CHAP. III.] NEW proof of rectangular systEM.

IV... Op=pp + cp = ẞSap + ẞ'Sa'p, with V... Vẞa+Vẞ'a' = 0,

699

as the condition (353, XXXVI.) of self-conjugation. With this condition we may then write,

VI... ẞ= Aa + Ba', ẞ' = A'a' + Ba;

and it is easy to see that no essential generality is lost, by supposing that a and a' are two rectangular vector units, which may be turned about in their own plane, if B and B' be suitably modified: so that we may assume,

VII... a2 = a22=-1, Saa'=0; whence VIII... Þa=- ß, $a' =- B', and IX. Vẞ'a' = Baa' =-Vẞa, Vẞa' = Aaa', Vẞ'a=- A'aa'.

..

(2.) The equation I., under the form,

X... VpÞp=0, is satisfied by XI... p=0, or XII... Vaa'p=0; and it cannot be satisfied otherwise, unless we suppose,

XIII. . . p = xa + x'a', and XIV. . . V (xß + x′ß′) (xa + x'a') = 0; that is, by IX.,

XV... B(x22 - x2) + (A − A′) xx′ = 0 :

while conversely the expression XIII. will satisfy I., under this condition XV. But this quadratic in x': x, of which the coefficients B and A-A' do not generally vanish, has necessarily two real roots, with a product == 1; hence there always exists, as asserted, a system of three real and rectangular directions, such as the following,

XVI. . . xa + x'a', x'a-xa', and aa' (or Vaa'),

which satisfy the equation I.; and this system is generally definite: which proves the first part of the Theorem.

(3.) The lines a, a' may be made by (1.) to turn in their own plane, till they coincide with the two first directions XVI.; which will give,

and therefore,

XVII... B=0, ẞ=Aa, ß' = A'a',

XVIII. . . p = − cp + AaSap + A'a'Sa'p

= (c + A) aSap + (c + A') a'Sa'p + caa'Saa'p;

and thus the scalar equation II. will take the form,

XIX... Spøp = (c+ A) (Sap)2 + (c + A′) (Sa'p)2 + c (Saa'p)2= const., which represents generally a central surface of the second order, with its three axes in the three directions a, a', aa' of p; and does not cease to represent such a surface, and with such axes, when for Spøp we substitute, as in III., this new expression:

XX... Spop- Cp2 = Spøp + C ((Sap)2 + (Sa'p)2 + (Saa'p)2) = C'= const. ;

the second surface being in fact concyclic (or having the same cyclic planes) with the first, and the new term, - Cp, in op, disappearing under the sign V.p: so that the second part of the Theorem is proved anew.

(4.) It would be useless to dwell here on the cases, in which the surfaces XIX., XX. come to be of revolution, or even to be spheres, and when consequently the directions of their axes, or of p in I., become partially or even wholly indeterminate. But as an example of the reduction of an equation in quaternions to the form I.,

without its at first presenting itself under that form, we may take the very simple equation,

XXI...` pipк = ɩpкp, with & not || 4,

ΧΧΙΙ. . . V. ρνιρκ = 0;

which may be reduced (comp. 354, (12.)) to

and which is accordingly satisfied (comp. 373, XXIX.) by the three rectangular directions,

XXIII... U-UK, Vik, Ui + Uk,

of the axes (abc) of the ellipsoid,

XXIV. . . T (ıp + px) = x2 − 12,

which is one of the surfaces of the concyclic system (comp. III.),

XXV... Siрkρ = Cp2 + C",

as appears from the transformations 336, XI., &c.

282, XIX.

(5.) In applying the theorem thus recently proved anew, we have on several occasions used the expression,

in which is a vector normal to a surface whereof p is the variable vector, and the function is treated as self-conjugate (363).

(6.) It is, however, important to remark that, in order to justify the assertion of this last property, the following expression of integral form,

XXVII... Svdp,

must admit of being equated to some scalar function of p, such as fp + const., without its being assumed that p itself is a function, of any determinate form, of a scalar variable, t. The self-conjugation of the linear and vector function & in XXVI., is the condition of the existence of the integral XXVII., considered as representing such a scalar function (comp. again 363).

(7.) There are indeed several investigations, in which it is sufficient to regard vas denoting some normal vector, of which only the direction is important, and which may therefore be multiplied by an arbitrary scalar coefficient, constant or variable, without any change in the results (comp. the calculations respecting geodetic lines, in the Section III. iii. 5, and many others which have already occurred).

(8.) And there have been other general investigations, such as those regarding the lines of curvature on an arbitrary surface, in which dv was treated as a selfconjugate function of dp, while yet (comp. 410, (17.)) the fundamental differential equation Sydvdp=0 was not affected by any such multiplication of by n.

(9.) But there are questions in which a factor of this sort may be introduced, with advantage for some purposes, while yet it is inconsistent with the self-conjugation above mentioned, unless the multiplier n be such as to render the new expression Savdp (comp. XXVII.) an exact differential of some scalar function of p. (10.) For example, in the theory of Reciprocal Surfaces (comp. 412, (21.)), it is convenient to employ the system of the three connected equations,

XXVIII... Svp=1, Svdp=0, Spdv=0;

373, L. LI.

but when the length of v is determined so as to satisfy the first of these equations, 1 being then the vector perpendicular from the origin on the tangent plane to the

CHAP. III.] CONdition of integRABILITY, FACTOR.

701

given but arbitrary surface of which p is the vector, while p-1 is the corresponding perpendicular for the reciprocal surface with v for vector, the differential dv loses generally its self-conjugate character, as a linear and vector function of do: although it retains that character if the scalar function fp be homogeneous, in the equation fp = const. of the original surface, as it is for the case of a central quadric,* for which v = op, dv = pdp, &c., as in former Articles.

(11.) In fact, the introduction of the first equation XXVIII. is equivalent to the multiplication of v by the factor n = (Svp)-1; and if we write (comp. 410, (16.)), XXIX... dfp = 2Svdp, dv = pdp, dn= = Sødp,

we shall have this new pair of conjugate linear and vector functions,

XXX... d. nv = ddp=nødp + vSodp, XXXI... d'dp=nødp + σSvdp; and these will not be equal generally, because we shall not in general have σ || v. But this last parallelism exists in the case of homogeneity (10.), because we have then the relations,

XXXII. . . 2Svp=rfp, d.n1=dSvp = rSvdp,

ifr be the number which represents the dimension of fp (supposed to be whole). (12.) On the other hand it may happen, that the differential equation Svdp = 0 represents a surface, or rather a set of surfaces, without the expression Svdp being an exact differential, as in (6.); and then there necessarily exists a scalar factor, or multiplier, n, which renders it such a differential.

(13.) For example the differential equation,

XXXIII... Sypdp=Svdp = 0, with XXXIV... v=Vyp, dv=Vydp = ¢dp, represents an arbitrary plane (or a set of planes), drawn through a given line y; but the expression Sypdp itself is not an exact differential, and the integral XXVII. represents no scalar function of p, with the present form of v, of which the differential dy is accordingly a linear function pdp, which is not conjugate to itself, but to its opposite (comp. 349, (4.)), so that we have heré p'dp = — pdp. (14.) But if we multiplyv by the factor,

XXXV... n = v-2 = (Vyp)-2, which gives XXXVI... dn= Sodp, σ = 2n2y Vyp, and therefore Syσ = 0, Spo = − 2n, then the new normal vector nv, or v-1, is found to have the self-conjugate differential,

XXXVII. . . d. nv = d. v ̄1 = — v-1Vydp. v-1 = ddp = d′dp ;

and accordingly the new expression,

do

XXXVIII... Snvdp = Sv-ldp = S

with y constant,

Vyp'

is easily seen to be an exact differential, namely (if Ty= 1), that of the angle which the plane of γ and ρ makes with a fixed plane through y: so that, when is thus

*It was for this reason that the symbol Tv was not interpreted generally as denoting the reciprocal, P-1, of the length of the perpendicular from the origin on the tangent plane, in the formula of 410, 412, 414: although, in several of those formulæ, as in an equation of 409, (3.), that symbol was so interpreted, for the case of a central surface of the second order.

changed to nv, the integral in XXVII. acquires a geometrical signification, which is often useful in physical applications, since it then represents the change of this angle, in passing from one position of p to another; or the angle through which the variable plane of yp has revolved.

(15.) In fact, the general formula 335, XV. for the differential of the angle of a quaternion gives, if we write

which contain the above-stated result, and can easily be otherwise established. (16.) In general, if the linear and vector function dy = pdp be not self-conjugate, and if the function d. nv = ódp be formed from it as in (11.), it results from that sub-article, and from 349, (4.), that we may write,

XLII. . . ( − ø′)dp=2Vydp, (6- 6′) dp = 2Vy,dp,

with the relation,

XLIII... 2y, = 2ny + Vvo;

where y, y, are independent of dp, although they may depend on p itself. If then the new linear function ddp is to be self-conjugate, so that y,=0, we must have

XLIV... 2ny + Vvo0, and therefore XLV... Syv = 0;

which latter very simple equation, not involving either n or σ, is thus a form, in quaternions, of the Condition of Integrability* of the differential equation Svdp = 0, if the vector y be deduced from v as above.

(17.) The Bifocal Transformation of Spøp, in 360, (2.), has been sufficiently considered in the present Section (III. iii. 7); but it may be useful to remark here, that the Three Mixed Transformations of the same scalar function fp, in the same series of sub-articles, include virtually the whole known theory of the Modular and Umbilicar Generations of Surfaces of the Second Order.

1, is the vector of an

(18.) Thus, in the formula of 360, (4.), if we make e = Umbilicar Focus of the surface fp = 1, and is the vector of a point on the Umbilicar Directrix corresponding; whence the umbilicar focal conic and dirigent cylinder (real or imaginary) can be deduced, as the loci of this point and line.

=

(19.) Again, by making e1 and es each 1, in the formula of 360, (6.), we obtain Two Modular Transformations of the equation of the same surface; £1, ₤3 being

*If the proposed equation be

Svdp=pdx+qdy +rdz = 0, so that 2=- − (ip+jq + kr),

we easily find that 2y = iP+jQ+kR, where

P=Dzq- Dyr, Q=Dar - D2p, R=Dyp - Drq; the condition of integrability XLV. becomes therefore here,

PP+qQ+rR= 0, which agrees with known results.

CHAP. III.] MODULAR AND UMBILICAR GENERATIONS.

703

vectors of Modular Foci, in two distinct planes, and 1, 3 being vectors of points upon the Modular Directrices corresponding: whence the modular focal conics, and dirigent cylinders (real or imaginary), are found by easy eliminations.

the equations 360, XVI., XVII. may be brought to the forms,

in which c1, c2, c3 are the three roots of a certain cubic (M=0), or the inverse squares of the three scalar semiaxes (real or imaginary) of the surface, arranged in algebraically ascending order (357, IX., XX.; 405, (6.), &c.): and m1, m3 are the two (real or imaginary) Moduli, or represent the modular ratios, in the two modes of Modular Generation* corresponding.

(21.) It is obvious that an equation of the form,

represents a central quadric, if op be any lineart and vector function of p, of the

* Mac Cullagh's rule of modular generation, which includes both those modes, was expressed in page 437 of the Lectures by an equation of the form,

T(p-a)=TV. yVẞp;

in which the origin is on a directrix, ẞ is the vector of another point of that right line, a is the vector of the corresponding focus, y is perpendicular to a directive (that is, generally, to a cyclic) plane, p is the vector of any point P of the surface, and Sẞy is the constant modular ratio, of the distance AP of P from the focus, to the distance of the same point P from the directrix OB, measured parallel to the directive plane. The new forms (360), above referred to, are however much better adapted to the working out of the various consequences of the construction; but it cannot be necessary, at this stage, to enter into any details of the quaternion transformations still less need we here pause to give references on a subject so interesting, but by this time so well known to geometers, as that of the modular and umbilicar generations of surfaces of the second order. But it may just be noted, in order to facilitate the applications of the formulæ L. and LI., that if we write, as usual, for all the central quadrics, a2 > 62 > c2, whether b2 and c2 be positive or negative, then the roots c1, c2, c3 coincide, for the ellipsoid, with a 2, b-2, c-2; for the singlesheeted hyperboloid, with c-2, a 2, b-2; and for the double-sheeted hyperboloid with b-2, c-2, a ̄2, (comp. page 651).

In page 664 the notation,

dp=28vdp=2Sppdp,

409, IV.

was employed for an arbitrary surface; but with the understanding that this function pp (comp. 363) was generally non-linear. It may be better, however, as a