sot SIXTH REPORT — 1836. 



the reasonings which are to be founded upon them, requires only 

 that some one such non-evanescent quantity ]). out of this set 



Poi Pi} ••• Pm-i should be made the denominator of a fraction 



P'i 



like (54), — = p, and that thus some one term q. x^ should be 



"i 



taken away out of the difference of the two polynomes p'o + p'] •*' 

 + ... and p {poX + Pi'V + ...) ; and it is so easy to make this 

 adaptation, whenever the occasion may arise, that I shall retain 

 in the present discussion, the asssumptions (54) (55), instead of 

 writing p. for p ,. 



o .Tz 1 rn.— \ 



The expression (57) for f{x), combined with the law (5) of 

 the composition of the coefficients A' and B', shows that these 

 two coefficients of the transformed equation in y may be ex- 

 pressed as follows, 



A'=(l +;^)A\o + A\i, (58.) 



and 



B'=(l+^)2B\o+(l+_^)B\i + B\,; . (59.) 

 A"^ J and B"^^ . being each a rational and integral function of the 

 2m - 1 quantities p^, p^, ... p^_^, q^, q^, ... q^_^, which is 



independent of the quantity p and of the form of the function 

 L, and is homogeneous of the dimension h with respect to 

 Po3 P\} ••• Pm—v ^"^ ^^ ^^^ dimension i with respect to 

 q^, qi, ... q^_2' Comparing these expressions (58) and (59) 

 with the analogous expressions (24) and (25), (with which they 

 would of necessity identically coincide, if we were to return 

 from the present to the former symbols, by substituting, for 

 PiPoiPiy-'Prn-V'-'loiqi} ••• 9„i_2' ^^^^'* values as functions of 

 A', A" A'", M', M", M'", M'v, deduced from the equations of 

 definition (54) (55) and (46) (47),) we find these identical equa- 

 tions : 



A',,0 = A"i^o ; A'o^, = p A",_„ + A\i ; . . (60.) 

 and 



B',,0 = B",_o ; B',,, = 2 p B"2,o + B",,, ; B',,,-1 



> . . (61.) 



= ;^"-B",,o + ;>B"i,, -HB",^^; / 



observing that whatever may be the dimension of any part of 

 A' or B', with respect to the m new quantities p, q^, q^, ... 

 9'^_2, the same is the dimension of that part, with respect to 

 the four old quantities M', M", M'", M'^. 



The system of the five conditions (26) (27) (28) (29) (30) may 

 therefore be transformed to the following system. 



