324 Transactions. 



If H be the circurncentre, the orthocentre, I the in-centre, and 

 Ii the ex-centre opposite to the vertex A of the triangle ABC, then 

 (1) if IH = 10 either A or B or C is 60°, and (2) if IjH = liO either 

 A is 60° or B or C is 120° 



(1.) The tripolar equation of a circle of radius p concentric with the 

 in-circle is 



a X + bY + cZ = {a + b + c) (2Rr + p 2 ). 



If this passes through H, X = Y = Z = R 2 ; hence 2Rr + P 2 = R 2 , 

 and the above equation becomes 



aX + bY + oZ = 2R 2 s. 

 If this circle passes through 0, then X = 4R 2 cos 2 A, Y = 4R 2 cos 2 B, 

 Z = 4R 2 cos 2 C, and we obtain 



a cos 2 A + b cos 2 B -\- c cos 2 C = ^s 

 4a (1 - sin 2 A) + 46 (1 - sin 2 B) + 4c (1 - sin 2 C) = a + b + c 

 i.e., sin 3 A + sin 3B + sin BG = o 



3A 3B 3C 



i.e., cos — cos — cos — - = o 



A A o 



whence A or B or C is 60°. 



(2.) The tripolar equation of the circle of radius p concentric with the 

 ex-circle opposite A is 



- aX + bY + cZ = (b + c -a) (p 2 - 2Rrj), 

 which, if it passes through H, reduces to 



- aX + bY + cZ = 2 (s - a) R 2 . 



Expressing that this circle passes through O we have 



— a cos 2 A + b cos 2 B -f c cos 2 C = ^ (s - a) 

 whence — sin 3A + sin 3B + sin 3C = o 



3A . 3B • 30 



i.e., cos — sin — sin — = o, 



2 2 2 



and therefore either A is 60° or B or C is 120°. 



Art. XLIX.^ — An Ancient Maori Stone-quarry. 



By H. D. Skinner. 



[Read before the Otago Institute, 17th March, 1910.] 



Plate XII. 



About nine miles from the Town of Nelson, on the old Maungatapu Track 

 that leads from the Maitai Forks into the valley of tbe Pelorus, is a well- 

 known Maori tool-manufactory or quarry. The present road from Nelson 

 follows the Maitai Valley to the Forks, where the old track strikes off 

 up the spur between the north and the south branch. (See map.) On 

 the spur, about a mile from the Forks, tbe track passes over a small 

 hummock, beyond which there lies a curious hollow in the ridge. This 

 basin encloses a shallow pool of water surrounded by a belt of rushes, 

 from which the place takes its modern name — the Rush Pool. The shores, 



