1901.] LOWELL — SUPPOSED SIGNALS FROM MARS. 173 



from which we find 0. To find r we have from the equation of the 

 ellipse 



sin 2 6 4- cos 2 



from which, knowing 0, the value of r follows. 



The distance / from the centre to the tip of the projection may 

 now be got by solving the plane triangle whose sides are d, rand /. 

 For d is given, r is now known and the angle included between 

 them is i8o° — % where 



x — Q — <t> 

 and this also is known. 



/ would give us the projected place upon the visible disk of the 

 tip of the projection, if the projection were on the surface of the 

 planetary sphere. As it is in reality raised above it, we must apply 

 a correction depending upon the height of the projection. It is 

 for this reason that the height must first have been found. Perhaps 

 the neatest way is the one adopted by Mr. Manson, who performed 

 the numerical computations, that of simple projection, which gives 



/, = 



h 



Knowing t and also the angle in the plane triangle opposite the 

 side d, which we may call D, we have a spherical triangle for the 

 determination of the latitude and longitude of the point on the 

 sphere directly under the projection. In this triangle we know the 

 side t, whose value in angular measure is cos t ; the side (90?/?), 

 which is the angle between the pole of the planet and the centre of 

 the disk ; and the angle between the two, which is 



C== 900 _ ( Q — 2700 _ P) _j_ e— D 

 = P—QJ r — D 



where P and Q have the meanings.of Crommelin's ephemeris for 

 the planet. 



We then have the latitude, / lt from 



cos /j = cos t x sin B -j- sin t x cos B cos C 



and the longitude, I, from 



sin (/I — X x sin t x 

 sin C sin 7j 



