Hogg. — On Isogonal Transformations. 305 



Y. " Transactions of tlie New Zealand Institute," vol. xxvii, 

 p. 273. 



W. " Westland : Geology of Tlokitika Sheet, North Quad- 

 rangle," 1906, p. 13. 



-X. " Zoologist," 1871, vol. xxix. 



Y. " Zoologist," 1881, p. 290. 



Z. " Zoologist," 1883, p. 276. 



EXPLANATION OF PLATE XV. 



-Map of the South Island of New Zealand, showing the Kea's 



Distribution. 

 No. L Places where keas have been seen to attack sheep and authentic 



accounts have been sent in. 

 No. 2. Places where keas have been reported to have attacked sheep 



but no accounts have been sent in. 

 No. 3. Place where keas have been reported to have been seen. 

 No. 4. Capital towns of the provinces. 



Art. xxix. — On Isogonal Transformations : Part I. 



By Evelyn G. Hogg, M.A., Christ's College, Christchurch. 



[Read before the Philosophical Institute of Canterbury, 5th December, 



1906.] 



1. " Two points P, P', which are such that lines drawn from 

 them to the summits of the triangle of reference are equally 

 inclined to the bisectors of its angles are called isogonal con- 

 jugates with respect to the triangle." — Casey. 



If the trilinear co-ordinates of P be (a/3y), those of P' will 



l^e (—7^ ); but as in what follows trilinear ratios will be 



\a (3 y! 



for the most part used, the co-ordinates of P' will be ( ~ 75 ~ ) ■ 



If the co-ordinates of P' be written {a'f3'y') we have aa' = 

 y3;8' = yy' = a constant : hence an isogonal transformation is 

 a species of inversion, and in the following paper isogonal 

 transformations will be described in the language of inversion. 

 The incentre and three excentres (1 + 1+1) of the triangle 

 of reference ABC are the only points which invert into them- 

 selves. The four points (a + ^±y) forming the vertices of 



a harmonic quadrangle invert into four points (- + ;; + -) 



forming the summits of another harmonic quadrangle. 



