Harxer—The Use of the Protractor in Field-Geology. 15: 
PQ (produced if necessary), place the edge of the protractor along 
PMQ, with its zero-point at M, and read off the angle between P 
and Q. This is the sum or difference of the readings of the 
tangent-scale at P and Q, according as these points are on opposite 
sides or the same side of M, and it gives the amount of the re- 
quired tilt. To see this, it is sufficient to imagine the protractor 
placed in an upright position beneath the line PMQ, with its zero- 
point at M. The following are some of the geological applications 
of this :— 
(1). Strata originally inclined receive a new tilt: given the 
original and the final dip of the strata, to find the 
direction and amount of the new tilt. 
(11). Strata originally of known inclination receive a new tilt 
of given direction and amount: to find their final dip. 
In this case draw OP to represent the original dip and a line- 
PQ in the direction of the given tilt: then draw OM perpendi- 
cular to this line, and placing the edge of the protractor along 
PM with, its zero-point at M, take the angle from P to Q equal to- 
the angle of the tilt: then OQ represents the final dip. 
(a1). Strata originally inclined have received a new tilt of 
known direction and amount, and their final dip is 
also known : to find their original dip. 
This is merely the preceding question reversed. 
(tv). Given the dip of a bed and the slope of a hill-side, to find 
the angle at which the bed meets the sloping surface. 
Here we have only to draw OP to represent the dip and OQ 
the slope, and to take the angle between P and Q in the manner 
described above, the zero-point of the protractor being placed at M. 
This result is merely approximate and for low angles, for the 
tilt was supposed to be about a horizontal axis. 
_ We pass on to another class of questions, of which the general 
problem is: given, in direction and amount, the inclinations to the. 
horizontal of two inclined planes, to find, in direction and amount, 
the inclination of their line of intersection. The construction is. 
simple. Draw OP and OQ (fig. 4 or 5) to represent the inclina- 
tions of the two planes: then OM, drawn perpendicular to PQ, 
represents the inclination of their line of intersection. To make- 
