32 On Equations of Condition for a Quadrilateral. [No. 121. 



some consideration it appeared that they were included in the expressions 

 for (E) and (F) : for if the point D in fig. I. be conceived to move along 

 the line B D till it comes within the line A C, the quadrilateral with its 

 diagonals is transformed into the triangle with its radial lines. The 

 figure may now be considered as a quadrilateral with a reentrant angle, 

 in which case the angles Ai and A3 exchange their designations, A2 re- 

 maining unchanged. 



The analogy of these figures may be otherwise apprehended by consi- 

 dering them as the perspective representations of a tetrahedron ; which 

 is a quadrilateral with its diagonals, when the apex is projected between 

 the exterior angles of the base ; and is a triangle with its radial lines, 

 when the apex is projected within the base, or within the vertical angles 

 formed by the sides of the base produced. 



These equations hold good in spherical as well as in plain figures, the 

 only change in the demonstration being to substitute the sines of the 

 spherical sides for the plane's sides as above. 



The equations marked (G) and (K) are obviously only particular 

 cases of a more general property given by W. Davies in his Supplement 

 to the spherical part of Young's Trigonometry, and there said to be due 

 to Professor Lowry. Lowry's Theorem is this: " Jf great circles be drawn 

 from the angular points of any spherical polygon to a point on the surface 

 of the sphere, the product of the sines of the alternate angles will be equal" 

 This theorem applies of course in plane as well as in spherical polygons, 

 and it is not unlikely, that if we substitute lines of shortest distance 

 (including at once both straight lines and great circles), it may be found 

 to apply on a spheroidal, as well as on a spherical or a plane surface. 



On farther consideration I find that the equations (E) and (H) are 

 also included in Lowry's theorem. In fig. I. D being the point to which 

 lines of shortest distance are drawn from the angles of the polygon A B 

 C Lowry's theorem gives at once the first of the equations (E) and 

 taking successively the points A B and C in like manner, the other 

 equations are evolved. 



Likewise on fig. 2 if C be the point to which the lines are drawn from 

 the angles of the polygon A B D, we have the first of the equations (H) 

 and the others by taking successively A and B as the point of drawing to 

 (i. e. attraction in its primary sense). 



The sinal equations furnish in each case four equations of condition for 



