Figure 13-20.-Random line method of locating intermediatestations.passes through both A and C. When you do this,the telescope is reversed, but the instrument isnot rotated. This means that if the telescope isreversed for backlighting on A, all sightings onA are made with the telescope reversed. Mark apoint on the ground directly under the instrument.Then, you continue to use this method with thetelescope direct for each backsight on A. Marka second point on the ground. The point you needon the line AC is then the midpoint between thetwo marked points.The method outlined above is usually timeconsuming. Even though the shifting head of theinstrument is used in the final instrumentmovements, you may have to pick up and movethe instrument several times. The followingmethod often saves time. After finding theapproximate position of the line between the twopoints, you mark two points B´ and B´´, (fig. 13-19,view C), 1 or 2 feet apart where you know theystraddle the line AC. Set up over each of thesetwo points in turn and measure the deflectionangles a and P. Also measure the horizontaldistance a, between points B´ and B´´. Then youcan find the position B on the line AC by usingthe following equation:in which a’ is the proportionate offset distancefrom B´ toward B´´ for the required point B, andaand P are both expressed in minutes or inseconds.RANDOM LINEIt is sometimes necessary to run a straight linefrom one point to another point that is not visiblefrom the first point. If there is an intermediatepoint on the line from which both end-points arevisible, this can be done by the balancing-inmethod described previously. If no such inter-mediate point exists, the RANDOM LINEmethod illustrated in figure 13-20, view A maybe used.The problem here is to run a line from A toB, B being a point not visible from A. It happens,however, that there is a clear area to the left ofthe line AB, through which a random line can berun to C; C being a point visible from A and B.To train a transit set up at A on B, you mustknow the size of the angle at A, You can measureside b and side a, and you can measure the angleat C. Therefore, you have a triangle in which youknow two sides and the included angle. You cansolve this triangle for angle A by (1) determiningthe size of side c by the law of cosines, thendetermining the size of angle A by the law of sines,(2) solving for angle A by reducing to two righttriangles.Suppose you find that angle A measures16°35´. To train a transit at A on B, you wouldsimply train on C and then turn 16°35´ to theright.You may also use the random line method tolocate intermediate stations on a survey line, Infigure 13-20, view B, stations 0 + 00 and 2 + 50,now separated by a grove of trees, were placedat some time in the past. You need to locatestations 1 + 00 and 2 + 00, which lie among thetrees.Run a line at random from station 0 + 00until you can see station 2 + 50 from somepoint, A, on the line. The transitman measuresthe angle at A and finds it to be 108°00´. Thedistances from A to stations 0 + 00 and 2 + 50are chained and found to be 201.00 ft and 98.30ft. With this information, it is now possible tolocate the intermediate stations between stations13-17