16 Scientific Proceedings, Royal Dublin Society. 
this evident, imagine the whole to receive a tilt equal and opposite 
to that represented by OM: then by means of (11) we see that the 
two planes will be made to dip in opposite (or like) directions, and 
therefore their line of intersection will become horizontal. The 
following, among other questions, can be solved by the aid of this 
principle. 
(v). Strata of known dip crop out on a hill-side of known slope: 
to find the direction of the outcrop. 
Here (in fig. 4 or 5) if OP represent the dip and OQ the slope, 
OM will be the direction of outcrop. 
(v1). Given the strike of the strata and the bearing of their 
outcrop on ground of known slope, to find the dip. 
In this case draw OQ to represent the given slope and the line 
QP at right angles to the bearing of the outcrop to meet OP 
drawn at right angles to the strike; this fixes P, and OP repre- 
sents the dip of the strata in amount as well as direction. 
(viz). Given the bearings of the outcrops of a 
bed at two spots on a hill, the slopes 
being different at the two spots and 
being known, to find the dip of the 
bed. 
Here (fig. 6) draw OQ and OR to represent 
the two slopes, and from Q and R draw lines Fic. 6. 
each at right angles to the bearing of the outcrop at the corre- 
sponding spot on the hill: these lines meeting in P, OP repre- 
sents the dip of the bed. Of course the same considerations 
apply throughout to a fault or lode as well as to a bed, and it is 
needless to restate the various questions to suit this case. It 
should be observed, however, that if the inclination of the fault 
or lode be expressed with reference to the vertical (7.e. as a hade), 
we must use the second or cotangent-scale on the protractor. 
(vit1). Given the dips on opposite sides of a fold with inclined 
axis, to find the direction and inclination of the axis. 
Here we regard the strata on opposite sides as inclined planes 
which meet in a line parallel to the axis of the fold. If OP and 
