By Campbell J.E.
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Of theorems about surfaces The classes there are the theorems which : are concerned with the surface regarded as a locus in space ; and there are the theorems about the surface regarded as a two-way space, and not as regards its position in a higher It is the latter type of theorems about which the space. ground form gives us all the information we require. 1) ^ ; where u We the colatitude and v the longitude. shall prove the fundamental formula is cos c and the formula = cos a cos b + sin a sin b cos (21 (7, .
Is consider the case not zero, but K when A 2 (K) and A This arises when the surface is not a constant and (K) are both functions applicable on a surface is of revolution. 2) 0, u though the had A 2 (u) ~- A (u) been (u) are both functions of reasoning would have held equally only assumed to bo a function of u. We . t t THE GROUND FORM WHEN H = 38 If then we take = 6 2 \idu^ = /zi^. 22 = a*0 and therefore ^ = a! (a 12 ^ + a v = alQ Similarly wo have = alv = a^v and therefore 2 = 14 v It follows that tf^ = v v = A (v), A (0, v) = 0.
2) , {rpih}. , and we know that they will be tensor components. 3) j^ will be a ten or component. what is called all We write the tensor components an Einstein space. A A ij ... space for which vanish TENSOR THEORY 24 where m is independent of the integers r, A, is called an extended Einstein space. We can form invariants from the associate tensor com- Thus ponents. is (18- 4 ) an invariant which we may denote by A. 5) write k Ap . We thus series of invariants A'P A*, A'P A*AI A'p A*A qr A r t .