#### Table I

Comparison of expanded and unexpanded solutions. α = 85°, λ = 180 Å, R = 50 cm, 1/σ = 11 520 cm^{−1}, |w| < 0.2 cm, |z| < 0.4 cm, Δr′ = 0. Column (i) gives the ray aberration for w = −0.2 cm, z = −0.4 cm. Column (ii) gives the ray aberration for w = +0.2 cm, z = +0.4 cm. These represent the extreme rays for each θ in this case.

θ = | 0° | 2° | 4° |
---|

| (i) | (ii) | (i) | (ii) | (i) | (ii) |
---|

Spherical Δλ |

Unexpanded | −0.0193 | −0.0436 | −1.004 | 1.686 | −2.47 | 5.53 Å |

Expanded | −0.0192 | −0.0435 | −1.032 | 1.656 | −2.59 | 5.43 Å |

Unexpanded-Expanded | −0.0001 | −0.0001 | −0.028 | 0.030 | 0.12 | 0.10 Å |

Cylindrical Δλ |

Unexpanded | −0.0042 | 0.0042 | −0.532 | 0.808 | −2.10 | 3.20 Å |

Expanded | −0.0035 | 0.0035 | −0.558 | 0.776 | −2.22 | 3.10 Å |

Unexpanded-Expanded | −0.0007 | 0.0007 | 0.026 | 0.032 | 0.12 | 0.10 Å |

#### Table II

Accuracy of approximate solution. α = 89°, λ = 10 Å, R = 400 cm, 1/σ = 11 520 cm^{−1}, |w| < 1.0 cm, |z| < 0.4 cm, Δr′ = 0.

| Δλ for unexpanded solution | Δλ for expanded solution |
---|

|θ|_{max} | Δλ_{spherical} | Δλ_{cylindrical} | Q | Δλ_{spherical} | Ratio of resolutions from unexpanded & expanded solutions |
---|

0° | 0.0107 Å | 0.0107 Å | 1.00 | 0.0104 Å | 1.03 |

${\scriptstyle \frac{1}{8}}\xb0$ | 0.0279 Å | 0.0271 Å | 1.03 | 0.0271 Å | 1.03 |

${\scriptstyle \frac{1}{4}}\xb0$ | 0.0796 Å | 0.0762 Å | 1.04 | 0.0772 Å | 1.03 |

#### Table III

Accuracy of approximate solution, cylindrical grating. α = 89.5°, λ = 3 Å, R = 400 cm, 1/σ = 11 520 cm^{−1}, |w| < 1.0 cm, |z| < 0.4 cm, Δr′ = 0.

θ_{max} | Δλ | Ratio of resolutions from unexpanded & expanded solutions | Line center shift unexpanded | Line center shift expanded |
---|

Unexpanded | Expanded |
---|

0° | 0.0227 Å | 0.0204 Å | 1.11 | 0.0028 Å | 0.0000 Å |

${\scriptstyle \frac{1}{8}}\xb0$ | 0.0583 Å | 0.0514 Å | 1.13 | 0.0082 Å | 0.0000 Å |

${\scriptstyle \frac{1}{4}}\xb0$ | 0.1652 Å | 0.1448 Å | 1.14 | 0.0243 Å | 0.0001 Å |