drawn in the sections correctly with the dips they would actually appear to have in a vertical cliff if one were formed along the line of section. Although the little trigonometry I ever possessed had long grown rusty from disuse, I yet contrived to puzzle out a formula which should express this correction, and from that calculated the table.
Subsequently, however, I lost the clue which had led me to the results, and became doubtful as to their correctness; I therefore applied to my friend Mr. Hopkins, then President of the Geological Society of London, and he, with his usual kindness, favoured me with the following solution of the problem, which I was glad to find gave the same result as that at which I had arrived by a more roundabout and empirical course.
Fig. 33.
Let O A be a horizontal line on the surface of a bed, it will be the direction of the strike: O C the direction of the section as given by the compass, O C being also horizontal.
Draw A C in the same horizontal plane as O A and O C, and at right angles to O A, A C will be the direction of the dip as given by the compass.
Draw C B, vertical, to meet the surface of the bed in B, and join A B and O B.
The angle C A B will be the real dip, and C O B the apparent dip, of the bed, as seen in the face of the supposed cliff or section.
Let O C A = x, the angle which the section makes with the direction of the dip,
C A B = y, the real dip,
C O B = z, the apparent dip,
then tan. y = BCAC
but B C = O B sin. z
and A C = O C cos. x
= OB cos. z cos. x
∴ tan. y = tan. zcos. x sec. x
or to radius r
(1) r tan. y—tan. z sec. x
and ∴ log. tan. y = log. tan. z + log. sec. x-10;
(2) or tan. z = tan. ysec. x
∴ log. tan. z = 10 + log. tan. y— log. sec. x.
(1) Giving the true dip if the apparent dip were observed in a cliff.
(2) Giving the apparent dip that ought to be drawn in the section when the true dip is known.
