12 Proceedings of the Royal Irish Academy. 



In the particular case in whicli ^i and ^2 are seli-conjugate, the 

 cones /and/' coincide, and every generator of the single cone is at 

 right angles to two other generators likewise mutually perpendicular. 



As a solid is strained from any configuration to any other by 

 gradually increasing a strain of given type, the locus of the unrotated 

 lines is a cubic cone. 



16. Homographic transformations. — Not only is the cone Sp(j}ip(f}2p = 

 the locus of axes of <^i + 1(]32, but it is also the locus of axes of the 

 system of functions 



{acjji + h(fi2 + c)-^ {a'^i + h'(jiz + c'), 



a, I, c, a', b', and e' beiag arbitrary scalars. 



Por the equation of the cone / may be written in the form 



>S(«<^i + hcji2 + c)p {a'^y + V4,2 + c')p {a"(jii + h"(f>2 + c")p = 0, 



and then in the form 



8p{a<j)i + J<^2 + c)-\a'(f>i + b'(}i2 + Op(«<^i + Hz + c)-\a"(j>^ + h"^)^ + c")p=0, 



or Spxl/,pij/2p = 0. 



To each cone/ corresponds an infinite number of cones /'. If 



>Ai = (<?<^i + H2 + o)~^ {a' 4^1 + h'cli. + c'), 

 its conjugate is 



1/^1' = («'0i' + h'^o' + c')(«</>i' + h4>^ + c)-^ ; 



but now the cone corresponding to /= Spif/^pif/^p = 0, with respect to 

 the functions ^1 and 1/^2, is Spil/^'pif/^'p = ; and this cone is not the 

 same as aSp</)/p02'p = 0, because a'cj>/ +b'<ji2' + c' and {acjiy,' + bcf^^' + c)-' 

 are not commutative in order of operation. 



17. Condition that two functions should be expressible in the form 

 ^3"^$i and $3-i<J>2. — Though an arbitrary cubic cone may be written 

 in the form S^.p^^p^^p = 0, in which the $ are self -conjugate func- 

 tions, it is not generally possible to express two functions by the 

 relations cf>, = ^f^^, and <^2 = $3-1 $2. If p„ p^, and ^3 are the axes 

 of <^i, and ffi, 02, and g^ its roots, then, if ^1 = $3-1 $j, 



^iPi = ffi^3pi = o,,g,rp2p„ 



^1P2 = g2^ip2 = ^^2^2 Vp^Pu 



