C. C. Briggs Center for Academic Computing, Penn State University, University Park, PA Tuesday, August 18, 1998

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1 SOME POSSIBLE FEATURES OF GENERAL EXPRESSIONS FOR LOVELOCK TENSORS AND FOR THE COEFFICIENTS OF LOVELOCK LAGRANGIANS UP TO THE 15 th ORDER IN CURVATURE (AND BEYOND) C. C. Briggs Center for Academic Computing, Penn State University, University Park, PA Tuesday, August 18, 1998 Astract. This paper presents some possile features of general expressions for Lovelock tensors for the coefficients of Lovelock Lagrangians up to the 15 th order in curvature ( eyond) in terms of the Riemann-Christoffel Ricci curvature tensors the Riemann curvature scalar for n-dimensional differentiale manifolds having a general linear connection. PACS numers: k, Cv, Fy The sheer size of many of these calculations oggles the mind. At times, several million terms are manipulated in such calculations, whereas the final result is often quite small. Boyle, A., B. F. Caviness, Future Directions for Research in Symolic Computation, Society for Industrial Applied Mathematics, Philadelphia, PA (1990), p. 42. This paper presents an assortment of possile features of general expressions for Lovelock tensors G (p)a the coefficients L (p) of Lovelock Lagrangians up to the 15 th order in curvature ( eyond) in terms of the Riemann- Christoffel Ricci curvature tensors R d ac R a, respectively, the Riemann curvature scalar R for n-dimensional differentiale manifolds having a general linear connection, the features having een otained, for 0 p 5, y inspection of complete expressions presented elsewhere 1, for p 6, y extrapolation of the results otained for 0 p 5. In accordance with various general definitions given y Müller-Hoissen 2 Verwimp, 3 general expressions for L (p) G (p)a are given y L (p) = 1, if p = 0 (2p)! i R 1 i 2 i 2 p [i1 i Ri3 3 i 4 2 i Ri2p 4 1 i 2p ] i 2p 1 i (1) 2p, ifp > 0 G (p)a = δ a, ifp = 0 (2p 1)! 2 p 1 p δ [a R i 1 i 2 i i 1 i Ri3 3 i 4 2 i Ri2 4 p 1 i 2 p ] i 2 p 1 i, (2) 2 p, ifp > 0 respectively, which formulas comprise (2p)! (2p 1)! covariant index permutations, respectively, where δ a is the Kronecker delta; R a cd = g ce R d ae ; g a is the contravariant metric tensor (the possily non-vanishing covariant derivative of which vanishes identically if the connection under consideration is Riemannian), which satisfies the equation g ac g c = δ a, where g a is the covariant metric tensor. Following Schouten, 4 the Riemann-Christoffel curvature tensor R d ac is defined y R d d d e e d ac 2( [a Γ ] c Γ [a e Γ ] c Ω a Γ e c ) (3) using anholonomic coordinates (of which holonomic coordinates are a special case), where a is the Pfaffian derivative, Γ a c the connection coefficient, Ω a c the oject of anholonomity; the Ricci curvature tensor R a y R a R c ca = R c ac = R c ca 2 ( [c Q c a] S d ca Q c d ), (4) where a is the covariant derivative, Q c a the nonmetricity tensor, S d ca the torsion tensor; the Riemann curvature scalar R y R R a a = R a a = R a a = R a a = R a a. (5) Now let a p = numer of terms in general expression for L (p), (6) c p = numer of partitions of p, (7) = a p c p, (8) p = numer of terms in general expression for G (p)a. (9) Values for a p, c p,, p for 0 p 5 determined empirically appear elow in Tales (1) through (4). TABLE (1). 0 th THROUGH 5 th DIFFERENCES OF a p FOR 0 p 5 a p a p a p a p a p a p TABLE (2). 0 th THROUGH 5 th DIFFERENCES OF c p FOR 0 p 5 c p c p c p c p c p c p... 4 TABLE (3). 0 th THROUGH 5 th DIFFERENCES OF = a p c p FOR 0 p TABLE (4). 0 th THROUGH 5 th DIFFERENCES OF p FOR 0 p 5 p p p p p p In Tale (3) for, it can e seen that 3 = 2 2 p 1 (10) where 0 p 2, which yields the recurrence 3 = p 1, (11) where 0 p 2. In addition, Eq. (10) yields the formula 4 = 3 3, (12) where 0 p 1, which yields the recurrence 4 = , (13) where 0 p 1. In Tale (4) for p, it can e seen that p 2 = 7 p 9 p, (14) where 0 p 3, which yields the recurrence p 2 = 7 p 1 7 p 9 p, (15) where 0 p 3. In addition, Eq. (14) yields the formula 2 p 2 = 7 3 p, (16) where 0 p 2, which yields the recurrence p 4 = 9 p 3 22 p 2 21 p 1 7 p, (17) where 0 p 1. 1 Briggs, C. C., A General Expression for the Quartic Lovelock Tensor, Preprint gr-qc/ ; A General Expression for the Quintic Lovelock Tensor, Preprint gr-qc/ Müller-Hoissen, F., Spontaneous Compactification with Quadratic Cuic Curvature Terms, Phys. Lett., 163B (1985) 106; From Chern-Simons to Gauss-Bonnet, Nucl. Phys. B, 346 (1990) Verwimp, T., On higher dimensional gravity: the Lagrangian, its dimensional reduction a cosmological model, Class. Quantum Grav., 6 (1989) Schouten, J. A., Ricci-Calculus, Second Edition, Springer-Verlag, Berlin, Germany (1954), pp

2 2 Tuesday, August 18, 1998 Z-15 Values for a p, c p,, p for 0 p 15 calculated y means as needs e of either of the foregoing recurrences for for p appear elow in Tales (5) through (8). TABLE (5). 0 th THROUGH 15 th DIFFERENCES OF a p FOR 0 p a p a p a p a p a p a p a p a p a p a p a p a p a p a p a p a p TABLE (6). 0 th THROUGH 15 th DIFFERENCES OF c p FOR 0 p c p c p c p c p c p c p c p c p c p c p c p c p c p c p c p c p TABLE (7). 0 th THROUGH 15 th DIFFERENCES OF = a p c p FOR 0 p TABLE (8). 0 th THROUGH 15 th DIFFERENCES OF p FOR 0 p p p p p p p p p p p p p p p p p

3 Z-15 Tuesday, August 18, In Tale (7) for it can e seen that 1. p 0, (mod 2) = 1 2 [1 ( 1)p(p 1)/2 ]; (18) 2. p 0, = 4 1 (p 2); (19) 3. p 0, (mod 10) = 2 p 1 2 [1 ( 1)p ], (20) whence 0, if p 0 (mod 10) 1, if p 1 (mod 10) 4, if p 2 (mod 10) 5, if p 3 (mod 10) 8, if p 4 (mod 10) ; (21) 9, if p 5 (mod 10) 2, if p 6 (mod 10) 3, if p 7 (mod 10) 6, if p 8 (mod 10) 7, if p 9 (mod 10) 4. p 0, 2 = 3 (p 1); (22) 5. p 0, 2 = ; (23) 6. p 0, 2 1, if 2p 0 (mod 4) ; (24) 3, if 2p 1 (mod 4) 7. p 0, 3 = 3 2 1; (25) 8. k 4 p 0, k = 3 k 1, (26) whence, q 0, k q = 3 q 1 k 1 ; (27) 9. k 3 p 0, k = 4 k 1, (28) whence, q 0, k q = 4 q 1 k 1 ; (29) 10. k 3 p 0, 6, if k 2p 0 (mod 4) k 8, if k 2p 1 (mod 4). (30) 4, if k 2p 2 (mod 4) 2, if k 2p 3 (mod 4) In Tale (8) for p it can e seen that 1. p 0, p (mod 2) = 1 2 [1 ( 1)p 1 ]; (31) 2. p 0, 1, if p 0 (mod 4) 5, if p 1 (mod 4) p ; (32) 9, if p 2 (mod 4) 9, if p 3 (mod 4) 3. k 2 (mod 4) p 0, 4, if p 0 (mod 4) k 4, if p 1 (mod 4) p ; (33) 0, if p 2 (mod 4) 2, if p 3 (mod 4) 4. k 3 (mod 4) p 0, 0, if p 0 (mod 4) k 6, if p 1 (mod 4) p ; (34) 2, if p 2 (mod 4) 2, if p 3 (mod 4) 5. 0<k 0 (mod 4) p 0, 6, if p 0 (mod 4) k 6, if p 1 (mod 4) p ; (35) 0, if p 2 (mod 4) 8, if p 3 (mod 4) 6. 1<k 1 (mod 4) p 0, 0 (= 1 1), ifp 0 (mod 4) k 4 (= 5 1), ifp 1 (mod 4) p ; (36) 8 (= 9 1), ifp 2 (mod 4) 8 (= 9 1), ifp 3 (mod 4) 7. k 0 p 0, k 2 p 2 = 7 k 3 p, (37) whence, 2q p, k 2 p 2 = 7 q 1 k q 3 p 2 q. (38) Values for the numers a p p together with some other features of the general expressions for L (p) G (p)a, such as the numers of covariant index permutations per Eqs. (1) (2) comprehended y, the overall sums of the magnitudes of the numerical coefficients of the terms appearing in, the expressions in question appear elow in Tales (9) (10). The 1 st a p 1 terms of general expressions for L (p) can e calculated directly (for p 1) y means of the inverse of the formula R L (p) = p L (p 1). (39) The initial terms (up to 10) of expressions for L (p) for 0 p 15 calculated y means as needs e of the inverse of Eq. (39) appear elow in Eqs. (40) through (55). L (0) = 1; (40) L (1) = R; (41) L (2) = R 2 4 R a R a R a cd R cd a ; (42) L (3) = R 3 12 R R a R a 3 R R a cd R cd a 16 R a R c R c a 24 R a c R d R cd a 24 R a e 2 R a cd R ef a 8 R a ce R cd af R ef d ; (43) L (4) = R 4 24 R 2 R a R a 6 R 2 R cd a R a cd 64 R R c a R a R c 96 R R a c R d R cd a 96 R R a e 8 R R ef a R a cd 32 R R cd af R ef d R a ce 48 R a R a 96 R d a R a R c, (44) there eing 15 additional terms, the 1 st of which is positive; L (5) = R 5 40 R 3 R a R a 10 R 3 R cd a R a cd 160 R 2 R c a R a R c 240 R 2 R a c R d R cd a 240 R 2 R a e 20 R 2 R ef a R a cd 80 R 2 R cd af R ef d R a ce 240 R R a R a 480 R R d a R a R c, (45) there eing 75 additional terms, the 1 st of which is negative; L (6) = R 6 60 R 4 R a R a 15 R 4 R cd a R a cd 320 R 3 R c a R a R c 480 R 3 R a c R d R cd a 480 R 3 R a e 40 R 3 R ef a R a cd 160 R 3 R cd af R ef d R a ce 720 R 2 R a R a 1440 R 2 R d a R a R c, (46)

4 4 Tuesday, August 18, 1998 Z-15 TABLE (9). SOME PROPERTIES OF L (p) FOR 0 p 15 OR- DER CURVATURE DEPENDENCE QUAN- TITY TERMS PERMUTATIONS COMPREHENDED p L (p) a p (2p)! OVERALL SUM OF THE MAGNITUDES OF THE NUMERICAL FACTORS (2p)! 2 p 0 Zero L (0) Linear L (1) Quadratic L (2) Cuic L (3) Quartic L (4) 25 40, Quintic L (5) 85 3,628, ,400 6 Sextic L (6) ,001,600 7,484,400 7 Septic L (7) ,178,291, ,080,400 8 Octic L (8) ,922,789,888,000 81,729,648,000 9 Nonic L (9) 19,458 6,402,373,705,728,000 12,504,636,144, Decic L (10) 77,727 2,432,902,008,176,640,000 2,375,880,867,360, Undecic L (11) 310,761 1,124,000,727,777,607,680, ,828,480,360,160, Duodecic L (12) 1,242, ,448,401,733,239,439,360, ,476,660,579,404,160, Tredecic L (13) 4,971, ,291,461,126,605,635,584,000,000 49,229,914,688,306,352,000, Quattuordecic L (14) 19,884, ,888,344,611,713,860,501,504,000,000 18,608,907,752,179,801,056,000, Quindecic L (15) 79,536, ,252,859,812,191,058,636,308,480,000,000 8,094,874,872,198,213,459,360,000,000 TABLE (10). SOME PROPERTIES OF G (p)a FOR 0 p 15 OR- DER CURVATURE DEPENDENCE QUAN- TITY p G (p)a 0 Zero G (0)a 1 Linear G (1)a 2 Quadratic G (2)a 3 Cuic G (3)a 4 Quartic G (4)a 5 Quintic G (5)a 6 Sextic G (6)a 7 Septic G (7)a 8 Octic G (8)a 9 Nonic G (9)a 10 Decic G (10)a 11 Undecic G (11)a 12 Duodecic G (12)a 13 Tredecic G (13)a 14 Quattuordecic G (14)a 15 Quindecic G (15)a TERMS PERMUTATIONS COMPREHENDED OVERALL SUM OF THE MAGNITUDES OF THE NUMERICAL FACTORS (2p 1)! p (2p 1)! 1 if p = 0; 2 p 1 p if p / / , ,916, , ,227,020,800 8,108,100 19,100 1,307,674,368, ,729, , ,687,428,096,000 86,837,751, , ,645,100,408,832,000 13,199,338,152,000 3,697,405 51,090,942,171,709,440,000 2,494,674,910,728,000 21,411,974 25,852,016,738,884,976,640, ,775,229,467,440, ,001,893 15,511,210,043,330,985,984,000, ,788,188,103,546,000, ,129,334 10,888,869,450,418,352,160,768,000,000 51,123,372,945,548,904,000,000 4,158,891,979 8,841,761,993,739,701,954,543,616,000,000 19,273,511,600,471,936,808,000,000 24,085,338,398 8,222,838,654,177,922,817,725,562,880,000,000 8,364,704,034,604,820,574,672,000,000

5 Z-15 Tuesday, August 18, there eing 308 additional terms, the 1 st of which is positive; L (7) = R 7 84 R 5 R a R a 21 R 5 R cd a R a cd 560 R 4 R c a R a R c 840 R 4 R a c R d R cd a 840 R 4 R a e 70 R 4 R ef a R a cd 280 R 4 R cd af R ef d R a ce 1680 R 3 R a R a 3360 R 3 R d a R a R c, (47) there eing 1224 additional terms, the 1 st of which is negative; L (8) = R R 6 R a R a 28 R 6 R cd a R a cd 896 R 5 R c a R a R c 1344 R 5 R a c R d R cd a 1344 R 5 R a e 112 R 5 R ef a R a cd 448 R 5 R cd af R ef d R a ce 3360 R 4 R a R a 6720 R 4 R d a R a R c, (48) there eing 4874 additional terms, the 1 st of which is positive; L (9) = R R 7 R a R a 36 R 7 R cd a R a cd 1344 R 6 R c a R a R c 2016 R 6 R a c R d R cd a 2016 R 6 R a e 168 R 6 R ef a R a cd 672 R 6 R cd af R ef d R a ce 6048 R 5 R a R a 12,096 R 5 R d a R a R c, (49) there eing 19,448 additional terms, the 1 st of which is negative; L (10) = R R 8 R a R a 45 R 8 R cd a R a cd 1920 R 7 R c a R a R c 2880 R 7 R a c R d R cd a 2880 R 7 R a e 240 R 7 R ef a R a cd 960 R 7 R cd af R ef d R a ce 10,080 R 6 R a R a 20,160 R 6 R d a R a R c, (50) there eing 77,717 additional terms, the 1 st of which is positive; L (11) = R R 9 R a R a 55 R 9 R cd a R a cd 2640 R 8 R c a R a R c 3960 R 8 R a c R d R cd a 3960 R 8 R a e 330 R 8 R ef a R a cd 1320 R 8 R cd af R ef d R a ce 15,840 R 7 R a R a 31,680 R 7 R d a R a R c, (51) there eing 310,751 additional terms, the 1 st of which is negative; L (12) = R R 10 R a R a 66 R 10 R cd a R a cd 3520 R 9 R c a R a R c 5280 R 9 R a c R d R cd a 5280 R 9 R a e 440 R 9 R ef a R a cd 1760 R 9 R cd af R ef d R a ce 23,760 R 8 R a R a 47,520 R 8 R d a R a R c, (52) there eing 1,242,843 additional terms, the 1 st of which is positive; L (13) = R R 11 R a R a 78 R 11 R cd a R a cd 4576 R 10 R c a R a R c 6864 R 10 R a c R d R cd a 6864 R 10 R a e 572 R 10 R ef a R a cd 2288 R 10 R cd af R ef d R a ce 34,320 R 9 R a R a 68,640 R 9 R d a R a R c, (53) there eing 4,971,141 additional terms, the 1 st of which is negative; L (14) = R R 12 R a R a 91 R 12 R cd a R a cd 5824 R 11 R c a R a R c 8736 R 11 R a c R d R cd a 8736 R 11 R a e 728 R 11 R ef a R a cd 2912 R 11 R cd af R ef d R a ce 48,048 R 10 R a R a 96,096 R 10 R d a R a R c, (54) there eing 19,884,260 additional terms, the 1 st of which is positive; L (15) = R R 13 R a R a 105 R 13 R cd a R a cd 7280 R 12 R c a R a R c 10,920 R 12 R a c R d R cd a 10,920 R 12 R a e 910 R 12 R ef a R a cd 3640 R 12 R cd af R ef d R a ce 65,520 R 11 R a R a 131,040 R 11 R d a R a R c, (55) there eing 79,536,629 additional terms, the 1 st of which is negative. In general, the initial terms (up to 10) of expressions for L (p) are given y L (p) = ( 1) p R p 4 ( 1) p 1 ( p p 2 ) Rp 2 R a R a ( 1) p ( p p 2 ) Rp 2 R cd a R a cd 16 ( 1) p ( p p 3 ) Rp 3 R c a R a R c 24 ( 1) p 1 ( p p 3 ) Rp 3 R a c R d R cd a 24 ( 1) p 1 ( p p 3 ) Rp 3 R a e 2 ( 1) p 1 ( p p 3 ) Rp 3 R ef a R a cd 8 ( 1) p ( p p 3 ) Rp 3 R cd af R ef d R a ce 48 ( 1) p ( p p 4 ) Rp 4 R a R a 96 ( 1) p 1 ( p p 4 ) Rp 4 R d a R a R c, (56) there eing a p 10 additional terms, the sign of the 1 st of which equals ( 1) p. The 1 st a p 1 terms (in addition to p 1 a p 1 of the last p a p of the remaining p a p 1 terms) of general expressions for G (p)a can e calculated directly (for p 2) y means of the inverse of the formula R G (p)a = (p 1) G (p 1)a. (57) The initial terms (up to 10) of expressions for G (p)a for 0 p 15 calculated y means as needs e of the inverse of Eq. (57) appear elow in Eqs. (58) through (73). G (0)a = δ a ; (58) G (1)a = 1 2 ( δ a R 2 R a ); (59) G (2)a = 1 4 (δ a R 2 4 δ a δ a R ef cd 4 R a R 8 R a c R c 8 R ad c 4 R ae cd R cd e ); (60) G (3)a = 1 6 ( δ a R 3 12 δ a R 3 δ a R R ef cd 16 δ a R e d R c e 24 δ a R e c R f d 24 δ a R fg de R ce fg 2 δ a R ef cd R gh ef R cd gh 8 δ a R eg cd R ch ef R df gh 6 R a R 2 24 R a c R R c ), (61) there eing 16 additional terms, the 1 st of which is negative; G (4)a = 1 8 (δa R4 24 δ a R2 6 δ a R2 R ef cd 64 δ a R R c e R e d 96 δ a R R e c R f d 96 δ a R R ce fg R fg de

6 6 Tuesday, August 18, 1998 Z-15 8 δ a R R cd gh R ef cd R gh ef 32 δ a R R ch ef R df gh R eg cd 48 δ a R e f R f e 96 δ a R c f R e d R f e ), (62) there eing 105 additional terms, the 1 st of which is positive; G (5)a = 1 10 ( δ a R5 40 δ a R3 10 δ a R3 R ef cd 160 δ a R2 R c e R e d 240 δ a R2 R e c R f d 240 δ a R2 R ce fg R fg de 20 δ a R2 R cd gh R ef cd R gh ef 80 δ a R2 R ch ef R df gh R eg cd 240 δ a R R e f R f e 480 δ a R R c f R e d R f e ), (63) there eing 586 additional terms, the 1 st of which is negative; G (6)a = 1 12 (δa R6 60 δ a R4 15 δ a R4 R ef cd 320 δ a R3 R c e R e d 480 δ a R3 R e c R f d 480 δ a R3 R ce fg R fg de 40 δ a R3 R cd gh R ef cd R gh ef 160 δ a R3 R ch ef R df gh R eg cd 720 δ a R2 R e f R f e 1440 δ a R2 R c f R e d R f e ), (64) there eing 3321 additional terms, the 1 st of which is positive; G (7)a = 1 14 ( δ a R7 84 δ a R5 21 δ a R5 R ef cd 560 δ a R4 R c e R e d 840 δ a R4 R e c R f d 840 δ a R4 R ce fg R fg de 70 δ a R4 R cd gh R ef cd R gh ef 280 δ a R4 R ch ef R df gh R eg cd 1680 δ a R3 R e f R f e 3360 δ a R3 R c f R e d R f e ), (65) there eing 19,090 additional terms, the 1 st of which is negative; G (8)a = 1 16 (δa R8 112 δ a R6 28 δ a R6 R ef cd 896 δ a R5 R c e R e d 1344 δ a R5 R e c R f d 1344 δ a R5 R ce fg R fg de 112 δ a R5 R cd gh R ef cd R gh ef 448 δ a R5 R ch ef R df gh R eg cd 3360 δ a R4 R e f R f e 6720 δ a R4 R c f R e d R f e ), (66) there eing 110,319 additional terms, the 1 st of which is positive; G (9)a = 1 18 ( δ a R9 144 δ a R7 36 δ a R7 R ef cd 1344 δ a R6 R c e R e d 2016 δ a R6 R e c R f d 2016 δ a R6 R ce fg R fg de 168 δ a R6 R cd gh R ef cd R gh ef 672 δ a R6 R ch ef R df gh R eg cd 6048 δ a R5 R e f R f e 12,096 δ a R5 R c f R e d R f e ), (67) there eing 638,530 additional terms, the 1 st of which is negative; G (10)a = 1 20 (δa R δ a R8 45 δ a R8 R ef cd 1920 δ a R7 R c e R e d 2880 δ a R7 R e c R f d 2880 δ a R7 R ce fg R fg de 240 δ a R7 R cd gh R ef cd R gh ef 960 δ a R7 R ch ef R df gh R eg cd 10,080 δ a R6 R e f R f e 20,160 δ a R6 R c f R e d R f e ), (68) there eing 3,697,395 additional terms, the 1 st of which is positive; G (11)a = 1 22 ( δ a R δ a R9 55 δ a R9 R ef cd 2640 δ a R8 R c e R e d 3960 δ a R8 R e c R f d 3960 δ a R8 R ce fg R fg de 330 δ a R8 R cd gh R ef cd R gh ef 1320 δ a R8 R ch ef R df gh R eg cd 15,840 δ a R7 R e f R f e 31,680 δ a R7 R c f R e d R f e ), (69) there eing 21,411,964 additional terms, the 1 st of which is negative; G (12)a = 1 24 (δa R δ a R10 66 δ a R10 R ef cd 3520 δ a R9 R c e R e d 5280 δ a R9 R e c R f d 5280 δ a R9 R ce fg R fg de 440 δ a R9 R cd gh R ef cd R gh ef 1760 δ a R9 R ch ef R df gh R eg cd 23,760 δ a R8 R e f R f e 47,520 δ a R8 R c f R e d R f e ), (70) there eing 124,001,883 additional terms, the 1 st of which is positive; G (13)a = 1 26 ( δ a R δ a R11 78 δ a R11 R ef cd 4576 δ a R10 R c e R e d 6864 δ a R10 R e c R f d 6864 δ a R10 R ce fg R fg de 572 δ a R10 R cd gh R ef cd R gh ef 2288 δ a R10 R ch ef R df gh R eg cd 34,320 δ a R9 R e f R f e 68,640 δ a R9 R c f R e d R f e ), (71) there eing 718,129,324 additional terms, the 1 st of which is negative; G (14)a = 1 28 (δa R δ a R12 91 δ a R12 R ef cd 5824 δ a R11 R c e R e d 8736 δ a R11 R e c R f d 8736 δ a R11 R ce fg R fg de 728 δ a R11 R cd gh R ef cd R gh ef 2912 δ a R11 R ch ef R df gh R eg cd 48,048 δ a R10 R e f R f e 96,096 δ a R10 R c f R e d R f e ), (72) there eing 4,158,891,969 additional terms, the 1 st of which is positive; G (15)a = 1 30 ( δ a R δ a R δ a R13 R ef cd 7280 δ a R12 R c e R e d 10,920 δ a R12 R e c R f d 10,920 δ a R12 R ce fg R fg de 910 δ a R12 R cd gh R ef cd R gh ef 3640 δ a R12 R ch ef R df gh R eg cd 65,520 δ a R11 R e f R f e 131,040 δ a R11 R c f R e d R f e ), (73) there eing 24,085,338,388 additional terms, the 1 st of which is negative. In general, the initial terms (up to 10) of expressions for G (p)a are given y G (p)a = 1 2p (( 1)p δ a R p 4 ( 1) p 1 ( p p 2 ) δ a R p 2 ( 1) p ( p p 2 ) δ a R p 2 R ef cd 16 ( 1) p ( p p 3 ) δ a R p 3 R e c R d e 24 ( 1) p 1 ( p p 3 ) δ a R p 3 R c e R d f R ef cd 24 ( 1) p 1 ( p p 3 ) δ a R p 3 R fg ce R de fg 2 ( 1) p 1 ( p p 3 ) δ a R p 3 R gh cd R ef gh 8 ( 1) p ( p p 3 ) δ a R p 3 R ef ch R gh df R cd eg 48 ( 1) p ( p p 4 ) δ a R p 4 R f e R e f 96 ( 1) p 1 ( p p 4 ) δ a R p 4 R f c R d e R e f ), (74) there eing p 10 additional terms, the sign of the 1 st of which equals ( 1) p.