234 Dr Baker, On the differential equations 



2£ G5 = 2 v z + C(2y + §X 3 ) - (2 V + ±\) R, 

 A* 



2 v ^=r.- ±\ lV + £(4^ + X 2 ) +(2^- 4£)P, 

 A 4, 



2^ = -X oV + ^X^+2yR ) 

 A* 



2%@™=- 2z£+ 2zP + {2y + |x 3 ) Q + (4£ - 4^ - X 4 ) i2, 

 A 1 



2^ £™ = \%g- ±\P + (4>z + X 2 ) Q - (22/ + JX,) JB, 

 A 6 



2^^- 1 =X r-X P + i\ 1 Q + 2^ 



- |\^ + (4* + X,) £ - 4yQ + (4* - 80 R + 2y ^ + (4£ - 2a?) &* 



-(4^ + iX 5 )^+(4f + X 6 )^ = 0, 



- 2\ oV + f U - 1X,P + X 2 Q -X 3 R+2z^ + (2y + |X 3 ) ^ 



A 4 A 4 



-(4^ + X 4 )^ 211 +iX 5 ^ = ; 

 A 4 P 



2A t - 2\ P + 2X,Q - \ 2 R - IX, ^ + (4s: + X 2 ) ^ 



A 4 A* 



-(2^+^3)^-2^ = 0, 



XoQ-IX^-iXo^ + iX^ +^^-2/^1=0. 



By the elimination of g) 222 and 2 from the first three of these 

 sixteen, x and y being expressed in terms of P, Q, g, rj, £, we 

 obtain a linear equation to express Q in terms of %, rj, £, P. From 

 another set of three we obtain an equation connecting P, Q, g, rj, £; 

 hence we obtain at once the fundamental equation connecting 

 f, 7), £, P. The algebraic expressions of u s , u 2 , u t are then found 

 from the equations 



dj; = fx (%du 3 + r]du 2 + ^du^), dn = /x (r)du 3 + Pdu 2 + Qdu^), 



d£= fi (£du s + Qdu 2 + Rdu t ). 



