### 9. Power Series Method

#### a. Equations

A more accurate way to compute spiral points is using infinite power series, Equations E-24 and E-25.

Equation E-24 | |

Equation E-25 |

δ_{i} in Equations E-24 and E-25 is in radians; it is computed with Equation E-18.

Although both equations have an infinite number of terms, each term is substantially smaller so only the first three need be used.

Radial chords are computed using Equation E-23 and deflection angles using Equation E-26.

Equation E-26 |

#### b. Example

Using the spiral of the previous example, set up and solve the spiral curve table.

Chord Num | l_{i} |
δ_{i}, radians |
x_{i}, ft |
y_{i}, ft |
c_{i}, ft |
a_{i} |

1 | 60.00 | 0.009425 | 60.00 | 0.19 | 60.00 | 0°10'48" |

2 | 120.00 | 0.037699 | 119.98 | 1.51 | 119.98 | 0°43'12" |

3 | 180.00 | 0.084823 | 198.87 | 5.09 | 179.87 | 1°37'12" |

4 | 240.00 | 0.150796 | 239.46 | 12.04 | 239.46 | 2°52'46" |

5 | 300.00 | 0.235619 | 298.34 | 23.47 | 298.34 | 4°29'52" |

Compare this curve table to the first table in Section 7e. The results of this "more exact" calculations are basically the same as those based on approximations.

Should a much longer spiral or one with a great radii difference be used, the approximations may be might introduce measurable error in which case the power series equations should be used.