330 ON NAPIER'S RULES. 



therefore to characterize the former triangle, we introduce a new 

 set of quantities p, such that a 1 +p 1 = a 2 +p^ = a 5 +p 5 = , 



the original triangle being characterized by p l9 p^ p 3 , p v p 5 the 

 secondary triangle is similarly characterized by p^ p, J9 5 , p lt p^ ; 

 and as the secondary triangle gives rise to a third, and so on, 

 we thus see that every right-angled spherical triangle is one of 

 a system of five such triangles. 



It is obvious, that the transformation just employed de- 

 pends upon the circumstance that the complement of the com- 

 plement of an angle is the angle itself, and would succeed 

 equally if the parts of the secondary triangle, which are the com- 

 plements of those of the first, had been any function (f) of them, 

 provided that/ 2 = 1. 



Keturning to the figure we observe, that if, instead of taking 

 a point .Z? in OB we had taken one, as^t, in OA, we should by a 

 similar construction have got another of the four spherical tri- 

 angles, which with the original one make up the system of 

 which we have been speaking. Further, if in BG we assume 

 any point as a first centre related to B as B to 0, and make a 

 similar assumption of a fifth centre in A C, the system will be 

 complete. But as a figure so drawn would be complicated, it is 

 better to adopt a different plan. Since BA is at right angles to 

 OA, and lies in the plane OAB, it is easy to represent the 

 corelate spherical triangles, to which the system of lines of which 

 we have been speaking gives rise. 



Let BA C (fig. 2) be the original triangle right-angled in A. 



Fig. 2. 



