14 Scientific Proceedings, Royal Dublin Society. 
way a line such as OP, laid down simply by the aid of the pro- 
tractor, may be taken to represent, both in direction and amount, 
the slope of a hill-side, the dip of a bed, or the co-hade? of a fault 
or lode. Ii we are given the inclination of a plane to the vertical, 
Pig. 2: 
¢.g. the hade of a fault, we use the cotangent- instead of the 
tangent-scale of the protractor. Of course the inclination of any 
plane to the horizontal is equal to the inclination of its normal to 
the vertical, and vice versd. By mentally supplying a picture like 
fig. 3 to each of the diagrams, all the following solutions will 
become evident. In actual practice the protractor is, of course, 
not placed in an upright position beneath the paper, but laid flat 
upon it. 
First we will suppose we have two inclined planes of given 
inclinations, and wish to know in what direction and through 
what angle one must be turned to bring it into the position of the 
P 12 
Q Oo 
Fie. 4. Fie. 5. 
other. By means of the protractor (tangent-scale) lay down OP 
and OQ (fig. 4 or 5) to represent the inclinations to the horizontal 
of the two planes: then the question is, what kind of tilt is 
required to bring CP into the position CQ. Draw PQ; this gives 
the direction of the required tilt. Draw OM perpendicular to ~ 
1 The term hade seems to be correctly used for the inclination of a fault to the 
vertical, and so the defect of this from 90° (co-hade) corresponds to dip, i.¢., inclination 
to the horizontal. Similarly, we may find it convenient to speak of the inclination of 
a bed to the vertical as its co-dip, equivalent to hade. 
