1878.] of coordinates. 183 



or the linear relations between {x, y, z) and {x^, ?/j, z^ are such 

 as to transform one of these quadric functions into the other : 

 the two quadrics in fact denote the squared distance from the 

 origin expressed in terms of the coordinates {x, y, z) and 

 {x^, 2/j, 5 J respectively. 



Since the nine cosines are connected by six equations, there 

 should exist values containing three arbitrary constants, and 

 satisfying these equations identically: but by what just precedes, 

 it appears that the problem to determine these values is in fact 

 that of finding the linear transformation between two given 

 quadric functions : the problem of the linear transformation of a 

 quadric function into itself has an elegant solution ; but it would 

 seem that this is not the case for the transformation between two 

 different functions. 



The foregoing equation 



/ir=(a, b, c, f, g, hj^y, ^,i)\ 



is a relation between X, /*, v, the cosines of the sides of a spherical 

 triangle, and (a, ^, 7) the cosines of the distances of a point P 

 from the three vertices: it can be at once verified by means of 

 the relation A + B + C= Stt, and thence 



1 — cos^ A — cos^ B — cos^ C+2cosA cos Bcos C = 0, 



which connects the angles A, B, C which the sides subtend at F : 

 writing a, h, c for \ /m, v, and /, g, h for a, /3, 7, the relation is 



1 - a^ - 6^ - c^ + 2ahc = (1 - a')/' + (1 - b') / + (1 - c') h' 



+ 2 {be- a) gh+2 {ca - h) hf+ 2 {ah - c) fg, 

 viz., this is 



l^a'-b'-c'-f-g'- h' + 2abc + 2agh + 2bhf + 2efg 



- ay - by - cVi' + 2bcgh + 2cahf +2abfg = 0; 



where {a, b, c,f, g, h) are the cosines of the sides of a spherical 

 quadrangle ; {a, b, c), {a, h, g), {h, b, /), {g, /,_ c) belong respect- 

 ively to sides forming a triangle, and the remaining sides (/, g, h), 

 {b, c,f),{c, a, g), {a, b, h) are sides meeting in a vertex. 



The equation 



AX==(a, b, c, f, g, hja, /3, 7) (a, /3', 7') 



is a relation between X, /a, v, the cosines of the sides of a spherical 

 triangle ; a, /3, 7, the cosines of the distances of a point P from 

 the three vertices ; a', /3', 7', the cosines of the distances of a point 

 Q from the three vertices ; and v^, the cosine of the distance P Q. 

 Vol. III. Pt. v. 14 



