Mr. Davies's Researches in Spherical Geometry, 367 



and the author has shown, that if the associated system of which 

 a b c is the fundamental triangle be formed, these are polar tri- 

 angles to the primary associated system, each to each. He also 

 establishes a considerable number of curious and interesting re- 

 lations between them. We shall transcribe one : If the sides of 

 the primary and polar triangles intersect in R R', S S', Q Q', these six 

 points will all lie in the same great circle of the sphere j and 

 if the arcs A a, B b, Cc, be drawn, they will intersect in the same 

 point, O, which will also be the pole of the circle Q R S Q' R' S'. 



Another property that struck us is, that if the tangents of all the 

 radii of the circles inscribed in and circumscribed about the eight 

 triangles forming the two associations are multiplied together, 

 their product is unity. A considerable number of other properties 

 of these radii, for the most part new, is given in this second chap- 

 ter, but we have not room to enumerate them. 



The third and last chapter relates to the spherical excess, and ex- 

 hibits this function in terms of data which we believe to be new. 

 $. new and concise investigation of Lhuillier's theorem is also 

 given ; but the majority of the inquiries are directed to the results 

 which flow from the application of that celebrated theorem to the 

 eight triangles above mentioned. There is also a curious analogy 

 to that theorem exhibited in the following property, expressing 

 one fourth of the perimeter in terms of the angles of the triangle: — 



tan* a + b + c = tan A + B + C =" tan A+B-C+I 



4 4 4 



tan A=?_±C +* tan -A + B+C+ir 

 4 4 



which analogy will be seen more distinctly by giving to the factors 

 of Lhuillier's theorem their unabbreviated form, thus 

 t a „,A±B + C+ 5= 

 4 



tan a + b + c tan S*£f tan ^±1 tan T^±*+f 



4 4 4 4 



where n always connects itself with the angles and never with the 

 sides. Vide page 245. 



One or two other curious analytical expressions may be here 

 mentioned : thus, if we denote by E the spherical excess of ABC, by 

 E / E // E /// the excesses of BA'C, CB'A, AC'B; and by E'E',E'„E', M 

 the excesses of the triangles respectively polar to these; then 

 (pp. 249, 250) 



F' F' F' F' F F F F 



tan 5 tan *J tan -» tan ~M = tan A tan ,' tan '' tan Sf« 

 4 4 4 4 4.44 4 



and again 



F F' F F' F F' F F' 



tan 1 ^ tan % = tan =^tan^ ' == tan SM tan ZJk = tan tUii tan ELmi 

 44 4 4 44 44 



But we must close our extracts, which we shall do with the fol- 

 lowing new expression for the area of a spherical triangle in terms 

 of its sines. It is not adapted in its present form to logarithmic 



