Prof. Cay ley on an Elementary Formula of Solid Geometry. 413 

 which is 



a \ a 



a p p 



b" 2 



bpp. 



H, 



c p p 



But p being greater than 0, since the numbers b', b n , . . . b p are 

 all of them taken out of the series 1, 2, . . . 0, some of these 

 numbers must necessarily be equal to each other, and we have 

 therefore 



6"2 

 - b P P- 



whence finally the commutant is =0. 



In the case where p = 6~% } we have for a determinant of the 

 order 2 the theorem 



2XX' , Xfi' + X'/i =- X, /a 2 # 



I Xfj+Vft, 2/ifji, 1 \', fit! ' 



and it is probable that there exists a corresponding theorem for 

 the commutant 



r A t.... w ' 



2 2 2 



where 



^rst..{ P Y 





pr* pi 



(P) 



fi i 



*2 , 



but I have not ascertained what this theorem is. 

 Cambridge, October 26, 1865. 



LIX. On the Signification of an Elementary Formula of Solid 

 Geometry. By Professor Cayley, F.R.S.* 



THE expression for the perpendicular distance of a point 

 {se } y, z) from a line through the origin inclined at the 

 angles [a, j3, y) to the three axes respectively, is 



j9 2 = # 2 -f- 2/ 2 -f £ 2 — (x cos « + y cos /3 + z cos y) 2 



= (y cos y— z cos y@) 2 

 H-(£cosa— a? cosy) 2 

 4- (a? cos @— y cos a) 2 ; 



* Communicated by the Author. 



