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Now a11A11= alla22a33...ann, wherein all is not to be changed, but the second suffixes in the product a 22 a 33 ...a nn assume all permutations, the number of transpositions necessary determining the sign to be affixed to the member.

00From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, ï¿½ï¿½ï¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.

00ï¿½ï¿½ a n1 - 1 a12 a 22 a32 ï¿½ï¿½ï¿½ an2 0 a13 a 23 a33 ï¿½ï¿½ï¿½ an3 0 a3n ï¿½ï¿½ ï¿½ a nn 0 0 0 ...

00anibll +an2b12+ï¿½ ï¿½ï¿½ +annbin a11b21+a12b22+ï¿½ï¿½ï¿½ +alnb2n, a21b21+a22b22+ï¿½ï¿½ï¿½ +a2nb2n, ï¿½ ï¿½ ï¿½ ani b21 + a n2 b 22 + ï¿½ ï¿½ ï¿½ +annb2n alib31+a12b32+ï¿½ï¿½ï¿½+ainb3n, a21b31+a22b32+ï¿½ï¿½ï¿½+a2nb3n, .ï¿½.a n lb 31 + a n2 b 32+ ï¿½ï¿½ï¿½ +annb2n a ll b nl + a 12 b n2+ ï¿½ï¿½ï¿½ + a ln b nn, a21bn1+a22bn2+ï¿½-ï¿½+a2nbnn, ï¿½ ï¿½ ï¿½ ani b nl + a n2 b n2 +ï¿½ ï¿½ ï¿½ +annbnn and all the elements of D become zero.

00In particular the square of a determinant is a deter minant of the same order (b 11 b 22 b 33 ...b nn) such that bik = b ki; it is for this reason termed symmetrical.

00The Adjoint or Reciprocal Determinant arises from A = (a11a22a33 ...a nn) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 ï¿½ï¿½ï¿½ A nn).

00If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) ï¿½ ï¿½;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4..

00As a teacher he had a remarkable power of kindling enthusiasm; and he sent out many distinguished pupils, among whom may be mentioned Hitzig, Schrader, Noldeke, Diestel and Dillma nn.

00Draw NR the same multiple of Nn and upwards.

00With urea it gives " /CH NN quinazolone I, and with mandelic nitrile and its H /C0 homologues it forms oxazole derivatives (S.

00The diagonal telescope nn is provided with cross hairs, and is used fcr the final centring of the instrument over an object.

00A set of stirring blades carried in the frame (b b), and driven by gearing, NN,, ï¿½,, .

00One important fact should m nn 23yz FIG.

00(1893) Lotsy, " Contributions to the Life-History of the genus Gnetum, "nn.

00nn, Nemacaulus or virgular tube.

00Conceiving the line NN' to be that of the nodes at any time, and the earth and lunar orbit to be moving in the direction of the straight arrows, the earth will be on one side of the ecliptic from M2 to M5, and on the other side from M6 to M 1, intersecting it at the nodes.

00Nj and nn give n mouillhe (ny in both old and modern spelling: senyor.

00nn, mn, nj, and sometimes to initial n: ao (a nn u m), dano (d a m n u m), nudo (n 0 d u m).

00From the theorem given above for the expansion of a determinant as a sum of products of pairs of corresponding determinants it will be plain that the product of A= (a ll, a22, Ã¯¿½Ã¯¿½Ã¯¿½ ann) and D = (b21, b 22, b nn) may be written as a determinant of order 2n, viz.

00Ã¯¿½Ã¯¿½ a n1 - 1 a12 a 22 a32 Ã¯¿½Ã¯¿½Ã¯¿½ an2 0 a13 a 23 a33 Ã¯¿½Ã¯¿½Ã¯¿½ an3 0 a3n Ã¯¿½Ã¯¿½ Ã¯¿½ a nn 0 0 0 ...

00anibll +an2b12+Ã¯¿½ Ã¯¿½Ã¯¿½ +annbin a11b21+a12b22+Ã¯¿½Ã¯¿½Ã¯¿½ +alnb2n, a21b21+a22b22+Ã¯¿½Ã¯¿½Ã¯¿½ +a2nb2n, Ã¯¿½ Ã¯¿½ Ã¯¿½ ani b21 + a n2 b 22 + Ã¯¿½ Ã¯¿½ Ã¯¿½ +annb2n alib31+a12b32+Ã¯¿½Ã¯¿½Ã¯¿½+ainb3n, a21b31+a22b32+Ã¯¿½Ã¯¿½Ã¯¿½+a2nb3n, .Ã¯¿½.a n lb 31 + a n2 b 32+ Ã¯¿½Ã¯¿½Ã¯¿½ +annb2n a ll b nl + a 12 b n2+ Ã¯¿½Ã¯¿½Ã¯¿½ + a ln b nn, a21bn1+a22bn2+Ã¯¿½-Ã¯¿½+a2nbnn, Ã¯¿½ Ã¯¿½ Ã¯¿½ ani b nl + a n2 b n2 +Ã¯¿½ Ã¯¿½ Ã¯¿½ +annbnn and all the elements of D become zero.

00The Adjoint or Reciprocal Determinant arises from A = (a11a22a33 ...a nn) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 Ã¯¿½Ã¯¿½Ã¯¿½ A nn).

00If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) Ã¯¿½ Ã¯¿½;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a symmetric function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4..

00A set of stirring blades carried in the frame (b b), and driven by gearing, NN,, Ã¯¿½,, .

00nllp p .m?mammlRUn1 !11 uu1p111MimmmMlumnuunn lluum11u?nlNnul?I I Imm11FUUUMMmIIlmmumNll1111m1m1111-1111tI111u11 c Nm Ih.nfl111111411?,q1mt1 n?It1?1111111111n111111U1 11111111111111111?IU 1?11111II OII?IIIIIIIII i IIIIII I 11U 4 I?I III I IIIIBIIIIII i I IInIIIIIIIIII I [[Iiiiiii?Iii Ii Iiiiiiiiiiiiiiii I Iiiiiiiiii I Iiiiiiiiiiiiiiiiiii Ii]] ' 'uml?ul?111?nn?11nnu.luuiui?.411?nllunnflunulrinnlm?nnunnl (p ?

001111111r11n1Ut11111Nn1n1111111U11n111hm1U1111-1111t4Nzm11111 11111rtlllu'mlla11?111h g 11P111111111NL111 'IIIIIO III IIIII' 11111 'N11111111111111J1?111'IIIIIIIIIIIIIIIIIIIIIt1111111111111N?mllllll111ro ' 'M11m111u11 11 Mmhm,ml1Nul?man?mhnnll?nN,r,ni??

00N 'n i nn eii n u ?;e?.?oow nci?,rn;' ??

00W a q anza s E 0_ nn JJ Po h oon ?'?.??,??e i I i L AY 122 D Longitude West of Greenwich A B;., :.

00and beauty will become a scorn " contains three such: 't175 mm 'D11 nn 51.t iir Ty 1 15p5 -rim nisi (see Charles, Apoc. Bar.

00NN: During the Challenger disaster, I had a friend at my house.

00NN: I had a company called Running in Motion, doing outreach to young people in the sciences.

00NN: Towards the end of the first season, I was getting offers from all over the country.

00

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