sponding equation of motion. The most elegant method of effecting this is to transform (2) and (7) simultaneously into their canonical forms. If аву be the coefficients of transformation, and if be the determinant formed by them, the terms involving the products of the variables will be destroyed by the conditions from the last two of which we have βιγ-βγ: β γ - βγ, : βy+βγ =Aa+Ha,+Ga,=(-)a+Da,+Gα, whence, 0 being a quantity to be determined, Proceeding to develope this expression, we have the term independent of =V2—(BC+CA+AB)S+S3—S' =A{VA—(B+C)S+S2}+H(VH+HS)+G(VG+GS) + =V(A2+H2+G2)+V +V(H2+B2+C2) + V§ +V(G2+F+C2)+V§ =V{A2+B2+C2+3(BC+CA+AB) — F2—G2— H3} =V(S'+). Hence (dividing throughout by V) (10) becomes 03+2.S02+(S2+)0+SS-V=0; It will be seen by reference to (9) that the values of 0 determined by this equation are equal to the ratios of the coefficients of the squares of the new variables respectively in the equivalents of (2) and (7). The coefficients of transformation are nine in number; if therefore to the six equations of condition (8) we add three more, the system will be determinate. Let three new conditions be then the variable terms of (2) will take the form of the sum of three squares, and the roots of (11) will be the coefficients of the transformed expression for (7). Or, if 0, 0, 0, be the roots of (11), (2) and (7) take the forms In order to determine the values of the coefficients of transformation a, α1, α, we have from (9), (A-S-A0)a+(H-H0)a+(-G0)α,=0, from the last two of which } a: BC−(B+G)S+S2−(BC+CB−B+C$)+BC02 (14) =a: VA++(B+CA+B+CC+2F)0+A02 =a: VA+AS+(27-HH-GG-AA+SA) 0+A02 =a: VA+AS+(V+SA)+A02 =a: V(A+0)+A(S+S0+02); or, writing for brevity +S9+02=T, the expression becomes a: V(A+0)+TA =a: FG-(FG+GF)0+FGO -CHF+HS+(CH+HS)0-CHO =a: VH+HS+SHO+Ho2 =a1: VH+TH =a2: VG+TG, whence the system a: a1: a2 1 =V(A+0)+TA = VH : VH : VG +TH= VG +TG +TH: V(B+0)+TB : VF +TF +TG: VF +TF: V(C+0)+TC,. with similar expressions for ß, B, Ba; Y, Y Y2, obtained by writing 0,, T1; 02, T, respectively for 0, T. 29 Returning to the equations of motion (1), and transforming by the formulæ (Aa+Ha2+Ga)p',=[—F(a,”—a ̧3)+(B−C)α,α2+Ha2α-Gaa1]p1 +H(aß+aß2) —G(aß1+α‚ß)]P11 +[.(H+B+F)–B(G6+F+Cß,)]* +[y2(Hy+By +Fy2)—y,(Gy+Fy1+Cy2)]r,2 +[62(Hy+By+Fy1)—ß1(Gy+Fy+Cy2) 1 +y(H+B+F,)−y(GB+FB+Cß,)]* +[a,(H+B+Fß,)– (G3+F3+C) +ß2(Ha+Ba2+Fa12)—ß1(Ga+Fa1+Ca2)]P11, (17) with similar expressions for the two other equations. Multiplying the system so formed by y, 7, 7, respectively and adding, the coefficients of P' q', will vanish, and that of r', will =1 in virtue of (12); and as regards 19 1 1 2 the right-hand side of the equation, the coefficient of p12 ={(S+0)0 | VH+TH VH+T2 | +V° | VH+T,L(S+0)H+K VG+TG VG+T,& VG+T,&(S+0)G+G +V0, (S+0)H+H VH+TH (S+0)G+ VG+TG ={(S+0)0\(T,−T)+V0(V−T2(S+0))+V0,(T(S+0)−V)}(HG-FG) But =V(0,−0){T(S+)0−V}(HG−HG). T(S+0) ▼=(S+0) (02+S0+§)−V=(S+0){(S+0)3—(S+0)+§}−V Similarly the coefficients of q, r,, and r, p, will be found to vanish; and lastly, the coefficient of p, q, which, by reference to (9), may be transformed into □ {(Aa+Ha,+Ga2)2+(Ha+Ba ̧ + Fa ̧)2+(Ga+Fa1+ Ca1); −(Aß+Hß,+G3⁄4 ̧)2+(H}+Bɓ1+Fß,)2+(Gß+Fß1+Cß ̧)2} =□ {(Aa2+Ba ̧2+Ca ̧a+2F¤‚ ̈ ̧+2Gx ̧x+2Haa ̧)S +(A—§)(a2 —ß3)+(B—§)(α,2—ß3 ̧3)+(C—§)(a, ̧2 —ẞ ̧3) in which the coefficient of S vanishes in virtue of (12); so that the coefficient of P1, 91 =□{(A—S,B—§, C—S, F, G, HXa, a1, a)' but, by (12), (A—S, B—S, C‒S, F, G, HXaa,a,)2=0, Hence the coefficient in question To find the value of in terms of A, B, C, F, G, H, we have from (12) Aa+Ha1+Ga ̧=□~1(ß1y.—ß ̧¥1), Gy+Fy1+Cy1=□−1(a‚3‚—a‚ß). And forming the determinant of each side of this system, there results |