444 MR. A. E. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 86. Similarly, had we taken 



ttg = U^, 



it would have appeared that the differential equation satisfied by o- is the same as 

 (24). Hence we derive the conclusion that, if a- and T be special solutions of (24), 

 the primitive of it is 



A + B<r + CT 

 = A' + B'r + C'r ' 



Now, if we consider these two special solutions cr and T to be known, we have 



= u^ + 37/V' + 3'X, 

 = wj r 1 " + 3u\ T" + 3u\ T 1 ; 

 and, therefore, 



O^T* - 



so that 



u \ (a*^ ^^) = constant. 



Hence we may take 



(o-V 



- 0*1*)-*, 



as three special linearly independent solutions of (22) ; they constitute a fundamental 

 system of integrals, and any other integral can be expressed in terms of them. 



87. It is not uninteresting to see how from these forms the case considered in 84 

 may be deduced ; we then have 



r=cr>, 



so that 



r 1 = 20-cr 1 , T" = 20-0^ + 2o-' \ 



whence, by neglecting a factor 2~ l , which may be absorbed in the quantities u, the 

 three special solutions are 



1 a- a 3 



Ul =-, u z = -> u 3 = -- 

 Taking u = I/a; we have 



the substitution of which in 



= Mjtr 1 " + Sw'jO* + 



