The main target this paper is to check a theory about dark matter nature, which was published by the author in previous papers. It was postulated it that dark matter density is a function which depend on E, gravitational field. Also were proposed several experimental test to check that theory. In this work has been calculated a new function for DM density for M31. Reader could think, why disturb me with a new DM density profile, called Bernoulli profile in this paper, whose values have relative differences with NFW ones below 10%? The reason is clear. This DM profile has been got starting from hypothesis that DM is generated by the own gravitational field. Therefore if DM Bernoulli profile fits perfectly to NFW DM profile then it is possible conclude that observational data supports author´s hypothesis about DM nature. To find reasons that author has to do so daring statement, reader can consult [1] Abarca,M.2014. Dark matter model by quantum vacuum. [2] Abarca,M.2015. Dark matter density function depending on gravitational field as Universal law. [3] Abarca,M.2015. A new dark matter density profile for NGC 3198 galaxy to demonstrate that dark matter is generated by gravitational field. [ 9] Abarca,M.2016.A New Dark Matter Density Profile as Power of Gravitational Field for Coma Cluster. Briefly will be explained method followed to develop this paper. Firstly are presented rotation curve and table with data about rotational velocity depending on radius inside M31 galaxy. These data come from [5]Sofue,Y.2015. In fourth epigraph, considering rotation curve of M31 from Sofue data, has been tabulate gravitational field inside a wide region of halo, from 40 kpc to 300 Kpc. In fifth epigraphs has been tabulated data of NFW DM density profile published by [5] Sofue, Y.2015. for M31. In sixth epigraph has been fitted data of NFW DM density profile as power of gravitational field, E. Particularly formula found is DM density= A*E^B. Where A = 0,0012004275 B= 1.878838501 and correlation coefficient r = 0,9996041653 into I.S. of units. In seventh epigraph it has been compared DM density as power of E and NFW profiles. Tables and plots show clearly that relative differences between both profiles are mainly below 6%. In eight epigraph it is considered derivative of gravitational field, E, in halo region where density of baryonic matter is negligible regarding DM density. As consequence it is got a Bernoulli differential equation whose solution allows to get a new DM density profile called hereafter Bernoulli density profile. In nineth epigraph Bernoulli and NFW DM density profiles have been compared. Its relative differences are below 10% for radius bigger than 100 kpc, and below 12% from 40 kpc to 100 kpc. This is a super result, speciallly if it is considered that Do parameter of NFW profile has 10,7 % as relative error. In addition Bernoulli density is rewritten with a similar formula to NFW profile thanks a variable change. In tenth chapter has been calculated total masses under radius 200 kpc, 385 kpc and 475 kpc through NFW profile and Bernoulli profile. Last one profile is very hard to integrate but thanks Wolfram alpha software on line it has been possible.