By Campbell J.E.
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The packets of MS. containing Chapters II and III, as submitted Jo the Delegates of the University Press, were numbered by the author in the reveiso order, and that order would probably have been made suitable, by some reai rangement of matter, had he lived to put the work in readiness It has seemed best, however, to revert to the order of a list for printing. ] 2843 E THE GROUND FORM WHEN n = 26 2 In connexion with the form ch 2 we = edu 2 + 2fdudv + gdv 2 egf 2 use h to denote the positive square root of h = a* = = ^5sina, where a The element of area on the hdudv surface a n a 22 : that a^.
2 , wo have We can therefore find or by the solution of Riccati's equation. To determine Z, in, n we have cr l = and + im f l. can now u + Ka n l and that We = im + m2 + 7i 2 = constant how, when we are given 2 : and we thus see functions, we can determine We l "" , I, m, the rotation n. at once verify that 0, A (1) I. l2 + Ka 12 l = 0, l. 2Z + Ka zz l = = K (m* + n = K (constant 2 ) 1 2 ). 5) have thus shown how a common integral of the combe obtained by aid of Riccati's equation. plete system can The I, m, n which we thus obtain will be the direction cosines of the normal to the surface, but we do not make any use of our knowledge of a third dimension in obtaining I, m, n.
The upper integers of the of tensors is product are the upper integers of the two factors, and the lower integers of the product are the lower integers of the two Two factors. tensors of the same character that is, with the same number can be of each kind of integers, upper and lower which have the if we the take together added, components same we They can integers. shall see. We also be combined in other ways, as form the tensor derivative of the product of two tensors by the same rule as in ordinary differentiation.