The “variance ratio distribution” refers to the distribution of the ratio of variances of two samples drawn from a normal bivariate correlated population. Today, we call this the bivariate normal distribution.
The Fisher-Snedicor F Distribution is sometimes called the “Variance Ratio” distribution because it is the distribution of the ratio of two independent variance estimates (S12/S22) . However, this is quite different from the variance ratio distribution in historical literature.
Variance Ratio Distribution History
Haight’s entry in  provides the formula:
The notation [e]2:65 refers to a 1935 article by Bose , titled On the Distribution of the Ratio of Variances of Two Samples Drawn from a Given Normal Bivariate Correlated Population and published in Sankhya, volume 2, 1935. The author, providing a solution to the question of “the distribution function of the ratio of variances obtained from two independent samples,” refers to Fisher’s earlier work 
Why the complicated (rarely seen in modern times) formula? The answer is the advent of the computer. Before the advent of the computer age (c. 1960s), mathematicians had to refer to tables for the variance ratio distribution F and – sometimes – equations with “great computational difficulty”. In the early days of computers, the distribution also required the “use of excessive amounts of computer time” .
Of course, nowadays, we just open our statistics software program and run an algorithm. That’s why you’ll rarely see the actual formula for the variance ratio distribution.
 Jolicoeur P. (1999) The distribution of the variance ratio, F = S12/S22. In: Introduction to Biometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4777-8_9(KENDALL, M. G. The Advanced Theory of Statistics , Volume 1 , London: Charles Griffin and Co., 1945.
 Bose, S., & Mahalanobis, P. C. (1935). On the Distribution of the Ratio of Variances of Two Samples Drawn from a Given Normal Bivariate Correlated Population. Sankhyā: The Indian Journal of Statistics (1933-1960), 2(1), 65–72. http://www.jstor.org/stable/40383720
 Fisher, R. (1924). On a distribution yielding the error function of well known statistics. Proceedings of the International Mathematical Congress, Toronto, 05-813.
 Box, M. & Box, R. (1969). Computation of the variance ratio distribution. Online: https://academic.oup.com/comjnl/article/12/3/277/363429