The paper considers a analytical problem fundamental the problem of positioning a electrical discharge discharge by the TOA arrangement for three stations. The mathematical question is formulated in this manner. Three points are fixed on the unit circle. Three spherical slices are introduced, arising out of the given points in the direction of few unknown point, whose position on the circle must be determined. The necessity must be met: the distance each point to an unknown point must add up to the sum of the distance of the segment arising out of this point, and some increment, the alike for all three segments. In the item, the conditions for the solvability of the question are established utilizing the methods of spherical arithmetic and vector arithmetic. It is proved that this lines problem is always explainable when they are fulfilled. The number of resolutions is two, except in rare cases place there is singular solution. An algorithm for the mathematical solution of the question of positioning a bolt discharge has been developed.

**Author(s) Details:**

**Yu. R. Shpadi,
**Institute of Space Techniques and Technologies, Almaty, Kazakhstan and Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan.

**A. S. Inchin,
**Institute of Space Techniques and Technologies, Almaty, Kazakhstan.

**A. Yu. Lozbin,
**Institute of Space Techniques and Technologies, Almaty, Kazakhstan.

**B. Aliyeva,
**Institute of Space Techniques and Technologies, Almaty, Kazakhstan.

**M. Yu. Shpadi,
**Institute of Space Techniques and Technologies, Almaty, Kazakhstan.

**G. Ayazbayev,
**Institute of Space Techniques and Technologies, Almaty, Kazakhstan.

**R. Bykayev,
**Institute of Space Techniques and Technologies, Almaty, Kazakhstan.

**Please see the link here: **https://stm.bookpi.org/RPST-V7/article/view/9951

**Keywords: **Meteorology, lightning discharge positioning, spherical trigonometry, vector algebra, algorithms