When four tangents are drawn from a point M on the Thomson cubic to the Thomson cubic itself, their anharmonic ratio is constant (Salmon) and, up to a permutation of these tangents, is equal to :
 which is obtained when the tangents are those drawn from K to the cubic namely AK, BK, CK, GK in this order. Now, given another pivotal cubic pK with pole Ω=p:q:r and pivot P=u:v:w, the tangents at A, B, C, P concur at P* and their anharmonic ratio is evaluated similarly. pK is said to be equivalent to the Thomson cubic if and only if these ratios are equal. This gives the condition :
 For a given pivot P, the pole Ω must lie on the circum-conic with perspector X512 x P^2. For example, any pK with pivot G must have its pole on the circum-conic through G and K. For a given pole Ω, the pivot P must lie on the diagonal conic passing through the square roots of Ω and whose center is Ω÷X512. For example, any isogonal pK must have its pivot on the Steiner hyperbola i.e. the diagonal conic through the in/excenters and G. This is the polar conic of G in the Thomson cubic. Any projection or linear transformation, any isoconjugation with pole Q transform the Thomson cubic into an equivalent cubic. For example, the barycentric product of the Thomson cubic by a point Q is the cubic pK(X6 x Q^2, Q), a cubic equivalent to the Thomson cubic for any point Q. Any such cubic is called a multiple of the Thomson cubic (yellow lines in the table). The following table gives a selection of such cubics equivalent to the Thomson cubic. See also Table 68.
 Ω P points on the cubic Xi for i = cubic 6 2 1, 2, 3, 4, 6, 9, 57, 223, 282, 1073, 1249 K002 6 20 1, 3, 4, 20, 40, 64, 84, 1490, 1498, 2130, 2131 K004 2 69 2, 4, 7, 8, 20, 69, 189, 253, 329, 1032, 1034 K007 37 8 1, 4, 8, 10, 40, 65, 72 K033 2 75 1, 2, 7, 8, 63, 75, 92, 280, 347, 1895 K034 394 69 2, 3, 20, 63, 69, 77, 78, 271, 394 K099 32 3 3, 6, 25, 55, 56, 64, 154, 198, 1033, 1035, 1436 K172 32 1 1, 6, 19, 31, 48, 55, 56, 204, 221, 2192 K175 76 14615 69, 75, 253, 264, 309, 322, 6527, 14615 K183 76 76 2, 69, 75, 76, 85, 264, 312 K184 81 86 1, 2, 7, 21, 29, 77, 81, 86 K317 1333 21 1, 3, 21, 28, 56, 58, 84, 1394, 2360 K318 1333 81 1, 6, 57, 58, 81, 222, 284, 1172, 1433 K319 6 63 1, 9, 19, 40, 57, 63, 84, 610, 1712, 2184 K343 81 8822 7, 20, 21, 27, 63, 84 K344 37 2 1, 2, 9, 10, 37, 226, 281, 1214 K345 1501 6 6, 25, 31, 32, 41, 184, 604, 2199 K346 213 1 1, 6, 33, 37, 42, 55, 65, 73, 2331 K362 321 75 2, 8, 10, 75, 307, 318, 321, 1441 K366 6 194 1, 194, 3224 K410 2207 4 4, 6, 19, 25, 33, 34, 64, 208, 393 K445 14585 3 3, 6, 48, 154, 184, 212, 577, 603, 2188 K576 2052 264 2, 4, 92, 253, 264, 273, 318, 342, 2052 K647 213 40 19, 40, 55, 64, 65, 71, 2357 K750 393 2 2, 4, 278, 281, 393, 1249 K879 37 329 4, 9, 72, 226, 329, 1490, 1903, 2184 K880 1427 5932 4, 223, 1439, 3182, 5930, 5932, 8807, 8808, 8809, 8810, 8811, 8812 K963 1427 7 1, 4, 7, 57, 65, 196, 226, 1439, 3668 K964 15 298 1, 2, 6, 298, 616 16 299 1, 2, 6, 299, 617 32 610 6, 19, 48, 198, 610, 1436, 2155 32 1498 6, 64, 154, 221, 1498, 2192 76 304 75, 85, 92, 304, 309, 312, 322 81 333 2, 27, 57, 63, 81, 189, 333, 1817 213 9 6, 9, 19, 37, 71, 198, 1400, 1903 213 1490 33, 64, 73, 198, 1490, 1903 220 8 1, 8, 9, 40, 55, 200, 219, 281 249 99 99, 110, 643, 648, 662, 1414 279 85 2, 7, 57, 77, 85, 189, 273, 279 279 348 2, 7, 278, 279, 347, 348, 1440 393 92 1, 4, 19, 92, 158, 196, 278, 281, 2184 393 1895 1, 4, 158, 1712, 1895 394 326 1, 63, 77, 78, 326 577 63 1, 3, 48, 63, 219, 222, 255, 268, 610 577 394 3, 6, 219, 222, 394, 1073, 1433, 1498 593 86 21, 27, 58, 81, 86, 285, 1014, 1790 762 10 10, 12, 37, 201, 210, 594, 756 1333 1817 3, 28, 57, 284, 610, 1436, 1817 1407 7 1, 7, 56, 57, 84, 222, 269, 278 1427 2 2, 57, 223, 226, 278, 1214, 1427, 2184 1427 347 1, 278, 347, 1214 1446 75 7, 75, 253, 273, 307, 347, 1441 1446 85 2, 7, 85, 92, 226, 342, 1441, 1446 1500 10 10, 37, 42, 65, 71, 210, 227, 1826 2052 75 75, 92, 158, 273, 318, 1895 2205 6 6, 31, 41, 42, 213, 607, 1400, 1409 2205 55 25, 31, 42, 55, 228, 1402, 2187, 2357 2205 198 25, 41, 198, 228, 1400, 2155 2207 1 1, 19, 33, 34, 204, 207, 1096, 2331 2207 1249 6, 19, 393, 1033, 1249, 2331 2207 1712 19, 204, 208, 1712 2287 333 2, 8, 9, 21, 63, 271, 333, 2287, 2322 2287 1043 1, 8, 20, 21, 29, 78, 280, 1043 14827 9 6, 9, 33, 41, 55, 198, 212, 220