308 SIXTH UEPORT — 1836. 



by satisfying another system, composed of eleven equations, 

 wliich are obtained by decomposing the condition (8) into three, 

 and the condition (14) into seven new equations, as follows. By 

 the law (5) of the formation of the four coefficients A', B', C, D', 

 and by the assumed expression (75), those four coefficients are 

 rational and integral and homogeneous functions, of the first, 

 second, third, and fourth degrees, of the twelve coefficients (77) ', 

 and therefore, when these latter coefficients are distributed into 

 three groups, one group containing A', A", A'", another group 

 containing M', M", M'", M'^, and the third group containing 

 N', N", N'", N''^, N^, the coefficient or function A' may be de- 

 composed into three parts, 



^ = -^ 1,0,0 + -^'0,1,0 + A'o,o,i> .... (83.) 



and the coefficient or function C' may be decomposed in like 

 manner into ten parts, 



^ ~ ^3,0,0 "^ ^2,1,0 + ^2,0,1 I 



+ c'j,2,o + c',,,,! + cv, r • ■ ■ ^^^'^ 



+ C/ 0,3,0 + ^0,2,1 + ^0,1,2 + Co,0,3>J 



in which each of the symbols of the forms A'^ . ^ and C'^ . j^ de- 

 notes a rational and integral function of the twelve quantities 

 (77) ; wliich function (A'^ ^ ^ or C'^^ . j^) is also homogeneous 

 of the dimension h with respect to the quantities A', A", A'", of 

 the dimension i with respect to the quantities M', M", M'", M^^, 

 and of the dimension k with respect to the quantities N', N", 

 N"', N^^, N^. Accordingly Mr. Jerrard decomposes the condi- 

 tions A' = and C = into ten others, which may be thus ar- 

 ranged : 



A',,0,0 = 0, C'3,0,0 = ; (85.) 



Ao,i,o = Oj C 2,1,0 = 0, C,_2 = 0; (86.) 



A'0,0,, = 0, C'2,0,, = 0, C'.,,,, = 0, C.o.c = 0; . . (87.) 



^0,3,0 + C 0,2,1 + Co, 1,2 + C g_o,3 = ; (88.) 



nine of the thirteen parts of the expressions (83) and (84) being 

 made to vanish separately, and the sum of the other four parts 

 being also made to vanish. He then determines the two ratios 

 (78), so as to satisfy the two conditions (85) ; the three ratios 

 (79), so as to satisfy the three conditions (86) ; the four ratios 

 (80), so as to satisfy the four conditions (87) ; and the ratio 

 (81), so as to satisfy the condition (88); all which determina- 

 tions can in general be successively effected, without its being 



