He passed his Master's examinations in 1893 and his Master's dissertation On certain modifications of Hamilton's principle and its application to the solution of problems of mechanics of solid bodies (Russian) (1903) contained his first really significant result. This result, which he had first announced in a publication in 1898, was the discovery of :-
... a new "integrated" case of motion for a top on a smooth surface, related to the turning of a solid body about a fixed point.The paper which announced this important result was On one case of the motion of a heavy solid body supported by a point on a smooth surface (Russian) (1898). We should note at this point that a Master's Degree in Russia at this time was equivalent to what today would be Ph.D. level.
In 1908 Kolosov began working on the theory of elasticity and his doctoral thesis (equivalent to the German habilitation standard) contains Kolosov's formulas expressing the components of the stress tensor and the displacement vector in terms of two analytic functions of one complex variable. However this deep and important piece of work did provoke some controversy as is explained in . Kolosov's thesis contained a formal solution of the plane problems of the theory of elasticity. After deriving these he successfully applied the formulas to solve some special cases. Steklov had been appointed to examine the thesis and at first he was not convinced that Kolosov's derivation of his formulas was rigorous. The paper  contains details of the interesting correspondence and a detailed analysis of Kolosov's work. Steklov was eventually became convinced that the derivation was satisfactory.
There are two interesting 'follow-ups' to these events. The first is that Steklov was right to question the work, but nevertheless it was correct. All doubts were finally removed when N I Muskhelishvili, a student of Kolosov, completely validated the results several years later. The second interesting comment is that Kolosov was not the first to discover the formulas now named after him . The first discoverer was Sergei Chaplygin who had derived them ten years earlier, but he had not thought them sufficiently important to publish - Chaplygin of course was quite wrong in this assessment and this fact alone would justify them now being named after Kolosov. The irony of this story is that Chaplygin had worked in the same department as Kolosov. The formulas were first published by Kolosov in his 1909 paper An application of the theory of functions of a complex variable to a planar problem in the mathematical theory of elasticity (Russian).
From 1893 Kolosov was employed both as the director of the mechanics laboratory at St Petersburg University, and as a teacher at the St Petersburg Institute of Communications Engineers. He worked at Yurev University from 1902 to 1913. If the name of this university is unfamiliar, then perhaps 'Tartu University' or 'Dorpat University' will be more familiar. All are names for the same institution founded in 1632 by Gustavus II Adolphus of Sweden. The Estonian name is Tartu, while the German and Swedish names for the same city (and university) are both Dorpat. However, between 1893 and 1918 the city was known as Yurev or Yuryev and it was during this period that Kolosov worked there. He was appointed in 1902 as a privatdozent then was promoted to professor during the eleven years he spent there.
In 1913 Kolosov returned to St Petersburg where he spent the rest of his career. He worked both at St Petersburg University, becoming head of the department of theoretical mechanics in 1916, and at the Electrotechnical Institute where he was appointed head of the department of theoretical mechanics immediately on his return to the city in 1913. Of course for much of this time (between 1914 and 1924), the city was known as Petrograd. After 1924 it became known as Leningrad. Kolosov worked in the famous city through a very difficult period since the Russian Revolution essentially began in St Petersburg in 1917. During the years of the civil war the city fell on very hard times and its population fell by two-thirds to around 700,000 over the three years 1917-1920. However after the civil war ended the city began to prosper again and by the time of Kolosov's death its population was nearly three million.
In addition to the important results we have mention above, we note that in 1907 Kolosov derived the solution for stresses around an elliptical hole. It showed that the concentration of stress could become far greater, as the radius of curvature at an end of the hole becomes small compared with the overall length of the hole. Engineers have to understand Kolosov results so that stresses can be kept to safe levels. Finally we mention his important text Application of a complex variable to the theory of elasticity (Russian) published in Moscow in1935.
Article by: J J O'Connor and E F Robertson