| For the flank, ρ − c = 2r sin θ | R |
| R − 2r |
| For the face, ρ′ − c = 2r sin θ | R |
| R + 2r |
(29)
For the proportions approved of by Willis, sin θ = 1⁄4 nearly; r = p (the pitch) nearly; c = 1⁄2p nearly; and, if N be the number of teeth in the wheel, r/R = 6/N nearly; therefore, approximately,
| ρ − c = | p | · | N |
| 2 | N − 12 |
| ρ − c = | p | · | N |
| 2 | N + 12 |
(30)
| Fig. 105. |
Hence the following construction (fig. 105). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab = ac − 1⁄2p. Draw radii bd, ce; draw fb, cg, making angles of 751⁄2° with those radii. Make bf = p′ − c, cg = p − c. From f, with the radius fa, draw the circular arc ah; from g, with the radius ga, draw the circular arc ak. Then ah is the face and ak the flank of the tooth required.
To facilitate the application of this rule, Willis published tables of ρ − c and ρ′ − c, and invented an instrument called the “odontograph.”