SURFACES IN HYPERSPACE. 



293 



and the last bracket becomes a single term repeated. Moreover the 

 first bracket may be changed by interchanging the indices i, j, k. For, 



a (a,-/) ^. d{aik) 



^ijkyriYijlsk -Z = -^ijklriysi-Ytk — 



dijk oijj 



since in either case the summation is over all values of j and k. Henqe, 



^ , , dh/i 



dXgdXr 



This somewhat cumbrous form may be simplified by introducing 

 the notation of the Christoffel symbols, 



/• s 

 t 



dcirt dttts da 



dXs dx 

 The above expression then becomes 



dxt 



(32) 



s r 



< iikHriy sjy tk 



^ 3 

 k 



d'Vi 



+ 2»v(«.;) T-^ To- 



OXaOXr 



To complete the solution for the second derivatives, multiply by Cti and 

 sum over t. Then 



^tCt 



r s 

 t 



— ^ijlfriysj 



I 



+ ^iian) 





Next multiply by (a^"^^'>) and sum over /. Then 



^tictiia^"''^) 



r .s 

 t 



= ^ijiyr^-^siia^"''') 



. I . 



+ 



dXrdXs ' 



and hence finally we have the expression 



d^ym 

 dXrdx. 



^tictiia^""'^) 



r s 

 t 



Si;77r.7.,(a ('"')) 



/ 



(33) 



