What is the derivation of cubic spline equations?

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The discussion focuses on the derivation of cubic spline equations, specifically the natural cubic spline interpolation method. The user seeks clarification on the origin of the equations S_{i}(x) and the associated system of equations. They reference a document from McGill University and express confusion regarding the integration of S''i(x), which results in products of x and subscripts. Additional resources, including a complete derivation from Physics Forums, are provided for further understanding.

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Ok, I have found a document that explains but there is something I don't understand. Heres a screenshot of the problem:

http://img412.imageshack.us/img412/2223/integraton22.th.jpg

When I integrate S''i(x), I get products of x and x_subscripts. For example, when I integrate once I get
[PLAIN]http://img717.imageshack.us/img717/8044/firstintegral.gif

Can anyone please help me?

(the document I found is here http://www.cs.mcgill.ca/~dtitle/cs350/notes/21_cubsplin.pdf)
 
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