The following calculations for ball diameter and dimension over balls are in two columns.
Left-Hand Column = for external gears with EVEN numbers of teeth.
Right-Hand Column = for external gears with ODD numbers of teeth.
Many calculations are the same for both odd and even numbers of teeth but are repeated to make the calculation procedure easier to follow.
Additional information: - Ball diameters for Internal Gears

Whereas:
From Gear Data
Z = number of teeth in gear (50)
M_n = normal module (8)
b = helix angle (15º)
a_n = normal pressure angle (20º)
x = profile shift coefficient (0)
Calculated
1. z_mn = virtual number of teeth (56.06031537)
2. a_vnm = normal pressure angle at Virtual C-Cylinder (20°)
3. inva_n = involute of pressure angle normal (0.014904384)
4. a_knm = pressure angle at contact (21.60541373°)
5. D_m = ball pin diameter (14mm) (13.5142mm)

Whereas:
From Gear Data
Z = number of teeth in gear (61)
M_n = normal module (8)
b = helix angle (15º)
a_n = normal pressure angle (20º)
x = profile shift coefficient (0)
Calculated
1. z_mn = virtual number of teeth (68.39358475)
2. a_vnm = normal pressure angle at Virtual C-Cylinder (20°)
3. inva_n = involute of pressure angle normal (0.014904384)
4. a_knm = pressure angle at contact (21.31591289°)
5. D_m = ball pin diameter (13) (13.48797374mm)

Formula: Ball diameter - D_m (EVEN number of teeth)
Calculation procedure:-
1. Virtual number of teeth: z_nm
= Z/((COS(b)^3.3))
= 50/((COS(15)^3.3))
= 50/((0.965925826)^3.3)
= 50/0.891896517
= 56.06031537

2. Normal pressure angle at virtual v-cylinder: a_vnm
= acos((z_nm*cos(a_n))/(z_nm+2*x))
= acos((56.06031537*cos(20))/56.06031537+2*0))
= acos((56.06031537*0.93969262)/56.06031537)
= acos(52.679463/56.06031537)
= acos(0.939692621)
= 20°

3. Involute of the normal pressure angle: inva_n
= tan(a_n)-a_n
= tan(20)-20
= 0.363970234-0.34906585(use a_n in radians)
= 0.014904384

4. Pressure angle at contact: a_knm
= tan(a_vnm)-inva_n+((Pi-4*x*tan(a_n))/(2*z_nm))
= tan(20)-0.014904384+((Pi-4*0*tan(20))/(2*56.06031537))
= (0.363970234-0.014904384)+(3.1415927/112.120637)
= 0.34906585+0.028019755
= 0.377085606
= 21.60541373°

5. Ball diameter: D_m (EVEN number of teeth)
= Int((z_nm*M_n*cos(a_n))*(tan(a_knm)-tan(a_vnm))+0.5)
= Int((56.06031537*8*cos(20))*(tan(21.60541373)-tan(20))+0.5)
= Int((56.06031537*8*0.93969262)*(0.3960373-0.363970234)+0.5)
= Int((421.43571*0.0320671)+0.5)
= Int(13.514221+0.5)
= Int(14.014221)
= 14mm (13.5142mm non-integer size)

Formula: Ball diameter - D_m (ODD number of teeth)
Calculation procedure:-
1. Virtual number of teeth: z_nm
= Z/((cos(b)^3.3))
= 61/((cos(15)^3.3))
= 61/((0.965925826)^3.3)
= 61/0.891896516
= 68.39358475

2. Normal pressure angle at virtual v-cylinder: a_vnm
= acos((z_nm*cos(a_n))/(z_nm+2*x))
= acos((68.39358475*cos(20))/68.39358475+2*0))
= acos((68.39358475*0.93969262)/68.39358475)
= acos(64.2689469/68.39358475)
= acos(0.939692621)
= 20°

3. Involute of the normal pressure angle: inva_n
= tan(a_n)-a_n
= tan(20)-20
= 0.363970234-0.34906585(use a_n in radians)
= 0.014904384

4. Pressure angle at contact: a_knm
= tan(a_vnm)-inva_n+((Pi-4*x*tan(a_n))/(2*z_nm))
= tan(20)-0.014904384+((Pi-4*0*tan(20))/(2*68.39358475))
= (0.363970234-0.014904384)+(3.141592654/136.7871695)
= 0.34906585+0.022967013
= 0.372032863
= 21.31591289°

5. Ball diameter: D_m (ODD number of teeth)
= Int((z_nm*M_n*cos(a_n))*(tan(a_knm)-tan(a_vnm))+0.5)
= Int((68.39358475*8*cos(20))*(tan(21.31591289)-tan(20))+0.5)
= Int((68.39358475*8*0.93969262)*(0.39020369-0.36397023)+0.5)
= Int((514.1515748*0.026233458)+0.5)
= Int(13.48797374+0.5)
= Int(13.98797374)
= 13mm (13.48797374mm non-integer size)

Whereas:
From Gear Data
Z = number of teeth in gear (50)
M_n = normal module (8)
b = helix angle (15º)
a_n = normal pressure angle (20º)
x = profile shift coefficient (0)

Calculated
1. a_t = transverse pressure angle (20.64689649º)
2. inva_t = involute of the transverse pressure angle (0.01645339)
3. b_d = base diameter (387.5126702mm)
4. b_b = base helix angle (14.07609542°)
5. inva_kt = involute of pressure angle in transverse section at circle through centre of ball (0.022283685)
6. a_kt = inverse of inva_kt (22.753668°)
7. d_k = diameter to centre of ball (420.21543mm)
8. M_dk = dimension over balls (434.2154mm)
9. A_md = change factor (2.5048006)
D_m = ball diameter (14mm)

Whereas:
From Gear Data
Z = number of teeth in gear (61)
M_n = normal module (8)
b = helix angle (15º)
a_n = normal pressure angle (20º)
x = profile shift coefficient (0)

Calculated
1. a_t = transverse pressure angle (20.64689649º)
2. inva_t = involute of the transverse pressure angle (0.01645339)
3. b_d = base diameter (472.7654577mm )
4. b_b = base helix angle (14.07609542°)
5. inva_kt = involute of pressure angle in transverse section at circle through centre of ball (0.019051628)
6. a_kt = inverse of inva_kt (21.641839°)
7. d_k = diameter to centre of ball (508.61935mm)
8. M_dk = dimension over balls (521.4507612mm)
9. A_md = change factor (2.6268242)
D_m = ball diameter (13)

2. Involute of the transverse pressure angle: inva_t
= tan(a_t)-(a_t)
= tan(20.64689649)-20.64689649
= 0.376809714-0.360356324
= 0.01645339

3. Base diameter: d_b
= M_n*(Z/cos(ß))*cos(a_t)
= 8*(50/cos(15))*cos(20.64689649)
= 8*(50/0.965925826)*0.9357712
= 8*51.76381*0.9357712
= 387.5126702mm

4. Base helix angle: b_b
= asin(sin(ß)*cos(a_n)
= asin(sin(15)*cos(20)
= asin(0.258819045*0.939692621)
= asin(0.243210347)
= 0.245674211radians (14.07609542°)

5. Involute of pressure angle in transverse section at circle through centre of ball: inva_kt
= (D_m/(d_b*cos(b_b)))-((Pi-4*x*tan(a_n))/(2*Z))+inva_t
= (14/(387.5126702*cos(14.07609542)))-((Pi-4*x*tan(20))/(2*50))+0.01645339
= (14/375.87705)-(3.141592654/100)+0.01645339
= 0.0372462-0.0314159+0.01645339
= 0.022283685

6. Inverse of inva_kt: a_kt
The evaluation of the inverse (i.e., the calculation of the angle which corresponds to a given involute value) can present a challenge. In many cases the term 'refer to tables' is used and for repeated manual calculations it is probably best to pre-calculate using a computer and forming a reference table.
This is not satisfactory for a computer program, however, so rapid convergence can be obtained from an iteration using:-
X_i+1 = X_i+[inv(a)-inv(X_i)]*cos²(X_i)
Using this method will provide the following:-
= 22.753668°

7. Diameter to centre of ball: d_k
= d_b/cos(a_kt)
= 387.5126702/cos(22.753668)
= 387.5126702/0.9221762
= 420.21543mm

8. Dimension over balls: M_dk
= d_k+D_m
= 420.21543+14
= 434.2154mm

9. Change factor: A_md .... see Change Factor
= cos(a_t)/(sin(a_kt)*cos(ß))
= cos(20.64689649)/(sin(22.753668)*cos(15)
= 0.9357712/(0.38677*0.9659258)
= 0.9357712/0.3735911
= 2.5048006

2. Involute of the transverse pressure angle: inva_t
= tan(a_t)-(a_t)
= tan(20.64689649)-20.64689649
= 0.376809714-0.360356324
= 0.01645339

3. Base diameter: d_b
= M_n*(Z/cos(ß))*cos(a_t)
= 8*(61/cos(15))*cos(20.64689649)
= 8*(61/0.965925826)*0.93577124
= 8*63.15184701*0.93577124
= 472.7654577mm

4. Base helix angle: b_b
= asin(sin(ß)*cos(a_n)
= asin(sin(15)*cos(20)
= asin(0.258819045*0.939692621)
= asin(0.243210347)
= 0.245674211radians (14.07609542°)

5. Involute of pressure angle in transverse section at circle through centre of ball: inva_kt
= (D_m/(d_b*cos(b_b)))-((Pi-4*x*tan(a_n))/(2*Z))+inva_t
= (13/(472.7654577*cos(14.07609542)))-((Pi-4*x*tan(20))/(2*61))+0.01645339
= (13/458.5699989)-(3.141592654/122)+0.01645339
= 0.028348998-0.025750759+0.01645339
= 0.019051628

6. Inverse of inva_kt: a_kt
The evaluation of the inverse (i.e., the calculation of the angle which corresponds to a given involute value) can present a challenge. In many cases the term 'refer to tables' is used and for repeated manual calculations it is probably best to pre-calculate using a computer and forming a reference table.
This is not satisfactory for a computer program, however, so rapid convergence can be obtained from an iteration using:-
X_i+1 = X_i+[inv(a)-inv(X_i)]*cos²(X_i)
Using this method will provide the following:-
= 21.641839°
Freebie to do the hard work for you

7. Diameter to centre of ball: d_k
= d_b/cos(a_kt)
= 472.7654577/cos(21.641839)
= 472.7654577/0.9295074
= 508.61935mm

8. Dimension over balls: M_dk
= d_k*cos(Pi/(2*z))+D_m
= 508.61935*cos(Pi/(2*61))+13
= 508.61935*cos(Pi/(122))+13
= 508.61935*0.999668468+13
= 521.4507612mm

9. Change factor: A_md .... see Change Factor
= cos(a_t)/(sin(a_kt)*cos(ß))
= cos(20.64689649)/(sin(21.641839)*cos(15)
= 0.9357712/(0.3688034*0.9659258)
= 0.9357712/0.3562367
= 2.6268242