METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. ol5 



which give 



?m-3+?'»*-3 = (!+?) ?m-3 + ^m-3»?m-2+?'m-2 ^ (128.) 

 = (l+?)?m-2> J 



we easily perceive that the three homogeneous functions (105) 

 (106) (107), of these m - \ sums q^ + q'o, ... qn-i^^m-i^ 

 may he expressed in the following manner : 



A"o,i,o + A'Vi = (1 + ?) A"'o,,,o + A'V, ; . . (129.) 



C"2,,,o + C%,i = (1 + ?) C'\,,o + C%,, ; . . (130.) 



C",..,o + C",,,,, + C'\,o,3 = (1 + qY C"',,,,o n ..... 



+ (i + ?)CV, + c'%,.;J ' • ■ ^ '' 



the synihol A'"^ ^ ^j. or C'"^ ^ ^ denoting here a rational and inte- 

 gral function of the 3 w — 3 quantities p^^ ... Pm—D 9o> ••• 

 S'm-SJ 'oj — ^»j-3» ^^^licli is, like the function A\,-_;t oi" C"h,i,k, 

 homogeneous of the dimension k with respect to^o> '"Pm— 1j ^^^ 

 of the dimension i with respect to q^, ... qm—'z^ I'^t is homo- 

 geneous of the dimension k with respect to r^, ... /"^^.s, and is 

 independent of q'^, ... 5f',„_2 and of^,y, g'; whereas A";^;;.. or 

 C"^ ! k '^^s homogeneous of the dimension k with respect to 

 ff'oj •••?'»»— 25 ^"<i ^^^s independent of r^,, ... r^_3. The three 

 identical equations (129) (130) (131) may he decomposed into 

 the seven following, which are analogous to (60) (and (61) : 



A 0,1,0 — A 0,1,0 J A 0^0,1 = S' A 'o,i,o + A QQ^■^^ ; . . . (132.) 



^ 2,1,0== C 2,1,0 5 C 2,0,1 = 9'^ 2,1,0 + ^ 2,0,1 J • • * (133.) 



C" — P'" • P" —On r'l' 4- P'" • "1 



1,2,0 — ^ 1,2,0 > ^ 1,1,1 — ■^ q ^ 1,2,0 ~ ^ 1,1,1 J I Cl«}4 



Cf _ ^2 cm 4- « P"' a. P"' • 1 ' 



1,0,2 — ? ^-^ 1,2,0 +9^ 1,1,1 + ^ 1,0,2 > -J 



and, in virtue of these, the seven conditions (112) and (113) may 

 be put under the forms, 



A'Vo = 0, C", 1,0 = 0, C'Vo=0, (135.) 



and 



A"'o,o,, = 0, C"'„o,i = 0, C"',,,,i = 0, C"',,o,2 = 0. . (136.) 



