?z—dzd*y) + (y—) (dzd’x—dad’z) + (z— y)(dxd?y—dyd’x) =0 (18) 
 Hence for the determination of the co-ordinates of the centre of the 
 circle, that is, the quantities « — «, y — 6, z — y, and the radius ¢, there 
 exist the three equations 
 
 
 
 dx dy dz 
 pana es wae — aCe z— ——— = 
 (v—a) "2 + y—B)@ + (e-E = 0 
 ge d’y @z 19 
 BE z—vy)__ = — | ( ) 
 (e—a) 7 + (y-A)ae t+ (2-9) 55 
 enn + (y—B)B + (z-y)C = 0 
 where 
 eo dy dz _ dzd’y 
 ds ds* ds ds* 
 2 odadiaen gay d. 2 L 
 ~ ds ds ds ds | (20) 
 
 Cy ee os ae 
 ds ds? ds ds* 
 M 
 
82 
 
 hence, in writing for convenience, 
 °= A’ + BP 4+ C (21) 
 we find by cross-multiplication from (19) 
 
 dz d 
 OF —a)\= Be 
 re) ds CT 
 
 (y—f) = cz - Af t (22) 
 
 OX(2—7) = AY — Be | 
 
 2 *Qe@] (28) 
 consequently 
 
 (2283) ~ (B+ GEEBY ~ CY ESB ~G) 
 But 
 
 But since by (7) 
 
 
 
 
 
 K K K 2 2 2 
 AEA) = EEF SIETE ESP = ae | 1 gales + (GN) + } (2s) 
 or neglecting infinitesimal quantities, or equating to zero the second 
 differential of s, which will come to the same thing, since s may be 
 taken as the independent variable, 
 
 Kran ee 
 
 
 
 (d(s+ds))? ds? (26) 
 hence neglecting also 
 (dx) (By)? (zz)? 
 in comparison with 
 dxd?x dyd’y dzd*z 
 (24) may finally be written 
 d:d'z . djdy ~uzde2e. 
 ds ds * ds dst ds de (27) 
 Hence (22) are reduced to 
 2 _@x 2 a a? 
 O?(2—a) = As OX y—B)= a OX(z—y) = — (28) 
 but since, also, by means of adding and subtracting the quantities 
 dx da \* dy d’y\? ‘dz CaN" 
 ds ds* ds ds? ds i 
 
 and keeping the condition Le in mind, it is found that 
 
 o- (3) +G@)+G) (29) 
 
83 
 
 (28) may also be written in the following form, 
 
 Pan dy 1 i¢z4 ~~ OF ao (80) 
 ds* ds* ds 
 
 hence also 
 
 =e) +GO+G @) 
 
 It may here be fed that by means of dividing throughout by ds? 
 we have been enabled to neglect terms involving d’s; if, however, this 
 had not been done, a formula somewhat similar would have been found, 
 as follows: instead of (19) we should have had 
 (z — a)dr + (y — B)dy + (z — y)dz = 0 
 (x — a)d’x + (y —B)d@’y + (z —y)@z = — a| 
 (7 — a)P + (y—B)Q+ (2 -—y)R=0 
 where 
 P = dyd’z — dzd’y Q = dzd*x — drd?z R = dxd*y — dyd@’x (33) 
 which would have given 
 0/(@—a) = ds*(Qdz—Rdy) = {ds*'d?x — dx(dad*x + dyd’y + dzd*z)} ds? 
 0?(y—f) = ds*(Rde— Pdz) = {dsd?y — dy(dad?x + dyd?y + case | 34) 
 0(z—y) = ds*(Pdy—Qdr) = {ds*d*z — dz(dad?x + dyd?y + dzd?z)\ds* 
 where 
 0 = P? + Q? + R? (35) 
 But 
 (d(a+dzx)) + (d(y+dy)) + (d(z+dz)) = (d(s+ds))? (36) 
 hence expanding and subtracting (7) from the result, and neglecting 
 infinitesimal quantities of higher orders in comparison with those of 
 lower, we find 
 dud*z + dyd?y + dzd?z = dsd’s (37) 
 and consequently (34) may be written also thus, 
 O?(«#—a) = (ds*d?x — daxdsd?s)ds* 
 O7(y— B) = (ds*d*y — dydsd* | 
 07?(z—y) = cs — dzdsd*s)ds* 
 Now from (34) 
 o(P? + Q? + R2) = ds?{(Qdz— Rady)? + (Rdx—Pdz)? + (Pdy—Qdz)*}* (29) 
 = ds’ {(P? + Q? + R?)ds? — (Pdx + Qdy + Rdz)*}* 
 
 M 2 
 
84 
 
 But 
 Pdz + Qdy + Rdz = 0 (40) 
 identically ; hence 
 ds® : 
 P = 7p? O?4 Re 
 (Pe Qe (41) 
 ds® 
 
 Again, from ae 
 0p = ds*{(dsd?x—dad?s)? + (dyd?y —dsd’s)? + (dsd’z—dzd’s)\* (42) | 
 or by means of (42) 
 
 8 
 gre {(dsd’x—daxd*s) + (dsd?y—dyd’s)? + (dsd*z—dzd’s) ; 
 p 
 
 = {ds*[(d2x)? + (dy)? + (d2z)2] — dsd?s(drd?x + dy@y + dzd?z}* ¢ (48) 
 = ds{ (da)? + (d*y)* + (d?z)? — (d’s)? ys 
 
 
 
 
 
 hence also 
 Pe e (44) 
 {(d'a)? + (@y)? + (dz) —(d’s)"}3 
 Again, (38) by means of (41) gives 
 de) | 
 dsd’x — dad’s i ds ) 
 oe APL Osan 
 dy 
 sosdiy — dyd’s _ iy Wi (45) 
 oY eae DE yer aaa 
 ds 
 
 
 
 dz 
 isdtz — deds =) 
 z—y=d* pry Oy P is 
 
 J 
 So that if a, b,c be the angles which the radius of curvature makes 
 with the three co-ordinate axes respectively 
 
 aaa 
 fy Se 
 
 
 
 
 
 
 
 Cos a = 
 p 
 al 
 Cos pis F—0 Ci t (46) 
 ds 
 d =) | 
 Coste E207 EP ae 
 p ds y 
 
85 
 
 S=14& a) + +(@9 yy + +(¢3) ne (27) 
 
 and if d’s be made to vanish either (38) and (44) or (46) and (47) will 
 give 
 
 and also 
 
 Cos a = fa Cos b = pe Cos c= see (48) 
 
 
 
 as before. 
 
 The above are the principal formule in use for determining the mag- 
 nitude and direction of the radius of curvature of curves of double 
 curvature. The corresponding formule for plane curves are easily 
 deducible from those given above by omitting terms involving z and its 
 differentials. 
 
 X.—MAXIMA AND MINIMA. 
 
 Wuen a function of any number of variables 
 
 Ley Ze rer es « 
 receives a particular real value which is greater than all neighbouring 
 real values, that is, all values obtained by increasing or decreasing the 
 values of x, y, v, . . . by infinitely small quantities, that particular 
 value is called a maximum. When the particular value of the func- 
 tion is real and less than all neighbouring real values, it is called a 
 minimum. 
 
 Thus, if 
 Ua f (oh Ys 25 oye 3) Ci 
 the condition that « may be a maximum will be that both 
 f(et+dz, ytdy, z+dz,...)—f(%,y, 2...) (2) 
 and — 
 SF (x—dz, y—dy, z—dz, .. .) —f (2%, y,2, .. -) (3) 
 be negative, or expanding by T'aylor’s Theorem, 
 2 1 3 
 Du + 7; Du +755 Dut... (4) 
 and. 
 —Dut ss Du -+ 55 D'u+... (5) 
 
 must both be negative. But since the orders of infinitesimals to which 
 the various terms of these series belong decrease by unity as we pass from 
 left to right, the sum of all the terms after the first can have no effect 
 
86 
 
 on the first term; hence in order that the condition may be fulfilled 
 the first term of (4) and (5) must be negative. This can evidently be 
 the case only when 
 Du = 0 (6) 
 So likewise, if when (6) is satisfied the second term of (4) and (5) also 
 vanish, it will be necessary that 
 D*x = 0 (7) 
 and so generally if the 2nth differential of w vanishes it will be necessary 
 that 
 Dyers (8) 
 A similar train of reasoning would show that if wu is to be a mmimum 
 the series (4) and (5) must both be positive; this will obviously lead to 
 the same system of conditions (6), (7), (8), as in the former case. 
 
 Hence if any system of values of the variables x,y, z,. . . make | 
 D2+ly, — 0 (9) 
 and 
 D*"t+Dy = negative quantity (10) 
 u will be a maximum. If on the other hand any system of values of 
 the variables x, y, z, . . . make 
 DY fie 0 (13) 
 and 
 D*%*+Dy = positive quantity (12) 
 u will be amimimum. If however it happens that as often as 
 D*+1y = 0 (13) 
 also 
 D2a+) — 9 (14) 
 
 then w will admit of neither a maximum or a minimum value. 
 
 From these considerations are to be deduced the conditions which the 
 differential coefficients of © must satisfy independently of the differentials 
 
 du, dy, dz,.... (15) 
 Now adopting a common notation 
 D?0 = (ude+w'dy+v'dz+... .)dzx 
 + (w'dxr+vdy+u'dz+... .)dy (16) 
 + (vdxr+u'dytwdz+... .)dz 
 
 which expression may be written also in any one of the n different ways 
 uda® + 2(v’'dz+w'dy+. . .)dx + vdy? + wdz* +... +2(u'dydz+.. .) 
 vdy? + 2(w'da+u'dz+. . .)dy + wdz* + ude? +... +42(v'dzdr+. . .) (17) 
 
 wdz* + 2(u'dy+vu'dz+.. .)dz + udx? + vdy? +... +2(w'drdy+.. .) 
 
 s ® 6.0 © ® es Je 
 
87 
 
 Now in order that the requisite conditions may be satisfied, we shall 
 derive as a condition from these expressions respectively 
 
 (v'dz +w'dy+.. .)? — ulvdy? + wd2z*+. . .+ 2(u'dydz+...)]< 0 
 (w'de+u'dz+. . .)?— v[wdz* + uda?+.. .+ Av'dzdr+...)]< 0 
 
 8 
 (wdy+v/dz+.. .? — wludu? + vdy? +. . .+ 2(w'dady+. ..)]) < 0 a. 
 which may be also written thus 
 (vo? —wu)du? + 2[(u'v’—ww')dy +. . .Jde + (u?-vw)dy?+...< 0 
 (w?—uv)dy? + 2[(v'w!—uu')dz +. . .|dy + (v?—wu)d2?7+...<0 19 
 (u’° —vw)dz? + Qf (w'u!—vv')de+. . .Jdz + (w?—wv)da*+...<0 (19) 
 Similarly these would give as the condition 
 [(u’v’—ww')dy +...) — (v?—wu)[(u*—vw)dy? +. . 7 < 0 
 [(v'w’ —uw')dz +...) — (w?—uv)[(v?—wu)d2? +... < 0 
 1 <0 (20) 
 
 [(w’u'—wv')dzr +... — (u?—vw)[(w?—uv)da? +... 
 and so on until by successive reduction we should at length obtain a 
 result independent of the differentials. 
 
 In the case of two variables (2, y) (18) would give as the criterion 
 w?*—uv <0 (21) 
 or 
 
 <0 (22) 
 
 
 
 C2 
 dady dx* dy? 
 In the case of three variables (2, y, z,) (20) would give as the criteria 
 
 (v’w’ — uu’)? — (v?—wu)(w?—uv) < 0 
 
 (w’u’ — vv’) — (w?— uv) (u?— vw) < 0 (28) 
 (u'v’ —ww')? — (u? —vw)(v? —wu) < 05 
 
 or, as they may be also written 
 u(uvw— uu’? — vy’? — ww" +2u'v'w’) > 0 
 v(uvw — uu’? —vv? —ww? + Qu'v'w’) > 0 (24) 
 w(uvw—uu'? — vv? — ww? + 2u'v'w’) > 0 
 
 But (19) require also, in order that the conditions may be all satisfied, 
 
 that 
 
 u’?—vw < 0 
 
 y?—wu< of (25) 
 w’*—uv < 0 
 
 hence replacing wu, v, w, u’, v’, w’, by their values in terms of the 
 
88 
 
 differential coefficients of ©, and adopting the notation of determinants, 
 the conditions finally become, 
 
 In the case of two variables 
 20 LO > 0 
 
 dx’ dady 
 : (26) 
 0, PO, 
 dydx’ dy* 
 In the case of three variables (2, y, z) the conditions will be 
 #0, FO 130 OD, LOA 0 QO QO |>0 
 dy’? — dydz dz*° dzdx dx?’ dady 
 PQ OQ d*0, QO dQ, PO, 
 dzdy dz | dae ce dydx dy? 
 
 and 
 #0 &O in &O > 0 
 dx? dx*’  dzxdy — dxdz 
 #0, @0, PD, 
 dydx’ dye? — dydz 
 #0, &O, @O, 
 dzdx dzdy dz 
 LO, LO LO, &O, > 0 
 dy? da”?  dady’ = dadz 
 &O, LO, LO, 
 dydx’ dy?’ — dydz 
 PO, &O, #2, 
 dzdx dzdy dz 
 2O, OQ LO, LO, > 0 
 dz? da? * dady dzdz 
 BO, &O, LO 
 dydx dy? : dydz 
 PO, LO, PQ 
 dzdx dzdy dz? 
 
 
 
 
 
 
 
 
 
 
 
89 
 
 Hence in general the final condition of a maximum or minimum will 
 be one of the following 
 
 
 
 
 
 
 
 (a) A@>0, 28@>0, €0>0 (27) 
 where 
 6 = A, tt. & 
 FF, B, D (28) 
 €, D, € 
 in which expressions @, 5, €, D, &, jf are of the form either 
 AQ’, BO’, CO’ (29) 
 where 
 or A, F, E 
 Hepa (30) 
 BK, DC 
 or ; 
 “| ge | en 
 
 and similarly A, B, C, D, E, F will be of the same form, and so on; or 
 
 the conditions will take the form 
 
 (8) = | A, 35 | >0 
 ub, @ 
 
 where @, &, €@ are of the form (29). 
 
 It is easily seen that, as the order of the final condition doubles itself 
 every time the number of the variables is increased by unity, the order 
 in the case of m variables will be 2”. 
 
 If all the differentials of the function ©, up to the (2nm+1)th inclu- 
 sively, vanish, it was seen above that 
 
 D2er+Do, 
 must always retain the same sign if there exists a maximum or minimum 
 value of 2. In order to find the conditions relative to this case a similar 
 course must be pursued ; thus writing for convenience 
 DD, = Ox, (33) 
 it will be necessary to find first the conditions that 
 
 D?0n 
 may remain always of the same sign; this may be done by the rules 
 given above. The order of the resulting condition will be as was seen 
 above of the order 
 
 (32) 
 
 Qm—-l 
 N 
 
90 
 
 where m is the number of the variables; but as Q,, is of the order 2n 
 with respect to the differentials, the order of the resulting condition 
 will be 
 2”"n (34) 
 with respect to the differentials. ‘This must be looked upon as a 
 quadratic, and the successive conditions deduced by degrees ; thus the 
 condition that 
 ax? + bx? + ...4 1 + ja +h 
 = (ax? + baP3 4+... 4) + jr +h 
 may not change sign is 
 jy? — Ak(aaP + bP +...4+ 7) < 0 
 which again may be written 
 Ak(aa’— + br? +...) 2* + Ahlx —7? > 0 
 and so on. 
 
So 
 
 On certain Formule made use of in Physical Astronomy. 
 
 
 
 Most of the following formule are given in the Cambridge Mathe- 
 matical Journal, vol. 1v., but the demonstrations are different. 
 
 By the principles of dynamics the equations of motion of a planet or 
 satellite, considered as a material particle, are 
 
 de _»x By _ ig &z _7 (1) 
 
 dt? dt? dt? 
 where X Y Z are the resolved parts of the accelerating force parallel 
 to the three axes of co-ordinates respectively. We shall find it convenient 
 to use polar co-ordinates in future approximations ; we will therefore 
 transform (1). Let then 
 P=resolved part of the accelerating force parallel to the radius 
 vector. 
 (= resolved part of the perpendicular force in the plane of the 
 orbit. 
 S = resolved part of the perpendicular force perpendicular to the 
 plane of the orbit. 
 Also let 
 
 
 
 ~ 
 a, 
 
 b 
 — 
 
 TUN Ysera Re 
 whence, by differentiation, 
 dx _ dr, di dy _ dr dm dz__ dr, dn 
 
 a ff ee = — i —— yee mee 3 
 Camda de fdene Uo geNe ce Wont aes de (3) 
 Pe _)@r | odldr , 
 dé dé Spr eS 
 
 dé aE dt dt = 
 d’z ne r dn dr Hs n 
 ae Osan ta tos 
 dem died es ade : 
 Now 
 d?l dm i =n 
 
 P=/X + mY eam Ng +h 72 (5) 
 
 ~@1(@) te )+(%)t 
 
92 
 
 as is found by differentiating the equation 
 P4+m+nt=!] 
 now from (3) 
 
 @)* 4) * GG) GG) Cee at 
 
 But if © be the angle through which the radius vector moves, we 
 
 have also 
 ds\? arn? dO? 
 =) =G)+"(@) (8) 
 hence 
 dl\?, (dm\?, (dn\?_ (d®@\?__, 
 G)a+G =(G) = (9) 
 
 where » represents the angular velocity of the radius vector. 
 
 But if 6 be the angle through which the radius vector moves, 
 measured along the orbit, and & that through which it moves, measured 
 perpendicular to the plane of the orbit, 
 
 Dia Dae a) 
 
 
 
 also 
 d_did@, dda 
 dt dO dt | dQ dt 
 dm __dmd@ , dmdQ 
 dt d0 dt | dQ dt (11) 
 ees | 
 dt d@ dt dQ « 
 so that 
 
 na (@)+@ (12) 
 (4) + GG) + (ay (18) 
 
 di dl dm dm dn dn 
 dd * @ao edo vl Co 
 
 and consequently also 
 
 hence we see that 
 
 di dm dn. dl dm dn 
 
 dé’ dé” do’ dQ? dQ dY 
 
 form the direction-cosines of a rectangular system ; viz. the radius 
 vector, the perpendicular to the radius vector in the plane of the orbit, 
 
 im, n; 
 
 
 
93 
 
 and the perpendicular to the same line perpendicular to the plane of 
 the orbit. 
 
 
 
 Hence 
 ir 
 i Oy 15 
 i en On (15) 
 Again, if we write 
 HN ee Se \ 
 Piroms “ee ao (16) 
 ie, : per dw _ 1d(ar?) _ 
 Q=EX + mV +L = 20 — +r = | - (17) 
 Similarly, if 
 dl dm _ ane 4 
 OM eae Oe oe 
 
 be the direction-cosines of a perpendicular to the plane of the orbit, and 
 if » represent the angular velocity of the plane of the orbit round the 
 line perpendicular to the radius vector, 
 
 (F) + (3 t) + (GZ) =" 2) 
 
 al d’m @n \ 
 
 hence 
 S=rAX + py ee bon 
 dndl | dudm , dvdn 
 dt dt dt dt ° dt al 
 taking account of the motion of the plane only, as is allowable by a 
 theorem given at the end of this paper. 
 
 (20) 
 
 Also since i he are the direction-cosines of a line perpendicular 
 - dx du. dy 
 to the plane containing the lines (/mn auv), so also are ee AE 
 
 direction-cosines of a line perpendicular to the same plane, and conse- 
 quently parallel to the first ; we shall for convenience measure the angles 
 so that the cosine of the angle between these lines is negative, hence 
 S=ros (21) 
 Again, to determine the motion of the plane of the orbit we have 
 
 dr dp dv 
 Boe ae ihe 
 dn. dp dy (22) 
 
 pew tae ae de 
 
oe 
 
 whence 
 
 2 ee 
 
 dt | dec (23) 
 
 my—no nry—vl lw—mr — 
 Since if 
 AL+ pm + vn = 0 
 then (24) 
 (my — nu)? + (rn — lv)? + (Iu— mdr)? = 1 
 
 also writing : 
 
 | me {+ GEY+G} es 
 
 the equations for determining the motion of the planet or satellite in 
 its orbit are 
 tas Shh? dh _ 
 ago aT x dey = 
 and those for the motion of the plane of the orbit 
 dx dp dv 
 he) dt dt Sr (28) 
 
 my—n nnu—lv Ip—mrn h 
 
 an (26) (27) 
 
 We will now calculate the expressions for P QS. 
 
 Let mm‘ m’ be a free system of three mutually attracting bodies ; we 
 may suppose one of them to become fixed (suppose m’’) if we apply to 
 the system forces equal and opposite to those which act on m’. Let 
 
 
 
 
 
 r= (m’’, m), y= (m’”, m’), 6= (m, m’), (29) 
 Then the forces which act on m are 
 me along mm” ie "along mm 
 Te 8 & 
 “along mm’ a parallel to mm 
 2 Pe) 
 Then 
 y 
 ea gl ie midis f+ Be) 
 ey. a m , 
 <a “y a pad as zy y) (30) 
 m +m , 
 Me ay 4 See a | 
 also 
 
 P=/(X+mY4+nZ, Q=IX+mY4+wZ, S=AX+pY +vZ (31) 
 
hence 
 —e ae 
 Ux’ + my! + nz! 
 —Q) = m! ( 2 
 a mee + py’ + v2’ 
 Li — 
 
 or writing 
 
 Now 
 
 and 
 
 hence 
 
 hence 
 
 99 
 
 
 
 4 Ee etm y— ¥) +n(z— 
 
 y (la! + my! + nz’ Ua—2’)+m(y—y’) +2(z—2)\ 
 a ( 7? 1 o ) 
 
 
 
 
 
 —-R= 
 
 
 
 mn oe we + yy -— 22’ m 
 
 & 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 7/3 “ao 
 po. tm dk dk dR 
 ide 4 dy eg dz 
 Cs dl dhgeon dh. dn dR 
 A d© dx * d@ dy * dO dz 
 pe dR dR dk 
 oe ts dy rads dz 
 dk _ dR dl dRdm . dRdn dR, 
 dx did dm dx dn dx dr 
 dR dR di _ dRdm , dRadn , dR» t 
 dy dl dy dm dy dn dy ar 
 dR_dRdl , dRdm , dRdv, dR, 
 dz didz dmdz dndz ar 
 damit di im dh. 4 
 dz r dy ip r dz r 
 dm_ ml dm_ +P dn _ om 
 dx r dy wy r dae'e r 
 dn _ __ nl dn _ _ nm dn _P + m* 
 AE NE dias mdz Mr J 
 ~R dR, (e ra ciety dik 
 dz ‘dl dl dm dn a) 
 pak dR dR dk dR pak 
 Pcaen GAME cick Cane dei dr 
 le = dR nk dR 
 jak se nee ieee 
 aa nts Fs wd ie i dn Cin 
 m+m dR dR dR _ m+ m’ dR 
 i + oe lias dy a dane + dr 
 
 sy f (32) 
 
 N(@—2’) + u(y—y’) +v(z—2’) 
 Meningie) 
 
 (33) 
 (34) 
 (35) 
 
 (36) 
 
 (37) 
 
 (38) 
 
 (39) 
 
 (40) 
 
96 
 
 
 
 also 
 oe di dR , dmdR | dn dR 
 ~ dOdx  dOdy  dOdz " 
 dR dl, dRdm , dR dn Ca 
 dd dmd® dndO 
 _1dR 
 —~r dO 
 similarly 
 di dR . dm dR dn dR 
 Pedi we Guede s 
 dR dl, dRdm , dR dn ; ae) 
 dl dQ. * dmdQ” dn dQ 
 _1dR 
 ~ rdQ 
 
 where dQ is the small angle through which the plane of the orbit turns — 
 round a line perpendicular to the radius vector. If we wish to express 
 this in terms of the partial differential co-efficient of R with respect to 
 the inclination (7) of the plane of the orbit to that of the ecliptic, it is 
 easily shown that if's be the angle between the radius vector and the 
 line of nodes we have merely to write 
 dQ, = sin ddr 
 for if 
 
 di Pie es 
 — = variation of inclination 
 dt 
 and if a, B, Fr; be the direction-cosines of the line of nodes, the com- 
 
 ponents of ; will be 
 
 att pe di 
 
 dé "de “de 
 
 Se . ae 
 
 and the resolved part in the direction of the line round which aS 
 measured will be 
 
 all! i a 155) 
 
 AAO dQ TIO dt 
 = sin we 
 
 where 3 is the angle between the radius vector and the line of nodes; so 
 that the expression for S becomes also 
 
 lpirdk 
 
 aoe! eed 43 
 rsind di Co 
 
Ua 
 
 also if 'T represent the force along the tangent 
 dz m+m’dr , dR _ dv 
 
 dx dy Ws 
 Bb sei anniiedase’) © asic ionrah was WMlaii ia (44) 
 
 and the results Pet give 
 ar kh? _ m’+m_dR dh dk 
 
 le EA NL Ni IRL ghee i ee 45) (46 
 Gee tina ei ade de) dO alae) 
 
 dn dp dv 
 
 dt di it Tak 4 
 a ee (47) 
 m—no nmr—Wv IIwW—mr h dQ 
 
 i =eitaet) dep of Eat (48) 
 
 We will now integrate ne equations of motion of an undisturbed 
 planet or satellite. In this case we have 
 
 mea) 2a O (49) 
 and consequently 
 dn du dy dh 
 dt ay dt dt : dt ou) 
 whence by integration 
 » = const. f& = const. vy = const. h = const. (51) 
 
 hence the plane of the orbit is fixed. Let @ be the value of © in this 
 case, and let 
 
 ~ 
 
 
 
 m+m’=Kk (52) 
 and in this case (21) and (40) become 
 Pee ee OL (53) 
 dt r 
 also writing for convenience 
 ae (54) 
 
 the second of (48) becomes by means of the first 
 
 2 
 2) ea Ga 
 
 h? ee, 
 
 which by integration gives 
 =;+5 + CHE. C08 (0—a) (56) 
 which is the equation to a conic section, consequently a planet or 
 
 satellite if undisturbed would move in a conic section. Now since the 
 O 
 
98 
 
 planets and satellites move approximately in ellipses, we will determine 
 the elements of the ellipse represented by (56) in terms of the con- 
 stants introduced by the integrations. Comparing the above equation 
 with the usual equation to an ellipse 
 
 1 é€ 
 
 “= za—e)t a (1-4) cos (@—7) (57) 
 we have 
 h? = xa (1—e?) C= —— (58) 
 
 conversely we may determine these quantities in terms of the initial 
 circumstances of motion, thus : 
 
 Let V be the initial velocity of the planet or satellite, 
 
 
 
 p =p, fx distance - - - 
 y - - angle between the directions of p and V ; 
 then 
 K 
 : T Qn 
 h=pVsiny C se a =k yp 
 P (59) 
 
 _ 1 (pv sin y\? (2«_ 
 Reese Gay) 
 It remains for us to determine the time of the body describing a given 
 arc; this may thus be found. We will have from (53) and (58) 
 
 a) =" Go) Gp eee Ge) ee) a 
 
 ir _, [ev eE= (61) 
 
 dt a or 
 
 whence 
 
 whence integrating and supposing 6 measured from perihelion 
 
 t=y/tfa cos 
 K 
 
 at aphelion r= a (1+e), and ¢ = 3 periodic time =5 
 
 
 
 Ler —V/ ee Z.. (a—rp } (62) 
 
 
 
 3 
 2 tra* 
 4 (63) 
 ie 
 
vo 
 
 We will now introduce some auxiliary quantities which are found 
 useful in Astronomy. In the ellipse 
 a (1—e?) 
 
 
 
  51B dcond ($5 
 6 is called the true anomaly ; let us assume 
 u=coss ~~", orr=a (1—e cos u) (65) 
 
 (this w is of course different from the w employed above in (54), &c.) 
 whence 
 (1 —ecos u) (1 + ecos#) = 1— 2 
 
 
 
 or 
 O06: ie 1—cos @_1+e 1—cos u 
 1 —ecosu Lt coa@ .L—2 Lt cos u 
 whence 
 0 l+e u 
 t —S fae ees t Thee : 
 oe een (66) 
 
 uw is called the eccentric anomaly ; it will be necessary to determine its 
 geometrical value. If the ellipse change form so that it becomes a 
 circle, then 
 
 a= asa te 
 that is § and w coincide ; but if 
 a Vn 
 — + etl) smale 1 
 a b? 
 
 be the equation to an ellipse, the equation to the circle described on its 
 
 major axis will be 
 2 
 vy 2 = 
 a a 
 
 te to 
 
 in other words, it is possible to pass from an ellipse to a circle described 
 on the major axis of the ellipse by increasing the ordinates in the ratio 
 a:6; or vice versa by decreasing the ordinates in the ratio d:a. But 
 besides this it must be remembered that when e = 0 the foci of the ellipse 
 fall into the common centre of the two figures. ‘These considerations 
 will enable us to pass at once from 4 to w; let C be the common centre 
 of the ellipse and circle, S the focus of the ellipse at which the origin 
 of co-ordinates is taken; then the rectangular co-ordinates of any point 
 P on the ellipse from the origin S will be SN, NP (suppose), and 
 the radius vector S P and 4 will be the angle N S P or its supplement ; 
 O27 
 
100 
 
 if then the ordinate NP be increased in the ratio a; 6 and become 
 N P’, P’ will be the point in the circle corresponding to the point P in 
 the ellipse; but as the origin is now removed from § to C (as was 
 remarked above) the co-ordinates of P’ will be CN, NP’, and the 
 radius vector C P’ and w will be the angle NC P’ or its supplement. 
 Thus the geometrical meaning of the eccentric anomaly is completely 
 determined. 
 
 dt Sy fe rdr ee Ve rdr = a (1—e cos u) du (67) 
 Kk Vae—(a—ry K acsmu W/xK 
 
 and supposing the time and 4 measured from perihelion 
 
 
 
 VK 
 git Wane sin w = nt suppose (68) 
 a 
 hence pee — — average angular velocity 
 
 and t= mean anomaly. 
 
 We will now give some expansions of these quantities in terms of one 
 another, deducing the actual series as far as a few terms only. 
 
 (1.) To expand the radius vector in terms of the true anomaly. 
 a{1—e?) 
 
 1+ecos 0 
 
 =a(1—e?)(1—e cos 9 + e? cos ?0—€? cos °8) 
 
 =a(1—e cos 0 +e’ cos ae cos re Cos ne 
 
 ip = 
 
 ) 
 
 
 
 2 
 =a(l—e cos 5 — F + Scost 8+ < c0s°0— * cos 39 +3 
 
 
 
 2 7 
 e” : 
 =a(1—F—(e~ £) cos 0+£ cos 20 —5 cos 38) (69) 
 te To expand the eccentric anomaly in terms of the mean. 
 =nt+esinu 
 =nt+ va 1.9 ae (sin 2nt) at 5 3 Fin nt) By Lagrange’s theorem. 
 
 (6 sin nt cos “nt — 3 sin ®né) 
 
 
 
 8 
 =nt+ 7 sin nt + 3 sin 2nt + a8 
 
 9 3 
 =nt+ 7 sin nt + is sin 2nt + 3 (6 sin nt—9 sin ®nt) 
 
 2 3 
 =nt + sin nt +75 sin 2n¢ + os G sin 3nt —? sin nt) 
 
 3 s 
 = nt + (> + =) sin né + oa sin 2nt +5 sin Sn (70) 
 
101 
 
 Again, by Lagrange’s theorem, 
 
 9 
 
 e~ 
 
 
 
 di (liven, 2h 
 5 aan nt—_ (cos nt) 
 
 Cos uw =cos nt + e sin nt (cos nt) +5 
 
 9 
 ; Gee de 
 =cos nt —e sin °nt ——. —— (sin ®nt) 
 
 1.2 dnt 
 =cos nt— 0a Et {a Snt — “cos nt} 
 ae ¢ mid é) cos nt + © cos 2nt + Bia cos dnt (71) 
 2 8 2 8 
 
 (3.) To expand the radius vector in terms of the mean anomaly. 
 " =1—e cos u 
 a 
 
 2 3 
 =1—ecos nt + = 5 (1—cos 2nt) —— (3 cos 3nt—3 cos ne) 
 
 mee Ub “—(¢-%) cos nt — © 
 
 (4.) To expand the mean anomaly in terms of the true. 
 
 8 
 cos 2nt — ae cos 3nt (72) 
 
 18 to 
 
 dt = rd = Ra capes 
 h WV Ka(1—e?) 
 7r°d0 RS dé 
 as meee of a) 7 Oi pereety 
 =(1-$¢ >) ( (1 —2¢ cos 6+ 8¢? cos *6 — 4° cos °0) 
 
 = 1—2e cos 6 + 3? cos 20 — 4e* cos oe e? + 83e? cos 9 
 =] —2e cos 045 e? cos 20 —e° cos 30 
 , SRT er. 
 = 6—2e sin Oe e* sin rig sin 30 (73) 
 To expand the true anomaly in terms of the mean. We have above 
 d=nt+e(2 sin g—" e sin 20 = sin 30) 
 =nt+ep(@) . 
 =nt+ ef(nt) + e { (nt f) Yat é a’ { (nt)}s 
 1.2 “i ut 1.2.3 dn tO) 
 
 =nt + (2¢ — =) sin nt al e? sin 2n¢ + ey e® sin 3nt (74) 
 4 4 12 
 
102 
 
 To expand the true anomaly in terms of the eccentric. 
 
 
 
 
 
 : a ; 1+ 5 
 In equation (66), writing for convenience i =k, and putting the 
 whole in an exponential form 
 22-1 _ | e2uv-1_y] 
 e26V—1 1 a 22-1 a 
 whence 
 ae (h+ Devo Be) wi pPeee. 
 (k+1)—-(k—lhev 1 — nervi— 
 let | 
 . — Z—HX 
 2u/—1 —. hae 20/—1 — 
 é =z é inc 
 whence taking logarithms 
 r 
 oe Lie 
 207 —-1=] 
 y °8 ‘a —Az 
 2 3 
 = Nog a a ee Ne 
 e hietins® 
 wee ae 2 
 = 2uW —14+27 —1(rsin Qu+ > sin 4u+ ...) 
 Ste Mee Mae 
 6=u+2Xrsin ute sin 2u +2 — sin 3u (75) 
 
 é 
 
 1—-/1—e 
 
 The problem of disturbed motion will be much simplified by the 
 following theorem. 
 
 The differential equations of the motion m expressed in rectangular 
 co-ordinates may be written thus, 
 
 C2 yer ay 
 ee ad 
 
 dy , wy _ dh 7 
 dt2 a ye mld dy (76) 
 ee ILC eee 
 
 ets 7 a) 
 
 or as they may be written 
 dc dR. d’y, VaR eae ae 
 
 GROW de ) “dei “pee ee ee a 
 < ¥ y a: ~ Sra ye (77) 
 
103 
 
 if the co-ordinates be changed by the formule 
 E=ar+ by+cz, n=ar+dyt+ez, C= ax + by + c’z (78) 
 
 in which the nine direction-cosines are subject to the six equations of 
 condition 
 
 aa De ce I a’a” + Bb" + ce” = 0 
 a? + $74 ¢2=1 aa+b’b 4+ cc=0 (79) 
 v?4+wp?4¢2=])1 aa’ + bb’ + cc’ = 0 
 whence also 
 Pape taher ee (30) 
 
 then using the factors a, 6, ¢; a’, b’, c’; a’, b”, c’, respectively, in the 
 equation (77), there results the following system, 
 
 
 
 
 
 VE pap ee pe _ aR 
 dt? dx faa oe | 
 E 
 ney lk re ARM, sik 
 d@ de dy ly de (81) 
 n 
 Gre m dR _ ya _ ak 
 = dt? dz dy Gz ape fh 
 c p° 
 where 
 p” =e 2 4 n? a e (82) 
 But 
 
 dR _ dR dx , dR dy , dR dz 
 
 dé ~ dx dé‘ dy d&~ dz dé 
 aR gh ak 
 
 dR _ dR a i dy , dRdz 
 
 dy dx dn dy dn dz dy 
 
 
 
 a a] 
 
 
 
 
 
 
 
 aR, dR, AR SS) 
 me Be ME en Pore 
 dR _dRdzx , dRdy , dRd& 
 dt dxdg& dy df dz d 
 ak ya, AR 
 Sei, it hoes 
 hence (81) becomes 
 @— dR dn_dR dg_dR 
 d& dé dt? dn dP dt be 
 ee mo er ees Fe 84 
 E n C p e 
 

 
 
 
 But since three of the nine direction-cosines are entirely arbitt 
 they may always be chosen so as to destroy the third of the expressi s 
 in (84), so that there will be only two equations of motion remaining. 
 Hence if the moon be first supposed to move in the plane of the eclipti c 
 the results so obtained will be true, when the inclination i is taken into 
 account. 
 
105 
 
 On the Calculus of Variations. 
 
 ],.—FUNDAMENTAL FORMULZ. 
 
 By means of the Calculus of Differentials the properties of given 
 functions of any variables may be investigated, and the changes of the 
 former traced through values of the latter differing from one another 
 by insensible quantities ; or, conversely, the values or a system of values 
 of the variables, where certain properties of the functions (represented 
 by given combinations of their differential coefficients, or of the dif- 
 ferentials of the variables) have any proposed values, may be found. 
 There exists, however, a more extensive class of problems, in which 
 the forms of the functions are themselves unknown, and consequently 
 become the subject of investigation. The method by which problems 
 of this kind are solved is called the Calculus of Variations. 
 
 This Calculus is analogous to that of Differentials; the operations of 
 the one are in form exactly similar to those of the other, so that certain 
 corresponding points in the two may be advantageously compared in 
 the course of the present investigations. In order to understand this 
 more clearly, consider the function 
 
 OC oh 0g we) 0, (1) 
 in which expression 
 
 ean oUt eis) Oh) fon, (cin 5 Zytah ew) 95 1 af (Ly Yy Zy 41+) oh ot, ooh 0(2) 
 the symbols ©, fi, fi, fs - -. representing any functions whatever. 
 Now the functions treated of in the Calculus of Differentials (or, that 
 which is nearly equivalent, the Differential Calculus) are such that 
 
 their forms are always given, and consequently in this case 
 Unrest ee on 10 as Chea ae (3) 
 where a, 6, c, . . . are constants made variable only in certain classes 
 of problems, such as envelopes, variation of elements of motion, &c. ; the 
 object of the former class (which is most to the present purpose) being 
 to eliminate w, y, z, . . . by means of the equations (3) and their dif- 
 ferentials, the original variables x, y, z, . . . being considered constant, 
 and the original constants a, b,c, ... . variable in the differentiations. 
 
 P 
 
106 
 
 But even in this case the form of the functions represented by (2) is 
 known, and it is possible to pass only from one individual to another of 
 the same class, and not from one class to another: Thus, for example, 
 in analytical geometry any surface under consideration is in the case of 
 envelopes, or ultimate intersections, supposed to change form according 
 to a certain law, the class to which it belongs remaining always the 
 same. As however in the Calculus of Variations the forms of the 
 functions must, when determined, hold good generally, 2. e. for all values 
 of the variables 7, y, z, . . ., it is clear that the changes of the two sets 
 of variables v, y, z, ... U,v,W, ... must be simultaneously taken 
 into account. In this case it will be convenient to introduce the new 
 symbol 8, and to represent the changes or (as they will be hereafter 
 called) thewariations of 0,1/ 42, et teil, ae SY 
 ox, Oy, 62, . .). 
 bu, Be OW, 5-0 i (4) 
 in the same way as their differentials were represented by 
 UO ss eats 
 du, dp, HN 4 } (5) 
 and similarly their successive variations by 
 OU, Oly Oreh ste ee 
 O74, "O"U, OW, 2 ne i 
 in the same way as their successive differentials were represented by 
 G32, i nl altel cs 
 a Gs, Goi. ‘I (7) 
 So also proceeding according to the rules above given, and represent- 
 ing the partial variations of any function ® by the symbols 
 OAD, 0, Ady” OD, te se (8) 
 in the same way as the partial differentials are represented by the 
 symbols 
 
 (6) 
 
 dO, One (9) 
 
 we shall have as the expression for the total variation of the same 
 function 
 
 
 
 Hyg. ES OMe) 
 80 = gt E yt “oe +... (10) 
 
 a formula analogous to the differential 
 
 Be 4 WO 7 ee (11) 
 a 
 
 dO 
 dx dy d. 
 
 
 
 
 
107 
 
 Before proceeding further it will be well to consider a particular case, 
 in order to throw as much light as possible upon the nature of variations. 
 Consider the case of a surface whose equation is 
 
 i (aay 2) 10 (12) 
 
 Now by means of the Differential Calculus we are enabled to pass 
 from one point of the surface to another, and also to discover certain 
 geometrical properties which it possesses. For instance, the direction 
 of the normal at any point may be determined by any two of the 
 equations 
 
 
 
 
 
 
 
 
 
 cosa= 6 C08 i= ' cosy = GS (13) 
 where 
 2 TW! 2 WY 2 
 * (Ga AG aes aa 
 and by assuming arbitrarily values for two of the variables as 
 Be aexer ay, OL 2 7 (15) 
 the remaining two can be determined by the equation 
 EB ra¥, 2) 0,0 0r h(x, 41.7 as 0; or F(x, y, 2) (16) 
 
 Whence we can find the values of the angles «, 8, y by means of the 
 equations (13), im which x, y, z are to be written for 2, y,2 after the 
 differentiations have been performed. Conversely we may find a point 
 where the normal has any given direction (provided that the surface 
 admits of such), that is, where two of the direction-cosines have any 
 arbitrary values, such as those given by any two of the equations 
 
 €08 i= beCOd O =, COS. = 7 (17) 
 by means of the equation to the surface, and the corresponding two of 
 the system 
 
 1 d,F Leagan 1 d,F 
 CO aMmcR ON Kuma Cigale (18) 
 Suppose however that we have a more general case, and consider any 
 equation of the form 
 F(u, v, w) = 0 (19) 
 where 
 U=fi(%, YZ) v= f(t YZ), w= f3(4s y 2) (20) 
 
 Now if uw, v, w be determinate functions of x, y, z, whose form does 
 
 not change, the case will be precisely similar to the former one ; but if 
 Py2 
 
108 
 
 the forms of u,v, w vary, we shall still proceed in a similar way, which 
 will give rise to the three expressions 
 | &F 8F 8F 
 Sx’ by? &z 
 or any other that may be required, ‘Thus taking the case before con- 
 sidered, the equations (13), (14), and (18) would retain the same form 
 as before ; but, by the principles of differentiation, 
 6,F on 6,F 6,u a OpLd,0) BOp tt j0 we. 
 ox ou oz ov Ox ow ox 
 oF OF OOF 0,0 0,,b 0,% 
 jy bu dy dv dy dw dy 
 Opt 20, F628 OMe OO ew Es OL 
 oz du oz dv 6z dw 6z 
 So that even in this case we have only three equations as before, by 
 means of which we may determine u,v, w; but since these quantities 
 are themselves functions of x, y, z, we shall not (as in the former case) 
 have determined a point, but a locus; which indeed is as it should be. 
 
 
 
 
 
 
 
 
 
 
 
 Since moreover by means of the operations indicated by the symbol 
 82 new quantities have been introduced, it will be allowable to assume 
 n relations involving them; let these be that the ratio of the partial 
 variations of any function with respect to any variable to the variation 
 of that variable are equal to the ratios of the corresponding differen- 
 tials; a definition which will have the double advantage of giving a 
 definite idea of the nature of the variations, and of leaving their absolute 
 magnitudes as arbitrary in this Calculus as those of the differentials were 
 in the Differential Calculus. The analytical expression of the above 
 
 definition is 
 12 40 40 
 dx dy dz 
 
 
 
 
 
 
 
 
 
 S07 50m hen eee (21) 
 ox by bz 
 or writing 
 8,0 8,0 8,0 
 ee OF “YY Boe Je DWV Oe 25 
 d,0, d,O, pea O ; Co 
 dx dy dz 
 Us lay oo ae ° a Ls (23) 
 so that the expression (10) will become 
 10.80 agi 0 <n 
 60 = We oz + hy eae ae oie (24) 
 
109 
 
 a formula which will enable us to calculate the variation of any given 
 function by means of the ordinary processes of the Differential Calculus. 
 
 It will be well to show that the variation indicated by the formula 
 (24) does really give rise to a change of form in the function © no less 
 than that indicated by (10) did; for in taking the differential (11) of 
 the function, the variables 7, y, z, . . . are by the principles of the 
 Differential Calculus supposed to receive increments consistent with their 
 given connexion; in other words, they are subject to the condition that 
 they must not violate the relation 
 
 ria; 
 
 and this would still be the case in (24) if the variations 62, dy, éz, .. . 
 were respectively equal to the differentials dr, dy, dz,...3; but by (23) 
 the former are shown to be equal tothe product of the latter and certain 
 functions of the variables. This must not be supposed to be arguing 
 in a circle ; for the expressions 
 
 6,2, 8,0, 6,0, . 
 are by definition different from 
 
 dM), 4,0, 4,0... 
 no less than 22 from dQ; so that although 
 
 w= Sfiz + or, y + dy, 2+ Oz, . « .) (25) 
 is the same function of 
 z+ 6x, y+ dy, 2+ 62z,.. 
 
 that 
 O,=f (a+dx, yt+dy, z+dz,...) (26) 
 is of 
 z+dzr,y+dy,z+dz,... 
 yet ,Q will differ from ©, inasmuch as (4) involve new functions of 
 Ly Yy Z% . + + Which do not appear in (5). It is necessary, however, in 
 order to retain processes in the Calculus of Variations similar to those 
 in that of differentials, to express ,Q in the two forms (25) and (26) ; 
 the latter might have been written also thus, 
 1,0 = f (e+ Udz, y+ Vdy, z+ Wadz, .. .) | (27) 
 = F(x+dz, yt+dy, z+dz,.. .) 
 where F denotes some other function different from f Hence a change 
 of form does actually take place. 
 Suppose however that the quantities are subject to both kinds of 
 operations, then by the principles already laid down. 
 d8x = §(a+ dx) —dx = bdx 
 
110 
 
 and so also generally 
 Bde 1 ode ee ee ao 
 — Sm—2 n—| — — jm—2 2 
 =l0t dol Ota Oa ae Oe (28) 
 
 S100 Van aa ee Ie eee ae 
 
 and similarly for the other variables y, z, . . . In order to show that 
 the same relation holds good with respect to any function © let 
 
 O, = f (« + dz, y+dy, z+dz,.. .) 
 
 OQ, = f(a+2dzx, yt+2dy, z+2dz, .. .) 
 
 O, =f (x+ndz, y+ndy, z+ndz,.. .) 
 and similarly let 
 10 = f(r@+dx, y+ Sy, 2+6z,. . .) 
 oD = f(x+26x, y+2dy, 2+26z, .. .) 
 O = f(e+vdx, ytvdy, +62, . - .) 
 then it is easily seen that 
 
 n(n—1) eS ene ae (n—i+l)o 
 
 
 
 
 
 
 
 
 
 
 
 d*Q = re — loo ga . 
 De On Sele te eae Pee 1.2.5 .08 ae 
 (-1) CONSE | 
 iC) ee v(iy— ~ as Peat Eee 
 ane [2 oe “lew 122 Seer aie 
 or writing for convenience 
 1.2.8...i=T(i+1),, 1.2.8...7 =T(j41) 
 1.2.3...n=T(n+1), 1.2.8...v=T(p+4l) 
 1.2.3... (n—i) =T(m—i+1), 1.2.8...(v—j) =T—j4+1) 
 so that 
 ; T(n+1) : T'(v+1) 
 a ece a te eae — eee Te See ee 
 n(n—1) (n—i+1) Tait)’ v(v—1)...(v—j+1) Tol) 
 es png T(n+1) . 
 een Teeplaaisy 
 T(v+1) 
 v0 = wn) ig oe ee 
 > ( TGA) Bosse) y 
 so that 
 a I(v+1) I(n+1) 
 a = — )v ris ; . > —j ra—?t 
 oee yj+1) Pv—j+1) P¢@+1) Cm—i4+}) [Ons] 
 aro = 23 (—)5, D(n+1) [~+1) OTe 
 
 (i+1) P(n—i+1) P(j+1) P—j+]) 
 
111 
 
 but 
 LO, =f {et (n—i)de+ (v—j)8(e+ (n—s)dr) 
 + yt (n—i)dy + (vj) 8(y + (n—7)dy) 
 + 2+ (n—i)dy + (v—j)d(z+ (n—i)dz) + ...} 
 = f{et+(v—j)Se+(n—D%(2 +(v—j)de) 
 + yt (v—s)ey + (n—i)8y + (v—s)dy) 
 + 2+ (v—j)d24+ (n—2)8(2+(v—s)dz) + ...} 
 = [52] n-1 
 hence also 
 SaO = PSO, (30) 
 as a particular case of this, which is in frequent use, we may here notice 
 mamas Yom b 
 and consequently 
 sf = fd (31) 
 
 I].—GEOMETRICAL INTERPRETATIONS. 
 
 In the case of three variables some of the above formule admit of 
 elegant geometrical interpretations. The three variables 2, y, z will 
 then be the three rectangular co-ordinates of a point on the surface 
 represented by the equation 
 
 
 
 Q(z, y, 2) =90 (1) 
 Then writing for convenience 
 dO OMe 1d. One 
 Ce asp i det 
 
 the fundamental formule give 
 Adz + Bdy + Cdz=0 
 Aéz + Boy + Coz = 0 
 whence 
 A B C 
 dy dz — dz Sy dz 8x — de Sz dx dy — dy dx 
 Then in the same manner that 
 dx dy dz 
 ce a 
 (where 
 ds* = dx? + dy? + dz’) 
 
 represent the direction-cosines of a line joining the points (2, y, 2) with 
 
112 
 
 the consecutive point (7+dr, y4dy, z4+dz) on the surface, that is, a 
 tangent line ; the expressions 
 
 dx by bz e 
 
 BOR 6) 
 (where 
 
 és? = 62° + dy? + 62°) 
 will represent the direction-cosines of a line joining the point (2, y, 2) 
 on the first surface, with a consecutive point (v+%2, y+%y, z+%z) on the 
 varied one. And the formulz (3) show that a transition is made from 
 a point on the first surface to one on the second lying in the tangent 
 plane to the first at the point x, y, 2; and that the normal to the first 
 surface is the intersection of two planes whose normals are the two 
 lines whose direction-cosines are (4) and (5) respectively. It is of 
 course obvious that the general theory cannot give the direction of the 
 normal to the second surface, as it would then determine the nature 
 of that surface, but merely shows the law according to which one 
 surface is arrived at from the other. Various other assumptions might 
 have been made instead of (21) of Sect. I. if they had been thought 
 convenient, such as 
 dx da + dy dy + dz 6z2 =0 
 
 in which case the line of transition from one surface to another would 
 always have been perpendicular to the tangent line to the first surface 
 at the point of starting. Or again, 
 
 Ue a dy weeds 
 
 which would have given as a result that the two lines which in the 
 former case were perpendicular should in this one be parallel, which 
 would be a particular case of the form which was actually chosen. 
 In the case of two variables, that is of plane curves, (3) becomes 
 Adz + Bdy = 0 
 
 whence 
 
 dr © by) de 62 
 which is equivalent to the third assumption in the case of three 
 variables. This result might of course have been anticipated a priori, 
 as in plane curves the tangent plane reduces itself to a straight line. 
 
 dx _ dy oe dy _ oy 
 
113 
 
 IIJ.—on THE VARIATION OF A DEFINITE MULTIPLE INTEGRAL, 
 
 Consider the multiple integral 
 
 Q=f fi fi -..%...biyd (1) 
 
 where ¥ is any function of the variables 
 
 ian eerak es (2) 
 and their differentials of any orders 
 re Gere eae Meee gets Cr tye Ge ney ke! « (3) 
 
 Now since, after the integrations indicated in (1) have been per- 
 formed, © will be a function of x, x’, y, y’,z,z,..., and since Q is a 
 function of Y, which although a function of the variables (2) and (3), 
 and consequently also of x, x’, y, y’, Z,z’, . . ., 18 still independent of 
 the latter set, we may at once consider © as a function of 
 
 VR BAO od Be aa oem (4) 
 hence by the principles laid down in the first section 
 
 
 
 
 
 
 
 SPOT LOtee dolented (0) 
 
 BT eR pe OT Sy eR! te 
 
 SMTi ah dant i 
 oa ON LG 
 x OX 50 : eon 
 eee a a ea 
 
 In order to be able to exhibit this expression explicitly, we may 
 remark, that after the integration with respect to x, Q becomes first a 
 function of x’ and then a similar function of x but affected with a 
 negative sign, and similarly with respect to the other variables. Sup- 
 pose then we represent the value of any function of x, ¢(x) for instance, 
 when «x receives a particular value x by the notation 
 
 | Lp(a) 
 so that the equation 
 
 os) = | $l) 6) 
 always holds good. ‘This being premised, the equation may be written 
 in any of the following forms: 
 
 D=T df? fl. 0. .dedy— Td ft fT. dedy 1 
 afi fe .T..dedr— | dy ff? .. 0. Aetht t (7) 
 STO SRST Pi cdyde— TO ff. 0. dy de | 
 
 eoee 
 
 Q 
 
114 
 
 It may also be remarked that 
 
 dx0yy a Od eee aad en | 
 = fefeafet oe ... dz dy dx t (8) 
 Lp ph fis ST eRe | 
 
 consequently, if the other differentiations indicated in (5) be performed 
 by means of the expressions (7), the final form which the value of #0 
 will take will be the following : 
 
 SSUES dz dy de 
 TSS EP delye TST SE Te dea 
 + fT, To. dedsdy — fr" a 1.0... dz du by 
 S eee .. dy dz &z ae ae aa .. dy dx &z 
 
 It is ne to be observed that the expression for 8Y may be thus 
 written : 
 
 se dT 
 éT = Tse a ie ai od—“x +. += be 
 my ae , 4a 
 + ty ody +-., ih gearrty od”—sy +. “a” 
 i APL ASL 
 + ears tf he - + mary OF ete + Ob 
 te a 
 mat a =m dT =n dT 
 
 = 2,29 farily parti, Oe t+ =0 pa, ee yt ,_ 0 qr—ttiz dn'8z + 
 
 where the symbol = represents that 7, s, ¢,... receive all values requisite 
 to make the quantities 
 l—r, m—s n—t... 
 
 coincide with the various orders of differentials involved in the 
 function Y. 
 
 Now it is easily seen by successive integration by parts that 
 ft fara so )e-Idt-1 Ld" + (—)-" f'd'"Lba | 
 
 JS Maney ee Sa ve )ede Md tarby ale (— yr f'd"—Mby t ( l 1) 
 
 J Naz = §(—y'd Nd" 782 + (=) +t fd Nbz J 
 
115 
 
 hence, using for convenience the following notation 
 oaex? 
 
 | P(x) = $(x’)—$(x) (12) 
 
 ~t=X 
 
 the general expression for 8 may be thus written 
 
 i q “4 a [Sea — ert a abe] + Tox i a dZ dy | 
 i x’ y= ie ‘Chine s ol Jo—l any eS ded 
 x ats 2a om ) eC patly y | + Toy Zax 
 vy! ih so fhe dv ra | 
 Y, op y %=z2 "ise pid =] ea} d (Ga,)4 Oz +- Toz ae dy dx 
 
 (13 
 f af his ae stl a 3S gon) | | 
 
 
 
 Hse) 
 
 +> (—yrar+( 5) |e +.. | 
 
 
 
 IV.—cONDITIONS OF MAXIMA AND MINIMA. 
 
 The expression (13) of § II. is of the form 
 @ + fib = 0 (1) 
 in which the term under the sign of integration by its very nature 
 (since it involves unknown functions) cannot be integrated ; but this 
 expression must hold good whatever be the form of the unknown 
 function ; 1. e. if & becomes successively ®,, ©, .. ., in other words 
 the equations 
 © + (dd = 0, © + {P\dh = 0, 0 + {[Pdp=0,... 
 must hold good; which obviously involve the conditions 
 @ = 0; Pr 0; (2) 
 Hence, in the case of a maximum or pee aa we must have 
 
 fk oo Ae 23 
 cif ae ee, 6 a ca ae —r+1. ae ‘bx a3 ]+tet. 2 dz dy 
 x’ y==y s=m ‘Soran eee : 
 +f. st is —)ege- (sony ya ey | + Tey } . dz da 
 eC - aS pa 
 Sad ye az seis ma —)*d (ex )a : bz] +78: } dy de 
 
 Q 2 
 
 | oy 
 
116 
 
 and 
 
 2 (ya GS) | | 
 Spade [ya (aa | 
 
 s=0 
 nelore Gtr 
 Py 
 
 the latter of whicheequations separates itself into the following system 
 
 iere Gay) sola eae 
 S(t =0, 20. 
 
 Suppose that besides the function © the variables are subject to — 
 certain conditions, such that 0 is to be a maximum or minimum for given 
 values of certain of the functions of the variables, as 
 
 (5) 
 
 Usviwe ce (6) 
 and let the given values be represented by the equations 
 bey oe VV at eee (7) 
 then whatever supposition makes © a maximum or minimum, 1. e. 
 O10 (8) 
 must also make 
 6U = 0, 6V =.0,.5W = 0, .. .«- (9) 
 
 But equations (3) and (4) are equivalent to the following 
 d0'g dO gD ih 
 ic Tage, Fin OF ee : | 
 
 dU dU 
 dx dy 
 
 dU 
 6 — ee = | 
 Yate ee 0 
 dV dV dV = t 
 de? ed ae ,.=0 
 
 (10) 
 dW, , dW, , dW 
 are Clg Sand aa a 
 
 Now by methods exactly similar to those employed in the differential 
 calculus will give us by means of Lagrange’s theory of indeterminate 
 multipliers the following equations (in which 4, »,»,... are the mul- 
 tipliers ), 
 
117 
 
 dO dU (dv dW 
 
 “th tee t+ et | 
 
 ah) Wale oad le dW 
 
 Soe te Need re i Tee 
 
 dy dye’ dy "dy a (11) 
 dQ”. dU dV dW 
 
 
 
 Xr eo ae 
 dz t dz he ea | ae ‘i 
 
 or multiplying these by 4, dy, dz, . . . respectively, and adding, the 
 general condition becomes 
 6Q + OU + wsV + 7dSW4+...=0 (12) 
 that is 
 6((O2+A0+pV+yW+...)=0 
 the usual formula for the investigation of Isoperimetrical Problems. 
 
 It remains for us to determine the conditions which will show whether 
 the result obtained by the above rules be a maximum or minimum; for 
 this purpose it will be necessary to have recourse to the second variation 
 of the function ©. 
 
 The conditions relative to the case in which the limits are subject 
 to variation presents some difficulty, we shall therefore for the present 
 content ourselves with determining the criteria when the variations of 
 the limits are not taken into account ; in this case we have the principles 
 
 already laid down, 
 
 xe vy’ oz! pal dv ey gel dv is 
 80 =f f f.--[2 Deal aN an tne 
 
 t—n dT 4, 
 Lp aeseris is d”—6z + a 
 
 (13) 
 
 so that the second variation may be written 
 
 =f fi SP nego 8) 
 
 eam ET m—s 
 als 2 a0 Gravee, (4 dy)? 
 t—n PT 
 2-0 gat, ( 
 
 &T 
 ele 2s, Pty ply 
 
 r=l da’?Tt 
 o>. 4, Day eri diol 
 
 s=m a7t 
 2 ie (or tly 
 
 +..| 
 
 doz)" 
 
 de—+l§z d+ 18x 
 
 dt 6x dat Oy 
 
 | 
 | 
 dass) Od 2 OZ | (14) 
 | 
 | 
 
118 
 
 that is of the form 
 JS (Aw + Bu? + Cw? + ... +2Fow+ 2Gwu+2Huv +...) (15) 
 
 Now if the element under the sign of integration remains always of 
 the same sign, the criteria will be found in the same way as those 
 relating to ordinary maxima and minima of many variables. But if this 
 be not the case, the element in question may be divided into certain 
 parts, one of which remains always of the same sign, and the remainder 
 are integrable; the integrals of the latter will contain the variations 
 dx, dy, 0z,... or their differentials as factors, which will vanish at the 
 limits; the integrals themselves will consequently also vanish, and the 
 whole will be reduced to an integral, the element of which, under 
 the sign of integration, remains always of the same sign. In the case 
 under consideration the part integrated, with respect to x, will be 
 generally of the form 
 
 au? + bv? +cew? +... +2fow+2Qgwu+2Qhuv+... 
 
 that with respect to y, 
 
 du +e? + cw? +... 4 2f'vw+ 2q/wu+2Qh’uv+ ..- 
 that with respect to 2, 
 
 awe + + we + we. FOF" w+ 2Qq'wut+Qh'uv+ ... 
 and soon. So that tiene the first of these with respect to sz, 
 the second with respect to y, the third with respect to z, and so on, and 
 subtracting the results from the element of the second variation of ©, 
 that is from (15), and writing for convenience 
 
 dada’ dal” db db’ db” de de’ dc’ 
 = —_ — B— ——————.... 
 Si dx dh Wy de cn dz dy hy) ode ara dx dy dz | 
 i Oe eee dg_ df dy” wy —7_dh_ dh’ dh” t (16) 
 g=P-f- FT"... 6-4. y= = 
 
 the element becomes 
 Ay? + Uv? + OQw?+ ... + 2 ffow + 2Ghwu+2WHuv+ ... =O (17) 
 
 which must remain always of the same sign. To determine the con- 
 ditions that this may be the case we will follow the course taken by 
 Lagrange in the Théorte des Fonctions Analytiques. Suppose it must 
 remain aways positive, it is evident that, for the contrary case, it will be 
 sufficient to take the quantities 4,%,@,..., ff, €, H, . . ., affected 
 
 with a negative sign. Since then it must never become negative, it 
 
Ig 
 
 follows that it should have a positive minimum ; and conversely, if it 
 
 has only positive minima it can never become negative. It remains 
 
 only to find the conditions that the quantity under consideration shall 
 
 have all its minima positive. Following the method of ordinary maxima 
 
 and minima, we shall find the first and second differential coefficients of 
 
 the quantity © with respect to one variable, as uw, and equate the first to 
 zero, and suppose the second positive. ‘That is 
 Au+ FHv+Giw+ ... =0 
 
 goa ty 
 
 Substituting the value of wu, deduced from the former of these 
 
 equations in the value of ©, and writing for convenience 
 
 a-u-B, g-c-f, a= 
 
 ope (19) 
 B=f-A a 
 
 (18) 
 
 the expression for © becomes 
 O= We’? + SMw? + Jr? + ... +23Bert+ 2@@rv4+Q@Mow+ ... (20) 
 In the same manner as before, finding the first and second differential 
 coefficients of this expression with respect to one variable, as v, the 
 conditions become 
 
 DY’ 4+Rwt+@r+...=0 \ (21) 
 
 L>0 
 Similarly, the expression for © may be again transformed by means of 
 these equations into another of the same form, such as 
 
 0 = Aw?+ P74 dAse+ ... QXrs-+2QWsw+QWwr+ ... (22) 
 and the conditions again become 
 GAw+ Zr+Ds+... =0 
 23 
 Gt > 0 j ) 
 
 and so on. Now it is easily seen that the last of these transformed 
 expressions, which will contain only one of the quantities wu, v,w, ... . 
 and which will be of the form e 
 will be the minimum of the quantity ©; hence the conditions that © 
 shall have a positive minimum will be 
 
 A>0, H>0, G>0,... B50 (24) 
 and since the equations which determine u,v,w, . . . are of the first 
 
120 
 
 degree, we may conclude that that will be the only minimum; hence 
 the problem is completely solved. 
 
 It may also be remarked that by the above transformations the value 
 of © becomes 
 
 (epee 
 
 +a(o+ ROL Grt... es 
 
 “ Gt ( w + wb i. 
 
 from which expression also it appears that the conditions (24) will - 
 ensure that the element (17) shall always remain positive. 
 
121 
 
 Problems in the Calculus of Variations. 
 
 
 
 The following problems are collected from various sources. 
 
 
 
 (I.) To find the shortest line in space from one given curve to 
 another. 
 
 We must find 
 
 Jds = a minimum. (1) 
 dfds = 0 (2) 
 but ds? = da? + dy’? + dz* (3) 
 Sqs — (dou t+ dydoy + dzddz (4) 
 ds 
 
 . _fiea= [ode + Loy 4S be] 
 
 ay!s! 
 
 es Uf (deat gt 5 yd hs Sed) =0 
 
 where wz’ y’ 2’ are the co-ordinates of one curve, and 2” y/’ 2’ those of 
 the other. And since the curves are entirely independent of one 
 another, the first part giyes 
 
 (5) 
 
 boll dy” yw, dz” Me 
 ee + ay + 7% =0 (6) 
 dz’ dy’ dz’ 
 Gy tt t+ aa ty + 82 = 0 (7) 
 J/ /} // 
 Now, - a are the direction-cosines of a tangent to one curve, 
 and. ce a aa those of a tangent to the other, therefore (6) and (7) 
 
 show that the required curve cuts both the given curves at right angles. 
 The second part, since tx, ty, 82 are now absolutely arbitrary, gives 
 
 dx d. dz 
 
 va eS doi Ce 
 
 d= 0 : d= 0 (8) 
 din" CM ey dz 
 
 a aoe at (9) 
 
 which shows that the line must be straight, its equations being 
 
122 
 
 (II.) If it were required to draw the shortest line from one given 
 point to another, the first part would equal zero of itself, and the second 
 would show that the line must be straight. 
 
 (III.) To find the shortest line on a given surface between two given 
 points. 
 
 Let u=0 (1) 
 be the equation to the surface. 
 
 Consequently, as before, we must have | 
 
 dfds = 0 (2) 
 and also 
 OF a0 + by + bz 
 
 — f° | ara + bya gy Y 4 bea | = 0 
 
 Now since 6, sy, ¢ must vanish at the fixed limits, therefore the 
 first part will equal zero of itself; and in order that the integral may 
 vanish we must have 
 
 ao ~ ) 8 +d(“)ey +a(4 = ) bz = 0 (4) 
 
 whatever be the values of 82, 8y, 8z, so that they satisfy the condition 
 that the point (x, y, z) is on the given surface ; viz. 
 
 wuz + 7pey + be = 0 (5) 
 
 (3) 
 
 consequently we must es 
 dO etn ee oe 
 
 ds PACs WIN higs 
 du ier ei (6) 
 dx dy dz 
 
 We will now show that the osculating plane to the curve is always a 
 normal plane to the surface, 
 
 Let 
 
 Q = dzd’?x — dzd*z 
 
 R = dad*y — dyd*x 
 then, the perpendicular on the osculating plane makes angles with the 
 co-ordinate axes, whose cosines are P, Q, R divided respectively by 
 
 VP Qa 
 
 P = dyd*z — dzd’y 
 ; 
 
123 
 
 Now, 
 qe — dsd’a—dad's _ ds*d?a — dadsd’s 
 dsp ds* a ds® 
 
 = a | (da? + dy? + dz*)d?x — dx(dxd*x + dyd’y + dzd*z) \ 
 = - | dz(dzd*x — dxd*z) — dy (dxd*y — dyd*x) \ 
 
 s 
 
 1 
 
 Similarly 
 
 dy _ 1 Paes bese 
 aY — qo Rae Paz | 
 
 dann io 
 ace = at Pdy Qae | 
 and therefore identically 
 
 pat + Qa + RaS = 0 (9). 
 
 and also by (6) 
 
 BtoF + Bea 0 (10) 
 i.e. the normal to the re 1S st cea to a perpendicular to the 
 osculating plane of the curve, i.e. coincides with the osculating plane. 
 
 Q. E. D. 
 
 (IV.) Find the curve which of all that can be drawn between two 
 given points contains between the evolute and the radii of curvature 
 at its extremities the greatest area. 
 
 Area = 3/pds 
 
 where 
 Ae Oe CB 
 
 Cuca dxd*y — dyd?x 
 hence 
 
 S(pdds + dsép) = 0 
 and 
 
 dés = © dB a5 Y aby 
 Sp = — £ (dad*by — daddy y — dyd*Sx + d?yd8x) + 70 dds 
 
 hence 
 
 if [4pd8s — © (dud*by — d2xddy — dyd?8x + dy yd) | = 0 
 
 R 2 
 
124 
 
 or 
 
 mp pe p 
 
 wit (ae TT pa’y) abe + £ (Ady + Pde) aby 
 + P dyd®8x — iw ded: ‘ty| = ='0 
 
 hence, integrating by parts, 
 
 [2 (4ar— dy) - a(f ody) |e + Fe (4ay+ P@x) + a( fa) |éy 
 
 + fe argue — " atnty 
 
 -f [1 aCe (4de— Pa? hy a (Fay) } be 
 
 e Pp ( & ] am 
 f | (Leas £ Shy ae ) fe i 
 
 hence 
 p P (Ade —P a? v) = a(f sy) = = 2a 
 p P (Ady +2 Le ‘a) " a(£ dx) = 2h 
 or 
 # & (Ade—22 “Paty) — dyad (€- >) = 2a 
 P (Ady + Pate) e dea(?-) — 2h 
 hence 
 
 2 
 Qpds + Fa (Tady—d?yde) = adzx + bdy = pds 
 2 
 
 that is, taking x as the independent variable, 
 
 
 
 Tee pay &: y 
 
 a dx dx? 
 
 Tey A 
 +E) GB 
 
 or integrating 
 x+b _, dy 
 ae ele Li Lae 
 t—-xX = 2 FAL TE 
 ree 
 
 which is not integrable ; but if a= 
 b dy” as \/ b : 
 Aa . ee 1 ie 
 *@—x aes et da 2¢—x 
 
 the equation to a cycloid. 
 
 
 
 
 
 
 
125 
 
 (V.) To find the relation between x and y, so that 
 
 S (2? +7’) Eds may be a minimum, 
 changing to polar co-ordinate 
 oe V dr? + rdé? = 0 
 ce jun lords + 7dds = 0 
 
 dae. eden Sie et © bdr +70 by +n 8dé 
 
 cs integrating by parts, we ee 
 
 oer rg rit | +f{ (rrr tds + i oo) br— a(rme) 90 | =0 
 
 ete the assigned limits, 
 nds mi pad(s =) a (1) 
 va af (2) 
 
 and 
 Ss 
 
 from either of which we obtain the equation 
 
 hdr 
 dll no Bo ae ch 
 r A pnt? __ f2 pinto £2 ea 9 
 
 which being rationalized and integrated give 
 m+! = h sec {n+1(0+C)} 
 whence we know the relation between w and y. 
 If m = O we have the equation to a straight line. 
 
 (VI.) To find the curve of quickest descent from one given curve to 
 another. 
 
 ey! ds 
 VQ Vh—z 
 
 
 
 
 
 eae 
 fla bth oS i } =0 
 fiz 7 (S ae + dy +2 Ze) ths goz } = 0) 
 
 therefore integrating by parts, 
 
 1 oY 5 
 Vh—z aa si a Ay 
 
 tf 12 C7 Viz) ‘ we 7) 
 
 da} it 
 d et a ee nae ee 3. g]ee} = —_ 0 
 "i [ ds sity i fae 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
126 
 
 The first part shows, as before, that the curve of quickest descent 
 must cut the given curves at right angles ; from the second part we get 
 
 ax * | dy ] 
 d (F =) an) d{“Z ) 10) 
 ds Vp—z ds V/h—z 
 
 
 
 
 
 
 
 da © LW Wty WL Pes, 
 ds /j,—z ds W/h—z 
 Y = 2 a constant quantity ; 
 
 consequently the required curve lies wholly in one plane. Let this be 
 the plane of x z. 
 
 d. ss 
 
 _ = a.Vh—z 
 are mney eee 
 da® — a*(h—Z) a dx? 
 
 dz* _ 1—a’h+a’z 
 
 “so a cycloid. 
 If the motion commences from the origin, instead of 
 h—z 
 we must write z 
 
 We have not written down the limits, as the method is the same as in 
 the other problems ; and the problem 
 
 (VII.) To find the curve of quickest descent from one given point to 
 
 another, being only a particular case of this, must be treated in the 
 same way. 
 
 (VIII.) A particle is in motion under the action of a central force, 
 find the nature of the orbit described by the principle of least action. 
 
 Jods = minimum 
 
 Now 
 v = C— 2fFdr 
 v= VW C—2/Fdr 
 
 as = /,2 dy* 
 r + 7 00 
 
 a u=f\/ (C-2/Far) (24) a0 
 
127 
 
 and taking the variation with respect to @ only, as the coefficients of each 
 of the variations 87 86 must be put = 0, we get 
 
 iu =0 = "| \/ (0-2/ Bar) (8 +) 6 
 
 
 
 
 
 
 
 
 
 __ dr? doo 
 + /C—2fFar. dé Tae st 
 therefore integrating by parts 
 dr? 
 (C-2/Far) (8+ a8) — de’ _ \s0 
 /( r)( Ve Ve me 
 —f{ (J (c-2sFar) (24%) — a VC—2 fFdr VC—2 fdr dy? 
 
 ie) Vee es a) + fF =0 
 e d{/ (c-2sFa) (14% 
 
 de? 
 
 
 
 _VC—2fFdr ne be ef 
 ry rr de? 
 + GF 
 
 dr? are far & dr® 
 + Pre se. =h 
 
 cae ae ose 
 Te 
 "6 PV C—2 fdr =h Ve +@ 
 
 C~2/Fdr = h? dr? 
 
 3 Pde 
 
 
 
 Vv C—Of Far (7* 
 
 Or let roma 
 u 
 c+osP = it(w+ e 
 which may be applied to eee cases, and is the usual formula 
 
 (1X.) A string is stretched over a given surface; show that it will 
 
 place itself along the shortest path, if the resistance of the surface be 
 the only acting force. 
 
 Let 
 
 be the equation to the surface. 
 
128 
 
 Let X, Y, Z be the resolved parts of the forces acting at the point 
 
 P(z,y, 2). 
 Let 
 
 T = tension at P 
 
 consequently the sum of the forces acting on PP’ parallel to the axis of 
 x will be 
 
 s pdx! pdx 
 
 Sf Xde+ T'S -TS = 0 
 and also when the element is indefinitely small the three components 
 will be 
 1 \ es 
 Xd+d(TS) =0 
 
 Yds+d @ 2) =0 
 Zds+d (‘T ze) =i) 
 
 and in the present case 
 R = pressure or resistance of surface. 
 
 Let 
 MuAh deve’. ep dine 
 v={Q+Q+Q hb 
 a | 
 a 
 .X=RZ 
 du 
 d 
 Y=R¥ 
 du 
 z-n& | 
 
 v 
 du 
 
 dy dy 
 
 Ray ds+d(T. 7) = 0 
 du 
 
 RS w+a(T%) = 
 
 du 
 RG a+a(T.2) = 0 | 
 
 | 
 Oo 
 EE 
 

 
 
 
 129 
 
 Multiplying by ‘a a a respectively, and adding, we get 
 dy , dy , dz ie dz\ _g 
 aT+1(F a+ Gass iam 
 but 
 dx? dy? | dz 
 =] 
 ds? a ds* " ds? 
 dx ,dx , dy ,dy , dz dz 
 o. d~+ —d—=0 
 ‘ds ds %, ds ds ds ds 
 dT = 0 
 T = constant. 
 du 
 dx dx 
 aes ds = — Td 
 du 
 y yy 
 v ds= — Td me 
 du 
 dz dz 
 v ds= — Td, | 
 dx dy dz 
 a et Oe 
 : ds i ds ds 
 du a Pe ise du 
 dz dy dz 
 
 which proves the proposition. 
 
 (X.) A particle moves on a 
 
 given surface, and is acted on by no 
 
 forces, but the resistance of the surface show that it will move along 
 
 the shortest path. 
 
 Let u = 0 be the equation to the surface, 
 R = resistance, and «fy the angles which it makes with the axes. 
 
 X = Reosa 
 
 v={(#) 
 
 du 
 
 Let 
 
 dx 
 Cosa = v 
 
 Y = Reosf Z = Reosy 
 du du 
 7 dz 
 Cos B= Cosy =F 
 
130 
 
 du 
 dx dx 
 qe hy a) 
 du 
 dy py 
 af a ie 
 du 
 d’z dz | 
 qa ky =9 
 Now, 
 dz _ de ds 
 Ut umtls eat 
 
 dx _@xds | d’sdz 
 dB, \dsidi* aval ads 
 but since the resistance of the surface is the only acting force, and this is 
 directed along the normal at each point, therefore the force along the - 
 tangent plane is equal to zero. 
 d’s 
 ae 
 oc 
 de 
 
 ay _ «) | 
 eee Coa 
 dz 2 
 ae i ) 
 
 
 
 whence 
 
 utes it ae Ea are the direction-cosines of the radius of curvature of 
 
 the curve described by the particle relative to the three axes, which 
 
 
 
 
 
 
 
131 
 
 radius lies in the osculating plane, therefore by what has been proved in 
 preceding problems the proposition follows. 
 
 (XI.) When a point moves freely in space or on a curve surface, 
 under the action of forces X, Y, Z, such that Xdx + Ydy + Zdz is a 
 complete differential, the path the body takes is such that 
 
 Jods is a minimum, 
 the integral being taken between the two points at which the motion 
 begins and ends. 
 
 Gene Ri ede 
 deh ih Miia 
 Cy. Te x, dE 
 cae a Mh as [ 
 Ca Ry, dF 
 faniiag a MLS de; 
 Let the equation to the surface be 
 F(zyz) = 
 
 di dif dF 
 and ae dy dz a 
 
 ICs ey +(¢ 
 
 dz d? dy da 
 ramos eats AY eh aR ait gti d's = 2{Xdx + Ydy + Zdz} 
 
 
 
 dt dt? dt dt* dt dt? 
 dx? 4 dy’ ee One » a yl 
 d? ae * ae” = 2| p(ay2) le : 
 if Xdz + Ydy + Zdz = d d(2yz) 
 
 dfvds = f(dsdv + véds) 
 CL by + bz z | 
 
 
 
 and .'. dvds = vovdt = ale 
 
 me ie or abe a iby +& dz 35, 
 
 dfvds =/ | Sfbe4 & oF eet “Yay + aby + © o82+ - az } 
 Te roe + Vey +b: \ 
 
 = oat +5 Ley +S = between the assigned limit, 
 
 ‘ es the variations of z, y, z at the limits = 
 . fvds is a minimum. 
 
 S 2 
 
132 
 
 (XII.) Find the equation to the curve along which, under the action 
 of given forces, a body will move from one point to another in the 
 shortest time. 
 
 
 
 ds = vdt as. fe is to be a minimum. 
 Sif Bap 
 v 
 6 ds a ds.bo a 
 v v 
 
 Let X, Y, Z represent the accelerating forces acting on the body. 
 Px Py _ dz 
 da ON dee ae 
 
 vdv = Xdx + Ydy + Zdz 
 vov = Xdx + Yoy + Zéz 
 
 and dds = = © bd f- au ody + “de 
 
 oe les 2 +i aby iat Ba, cy =% 
 
 Lie atte} fab ie Ha Biya) 
 
 oes + Ydy + Zdz) = 0 
 
 ¢(= =) Sty ty 
 
 a(* @) +SY=0 
 
 1 dz ds 
 a(; =) tae Te 
 
 1 dv dz 1d eee 
 W ig? dade @ eaeas ie 
 ldu dyin lida eee 
 engi, a we -—=0 
 v? ds ds ni v ds* cf v 
 lide dz) Wz a7 
 Betufs tao — —0 
 ~ 2 ds ds . iit we 
 
 7 aaa (Gal) eke reer) 
 
 2 2 2 2 2 2 2 
 alee {> 4 ay +ot 5 Ol xt avis 2h 4% 4+ Y247 
 
 
 
 ~ oe ds?) ds? ds®_ ~— ds? vw ds yt 
 
133 
 
 
 
 vt otlv? 9 2 
 = — Dy*_ + X? 4+ Y? + 2? 
 p ~ as oe i2 + Sais: 
 odv" 
 ds? 
 
 v* 
 
 tee APR 2 ok 
 
 o 
 
 = X?4 Y2+72— {Xa +e + Ca 
 eye | ae FON dx x dy Z dz \ 
 mar { : R ‘ds 2H ds an =) 
 
 putting X?+ ¥7+ 7? = R? 
 and since = eh £ are the direction-cosines of the resultant of the forces, 
 dx dy dz 
 
 ds’ ds’ ds 
 angle between the resultant of the forces and the tangent to the curve, 
 
 and of the element of the curve, therefore if ¢ is equal to the 
 
 4 
 han R? sin * 
 = R sing 
 
 consequently the pressure on the curve arising from the centrifugal force 
 is equal to that which arises from the impressed forces, then time spent 
 in the description of the trajectory isa minimum. Q. E. F. 
 
 (XIII.) Find the curve of a given length, which being suspended 
 from two given points may have its centre of gravity the lowest 
 possible. 
 
 Take the axis of x vertical, 
 
 consequently we have 
 JS (@ + &) ds a minimum 
 
 df (x + kh) ds = f {dxds + (x + k) dds = 0 
 
 ie (w+ Wy | Hon + Loy | + f'[{-a(e42%) \ 2—a(eF7aZ) ay] =0 
 
 between the given limits, 
 
 
 
 
 
 
 
 d 
 i ds—d(z+ he =40 d(z+h%) =0 
 Oe dy 
 eae eee laa “7—) 
 t+ he ats x ha. 
 dx __a+s 
 
 dy ob 
 The equation to the catenary. 
 
 
 
 (cx +h)? = (at+s)?+0? 
 
134 
 
 If the curve was to hang from some point in a given curve to some 
 point in another given curve, the former part of the resulting expression 
 shows that the curve cuts both the given curves at right angles. 
 
 (XIV.) To find the form of a curve which a string of a given length 
 assumes when the enclosed area is a maximum. 
 Sf fydx + hds} = 0 
 
 S {ya ae + h(a + dy) | — 0 
 
 (y+ 1D) a0 4 Yay sf [ fae — 2a) | by — { ay + 2a | ae = 
 
 i =o +a i = —y—b 
 
 = (etay + (y+by 
 equation to a circle, of which the radius equals k. 
 
 (XV.) Find the form of the solid of evolution of given contents which 
 exercises the greatest attraction on a particle in its axis, law varying 
 1 
 distance 
 
 
 
 DY Oe (tli eee + hy?) de =0 
 vA ( TERE y 
 
 otis veal 
 (e+ 9°)" 
 
 a + Qh (22+y2)? = 0 
 the equation to the curve which generates the surface of the solid. 
 
ag 
 ee po na OO ce 
 
 eo 
 
 a 
 
 135 
 
 Note on Lagrange’s Condition for Maxima and Minima of 
 Two Variables. 
 
 THE conditions that 
 
 z=f(%y) (1) 
 have a maximum or minimum value are 
 dz dz 
 —)=0 (=) a 2 
 ) 2 dy (2) 
 and the discriminating condition is 
 22 dz d?z \? 
 ee Ny 5S eee ee 3 aaa 0 * 
 du* dy? oe) Saige as (3) 
 
 in the first of which cases the value determined by (2) will be a 
 minimum, in the last a maximum, and in the second neither one nor 
 
 the other. 
 If in (3) we substitute 
 
 d°z d*z dz 
 
 de” apo” dedy° (4) 
 the condition will depend upon the form of the surface 
 
 Ey — OP = (5) 
 
 where 4 is positive in the case of a minimum, negative in that of a 
 maximum, and zero in the case of neither a maximum nor minimum. 
 Now (5) is the equation to a central surface of the second order, 
 having its origin at the centre. To refer this to its principal axes, we 
 have merely to substitute in the usual cubic 
 (P—A)(P—A’)(P— A”) —B*(P— A)— B”(P— A’) B’2(P —A”)2BB’B’=0 (6) 
 the three roots of which are the coefficients of the squares of the 
 variables in the transformed equation ; but since 
 
 Aaa. 0 ey et 
 > 3 7 
 B=0, B/=0, B’ =! (7) 
 
136 
 
 (6) becomes 
 .(P4+1)(P?—1) =0 (8) 
 
 the roots of which are —1, —1, and 1, hence the transformed equals 
 becomes 
 
 2+y—2=— 6 (9) 
 which is consequently i in the case of a 
 
 minimum, a hyperboloid of revolution of two eifbets, 
 
 maximum, . + 4 one sheet, 
 neutral result, a cone. 
 

 
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