1915-] SEE— THE EULER-LAPLACE THEOREM. 345 



Ji^a (^i — c) = perihelion distance, 



g^=a{i — c^) ^=p^^\3.tns rectum of the orbit, 



3; = r = radius vector of the planet, 



Z = ^ = the eccentricity of the orbit, 



f = true anomaly =: z^^ in the notation now commonly used, 



j = arc of the orbit, reckoned from perihelion, 



c = sun's mean distance, 



=^ fjia, where a is the earth's equatorial semi-diameter, and 



/x a number which expresses the sun's mean distance in this unit. 

 Euler uses a solar parallax of 13", and takes c=: 15866a. With 

 the values now adopted in astronomy we have about c = 234450!. 

 In some of his numerical work Euler uses c=g = a (1 — e-), 

 which is admissible when we neglect the square of the eccentricity. 

 Euler also uses a small angle of deviation due to the angular 

 efifects of resistance, s = 6, such that tan s = 2g/'^c ; and then takes 

 the equation for the Keplerian elHpse 



a(i — e^) 



Y = 



I + e cos V ' 

 to have the form of an ellipse modified by resistance 



11^ . I e cos V 



- = - (i + e cos zO = T + — .,— + P, 

 r p P P 



where P is function of the time, but modified by a very small 

 quantity depending on the effects of the secular action of the resist- 

 ing medium. 



From the equations of the disturbed ellipse, in his notation, 



- = - + - cos / + P, 



y g g 



P = -(t - s'mt - ^^s'mt + ^t^cos^ + frr^-frrsin^-2\i"i-sin20, 



