** Next:** Examples of Applications
** Up:** Track Fitting in r-z
** Previous:** Pivot Transformation

Now let us turn our attention to the *r*-*z* track fitting under
the influence of multiple scattering.
The treatment here provides a way to calculate the *E*_{0}
of the last subsection including
multiple scattering effects in track fitting.
From now on, we take (*y*_{0},*z*_{0}) = (0,0) and .The multiple scattering modifies the track parameter of the particle as

| |
(90) |

where
| |
(91) |

represents the change of the dip angle due to the multiple
scattering as a function of *y*.
This results in
| |
(92) |

where the parmeters with the suffix are those at the
initial point before multiple scattering.
Now, let us assume that we are given
(*n*+1) hit points, ,measured at fixed *y* positions, .Then the minimization of the :

| |
(93) |

The parameter vector that gives the minimum is
| |
(94) |

with
| |
(95) |

Through the above equation, the change in
due to the multiple scattering induces a change in :
| |
(96) |

From this we obtain the full error matrix including multiple scattering:
| |
(97) |

Again the problem reduces to the evaluation of the
correlation matrix in the coordinate measurements:

| |
(98) |

with
| |
(99) |

where the last expression is valid for .The full error matrix is then written in the form:
| |
(100) |

where the first term is from the coordinate measurement errors while the
second term from the multiple scattering:
| |
(101) |

In the case of equal space sampling with the same position resolution,
*E*_{M} and *E*_{MS} reduce to

| |
(102) |

with
| |
(103) |

in the large *n* limit.

** Next:** Examples of Applications
** Up:** Track Fitting in r-z
** Previous:** Pivot Transformation
*Keisuke Fujii*

*12/4/1998*