Whence are derived 



TRANSACTIONS OF THE SECTIONS. 3 



K S be the area of the triangle ABO, and p^p.^ p^ the pei-pendiculars from A, B, C, 

 on a, b, c, 



2S=ap^ = bp2=cp3; and S = rs=t\ (s — a)=r.-, (s — b) = i-3 (s—c). 

 Hence, if the plane of reference be perpendicular to the plane of the triangle, and 

 intersect that plane in a tangent to any of the circles 7; r^, r.-,, r^, so that b=r for 

 instance, then 2S = rta+6/3+cy ; a, /3, y having the proper algebraical sig-ns for each 

 case. The above equations are now readily transformed into 



r Pi Ih "Pz ^1 Pi Pi Pz' 



-r-i Px /»2 i'3' r^ Pi P2 pi 



r r^ To r^' p, r^ r^' 



P2 r^ r,' p^ r, r/ 



If the plane of reference is parallel to the plane of the triangle, 



cc=^=y = 8 = 8,^B, = 8, ; (A) 



and the above eight equations give con-esponding plane theorems. When the ordi- 

 nates are referred to the plane which is tangential to the three spheres of which the 

 centres are 0„ O^, O3, and radii r^, r^, r^, we have 



.-. 8 =3r, a, =p^, ^=p.„ and 7=^3 J ^ ^ 



The plane (B) is therefore tangential to the seven spheres of which A, B, C, Oj, O2, O3, 

 and are the centres, imdpj,p.^,p^, t\, r,, r^, and 3r the radii. 



Let Q be the centre of the circle circiunscribing the triangle A B C, R its radius, 

 and A the distance of Q from the plane of reference. Then, proceeding as before, it 

 is found that 



4 sin A sin B sin C . A = a sin 2A+ j3 sin 2B+y sin 2C ; 



^=-co8 A+^cosB+IcosC; 



E, p, p.-^ p. 



X P+ J\= « cosHA+^ cosHB+Xcos^AC ; 

 \r E/ p, p^ ps 



i f?_A\=£sin^JA+^sin=iB+3^sin^iC. 

 \r R/ j,^ p^ 2h 



For the plane (B) the second equation gives A=R+r. 



The cu'cimiscribing cu'cle bisects 0^ O3, O3 Oj, 0^ 0^ in A', B', C'j and if «' /3' y' be 



for A' B' C what cc^y ai-e for ABC, 2cc'=8,_+d^ ; also Z A'=Z 0^=1 (tt- A). 



Applying the first of the above theorems to the triangle A' B' C, and reducing, we 



get 



and if this be referred to the plane (B), we have the well-known plane theorem, 



4R=-r-f-ri + »'2+»'3- 

 Applying the same general theorem to the triangle Oj O2 O3, we have A'+5 = 2A, 

 where A' is for 0^ 0^ O3 what A is for ABC. Also 



^=^. + §2 + ^. 

 »• Pi P2 2h' 

 if this be refen'ed to the plane (B), for which A'=2R— r, we get the plane theorem 



»• Pl P2 P3 



Let 8' be the distance from the plane of rtfereuce of the point of intersection of 



1* 



