594 Proceedings of the Royal Society 
Calling the vector of orthocentre (X) w, we have from (6) 
w = sin + sin (p 8 -g 3 ) „ + sin (p 8 -g ;i ) 
sin sin sin p 3 ' ’ 
therefore 
Sour = cos E ( Sin Sac' + m' + Sin( . P r ?3) ST7 'l . 
\ sm*^ sin 7^3 J 
therefore 
cos XQ = cos B f sin fa ~ ^ + sin fe~g-^ + !L n _ fe-g 8 ) y 
v sin Bin p 2 sin p 3 / 
sin 
Again, from second form of (6) 
cos q, 
> = - — ^ c 
sin 
cos & w + CQS ?3 
sinp 2 ^ sin p 3 
therefore 
So* = sin r ( 22Ml Saa' + S fifT + ^ Syy' ) , 
\ sinV sm 2 p 2 em 2 ®, / 
1 Pi 
cos XP = sin r 
/cos_2i 
Vsin^ 
' ' sm z p 3 
£i + cos & + cos Z» ) t 
sin p 2 sin p 3 / ’ 
Similarly from (2) (6) we find 
cos XP 
1 = B * n r i(^~ 
cos q x cos $ 2 COS q.f 
sin pj sin p 2 sinp 3 / 
and similar expressions for cos XP 2 , cos XP 3 . 
Ex. (3.) To find the volumes of pyramids OP^Pg, OP^P, &c., 
where 0 is the centre of the sphere, in terms of the volume of the 
pyramid OABC. 
We have 
therefore multiplying these, and taking the vector of each side, 
we have 
