158 RELATIONS BETWEEN SEVERAL [BOOK I. 



centric longitudes and latitudes of the earth whatever, are substituted for two 

 geocentric. If two longitudes and latitudes entering into the same expression are 

 only interchanged with each other, the corresponding numbers should also be 

 interchanged ; but the value is not changed from this cause, but it only becomes 

 negative from being positive, or positive from negative. Thus, for example, we 

 have 



(0.1.2)= (0.2. !) = (!. 2.0) = (1.0.2) = (2. 0.1) = (2. 1.0). 

 All the quantities, therefore, originating in this way are reduced to the nineteen 

 following : 

 (0.1.2) 



(0.1.0), (0.1. 1.),. (0.1. II.), (0.0.2), (0.1.2), (O.H.2), (0.1.2), (1. 1.2), (H 1.2), 

 (0. 0. 1.), (0. 0. II), (0. 1 tt), (1. 0. 1.), (1. 0. II), (1. 1. II.), (2. 0. L), (2. 0. II), 

 (2. 1. II.), 

 to which is to be added the twentieth (0. 1. II.). 



Moreover, it is easily shown, that each of these expressions multiplied by the 

 product of the three cosines of the latitudes entering into them, becomes equal 

 to the sextuple volume of a pyramid, the vertex of which is in the sun, and the 

 base of which is the triangle formed between the three points of the celestial 

 sphere which correspond to the places entering into that expression, the radius 

 of the sphere being put equal to unity. When, therefore, these three places lie in 

 the same great circle, the value of the expression should become equal to ; and 

 as this always occurs in three heliocentric places of the earth, when we do not 

 take account of the parallaxes and the latitudes arising from the perturbations of 

 the earth, that is, when we suppose the earth to be exactly in the plane of the 

 ecliptic, so we shall always have, on this assumption, (0. 1. II.) = 0, which is, in 

 fact, an identical equation if the ecliptic is taken for the third plane. And fur 

 ther, when B, B , B&quot;, each, = 0, all those expressions, except the first, become 

 much more simple ; every one from the second to the tenth will be made up of 

 two parts, but from the eleventh to the twentieth they will consist of only one 

 term. 



