| PH² ∶ PN² ∷ NO ∶ OP, | |
| then PN²- PH² = HN² | |
| and PO - NO = NP, | |
| therefore | HN² ∶ PN² ∷ NP ∶ PO, |
| and finally, PO = PN³HN². |
| Then put FP = HC = FN = ρ; HN | = | ρ - z; then as FP² - FH² = PH² = ρ² - z² |
| PN² = PH² + HN² | = | (ρ² - z²) + (ρ² - 2 ρ z + z²) |
| = | 2ρ² - 2 ρ z | |
| PN | = | √2 ρ (ρ - z) |
| Therefore PO | = | √{2 ρ (ρ - z)}³ (ρ - z)² |
| = | √{2 ρ (ρ - z)}³ (ρ - z)⁴ | |
| and finally, PO | = | 2√2 √ρ³ ρ - z |
Fig. 67.
To find the versed sine of the curvature (which may be useful in the examination of the mirrors by a mould) we may proceed (see [fig. 67]) to
put AG = f; BE = C; AC = R
then BG² = AG . GD
4 BG² = BE² = 4 AG . GD
C² = 4 f . (2 R - f)
C² = 8 f R - 4 f²
From which equation,
2 f - 2 R = ± √4 R² - C² = -2 R + C²4 R - C⁴64 R³ &c.
2 f = C²4 R - C⁴64 R³
f = C²8 R - C⁴128 R³.
Fig. 68.
