192 THE ROYAL SOCIETY OF CANADA 



(40) iJ8=C8+<^4C4+08Co, 



where the invariants 04, 08, are defined by the equations 



. . 0^-4^804+216^^3 = 0, 



^ ^ 024-2(08+^8) =0. 



Let (l>i=Qd^3V. The first equation of (41) then reduces to 



(42) v^-^^v-\-l = 0, 



9 eh 



or, after reducing by means of (24), 



î'''-^V20 2^+1 = 0. 



— 5 4 1 



The roots of this equation are v = — =, — =_-, — =. From (22) and 



V20 V20 V20 

 (18) we find _ 



X''^^3 = 20V20/'. 

 Hence 



X^0i^^= -600/^ 



(43) XVf = 480/2, 

 XVf ^ = 120 l\ 



Also, from (23), (43), (18), 



(44) i^80r= 11-3600 /S 

 ^808^^= 19-3600 /I 



Substituting (39), (43), (44), in (40), we obtain, after a rather 

 tedious reduction 



m^i^s^^^' = 36-1200 l'iA+Wt+SCf'+Dt') 



where 



A = "i^ {la + 2mb + nc) = ~ {lb + 2mc + nd), 

 ôP I 



B = ~ {la + 2mb + nc) = aÔ(lb-^2mc + nd), 



C= aô {la-\-2mb-\-nc), 



D=^-^ (la-\-2mb-\-nc) = 8l(3l3l-{-2am)a-\-ian-3yl)b-\-ôlc]. 



That is, 



w^X^i^i^^ = 36- 1200/4 r 



_8P I n 



^^ nPKm'i^ = 36-1200 aH'n {lb+2mc-\-nd), 

 according as 5 is, or is not, different from zero. 



(45) w2X«i7ji) = 36-1200/4 r^V^^^+3a5/2 + ^'l (/a + 2mô + wc) 



15/2/ w J 



