70 Proceedings of the Royal Irisli Academy. 



last expression a perfect square. The condition for this is at once found 

 to be 



(1j - Cj n; - (-,-. 

 which may be written in the form 



or again, 



/• ' *\ ( 1 l 



A Mil I- - = /1,-d — I 





(3) 



A,/-, -/a K/n-Hi-1 



Hence, for the fire) pan of H',' 7 , 



... '',' ( ■> * A>, - ;i,- , KjTi - )t ; \ 



- " " < /> ,'V - /', '' .' / - /'; , / 



It i> conveuienl i" write 



A',' - A',/-, - /(,.,, A'." - A",/-, - /(; , 



and then tli" complete expression "i (/',•' becomi 



W ....... | M A' ■-...-,«'; .,/iV >'»(,.,»,• A'/- „,,«,, A-/')' ) . 



■ A. 



In the Bame circumstance H Jiily simplified, and becomes 



for the necess d ifl 



and this i | (3). 



3. Instead of the conjugate planes through <',.,(>,.,, 0, 0) and C'.fr,-, 0, 0) 

 let auother pair he taken through (','.,<■,',. 0, 0) and €','(</, 0, 0). If, for 

 the same ray, I', correspond to If',, in the same way 



Vi " = 97~ r< \ t't- '■ " ' ~ vt-i m H L i"y + lw"A' " "■ ■ "■ . / "") J !> 



- <"i - /li-l) A 1'i 



~8rT~ "••• (4) 



