332 



SIXTH REPORT 1836. 



A(-) = A<*) ; A(«) = a A<'-) + A<«) ; 



1,0 1,0 0,1 1,0 0,1 ' 



B(^) = B<^) ; B^'^) = 2 a B<^) + B^(^) : 



2,0 2,0 ' 1,1 2,0 1,1 ' 



^/,o~ ^^,0' ^/-i.i-^^^^.o"^ ^^-1,1' J.. (227.) 



^/-2,2 2 ^0^^ ^^"^^-1,1 



+ ^ if - 2, 2 ' 



0,< ^,0 (--l.l 0,/' 



so that the first system of auxiliary equations, (206) . . . (209), 

 which are of the forms 



<-) 



1^^ = 



(228) 



tW = 0, . . (229.) 



AS = 0,Bg = 0,C^^ = 0,.. 



may be replaced by the system 



A'S = «.B* = 0,C«=0,. 



the change, so far, being only a change of notation ; and, after 

 satisfying this system by a suitable selection of the ratios of the 

 quantities (204), the second system of auxiliary equations, (210) 

 ... (213), may then be transformed, with a real simplification, 

 (which consists in getting rid of the arbitrary multiplier a, 

 and in diminishing the number of the quantities whose ratios 

 remain to be disposed of,) to another system of equations of the 

 forms. 



T 



A^W = o,B<^ = 0,C^^^) = 0, 



<2 = 0'^t2 = «'---'r1- 2,2 = 0; 



C<^ = 0,...rW 3^3 = 0; 



W ^n.1 



^-1,1 



0; 



. (230.) 



A-) 



T- = 0; 



i 



