STATEMENT: In a isosceles triangle, the ratio of the perpendiculars from the vertex on their bases is constant.

Let `ABC` and `AB'C'` be two isosceles triangles whose bases are `BC` and `B'C'` and their vertex is at A. If `AO` and `AO'` are perpendiculars to their respective bases, then

`(AO)/(BC)=(AO')/(B'C')=1/2*cot(alpha/2)`

        `=Constant`

    Where, `alpha =`apex angle.

STATEMENT: In a isosceles triangle, the ratio of the perpendiculars from the vertex on their bases is constant.

Let `ABC` and `AB'C'` be two isosceles triangles whose bases are `BC` and `B'C'` and their vertex is at A. If `AO` and `AO'` are perpendiculars to their respective bases, then

`(AO)/(BC)=(AO')/(B'C')=1/2*cot(alpha/2)`

        `=Constant`

    Where, `alpha =`apex angle.

The stadia method of traversing is useful in following situations:

1. In differential levelling, the backsight and foresight distances are balanced conveniently if the leval is equipped with stadia hairs.
2. In profile levelling and cross sectioning, stadia is convenient means of finding distance from level to points on which rod readings are taken.
3. In rough trignometric, or indirect levelling with the transit, the stadia method is more rapid than any other method.
4. For traverse surveying of low relative accuracy, where only horizontal angles and distances are required, the stadia method is a useful rapid method.
5. On surveys of low relative accuracy-particularly topographic surveys where both the relative locations of points in a horizontal plane and the elevation of these points are desired, stadia are useful.

The stadia method of traversing is useful in following situations:

1. In differential levelling, the backsight and foresight distances are balanced conveniently if the leval is equipped with stadia hairs.
2. In profile levelling and cross sectioning, stadia is convenient means of finding distance from level to points on which rod readings are taken.
3. In rough trignometric, or indirect levelling with the transit, the stadia method is more rapid than any other method.
4. For traverse surveying of low relative accuracy, where only horizontal angles and distances are required, the stadia method is a useful rapid method.
5. On surveys of low relative accuracy-particularly topographic surveys where both the relative locations of points in a horizontal plane and the elevation of these points are desired, stadia are useful.

Case I:When the line of sight is inclined and staff is held vertical:

a) when the line of sight is inclined upwards and staff is held vertical:

let `theta` is the angle of elevation of the line of sight. Let us draw an intercept `A'B'` through `C` Perpendicular to `OC`.

Now,`ACA'=theta` and `A AC'` may be taken as `90 degree`.

From triangle `A A'C`, we have,

`cos(theta)=(A'C)/(AC)`

Thus, `A'B'=2*A'C` and `AB=2*AC`

so,`A'B'=S'=AB*cos(theta)=Scos(theta)`

Now,

`D=K*S'+C`

Or,`D=K*Scos(theta)+C`......(i)

Again, from triangle GCF,

`cos(theta)=(GF)/(GC)=H/D`

Or, `H=D*cos(theta)`

Or, `H=K*S*cos^2theta+C*costheta`.....(ii)

From (ii), the horizontal diatance between the staff held vertical and instrument station can be computed.

For determining the elevation difference between P and Q, it is necessary to determine the value of `V` i.e, `FC`, which is the difference of levels between the collimation plane and the central hair reading on the staff.

Again, from triangle CFG,

`sin(theta)=(CF)/(GC)=V/D`

Or, `V=D*Sin(theta)`

Or, `V=K*Scos(theta)*sin(theta)+C*sin(theta)`

Or, `V=K*S/2Sin2theta+Csintheta`

Let,`h=QC`, the central hair reading, then the level difference between G and Q for an angle of elevation is given by

`FQ=C-h`

If HI be the height of instruments, the reduced level is given by:

RL of `Q=HI+V-h`

Thus, if the RLs of instrument station and height of instrument are known,then

RL of Q=RL of instrument station +HI+V-h

b) when the line of sight is inclined downwards and staff is held vertical:

In this case, we can derive the expression of H and V following the above procedures.

`H=K*S*cos^2theta+C*cos(theta)`.......(iii)

`V=K*S/2Sin2theta+Csintheta`........(iv)

Similarly, If RL of instrument axis is known,

Elevation of Q=elevation of instrument axis (P')-V-h

Again, If the RL of instrument station and height of instrument is known,

RL of Q=RL of instrument station(P)+ height of instrument-V-h

Thus,combining both cases i.e (a) and (b),

RL of staff station (Q)=RL of instrument axis `+-V-h`

RL of staff station (Q)=RL of instrument station `+-V-h`

where positive sigh is used for angle of elevation and negative sign is used for angle of depression.

Case(ii): When the line of sight is inclined and the staff is held normal to the line of sight:

a)When the line of sight is inclined upwards and the staff is held normal to the line of sight i.e,angle of elevation:

Let `theta` is the angle of elevation of the line of sight.

    `AB` is the slope intercept `S`.

    `CQ` is the central hair reading `h`.

 Now, from right angled triangles `CQC'`,

 `sin theta=(C'C)/(CQ)=(C'C)/h`

 Or, `C'C=h*sin theta`

 Also,

 `cos(theta)=(C'Q)/(CQ)=(C'Q)/h`

 Or, `C'Q=h*cos theta`

 From the basic principle of stadia system,

 `D=K*S+C`

 We know,`H=GF'+FF'`

 Again, from triangle GCF',we have,

 `cos(theta)=(GF)/(GC)=(GF)/D`

 Or, `H=GF'+CC'`

 Or, `H=D*cos theta+CC'`

 Or, `H=D*cos theta+h*sin theta`

 Or, `H=(K*S+C) cos theta+h*sin(theta)`

 Similarly,

 `V=F'C=D*sin theta`

 Or,`V=(K*S+C)sin theta`

 Now, for angle of elevation,

`FQ=FC'=D sin theta`

Or,`FQ=V-h*cos theta`

If HI is the height of instruments above datum,

RL of Q`=HI+V-h*cos theta`

Or,RL of Q= RL of instrument station`+HI+(K*S+C)sin(theta)-h cos theta`

b) When the line of sight is inclined downwards and staff is held normal to the line of sight i.e, angle of depression.

Let `theta` is the angle of depression of the line of sight.

We can derive,

`H=(K*S+C)cos(theta)-h*sin theta`

`V=(K*S+C)sin theta`

Now,for angle of depression,we have,

`FQ=V+h*cos theta`

and RL of Q`=HI-V-h cos theta`

Thus,combining both cases,

RL of staff station (Q)=RL of instrument station `(+-)(KS+C)sin theta-h costheta`

where positive sigh is used for angle of elevation and negative sign is used for angle of depression.

Case I:When the line of sight is inclined and staff is held vertical:

a) when the line of sight is inclined upwards and staff is held vertical:

let `theta` is the angle of elevation of the line of sight. Let us draw an intercept `A'B'` through `C` Perpendicular to `OC`.

Now,`ACA'=theta` and `A AC'` may be taken as `90 degree`.

From triangle `A A'C`, we have,

`cos(theta)=(A'C)/(AC)`

Thus, `A'B'=2*A'C` and `AB=2*AC`

so,`A'B'=S'=AB*cos(theta)=Scos(theta)`

Now,

`D=K*S'+C`

Or,`D=K*Scos(theta)+C`......(i)

Again, from triangle GCF,

`cos(theta)=(GF)/(GC)=H/D`

Or, `H=D*cos(theta)`

Or, `H=K*S*cos^2theta+C*costheta`.....(ii)

From (ii), the horizontal diatance between the staff held vertical and instrument station can be computed.

For determining the elevation difference between P and Q, it is necessary to determine the value of `V` i.e, `FC`, which is the difference of levels between the collimation plane and the central hair reading on the staff.

Again, from triangle CFG,

`sin(theta)=(CF)/(GC)=V/D`

Or, `V=D*Sin(theta)`

Or, `V=K*Scos(theta)*sin(theta)+C*sin(theta)`

Or, `V=K*S/2Sin2theta+Csintheta`

Let,`h=QC`, the central hair reading, then the level difference between G and Q for an angle of elevation is given by

`FQ=C-h`

If HI be the height of instruments, the reduced level is given by:

RL of `Q=HI+V-h`

Thus, if the RLs of instrument station and height of instrument are known,then

RL of Q=RL of instrument station +HI+V-h

b) when the line of sight is inclined downwards and staff is held vertical:

In this case, we can derive the expression of H and V following the above procedures.

`H=K*S*cos^2theta+C*cos(theta)`.......(iii)

`V=K*S/2Sin2theta+Csintheta`........(iv)

Similarly, If RL of instrument axis is known,

Elevation of Q=elevation of instrument axis (P')-V-h

Again, If the RL of instrument station and height of instrument is known,

RL of Q=RL of instrument station(P)+ height of instrument-V-h

Thus,combining both cases i.e (a) and (b),

RL of staff station (Q)=RL of instrument axis `+-V-h`

RL of staff station (Q)=RL of instrument station `+-V-h`

where positive sigh is used for angle of elevation and negative sign is used for angle of depression.

Case(ii): When the line of sight is inclined and the staff is held normal to the line of sight:

a)When the line of sight is inclined upwards and the staff is held normal to the line of sight i.e,angle of elevation:

Let `theta` is the angle of elevation of the line of sight.

    `AB` is the slope intercept `S`.

    `CQ` is the central hair reading `h`.

 Now, from right angled triangles `CQC'`,

 `sin theta=(C'C)/(CQ)=(C'C)/h`

 Or, `C'C=h*sin theta`

 Also,

 `cos(theta)=(C'Q)/(CQ)=(C'Q)/h`

 Or, `C'Q=h*cos theta`

 From the basic principle of stadia system,

 `D=K*S+C`

 We know,`H=GF'+FF'`

 Again, from triangle GCF',we have,

 `cos(theta)=(GF)/(GC)=(GF)/D`

 Or, `H=GF'+CC'`

 Or, `H=D*cos theta+CC'`

 Or, `H=D*cos theta+h*sin theta`

 Or, `H=(K*S+C) cos theta+h*sin(theta)`

 Similarly,

 `V=F'C=D*sin theta`

 Or,`V=(K*S+C)sin theta`

 Now, for angle of elevation,

`FQ=FC'=D sin theta`

Or,`FQ=V-h*cos theta`

If HI is the height of instruments above datum,

RL of Q`=HI+V-h*cos theta`

Or,RL of Q= RL of instrument station`+HI+(K*S+C)sin(theta)-h cos theta`

b) When the line of sight is inclined downwards and staff is held normal to the line of sight i.e, angle of depression.

Let `theta` is the angle of depression of the line of sight.

We can derive,

`H=(K*S+C)cos(theta)-h*sin theta`

`V=(K*S+C)sin theta`

Now,for angle of depression,we have,

`FQ=V+h*cos theta`

and RL of Q`=HI-V-h cos theta`

Thus,combining both cases,

RL of staff station (Q)=RL of instrument station `(+-)(KS+C)sin theta-h costheta`

where positive sigh is used for angle of elevation and negative sign is used for angle of depression.

There are basically three systems of tacheometric leveling:

•  Stadia hair system
•  Tangential system
•  Subtense bar system

THE STADIA HAIR METHOD:

 This is the most extensively used system of tacheometry, particularly for detailed work, such as those required in engineering surveys. Stadia hair system may be of two types:

•  The fixed hair method
• The movable hair method

FIXED HAIR METHOD:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system is fixed. The staff intercept `s`,therefore changes according to the distance `D` and vertical angle `theta`. when the staff intercept is more than the length of the staff, only half intercept is read, which is equal to the length of the difference between central stadia hair reading and the lower/upper stadia hair reading.

fig: Fixed hair method

This methods can be useful even when horizontal sights are not possible. For inclined sights, reading may be taken by holding the staff either vertical or normal to the line of sight.some important features of this method are:

• Distance between hairs is constant.
• The staff intercept varies.
• The apex angle is constant.
• The distance varies with the staff intercept.

Movable Hair Method:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system can be varied.The staff intercept,`s` is kept fixed. The stadia hairs can be moved vertically up and down by using micrometer screws. Movable hair method is not in common use due to difficulties in determining the stadia interval accurately. Some important features of this method are:

• Top and bottom hairs are movable.
• Staff intercept is kept constant.
• This method is rarely used.

There are basically three systems of tacheometric leveling:

•  Stadia hair system
•  Tangential system
•  Subtense bar system

THE STADIA HAIR METHOD:

 This is the most extensively used system of tacheometry, particularly for detailed work, such as those required in engineering surveys. Stadia hair system may be of two types:

•  The fixed hair method
• The movable hair method

FIXED HAIR METHOD:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system is fixed. The staff intercept `s`,therefore changes according to the distance `D` and vertical angle `theta`. when the staff intercept is more than the length of the staff, only half intercept is read, which is equal to the length of the difference between central stadia hair reading and the lower/upper stadia hair reading.

fig: Fixed hair method

This methods can be useful even when horizontal sights are not possible. For inclined sights, reading may be taken by holding the staff either vertical or normal to the line of sight.some important features of this method are:

• Distance between hairs is constant.
• The staff intercept varies.
• The apex angle is constant.
• The distance varies with the staff intercept.

Movable Hair Method:

In this method, the distance between the upper hair and lower hair i.e, stadia interval `i`, on the diaphragm of the lens system can be varied.The staff intercept,`s` is kept fixed. The stadia hairs can be moved vertically up and down by using micrometer screws. Movable hair method is not in common use due to difficulties in determining the stadia interval accurately. Some important features of this method are:

• Top and bottom hairs are movable.
• Staff intercept is kept constant.
• This method is rarely used.

Let us consider the external focusing type of telescope and let us consider the line of sight to be horizontal. Let,

`f=`focal length of the object glass.

`i=`stadia hair interval `=ab`

 `S=`Staff intercept

  `=AB`

`c=`siatance from O to the vertical axis of the instrument

`d=`distance from O to the staff

`d'=`distance from O to the plane of the diaphragm

`D=`horizontal axis from the vertical axis to the staff.

Now, from triangle AOB and aOb, we have,

`d/d'=u/v=S/i`.......(i)

Again, from lens formula, we have,

`1/f=1/u+1/v`

Multiplying both sides by `uf`, we get,

`(uf)/f=(uf)/u+(uf)/v`

or, `u=(uf)/v+f`....(ii)

Now, from (i) and (ii), we get,

`u=f+f/(i/S)`

 `=f+(fS)/i`

Adding a constant `c` on both sides, we get,

`u+c=(fS)/i+(f+c)`

Again from figure,

`D=u+c`

Thus,`D=(fS)/i=(f+c)

Also, let `K=f/i` and `C=f+c`,

then the above expression becomes,

`D=K*S+C`.....(iii)

This is the standard equation used to explain the principle of stadia hair system.

Where `K` is the multiplying constant and `C` is the additive constant.

Now, 

for analytic lens, `K=100` and `C=0`

for external focusing telescope, `C=0.3` to `0.61`

for internal focusing telescope, `C=0.08` to `0.2`

Let us consider the external focusing type of telescope and let us consider the line of sight to be horizontal. Let,

`f=`focal length of the object glass.

`i=`stadia hair interval `=ab`

 `S=`Staff intercept

  `=AB`

`c=`siatance from O to the vertical axis of the instrument

`d=`distance from O to the staff

`d'=`distance from O to the plane of the diaphragm

`D=`horizontal axis from the vertical axis to the staff.

Now, from triangle AOB and aOb, we have,

`d/d'=u/v=S/i`.......(i)

Again, from lens formula, we have,

`1/f=1/u+1/v`

Multiplying both sides by `uf`, we get,

`(uf)/f=(uf)/u+(uf)/v`

or, `u=(uf)/v+f`....(ii)

Now, from (i) and (ii), we get,

`u=f+f/(i/S)`

 `=f+(fS)/i`

Adding a constant `c` on both sides, we get,

`u+c=(fS)/i+(f+c)`

Again from figure,

`D=u+c`

Thus,`D=(fS)/i=(f+c)

Also, let `K=f/i` and `C=f+c`,

then the above expression becomes,

`D=K*S+C`.....(iii)

This is the standard equation used to explain the principle of stadia hair system.

Where `K` is the multiplying constant and `C` is the additive constant.

Now, 

for analytic lens, `K=100` and `C=0`

for external focusing telescope, `C=0.3` to `0.61`

for internal focusing telescope, `C=0.08` to `0.2`

Subtense bar is an instrument used for measuring horizontal distances in the areas where direct chaining becomes difficult due to undulation or other obstructions.

It consists of a metal tube of length varying from 3m to 4 m. Two disc 20 cm in diameter painted either black or red on one side and white on the other, each with a 7.5m white or black center are placed 3m apart. Red or black faces of discs are kept towards the theodolite.

At the center of the bar,an alidade perpendicular to the axis of the bar is attached. The bar is mounted on special tripod.

COMPUTATION OF SUBTENSE BAR DISTANCE:

Here, from triangle `OAC` and `OCB`,

`D*tan (theta/2)=S/2`

`D=S/2 cot(theta/2)`

Or, `D=S/(2tan(theta/2))`

Now, for an small angle `theta`,

`tan(theta/2)=theta/2`, where `theta` is in radian.

      `=1/2*theta/(206265)`, where `theta` is in second..

 Thus, we have,

 `D=S/(2*(1/2*theta/206265))`
Or,`=(S*206265)/theta`, Where `theta` is in second.

 EFFECT OF ANGULAR ERROR ON HORIZONTAL DISTANCE:

 From figure,

 `tan(theta/2)=((S/2)/D)`

 `S/2=D*tan(theta/2)=D*theta/2`

Or,`S=D*theta`.......(i)

 We know,

 `D=(S*206265)/theta`

 Let the negative error in `theta` be `delta theta` and the positive error in `D` be `delta D`then,

 `S=(D+delta D)(theta- delta theta)`......(ii)

 Or,`(D+delta D)/D=theta/(theta+delta theta)`

 On cross multiplication,

 `D theta-D delta theta-theta*D delta theta=D theta`

Or,`delta D(theta - delta theta)=D*delta theta`

Or,`delta D=(D delta theta)/(theta + delta theta)`

Similarly, if `delta theta` is positive error and `delta D` is the negative error.,

`delta D=(D delta theta)/(theta+delta theta)`

However,`delta theta` is too small as compared to `theta`,so,

`delta theta=(D delta theta)/theta`

Subtense bar is an instrument used for measuring horizontal distances in the areas where direct chaining becomes difficult due to undulation or other obstructions.

It consists of a metal tube of length varying from 3m to 4 m. Two disc 20 cm in diameter painted either black or red on one side and white on the other, each with a 7.5m white or black center are placed 3m apart. Red or black faces of discs are kept towards the theodolite.

At the center of the bar,an alidade perpendicular to the axis of the bar is attached. The bar is mounted on special tripod.

COMPUTATION OF SUBTENSE BAR DISTANCE:

Here, from triangle `OAC` and `OCB`,

`D*tan (theta/2)=S/2`

`D=S/2 cot(theta/2)`

Or, `D=S/(2tan(theta/2))`

Now, for an small angle `theta`,

`tan(theta/2)=theta/2`, where `theta` is in radian.

      `=1/2*theta/(206265)`, where `theta` is in second..

 Thus, we have,

 `D=S/(2*(1/2*theta/206265))`
Or,`=(S*206265)/theta`, Where `theta` is in second.

 EFFECT OF ANGULAR ERROR ON HORIZONTAL DISTANCE:

 From figure,

 `tan(theta/2)=((S/2)/D)`

 `S/2=D*tan(theta/2)=D*theta/2`

Or,`S=D*theta`.......(i)

 We know,

 `D=(S*206265)/theta`

 Let the negative error in `theta` be `delta theta` and the positive error in `D` be `delta D`then,

 `S=(D+delta D)(theta- delta theta)`......(ii)

 Or,`(D+delta D)/D=theta/(theta+delta theta)`

 On cross multiplication,

 `D theta-D delta theta-theta*D delta theta=D theta`

Or,`delta D(theta - delta theta)=D*delta theta`

Or,`delta D=(D delta theta)/(theta + delta theta)`

Similarly, if `delta theta` is positive error and `delta D` is the negative error.,

`delta D=(D delta theta)/(theta+delta theta)`

However,`delta theta` is too small as compared to `theta`,so,

`delta theta=(D delta theta)/theta`

• Reconnaissance
• Establishment of tacheometric closed traverse as per field requirement ensuring each instrument station should command a wide area. If the area is too large,link traverse and addition of triangles may also be prederred.

• A tacheometer is set up at the starting station. It is centered and levelled accurately. Height of instruments is Measured.
• The instrument is oriented with reference to any predetermined station (BS) by taking it's bearing.Horizontal angles of all nearby details are recorded in a field book. Index map and reference sketches are also prepared in field.

• A back sight reading is taken on a nearby benchmark if available.If not,fly levelling should be done to know the RL of the instrument station. After completing the loop of fly levelling between bench mark and any nearby traverse station, loop levelling is conducted along the traverse to know the RL of all stations.
• Similarly other stations are connected and above steps are repeated.
• Horizontal distance, vertical distance, bearings are computed from field book. RLs of all points are calculated in a separate field book for the purpose of plotting. The points are plotted in the maps with suitable scale and Rls of the respective points are noted by taking the distances and RLs from the prepared table.The contour lines may be drawn by the method of interpolation.

• Reconnaissance
• Establishment of tacheometric closed traverse as per field requirement ensuring each instrument station should command a wide area. If the area is too large,link traverse and addition of triangles may also be prederred.

• A tacheometer is set up at the starting station. It is centered and levelled accurately. Height of instruments is Measured.
• The instrument is oriented with reference to any predetermined station (BS) by taking it's bearing.Horizontal angles of all nearby details are recorded in a field book. Index map and reference sketches are also prepared in field.

• A back sight reading is taken on a nearby benchmark if available.If not,fly levelling should be done to know the RL of the instrument station. After completing the loop of fly levelling between bench mark and any nearby traverse station, loop levelling is conducted along the traverse to know the RL of all stations.
• Similarly other stations are connected and above steps are repeated.
• Horizontal distance, vertical distance, bearings are computed from field book. RLs of all points are calculated in a separate field book for the purpose of plotting. The points are plotted in the maps with suitable scale and Rls of the respective points are noted by taking the distances and RLs from the prepared table.The contour lines may be drawn by the method of interpolation.

TACHEOMETER

 A tacheometer is a transit theodolite having a stadia telescope fitted with two horizontal hairs called Stadia hairs(stadia lines) in addition to the usual cross hairs. The stadia diaphragm has three horizontal hairs: a central horizontal hair, and upper and lower stadia hairs. The upper and lower stadia hairs are equidistant from the central horizontal hairs.

fig:  Tacheometer

 Characteristics of tacheometer:

•  The value of multyplying constant should be 100.
•  The value of additive constant should be zero.
•  The telescope is fitted with analytic lens.
•  The magnification of telescope should be 20 to 80 diameter.
• Magnifying power of eye piece is kept high.

 Stadia rods/Levelling staffs:

 The stadia rod is about 3m to 5 m long. For short sights, an ordinary levelling staffs may be used. For long sights, special staff called stadia rods is generally used.

TACHEOMETER

 A tacheometer is a transit theodolite having a stadia telescope fitted with two horizontal hairs called Stadia hairs(stadia lines) in addition to the usual cross hairs. The stadia diaphragm has three horizontal hairs: a central horizontal hair, and upper and lower stadia hairs. The upper and lower stadia hairs are equidistant from the central horizontal hairs.

fig:  Tacheometer

 Characteristics of tacheometer:

•  The value of multyplying constant should be 100.
•  The value of additive constant should be zero.
•  The telescope is fitted with analytic lens.
•  The magnification of telescope should be 20 to 80 diameter.
• Magnifying power of eye piece is kept high.

 Stadia rods/Levelling staffs:

 The stadia rod is about 3m to 5 m long. For short sights, an ordinary levelling staffs may be used. For long sights, special staff called stadia rods is generally used.

The stadia constant can be determined by:

• Laboratory measurement method
• Field measurement method
• Semi field method

Laboratory measurement method:

In this method,

• The stadia intercept (i) can be measured from the diaphragm by means of using vernier caliper.
• The distance `c` between the optical center and the vertical axis of the tacheometer can also be measured.
• The focal length can be determined using lens formula as:

`1/f=1/u+1/v`

Thus, after measuring `f`,`i` and `c`, the constants `K` and `C` can be calculated as:

`K=f/i` and `C=f+c`

Field measurement method:

In this method,

A fairly leveled ground is selected and the tacheometer is set up at point `A` and another point, say `A_4` is also selected at a known distance, say 200m.The pegs are then driven at a regular intervals,say `50m` and let the points be `A_1`,`A_2`, and `A_3`.Now, holding the staff at `A_1`,`A_2`, and `A_3`, the staff intercept `S_1`,`S_2`,`S_3`, and `S_4` are taken.

Let,

`A A_1=D_1`

`A A_2=D_2`

`A  A_3=D_3`

`A A_4=D_4`

Now, we know that,

`D_1=K*S_1+C`.....(i)

`D_2=K*S_2+C`.....(ii)

Thus, `D_2-D_1=K*S_2-K*S_1`

Or,`K=(D_2-D_1)/(S_2-S_1)`....(iii)

Where, `D_1`,`D_2`, `S_1`, `S_2` are known and hence `K` can be determined.

Now, from (i), we get,

`C=D_1-K*S_1`

Or, `C=D_1-(D_2-D_1)/(S_2-S_1)*S_1`

Or, `C=((D_1*S_2)-(D_2S_1))/(S_2-S_1)`

Similarly,

`k=(D_4-D_3)/(S_4-S_3)` and

`C=(D_3*S_4-D_4-S_3)/(S_4-S_3)`.....(iv)

Thus, the mean value of `K` and ` C` can be determined from the above equation.

Semi-Field Method:

In this method, the value of `K` is measured from field measurement method and the value of `C=f+c` is measured by means of measuring `f` and `c` in lab.

The stadia constant can be determined by:

• Laboratory measurement method
• Field measurement method
• Semi field method

Laboratory measurement method:

In this method,

• The stadia intercept (i) can be measured from the diaphragm by means of using vernier caliper.
• The distance `c` between the optical center and the vertical axis of the tacheometer can also be measured.
• The focal length can be determined using lens formula as:

`1/f=1/u+1/v`

Thus, after measuring `f`,`i` and `c`, the constants `K` and `C` can be calculated as:

`K=f/i` and `C=f+c`

Field measurement method:

In this method,

A fairly leveled ground is selected and the tacheometer is set up at point `A` and another point, say `A_4` is also selected at a known distance, say 200m.The pegs are then driven at a regular intervals,say `50m` and let the points be `A_1`,`A_2`, and `A_3`.Now, holding the staff at `A_1`,`A_2`, and `A_3`, the staff intercept `S_1`,`S_2`,`S_3`, and `S_4` are taken.

Let,

`A A_1=D_1`

`A A_2=D_2`

`A  A_3=D_3`

`A A_4=D_4`

Now, we know that,

`D_1=K*S_1+C`.....(i)

`D_2=K*S_2+C`.....(ii)

Thus, `D_2-D_1=K*S_2-K*S_1`

Or,`K=(D_2-D_1)/(S_2-S_1)`....(iii)

Where, `D_1`,`D_2`, `S_1`, `S_2` are known and hence `K` can be determined.

Now, from (i), we get,

`C=D_1-K*S_1`

Or, `C=D_1-(D_2-D_1)/(S_2-S_1)*S_1`

Or, `C=((D_1*S_2)-(D_2S_1))/(S_2-S_1)`

Similarly,

`k=(D_4-D_3)/(S_4-S_3)` and

`C=(D_3*S_4-D_4-S_3)/(S_4-S_3)`.....(iv)

Thus, the mean value of `K` and ` C` can be determined from the above equation.

Semi-Field Method:

In this method, the value of `K` is measured from field measurement method and the value of `C=f+c` is measured by means of measuring `f` and `c` in lab.

Instrumental errors:

1. Permanent adjustment not perfect.
2. Graduation of staff not uniform.
3. The value of multiplying constant not correct.
4. Error due to imperfect adjustment of the plate.
5. Idex erroe i.e, error due to imperfect adjustment of vertical circle.
6. Error due to horizontal axis not being perpendicular to the vertical axis.

Observational errors:

1. Error due to incorrect centering or levelling.
2. Error due to Staff not being perpendicular.
3. Error due to imperfect focusing.i.e, parallex error causing incorrect readings.
4. Error due to taking very long sights.
5. Error due to slip.
6. Error due to inaccurate bisection to targets.
7. Mistakes.

Natural Errors:

1. High winds.
2. Very high temperature causing expansion.
3. Error due to glare of sun.
4. Poor visibility.
5. Error due to unequal settlement of tripod legs.
6. Error due to unequal refraction.

Instrumental errors:

1. Permanent adjustment not perfect.
2. Graduation of staff not uniform.
3. The value of multiplying constant not correct.
4. Error due to imperfect adjustment of the plate.
5. Idex erroe i.e, error due to imperfect adjustment of vertical circle.
6. Error due to horizontal axis not being perpendicular to the vertical axis.

Observational errors:

1. Error due to incorrect centering or levelling.
2. Error due to Staff not being perpendicular.
3. Error due to imperfect focusing.i.e, parallex error causing incorrect readings.
4. Error due to taking very long sights.
5. Error due to slip.
6. Error due to inaccurate bisection to targets.
7. Mistakes.

Natural Errors:

1. High winds.
2. Very high temperature causing expansion.
3. Error due to glare of sun.
4. Poor visibility.
5. Error due to unequal settlement of tripod legs.
6. Error due to unequal refraction.